Monodromy theorems in the affine setting... · 2019. 2. 13. · This connection is equivariant...

Monodromy theorems

in the affine setting

A dissertation presented

by

Andrea Appel

to

the Department of Mathematics

In partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in the field of

Mathematics

Northeastern University

College of Science

Boston, MA

June 21, 2013

Monodromy theorems

in the affine setting

by

Andrea Appel

Abstract of Dissertation

Submitted in partial fulfillment of the requirements

for the degree of Doctor of Philosophy in Mathematics

in the College of Science of Northeastern University

June 21, 2013

Abstract

My doctoral work revolves around a principle which first appeared in the

work of T. Kohno and V. Drinfeld and states, roughly speaking, that quan-

tum groups are natural receptacles for the monodromy representations of cer-

tain flat connections of Fuchsian type. It stems more specifically from the

monodromic interpretation of quantum Weyl groups obtained by V. Toledano

Laredo. This shows that these operators describe the monodromy of the

Casimir connection, a flat connection with logarithmic singularities on the

root hyperplanes of a semisimple Lie algebra g, and values in any finite di-

mensional g-modules. The main goal of this thesis is to extend these results

to the case of an affine Kac-Moody algebra g.

The method we follow is close to that developed by Toledano Laredo in the

semisimple case, and relies on the notion of a quasi–Coxeter quasitriangular

quasibialgebra (qCqtqba), which is informally a bialgebra carrying actions of a

given generalized braid group and Artin’s braid groups on the tensor products

of its modules. A cohomological rigidity result shows that there is at most one

such structure with prescribed local monodromies on the classical envelop-

ing algebra Ug[[~]]. It follows that the generalized braid group actions arising

from quantum Weyl groups and the monodromy of the Casimir connection are

5

6 ABSTRACT

equivalent, provided the quasi–Coxeter quasitriangular quasibialgebra struc-

ture responsible for the former can be transferred from U~g to Ug[[~]]. The

proof of this fact relies on a modification of the Etingof–Kazhdan quantization

functor, and yields an isomorphism between (appropriate completions of) U~g

and Ug[[~]] preserving a given chain of subalgebras.

Acknowledgements

I wish to express my deepest gratitude to my advisor Prof. Valerio Toledano

Laredo for the large doses of guidance, endless patience and encouragement he

has shown during my years at Northeastern, and for being a constant source

of inspiration to me.

I would like to thank Professors Pavel Etingof, Venkatraman Lakshmibai,

and Ben Webster for being part of the Dissertation Committee. I am par-

ticularly grateful to Pavel Etingof for his interest in this work and for many

enlightening discussions.

This work would not have been possible without the constant help of my

friend and ”academic brother” Sachin Gautam: for the innumerable discus-

sions, classes and seminars we had together, as well as beers, glasses of scotch

and movies (korean, obviously).

I am grateful to my teachers at Northeastern University, Profs. Marc

Levine, Alexander Martsinkovsky, Alex Suciu, Gordana Todorov, Jonathan

Weitsman, Jerzy Weyman, and, in particular, Andrei Zelevinsky for offering

several excellent courses and seminars over the years.

7

8 ACKNOWLEDGEMENTS

I would like to thank Tiffany Brown, Donika Kreste, Elizabeth Qudah,

and Logan Wangsgard for their constant help and support in my endless fight

against bureaucracy. I am also very grateful to Pam Mabrouk and Phil He,

for all the support that I received by the College of Science and the Provost

Office, especially during my last year.

These five year in Boston would not have been so enjoyable without all my

friends and colleagues, Federico, Jeremy, Jason, Giulia, Martina, Luis, Sasha,

Roberto, Giulio, Francesco, David, Martina, Thomas, Carl, Nate, Thomas,

Floran, Yaping, Gufang, and many others.

I owe special thanks to my friends and colleagues, my actual family in

Boston: Giorgia and Salvo. Five years with them, full of joy, sorrow and frus-

tration, cannot be summarized in few lines. Thanks.

I would like to thank my family, my parents, my brother and sister, my

little niece. And finally I would like to thank one more time, for her endless

love and patience, Valentina, to whom I dedicate this thesis.

Contents

Abstract 3

Acknowledgements 7

Contents 9

Chapter 1. Introduction 11

1. Monodromy theorems in the affine setting 11

2. Results - Part I 17

3. Results - Part II 22

4. Results - Part III 30

5. Towards a complete solution 36

6. Outline of the dissertation 38

Chapter 2. Quasi–Coxeter Categories 41

1. Diagrams and nested sets 41

2. Quasi–Coxeter categories 46

Chapter 3. An equivalence of quasi–Coxeter categories 63

1. Etingof-Kazhdan quantization 64

2. A relative Etingof–Kazhdan functor 78

3. Quantization of Verma modules 101

4. Universal relative Verma modules 106

9

10 CONTENTS

5. Chains of Manin triples 117

6. The equivalence of quasi–Coxeter categories 124

Chapter 4. A rigidity theorem for quasi–Coxeter categories 141

1. The Dynkin-Hochschild bicomplex 142

2. Dynkin–Hochschild complex of monoidal functors 145

3. Uniqueness of the relative twist 153

4. The main theorem 162

Appendix A. 181

1. The completion of Ug in category O 181

2. The isomorphism between Ug[[h]] and U~g 189

3. A braided Hopf algebra in DΦD(gD) 197

Bibliography 205

CHAPTER 1

Introduction

My doctoral work revolves around a principle which first appeared in the

work of T. Kohno and V. Drinfeld and states, roughly speaking, that quantum

groups are natural receptacles for the monodromy representations of certain

flat connections of Fuchsian type [Ko, D2]. It stems more specifically from

the monodromic interpretation of quantum Weyl groups obtained by Toledano

Laredo in [TL2, TL4]. This shows that these operators describe the mon-

odromy of the Casimir connection, a flat connection with logarithmic singu-

larities on the root hyperplanes of a semisimple Lie algebra g, and values in

any finite dimensional g-modules. The main goal of my doctoral work is to

extend these results to the case of an affine Kac-Moody algebra g.

1. Monodromy theorems in the affine setting

1.1. Motivation: the Knizhnik–Zamolodchikov equations. Around

1990, Kohno [Ko] and Drinfeld [D2] proved a rather astonishing result, now

known as the Kohno-Drinfeld Theorem. The theorem states that quantum

groups can be used to describe the monodromy of certain first order Fuchsian

PDE’s known as the Knizhnik-Zamolodchikov (KZ) equations. Given a sim-

ple Lie algebra g, a representation V of g and a positive integer n, the KZ

11

12 1. INTRODUCTION

equations are the following system of PDEs

∂Φ

∂zi= ~

∑j 6=i

Ωij

zi − zjΦ

where

(i) Φ is a function on the configuration space of n ordered points in C

Xn := (z1, . . . , zn) ∈ Cn | zi 6= zj

with values in V ⊗n,

(ii) Ω ∈ g⊗ g is the Casimir tensor,

(iii) ~ is a complex deformation parameter.

This system is integrable and equivariant under the natural Sn-action and

hence defines a one-parameter family of monodromy representations of Artin’s

braid group Bn = π1(Xn/Sn) on V ⊗n. The Kohno-Drinfeld theorem asserts

that this representation is equivalent to the R-matrix representation of Bn on

V⊗n arising from the quantum group U~g. Here V is a quantum deformation

of V , i.e., a U~g–module such that V/~V ' V .

1.2. The Casimir connection. In subsequent work, J. Millson and V.

Toledano Laredo [MTL, TL2], and independently C. De Concini (unpub-

lished) and G. Felder et al. [FMTV], constructed another flat connection

∇C , now known as the Casimir connection, which is described as follows. Let

h ⊂ g be a Cartan subalgebra, and hreg be the complement of the root hyper-

planes in h, that is

hreg = h \⋃α∈Φ

ker(α)

1. MONODROMY THEOREMS IN THE AFFINE SETTING 13

where Φ = α ⊂ h∗ is the set of roots of g. For a finite–dimensional g-module

V , ∇C is the following connection on the trivial vector bundle hreg×V → hreg:

∇C = d− ~2

∑α∈Φ+

dα

αCα

where

(i) the summation is over a chosen system of positive roots Φ+ ⊂ Φ;

(ii) Cα is the Casimir operator of the sl2-subalgebra of g corresponding

to the root α.

This connection is equivariant under the action of the Weyl group W of g, and

gives rise to a one-parameter family of monodromy representations on V of

the generalized braid group

Bg = π1(hreg/W )

If V is a quantum deformation of V , the quantum Weyl group operators

Ti ∈ U~g defined by G. Lusztig, A.N. Kirillov–N. Reshetikin and Y. Soibel-

man, [L, KR, S] define a representation of Bg on V . In this setting a Kohno–

Drinfeld theorem was obtained by V. Toledano Laredo, stating the equivalence

of the above two representations of Bg [TL1, TL2, TL3, TL4, TL5].

The proof of this theorem is inspired by Drinfeld’s proof of the Kohno–

Drinfeld theorem [D2]. It proceeds along the following lines.

14 1. INTRODUCTION

(i) The notion of quasi-Coxeter algebras is introduced. Informally speak-

ing, these are algebras which carry a natural Bg-action on their finite-

dimensional modules. They replace the role of Drinfeld’s quasitrian-

gular quasibialgebras with respect to Artin’s braid group Bn. Just

as quasitriangular quasi–Hopf algebras are related to the Deligne–

Mumford compactification of the configuration space of n points in

C, quasi–Coxeter algebras are related to the compactifications of hy-

perplane arrangements constructed by De Concini–Procesi [DCP1,

DCP2].

(ii) It is shown that the monodromy of the Casimir connection and the

quantum Weyl group representations of Bg arise from suitable quasi-

Coxeter algebra structures on Ug[[~]] and U~g respectively.

(iii) It is shown that the quasi-Coxeter structure on U~g can be cohomo-

logically transferred to one on Ug[[~]].

(iv) An appropriate deformation cohomology for quasi-Coxeter algebras

(called Dynkin diagram cohomology) is introduced. It shows however

that quasi-Coxeter algebras on Ug[[~]] are not rigid. To remedy this,

it is necessary to also take into account the quasitriangular quasib-

ialgebra structure on Ug[[~]].

(v) The notion of quasi-Coxeter algebras is extended to that of quasi-

Coxeter quasitriangular quasibialgebras, whose deformation theory

1. MONODROMY THEOREMS IN THE AFFINE SETTING 15

is controlled by a bicomplex, namely the Dynkin-Hochschild bicom-

plex. Finally, one proves the uniqueness of quasi-Coxeter quasitrian-

gular quasibialgebra structures on Ug[[~]].

1.3. Affine extension of rational Casimir connection. It is natural

to ask if it is possible to generalize the previous construction to the affine

setting. If l is a symmetrisable Kac-Moody algebra, and U~l is its quantized

universal enveloping algebra, the corresponding quantum Weyl group opera-

tors give an action of the braid group of type l on any integrable U~l–module.

In particular, if l = g is the affinization of a semi–simple Lie algebra g, we ob-

tain an action of the affine braid group Bg of type g on integrable g-modules.

The purpose of my thesis is to solve the following

Problem. Give a monodromic description of the quantum Weyl group

action of the generalized affine braid group Bg on integrable g-modules in

category O.

The strategy to attack this problem consists in adapting the machinery

developed by Toledano Laredo in [TL4] to the affine setting.

Two major tools involved in the proof of the main theorem in [TL4] are still

applicable to the affine case. The theory of quasi-Coxeter algebras refers to a

general connected graph and it is therefore possible to consider affine Dynkin

diagrams as well. In addition, the Kohno-Drinfeld theorem was extended

to all symmetrizable Kac-Moody algebras by P. Etingof and D. Kazhdan in

16 1. INTRODUCTION

[EK1, EK2, EK6].

We proceed along the following steps, according to the principle that the

quantum Weyl group action is encoded by an appropriate quasi-Coxeter struc-

ture.

(i) Show that the quantum Weyl group representation of Bg and the

R-matrix representation of Bn arise from a quasi-Coxeter quasitrian-

gular quasibialgebras structure on the quantum affine algebra U~g.

(ii) Show that this structure on U~g can be transferred to U g[[~]].

(iii) Prove the rigidity of quasi-Coxeter quasitriangular quasibialgebra

structures on U g[[~]].

(iv) Define an affine extension of the rational Casimir connection, whose

monodromy representations are encoded by a quasi–Coxeter struc-

ture on U g[[~]]

The main difficulty in carrying out the above program lies in step (ii).

This step is carried out in the semisimple case by relying on the vanishing of

Hochschild cohomology groups, which does not hold for affine Lie algebras. An

alternative solution which relies instead on the Etingof-Kazhdan quantization

functor is explained in Sections 2 and 3 of this chapter. The ongoing work on

the solution to steps (iii) and (iv) is presented in Sections 4 and 5, respectively.

The step (i) is trivial.

2. RESULTS - PART I 17

2. Results - Part I

In order to correctly state the problem in the infinite–dimensional setting,

we first need to rephrase the notion of transfer of the quasi–Coxeter structure

in a category theoric way.

We first outline the definition of quasi–Coxeter algebras, following [TL4].

We then give the description of their categorical counterpart.

2.1. Quasi-Coxeter algebras.

2.1.1. Let D be a connected diagram, that is an undirected graph, with-

out loops or multiple edges, SD(D) the collection of subdiagrams of D and

CSD(D) the collection of connected subdiagrams. The following combinato-

rial definitions are due to De Concini–Procesi [DCP1, DCP2]. Two elements

B,B′ ∈ CSD(D) are compatible if either B ⊆ B′, B′ ⊆ B or B ⊥ B′, where the

latter means that B and B′ are disjoint and that no two vertices α ∈ B,α′ ∈ B′

are connected by an edge of D. A maximal nested set on D is a maximal col-

lection of pairwise compatible elements in CSD(D). If D is the Dynkin diagram

of type An−1, this notion is equivalent to that of complete bracketing on the

non-associative monomial x1 . . . xn, by mapping the connected diagram [i, j]

to the bracketing x1 . . . xi−1(xi . . . xj+1)xj+2 . . . xn. We denote by Mns(D) the

collection of maximal nested sets on D and by ND the poset of nested sets on

D, ordered by reverse inclusion.

2.1.2. AD-algebra A is an algebra A endowed with a family of subalgebras

ABB∈CSD(D) satisfying

AB ⊆ AB′ if B ⊆ B′ and [AB, AB′ ] = 0 if B ⊥ B′

18 1. INTRODUCTION

For any B ∈ SD(D) with connected components Bk, we denote by AB the

subalgebra generated by ABk.

A naıve morphism of D–algebras Ψ : A → A′ is a homomorphism such

that Ψ(AB) ⊆ A′B for any B ∈ SD(D). This notion is too restrictive for appli-

cations however. The correct notion of morphism of D-algebras Ψ : A → A′

is a collection of homomorphisms ΨF : A→ A′ indexed by F ∈ Mns(D), such

that ΨF(AB) ⊂ A′B for any B ∈ F .

2.1.3. A D–bialgebra A is a D–algebra and a bialgebra with coproduct

∆ such that ∆(AB) ⊆ AB ⊗ AB. A D-quasitriangular quasibialgebra A is a

D-bialgebra in which every AB is equipped with an additional quasitriangular

quasibialgebra structure of the form (AB,∆, RB,ΦB), so that the category

Rep(AB) of AB-modules is a braided tensor category. Moreover, for each

inclusion B ⊆ B′, the restriction functor Rep(AB′) → Rep(AB) is endowed

with a tensor structure. The latter depends upon the additional choice of a

chain:

B′ = B0 ⊃ B1 ⊃ B2 · · · ⊃ Bn = B (2.1)

where |Bi| = |B′| − i. These tensor structures compose in the obvious way

under the concatenation of chains. For any F ∈ Mns(D), we denote by

JF∈ A⊗2 the structural twist defining the tensor structure on the forgetful

functor Rep(A)→ Vect = Rep(A∅).

2.1.4. A quasi-Coxeter quasitriangular quasibialgebra (qCqtqb) of type D

is a D-quasitriangular quasibialgebra with two additional data. First, for any

G,F ∈ Mns(D), we are given a gauge transformation ΦFG ∈ A relating the


corresponding twists, i.e.,

JG = Φ⊗2GF · JF ·∆(ΦGF)−1

These satisfy additional relations, in particular,

Orientation: for any F ,G ∈ Mns(D),

ΦFG = Φ−1GF

Transitivity: for any F ,G,H ∈ Mns(D),

ΦFH = ΦFG · ΦGH

In the following, we shall refer to the gauge transformation ΦFG as Casimir

associators.

We recall that a quasi–Coxeter algebra admits an equivalent description

in which the relations satisfied by the elements ΦFG are labeled by the two–

dimensional faces of the De Concini–Procesi associahedron, i.e., the regular

CW–complex whose poset of nonempty faces is ND [TL4, 3.8].

Assume now that D is labeled,i.e., an integer mij ∈ 2, 3, ...,∞ is given

for any pair αi, αj of distinct vertices of D, such that mij = mji and mij = 2 if

and only if αi and αj are orthogonal. The second piece of data is given by the

local monodromies Sαα∈D, which define the action of the generalized braid

group BD attached to D and its labeling. For any vertex α ∈ D, Sα is not

an element of Aα, but rather an element of the completion of Aα with respect

to a fixed full subcategory Cα ⊂ Rep(Aα). In [TL4], Cα is the subcategory of

20 1. INTRODUCTION

finite-dimensional representations. Moreover, the elements Sα and ΦG,F

satisfy the following

Braid relations: for any pair of vertices αi 6= αj ∈ D such that

2 < mij <∞, and for any G,F ∈ Mns(D) such that αi ∈ F , αj ∈ G,

Ad(ΦGF)(Sαi) · Sαj · · · = Sαj · Ad(ΦGF)(Sαi) · · ·

where the number of factors on both sides is equal to mij.

The local monodromies are not assumed to preserve the structural twists.

Rather, they satisfy the coproduct identity

∆JF (Sα) = (R21α )JF · (Sα ⊗ Sα)

for any F ∈ Mns(D) such that α ∈ F .

2.2. Categorical analogue of quasi–Coxeter algebras. It is clear

from the previous section that every ingredient in the definition of a quasi–

Coxeter quasitriangular quasibialgebra has a categorical interpretation. More-

over, the definition of the local monodromies seems more natural in the cat-

egorical setting. Implicitly, we are using this approach to describe a quasi–

Coxeter structure on the completion of a D–bialgebra with respect to a fixed

compatible subcategory of representations. A detailed description of quasi–

Coxeter categories is presented in Chapter 2.

A quasi–Coxeter braided monoidal category is to a quasi–Coxeter quasitri-

angular quasibialgebra what a braided monoidal category is to a quasitriangu-

lar quasibialgebra. Following this principle, we define a quasi–Coxeter braided


monoidal category of type D as a tuple

C = ((CB,⊗B,ΦB, βB), (FBB′ , JFBB′), ΦFG, Si)

where

• (CB,⊗B,ΦB, βB) are braided monoidal categories indexed by sub-

diagrams of D;

• (FBB′ , JFBB′) : CB′ → CB are monoidal functors, corresponding to

the inclusion B′ ⊂ B, with a tensor structure depending upon the

choice of a maximal nested set in Mns(B,B′);

• A collection of invertible endomorphisms Si ∈ End(Fi)i∈D and nat-

ural transformations

ΦFG ∈ Nat⊗((FBB′ , J

GBB′), (FBB′ , J

FBB′)

)×for any F ,G ∈ Mns(B,B′), satisfying analogous relations to those

described in the previous section.

We then obtain the description of morphisms of quasi–Coxeter structure

as the datum of

• for any B ⊆ D a functor HB : CB → C ′B;

• for any B ⊆ B′ and F ∈ Mns(B,B′), a natural transformation

CB′HB′

//

FBB′

C ′B′

F ′BB′

γFBB′

z

CBHB

// C ′B

22 1. INTRODUCTION

satisfying the obvious compatibility condition with respect to the quasi–Coxeter

braided monoidal structure.

Interestingly perhaps, these notions can be concisely rephrased in terms of a

2–functor from a combinatorially defined 2–category qC(D) to the 2–categories

Cat⊗ of tensor categories. The objects of qC(D) are the subdiagrams of the

Dynkin diagram D of B and, for two subdiagrams D′ ⊆ D′′, HomqC(D)(D′′, D′)

is the fundamental 1–groupoid of the De Concini–Procesi associahedron for the

quotient diagram D′′/D′.

3. Results - Part II

For a semisimple Lie algebra g, the transfer of structure ultimately rests on

the vanishing of the first and second Hochschild cohomology groups of Ug[[~]],

and in particular on the fact that U~g and Ug[[~]] are isomorphic as algebras,

a fact which does not hold for affine Kac–Moody algebras. Rather than the

cohomological methods of [TL4], we use instead the Etingof–Kazhdan (EK)

quantization functor [EK1, EK2, EK6], which yields a canonical isomor-

phism

ΨEK : U~g∼−→ Ug[[~]]

between the completions of U~g and Ug[[~]] with respect to category O.

Surprisingly perhaps, and despite its functorial construction, the isomor-

phism ΨEK does not preserve the inclusions of subalgebras

U~gD ⊆ U~g and UgD[[~]] ⊆ Ug[[~]]

determined by a subdiagram D of the Dynkin diagram of g, something which

is clearly required by the transfer of structure.

3. RESULTS - PART II 23

The isomorphism ΨEK has a categorical origin. Namely, it comes from

the equivalence between the category O of Ug[[~]] and U~g established by the

Etingof–Kazhdan functor. Similarly, the compatibility conditions with the

subalgebras gD can be expressed in terms of natural relations with the corre-

sponding restriction functors. It follows that the desired isomorphism needs

to be obtained from an equivalence of quasi–Coxeter categories, as discussed

in the previous section. Such equivalence is constructed in Chapter 3.

3.1. To outline our construction, which works more generally for an in-

clusion (gD, gD,−, gD,+) ⊂ (g, g−, g+) of Manin triples over a field k of char-

acteristic zero, recall first that the main steps of the EK construction are as

follows (cf. Ch.3, Section 1).

(i) One considers the Drinfeld category DΦ(g) of (deformation) equicon-

tinuous g–modules, with associativity constraints given by a fixed Lie

associator Φ over k. This category can be thought of as a topological

analogue of category O when g is the Manin triple associated to a

Kac–Moody algebra. It can equivalently be described as the category

of Drinfeld–Yetter modules over the Lie bialgebra g−.

(ii) One constructs a tensor functor F fromDΦ(g) to the category Vectk[[~]]

of topologically free k[[~]]–modules. The algebra of endomorphisms

H = End(F ) is then a topological bialgebra, i.e., it is endowed with

a coproduct ∆ mapping H to a completion of H ⊗H.

(iii) Inside H, one constructs a subalgebra U~g− such that ∆(U~g−) ⊂

U~g− ⊗ U~g−, and which is a quantization of Ug−. The quantum

group U~g is then defined as the quantum double of U~g−.

24 1. INTRODUCTION

(iv) By construction, Ug− acts and coacts on any F (V ), V ∈ DΦ(g),

so that the functor F lifts to F : DΦ(g) → Rep(U~g) where, by

definition, the latter is the category of Drinfeld–Yetter modules over

U~(g−).

(v) Finally, one proves that F is an equivalence of categories.

Since F is isomorphic to the forgetful functor f : DΦ(g) → Vectk[[~]] as

abelian functors, we obtain the following diagram

DΦ(g)

f

DΦ(g)F

//

F

Rep(U~g)

f~

ks

Vectk[[~]] Vectk[[~]] Vectk[[~]]

where f~ : Rep(U~g)→ Vectk[[~]] is the forgetful functor. The EK isomorphism

ΨEK is then given by the identifications

Ug[[~]] := End(f) ∼= End(F ) = End(f~ F ) ∼= End(f~) =: U~g

It is easy to show that in the Kac–Moody case the construction is compatible

with the weight decomposition. Therefore, by restriction of the above diagram

to the category O, we get the desired isomorphism.


3.2. Overlaying the above diagrams for an inclusion iD : gD → g of

Manin triples shows that constructing an isomorphism U~g∼−→ Ug[[~]] com-

patible with iD may be achieved by filling in the diagram

DΦ(g)i∗D

f

DΦ(g)Γ

F//

F

Rep(U~g)i∗D,~

yy

f~

DΦ(gD)fD

##

DΦ(gD)FD//

FD

##

Rep(U~gD)fD,~

%%

Vectk[[~]] Vectk[[~]] Vectk[[~]]

where fD, fD,~ are forgetful functors, FD the EK functor for gD, and iD,~ :

U~gD → U~g is the inclusion derived from the functoriality of the quantization.

To do so, we first construct a relative fiber functor, i.e., a (monoidal)

functor Γ on DΦ(g) whose target category is DΦ(gD) rather than Vectk[[~]], and

which is isomorphic as abelian functor to the restriction i∗D. We then show

the existence of a natural transformation between the composition FD Γ

and i∗D,~ F . Our constructions do not immediately yield a commutative

diagram, i.e., the two factorizations F ∼= FD Γ deduced from f = fD i∗Dand fh = fD,~ i∗D,~ do not coincide, but this can easily be adjusted by using

a different identification F ∼= f, which amounts to modifying the original EK

isomorphism.

3.3. The construction of the functor Γ (Ch.3 Section 2) is very much

inspired by [EK1]. The principle adopted by Etingof and Kazhdan is the

following. In a k-linear monoidal category C, a coalgebra structure on an

26 1. INTRODUCTION

object C ∈ Obj(C) induces a tensor structure on the Yoneda functor

hC = HomC(C,−) : C → Vectk

If C is braided and C1, C2 are coalgebra objects in C, then so is C1 ⊗ C2, and

there is therefore a canonical tensor structure on hC1⊗C2 .

If g is finite–dimensional, the polarization Ug 'M− ⊗M+, where M± are

the Verma modules Indgg∓ k, realizes Ug as the tensor product of two coalgebra

objects in DΦ(Ug[[~]]). This yields a tensor structure on the forgetful functor

hUg : DΦ(Ug)[[~]]→ Vectk[[~]]

Our starting point is to apply the same principle to the (abelian) restriction

functor i∗D : DΦ(Ug) → DΦ(UgD[[~]]). We therefore factorize Ug as a tensor

product of two coalgebra objects L−, N+ in the braided monoidal category of

(g, gD)–bimodules, with associator (Φ ·Φ−1D ), where Φ−1

D acts on the right. Just

as the modules M−,M+ are related to the decomposition g = g−⊕g+, L− and

N+ are related to the asymmetric decomposition

g = m− ⊕ p+

where m− = g− ∩ g⊥D and p+ = gD ⊕ m+. This factorization induces a tensor

structure on the functor Γ = hL−⊗N+ , canonically isomorphic to i∗D through

the right gD–action on N+. As in [EK1, Part II], this tensor structure can

also be defined in the infinite–dimensional case.


3.4. To construct a natural transformation making the following diagram

commute

DΦ(g)F

//

Γ

Rep(U~g)

v~

(iD)∗~

DΦ(gD)FD

// Rep(U~gD)

we remark, as suggested to us by P. Etingof, that a quantum analogue Γ~ of

Γ can be similarly defined using a quantum version L~−, N~+ of the modules

L−, N+. The functor Γ~ = HomU

EK~ g

(L~− ⊗ N~+,−) is naturally isomorphic

to (iD)∗~ as tensor functor, since there is no associator involved on this side.

Moreover, an identification

FD Γ ' Γ~ F

is readily obtained, provided one establishes isomorphisms of (UEK

~ g, UEK

~ gD)–

bimodules

FD F (L−) ' L~− and FD F (N+) ' N~+

3.5. While for M± it is easy to construct an isomorphism between F (M±)

and the quantum counterparts of M±, the proof for L−, N+ is more involved

and is presented in Ch.3, Section 4. It relies on a description of the quan-

tization functor F EK in terms of Prop categories (cf. [EK2, EG]) and the

28 1. INTRODUCTION

realization of L−, N+ as universal objects in a suitable colored Prop describ-

ing the inclusion of bialgebras gD ⊂ g. This yields in particular a relative

extension of the EK functor with input a pair of Lie bialgebras a, b which is

split, i.e., endowed with maps aip b such that p i = id.

3.6. The previous construction leads to the following

Theorem. Let g, gD be Manin triples with a finite N–grading on g−, gD,−

and iD : gD ⊆ g an inclusion of Manin triples compatible with the grading.

Then, there exists an algebra isomorphism

Ψ : UEK

~ g→ Ug[[~]]

restricting to ΨEKD on U

EK

~ gD, where the completion is given with respect to

Drinfeld–Yetter modules.

This result naturally extends to a chain of Manin triples of arbitrary length

(Ch.3, Section 5).

Theorem.

(i) For any chain of Manin triples

C : g0 ⊆ g1 ⊆ · · · ⊆ gn ⊆ g

there exists an isomorphism of algebras

ΨC : UEK

~ g→ Ug[[~]]

such that ΨC(UEK

~ gk) = Ugk[[~]] for any gk ∈ C.


(ii) Given two chains C,C′, the natural transformation

ΦCC′ := γ−1C γC′ ∈ Aut(F EK)

satisfies

Ad(ΦCC′)ΨC′ = ΨC

A straightforward computation shows that the natural transformations Φ

satisfy the properties of orientation and transitivity required by the Casimir

associators.

3.7. Given a maximal nested set F , by iteration of the previous pro-

cedure, we obtain an algebra isomorphism ΨF : U~g → Ug[[~]] such that

ΨF(U~gD) = UgD[[~]] for any D ∈ F . This isomorphism of Dg-algebras

transfers the quasi-Coxeter quasitriangular quasibialgebra structure of U~g to

Ug[[~]].

Theorem. Let g be a symmetrizable Kac–Moody algebra with a fixed Dg–

structure and U~g the corresponding Drinfeld–Jimbo quantum group with the

analogous Dg–structure. For any choice of a Lie associator Φ, there exists an

equivalence of quasi–Coxeter categories between

O := ((OintgB,⊗B,ΦB, σRB), (ΓBB′ , JBB

′

F ), ΦGF, Si,C)

and

O~ := ((OintU~gB,⊗B, id, σR~B), (Γ~BB′ , id), id, S~i )

30 1. INTRODUCTION

where ⊗B denotes the standard tensor product in OintgB

and

Si,C = si exp(h

2· Ci)

ΦB ≡ 1 mod ~2

RB = exp(~2

ΩD)

Alt2JBB′

F ≡ ~2

(rB′ − r21B′

2− rB − r21

B

2

)mod ~2

and ΦGF , JBB′

F are weight zero elements.

We note that Kac–Moody algebras of finite, affine and hyperbolic type are

easily endowed with a D–structure. This is canonical in finite and affine type

and depends upon an initial choice in the hyperbolic case. For an arbitrary

Kac–Moody is not always possible to define a D–structure, but it is still un-

clear how to characterize this special class of Lie algebras (cf. Ch.3, Section 6).

The results of Chapters 2 and 3 are contained in the preprint [ATL1].

4. Results - Part III

4.1. The quasi–Coxeter braided monoidal category naturally associated

to U~g admits an equivalent counterpart for Ug[[~]], as illustrated in the previ-

ous section. The next goal is to show that there is at most one quasi–Coxeter

braided monoidal category for the classical enveloping algebra Ug[[~]] with

prescribed local monodromies (Chapter 4). This result is needed to show that

the quasi–Coxeter braided monoidal category naturally attached to the quan-

tum affine algebra U~g is equivalent to the one constructed for Ug[[~]], which

underlies the monodromy of the KZ and Casimir connections of g.

4. RESULTS - PART III 31

4.2. In the semisimple case, a central role is played by the Dynkin–

Hochschild bicomplex of a D–bialgebra A, which controls the deformations

of quasi–Coxeter quasitriangular quasibialgebra structures on A. In Chap-

ter 4 Section 2, we extend the notion of the Dynkin–Hochschild bicomplex

to the case of monoidal D–categories, using the language of endomorphisms

algebra of monoidal functors and the corresponding cohomology theory. In

analogy with the semisimple case, this bicomplex controls the deformations of

the quasi–Coxeter structures on the D–category.

4.3. Let g be a Kac–Moody algebra of affine type with extended Dynkin

diagram Dg. For any proper connected subdiagram D ⊂ Dg, let gD ⊂ lD ⊂ g

be the corresponding simple and Levi subalgebras, with decomposition lD =

gD ⊕ cD. Denote by

ΩD = xa ⊗ xa, CD = xa · xa and rgD =∑α0:

supp(α)⊆D

(α, α)

2· eα ∧ fα

where xaa, xaa are dual basis of gD with respect to (·, ·), the correspond-

ing invariant tensor, Casimir operator and standard solution of the modified

classical Yang–Baxter equation for gD respectively.

We then consider two quasi–Coxeter braided monoidal categories of type

D of the form:

(Oint

g [[~]], OintgD

[[~]],⊗, RD,ΦKZD , Si,c, Φk

(D;αi,αj), F(D;αi), J

k(D;αi)

)

32 1. INTRODUCTION

for k = 1, 2, where ⊗ is the standard tensor product in Og, (F(D;αi), Jk(D;αi)

)

are fiber functors,

Si,c = si · exp (~/2 · Ci) (4.1)

RD = exp(~/2 · ΩD) (4.2)

Alt2 J(D;αi) ≡~2

(rgD − rgD\αi

)mod ~2 (4.3)

and Φk(D;αi,αj)

, Jk(D;αi)are of weight 0, i.e., they properly extend to the corre-

sponding Levi’s subalgebras.

We are explicitly assuming that the braided monoidal structure on OintgD

is induced in both categories by the KZ equations for gD. This assumption

is not strictly necessary and can be easily dropped by Drinfeld’s uniqueness

theorem [D3, Prop. 3.5]. The proof of the rigidity property amounts to match

the relative twists Jk(D;αi) and the Casimir associator Φk

(D;αi,αj).

4.4. The procedure matching the relative twists relies on the essential

uniqueness (up to gauge transformations) of solutions of the equation

F (ΦVWZ) JV⊗W,Z (JV,W ⊗ idZ) = JV,W⊗Z (idV ⊗JW,Z) (ΦD)F (V )F (W )F (Z)

in HomUgD(F (V ) ⊗ F (W ) ⊗ F (Z), F (V ⊗ W ⊗ Z)), where F = FDgD and

V,W,Z ∈ Ointg .


In the semisimple case (cf. [TL3, Thm. 6.1]), it is possible to construct

elements u ∈ (Ug)gD [[~]] and λ ∈∧2 cD[[~]] such that

expλ · J2 = u ? J1

where u?J = ∆(u−1)·J ·u⊗u, i.e., the lack of uniqueness of such solutions (up

to gauge transformation) is measured by an element λ ∈∧2 cD. We illustrate

two cases in which is possible to eliminate λ and obtain the gauge equivalence.

When g is a finite type Lie algebra and |Dg \ D| = 1, the space cD is

one dimensional and λ = 0. Therefore, the twists J(D;αi) are uniquely deter-

mined up to gauge transformation. If we try to apply a similar result when

g is an affine Kac–Moody algebra, we can assert the uniqueness of the twists

J(D;αi) only for proper subdiagrams D ⊂ Dg. Indeed, when D = Dg and D′ is

a subdiagram satisfying |Dg\D′| = 1, dim cD′ = 2 and λ is not necessarily zero.

4.5. Alternatively, we can take into the account the behavior of the rel-

ative twists with respect to the Chevalley involution θ of g. It is easy to show

that if the relative twists satisfy the additional condition

Jθ = J21

then λ can be chosen to be zero and it is possible to determine a unique gauge

transformation u satisfying uθ = u.

34 1. INTRODUCTION

Once again, this condition cannot be directly applied to our case. First of

all, we have to make sense of the identity in the setting of category O, where

θ does not define an involution of the category. This can be easily rephrased

in terms of a natural compatibility of the monoidal functor with the duality

functor in category O.

More importantly, we have to make sure that the relative twists, described

in the previous section, satisfy (up to gauge equivalence) the compatibility con-

dition with the Chevalley involution. The solution to this problem presents

several difficulties. The current results are presented in Chapter 4 Section 3.

4.6. An easy computation shows that this is true for the Etingof–Kazhdan

twist in its finite–dimensional formulation i.e., as a tensor structure on the

functor hM+⊗M− . This is proved by identifying Jθ and J21 as tensor structures

on hM−⊗M+ given by the same formulae. This case is extremely special and it

can be shown, working modulo ~3, that this condition is not satisfied by the

twist JEK, in its universal form, or by the relative twist. We observe that in

both cases the symmetry between M+ and M− is broken.

We then observe that the condition Jθ = J21 can be weakened, assuming

there exists a natural transformation χ satisfying χθ = χ−1 and χ ? Jθ = J21.

Since the relative twist is gauge equivalent to a composition of twists of

Etingof–Kazhdan type, we reduced the problem to find a suitable χ for JEK.


This element is easily constructed in the finite–dimensional case, but its exis-

tence is not yet proven in the infinite–dimensional case.

We also have to remark here that the construction of the elements u and

λ above strongly relies on the assumption that the Hochschild cohomology of

the completion of Ug with respect to category O can be described in terms

of the exterior algebra of g, possibly completed. The result is well–know for

Ug [D3, 2.2], but it is not present in the literature for completed universal

enveloping algebras.

4.7. It remains to construct the natural transformation matching the

Casimir associators Φk(D;αi,αj)

and preserving the local monodromies. This

amounts to solving the following problem. Given a Cartan 2–cocycle in the

Dynkin complex, i.e., a special set of elements ϕ = ϕ(D;αi,αj) ⊂ hD, there

exists a Cartan 1–cochain a such that dDa = ϕ. The construction is recur-

sive and the initial choice a(αi,αi) = 0 allows to preserve the local monodromies.

The original proof of this fact in [TL4] is actually incomplete, since it is

implicitly assumed that the fundamental coweights λ∨i,D ⊂ hD do not de-

pends upon D or, in other words, that λ∨i,D′ ∈ Cλ∨i,D for D′ ⊂ D. We then

present a corrected proof of the semisimple case and we extend it to the affine

case. Since the proof strongly relies on the combinatorics of the Dynkin dia-

gram of g, it is actually necessary to go through the entire computation for

an affine Dynkin diagram. It is unclear at present whether such result holds

for an arbitrary symmetrizable Kac–Moody algebra. The results of Chapter 4

36 1. INTRODUCTION

are available in [ATL2].

5. Towards a complete solution

To complete the solution to Problem 1.3, we still need two steps, namely,

an affine extension of the rational Casimir connection, and the construction of

a monodromic quasi–Coxeter braided monoidal structure on the category of

integrable modules in category O of Ug[[~]].

5.1. The affine extension of the Casimir connection. A rational

Casimir connection was defined for any symmetrisable Kac–Moody algebra in

[FMTV]. It has the form

∇affC = d− ~

∑α>0

dα

α:Kα:

where α ranges over all positive (real and imaginary) roots, Kα is the truncated

Casimir operator corresponding to α, and

:Kα:=

mult(α)∑i=1

2e(i)−αe

(i)α

is its Wick ordered form.

This ordering makes the sum over α locally finite on a module in category

O, but breaks the equivariance of ∇affC with respect to the Weyl group of g.

If g is affine, this equivariance can however be restored by adding to ∇affC a

closed one–form A with values in the Cartan subalgebra h ⊂ g [TL6]. The

monodromy of the connection ∇affC + A then gives rise to a one–parameter

family of representations of the generalized affine braid group on V .

5. TOWARDS A COMPLETE SOLUTION 37

5.2. The Fusion operator. In the semisimple case, the quasi-Coxeter

quasitriangular quasibialgebra structure on Ug[[~]] underlying the monodromy

of the KZ and Casimir connections is obtained as follows [TL5]. The Dg–

algebra structure is the standard one. For any D ⊆ Dg, the quasitriangular

quasibialgebra structure on UgD[[~]] is given by the associator of the KZ equa-

tions for gD and the corresponding R–matrix. The gauge transformations ΦGF

and local monodromies are given by the De Concini–Procesi associators of the

Casimir connection, and its local monodromies. Finally, the twists JF are ob-

tained as suitable limits of a joint solution with prescribed asymptotics of the

system (cf. [FMTV])

dµJ =~2

∑α>0

dα

α

[∆(Kα)J − J(K(1)

α +K(2)α )]

+ (z1 ad(dµ(1)) + z2 ad(dµ(2)))J

dzJ = ~d(z1 − z2)

z1 − z2

Ω12J + (ad(µ(1))dz1 + ad(µ(2))dz2)J

where (z, µ) ∈ C2 × h and J has values in Ug[[~]]⊗2.

Following [Ka], we write

Kα =:Kα: + mult(α) · ν−1(α)

where ν : h → h∗ is the isomorphism induced by the non–degenerate bilinear

form on h. Since J is of weight zero, the above equations can be written with

the Wick ordered :Kα: instead of Kα, and therefore make sense for g. Moreover,

preliminary computations show that the construction of the joint solution J

carries over to affine Lie algebras and, in fact, arbitrary symmetrizable Kac–

Moody algebras.

38 1. INTRODUCTION

6. Outline of the dissertation

6.1. Chapter 2 is devoted to rephrasing the definition of quasi–Coxeter

quasitriangular quasibialgebra in categorical terms. This yields the notion

of a quasi–Coxeter category, which is to a generalized braid group B what

a braided tensor category is to Artin’s braid groups, and of a quasi–Coxeter

braided monoidal category. Both notions will be concisely rephrased in terms

of a 2–functor from a combinatorially defined 2–category qC(D) to the 2–

categories Cat,Cat⊗ of categories and monoidal categories respectively.

6.2. In Chapter 3, we establish an equivalence of quasi–Coxeter braided

monoidal categories for Ug[[~]] and U~g, where g is a Kac–Moody algebra

endowed with a D–algebra structure.

In Section 1, we start with a short review the construction of the Etingof–

Kazhdan quantization functor and the associated equivalence of categories,

following [EK1, EK6]. In Section 2, we modify this construction by using

generalized Verma modules L−, N+, and we obtain a relative fiber functor

Γ : DΦ(g) → DΦD(gD). In Section 3, we define the quantum generalized

Verma modules L~− and N~+. Using suitably defined Props we then show, in

Section 4 that these are isomorphic to the EK quantization of their classical

counterparts. In Section 5, we use these results to show that, for any given

chain of Manin triples ending in a given g, there exists a natural transformation

that preserves the given chain. Finally, in Section 6, we apply these results

to the case of a Kac–Moody algebra g and obtain the desired tranport of its

quasi–Coxeter quasitriangular quasibialgebra structure to the completion of

Ug[[~]] with respect to category O, integrable modules.

6. OUTLINE OF THE DISSERTATION 39

6.3. The goal of Chapter 4 is to show that, under appropriate hypotheses,

there is at most one quasi–Coxeter braided monoidal category for the classical

enveloping algebra Ug[[~]] with prescribed local monodromies (Thm 4.2).

In Section 1, we review a number of basic notions from [TL4], in particular

that of the Dynkin–Hochschild bicomplex of a D–bialgebra A which controls

the deformations of quasi–Coxeter quasitriangular quasibialgebra structures

on A. In Section 2, we extend the notion of the bicomplex to the case of

completion of algebras, adopting the language of endomorphisms algebra of

monoidal functors and the corresponding cohomology theory. In Section 3

we study invariant properties of the relative twist under the Chevalley invo-

lution of g and we prove a uniqueness property up to gauge transformation.

Section 4 contains the rigidity result, consisting in the construction of a natu-

ral transformation matching the Casimir associators and preserving the local

monodromies.

6.4. Finally, in Appendix 1 we discuss the isomorphism between the com-

pletion of the universal enveloping algebra described in [D5, EK6] and the

algebra of the endomorphism of the forgetful functor in category O. In Ap-

pendix 3 we use the relative twist to construct an Hopf algebra object in the

Drinfeld category of gD. The corresponding Radford algebra realizes a partial

quantization of Ug.

CHAPTER 2

Quasi–Coxeter Categories

This chapter is devoted to rephrasing the definition of quasi–Coxeter qua-

sitriangular quasibialgebra given in [TL4, Ch. 6] in categorical terms. This

yields the notion of a quasi–Coxeter category, which is to a generalized braid

group B what a braided tensor category is to Artin’s braid groups, and of

a quasi–Coxeter braided monoidal category. Interestingly, both notions can

be concisely rephrased in terms of a 2–functor from a combinatorially defined

2–category qC(D) to the 2–categories Cat,Cat⊗ of categories and monoidal cat-

egories respectively. The objects of qC(D) are the subdiagrams of the Dynkin

diagram D of B and, for two subdiagrams D′ ⊆ D′′, HomqC(D)(D′′, D′) is

the fundamental 1–groupoid of the De Concini–Procesi associahedron for the

quotient diagram D′′/D′ [DCP2, TL4].

1. Diagrams and nested sets

We review in this section a number of combinatorial notions associated to

a diagram D, in particular the definition of nested sets on D and of the De

Concini–Procesi associahedron of D following [DCP2] and [TL4, Section 2].

1.1. Nested sets on diagrams. By a diagram we shall mean an undi-

rected graph D with no multiple edges or loops. We denote the set of vertices

of D by V(D) and set |D| = |V(D)|. A subdiagram B ⊆ D is a full subgraph

of D, that is, a graph consisting of a subset V(B) of vertices of D, together

41

42 2. QUASI–COXETER CATEGORIES

with all edges of D joining any two elements of V(B). We will often abusively

identify such a B with its set of vertices and write i ∈ B to mean i ∈ V(B).

We denote by SD(D) the set of subdiagrams of D and by CSD(D) the set of

connected subdiagrams.1

The union B1 ∪ B2 of two subdiagrams B1, B2 ⊂ D is the subdiagram

having V(B1)∪ V(B2) as its set of vertices. Two subdiagrams B1, B2 ⊂ D are

orthogonal if V(B1)∩V(B2) = ∅ and no two vertices i ∈ B1, j ∈ B2 are joined

by an edge in D. B1 and B2 are compatible if either one contains the other or

they are orthogonal.

Definition. A nested set on a diagram D is a collection H of pairwise

compatible, connected subdiagrams of D which contains the empty set and

the connected components D1, . . . , Dr of D.

1.2. The De Concini–Procesi associahedron. LetND be the partially

ordered set of nested sets on D, ordered by reverse inclusion. ND has a unique

maximal element, consisting of the connected components Di of D and the

empty set. Its minimal elements are the maximal nested sets. We denote the

set of maximal nested sets on D by Mns(D). Every nested set H on D is

uniquely determined by a collection Hiri=1 of nested sets on the connected

components of D. We therefore obtain canonical identifications

ND =r∏i=1

NDi and Mns(D) =r∏i=1

Mns(Di)

1The definition of diagram, provided in [TL4, 2.1], did not allow empty graphs. InSection 2, we need to include the empty set in SD(D) and we are therefore removing thenon–emptiness condition.

1. DIAGRAMS AND NESTED SETS 43

The De Concini–Procesi associahedron AD is the regular CW–complex

whose poset of (nonempty) faces is ND. It easily follows from the definition

that

AD =r∏i=1

ADi

It can be realized as a convex polytope of dimension |D|−r. For any H ∈ ND,

we denote by dim(H) the dimension of the corresponding face in AD.

1.3. The rank function of ND. For any nested set H on D and B ∈ H,

we set iH(B) =⋃mi=1 Bi where the Bi’s are the maximal elements ofH properly

contained in B.

Definition. Set αBH = B \ iH(B). We denote by

n(B;H) = |αBH| and n(H) =∑B∈H

(n(B;H)− 1

)An element B ∈ H is called unsaturated if n(B;H) > 1.

Proposition. .

(i) For any nested set H ∈ ND,

n(H) = |D| − |H| = dim(H)

(ii) H is a maximal nested set if and only if n(B;H) = 1 for any B ∈ H.

(iii) Any maximal nested set is of cardinality |D|+ 1.

For any F ∈ Mns(D), B ∈ F , iF(B) denotes the maximal element in F

properly contained in B and αBF = B \ iF(B) consists of one vertex, denoted

αBF .


For any F ∈ Mns(D), B ∈ F , we denote by FB ∈ Mns(B) the maximal

nested set induced by F on B.

1.4. Quotient diagrams. Let B ⊆ D be a subdiagram with connected

components B1, . . . , Bm.

Definition. The set of vertices of the quotient diagram D/B is V(D) \

V(B). Two vertices i 6= j of D/B are linked by an edge if and only if the

following holds in D

i 6⊥ j or i, j 6⊥ Bk for some k = 1, . . . ,m

For any connected subdiagram C ⊆ D not contained in B, we denote by

C ⊆ D/B the connected subdiagram with vertex set V (C) \ V (B).

1.5. Compatible subdiagrams of D/B.

Lemma. Let C1, C2 * B be two connected subdiagrams of D which are

compatible. Then

(i) C1, C2 are compatible unless C1 ⊥ C2 and C1, C2 ⊥Bi for some i.

(ii) If C1 is compatible with every Bi, then C1 and C2 are compatible.

In particular, if F is a nested set on D containing each Bi, then F = C,

where C runs over the elements of F such that C * B, is a nested set on

D/B.

Let now A be a connected subdiagram of D/B and denote by A ⊆ D the

connected sudbdiagram with vertex set

V (A) = V (A)⋃

i:Bi⊥V (A)

V (Bi)

1. DIAGRAMS AND NESTED SETS 45

Clearly, A1 ⊆ A2 or A1 ⊥ A2 imply A1 ⊆ A2 and A1 ⊥ A2 respectively, so the

lifting map A→ A preserves compatibility.

1.6. Nested sets on quotients. For any connected subdiagrams A ⊆

D/B and C ⊆ D, we have

A = A and C = C⋃

i:Bi⊥C

Bi

In particular, C = C if, and only if, C is compatible with B1, . . . , Bm and not

contained in B. The applications C → C and A→ A therefore yield a bijection

between the connected subdiagrams of D which are either orthogonal to or

strictly contain each Bi and the connected subdiagrams of D/B. This bijection

preserves compatibility and therefore induces an embedding ND/B → ND.

This yields an embedding

ND/B ×NB = ND/B ×(NB1 × · · · × NBm

)→ ND

with image the poset of nested sets on D containing each Bi. Similarly, for

any B ⊆ B′ ⊆ B′′, we obtain a map

∪ : NB′′/B′ ×NB′/B → NB′′/B

The map ∪ restricts to maximal nested sets. For any B ⊂ B′, we denote by

Mns(B′, B) the collection of maximal nested sets on B′/B. Therefore, for any

B ⊂ B′ ⊂ B′′, we obtain an embedding

∪ : Mns(B′′, B′)×Mns(B′, B)→ Mns(B′′, B)


such that, for any F ∈ Mns(B′′, B′),G ∈ Mns(B′, B),

(F ∪ G)B′/B = G

1.7. Elementary and equivalent pairs.

Definition. An ordered pair (G,F) in Mns(D) is called elementary if G

and F differ by one element. A sequence H1, . . . ,Hm in Mns(D) is called

elementary if |Hi+1 \ Hi| = 1 for any i = 1, 2, . . . ,m− 1.

Definition. The support supp(F ,G) of an elementary pair in Mns(D) is

the unique unsaturated element of F ∩ G. The central support z supp(F ,G) is

the union of the maximal elements of F ∩G properly contained in supp(F ,G).

Thus

z supp(F ,G) = supp(F ,G) \ αsupp(F ,G)F∩G

Definition. Two elementary pairs (F ,G), (F ′,G ′) in Mns(D) are equiva-

lent if

supp(F ,G) = supp(F ′,G ′)

αsupp(F ,G)F = α

supp(F ′,G′)F ′ α

supp(F ,G)G = α

supp(F ′,G′)G′

2. Quasi–Coxeter categories

The goal of this section is to rephrase the notion of quasi–Coxeter quasi-

triangular quasibialgebra defined in [TL4] in terms of categories of represen-

tations.

2. QUASI–COXETER CATEGORIES 47

2.1. Algebras arising from fiber functors. We shall repeatedly need

the following elementary

Lemma. Consider the following situation

C

F **

H// D

G

α

w

A

where A, C,D are additive k–linear categories, F,G,H functors, and α is

an invertible transformation. If H is an equivalence of categories, the map

End(G) −→ End(F ) given by

gW 7→ Ad(α−1V )(gH(V ))

is an algebra isomorphism.

2.2. D–categories. Recall [TL4, Section 3] that, given a diagram D,

a D–algebra is a pair (A, ABB∈SD(D)), where A is a unital associative k–

algebra and ABB∈SD(D) is a collection of subalgebras indexed by SD(D) and

satisfying

AB ⊆ AB′ if B ⊆ B′ and [AB, AB′ ] = 0 if B ⊥ B′

with A∅ = k. Moreover, if B ∈ SD(D) has connected components Bk, then

AB is generated by the subalgebras ABk.

The following rephrases the notion of D–algebras in terms of their category

of representations.

Definition. A D–category C = (CB, FBB′) is the datum of


• a collection of k–linear additive categories CBB⊆D

• for any pair of subdiagrams B ⊆ B′, an additive k–linear functor

FBB′ : CB′ → CB1

• for any B ⊂ B′, B′ ⊥ B′′, B′, B′′ ⊂ B′′′, a homomorphism of k–

algebras

η : End(FBB′)→ End(F(B∪B′′)B′′′)

satisfying the following properties

• For any B ⊆ D, FBB = idCB .

• For any B ⊆ B′ ⊆ B′′, FBB′ FB′B′′ = FBB′′ .

• For any B′ ⊥ B′′ and any B ⊂ B′ and any B′′′ ⊃ B′, B′′, the following

diagram of algebra homomorphisms commutes:

End(FBB′)id⊗1

tt

1⊗η++

End(FBB′)⊗ End(FB′B′′′)

**

End(FB(B∪B′′))⊗ End(F(B∪B′′)B′′′)

ss

End(FBB′′′)

Remark. We will usually think of C∅ as a base category and at the functors

F as forgetful functors. Then the family of algebras End(FB) defines, through

the morphisms α, a structure of D–algebra on End(FD). Conversely, every

D–algebra A admits such a description setting CB = RepAB for B 6= ∅ and

C∅ = Vectk, FBB′ = i∗B′B, where iB′B : AB ⊂ AB′ is the inclusion.

1We adopt the convention FB = F∅B .


Remark. The conditions satisfied by the maps η imply that, for any B′ ⊥

B′′ and any B ⊂ B′ and B′′′ ⊃ B′, B′′, the images in End(FBB′′′) of the maps

End(FBB′)→ End(FBB′FB′B′′′) = End(FBB′′′)

End(FB(B∪B′′))→ End(FB(B∪B′′)F(B∪B′′)B′′′) = End(FBB′′′)

commute. In particular, given B′′′ =⊔rj=1Bj, with Bj ∈ SD(D) pairwise

orthogonal, the images in End(FB′′′) of the maps

End(FBj)→ End(FBjFBjB′′′) = End(F ′′′B )

pairwise commute. This condition replaces the D–algebra axiom

[AB′ , AB′′ ] = 0 ∀ B′ ⊥ B′′

which is equivalent to the condition that AB′ ⊂ AB′′

B , for any B ⊃ B′, B′′.

Remark. It may seem more natural to replace the equality of functors

FBB′ FB′B′′ = FBB′′ by the existence of invertible natural transformations

αB′

BB′′ : FBB′ FB′B′′ ⇒ FBB′′ for any B ⊆ B′ satisfying the associativity

constraints αB′

BB′′′ FBB′(αB′′

B′B′′′) = αB′′

BB′′′ (αB′

BB′′)FB′′B′′′ for any B ⊆ B′ ⊆

B′′ ⊆ B′′′. A simple coherence argument shows however that this leads to a

notion of D–category which is equivalent to the one given above.

Remark. The above definition of D–category may be rephrased as fol-

lows. Let I(D) be the category whose objects are subdiagrams B ⊆ D and

morphisms B′ → B the inclusions B ⊂ B′. Then a D–category is a functor

C : I(D) → Cat which is compatible with the disjoint union of orthogonal

subdiagrams.


2.3. Strict morphisms of D–categories. The interpretation ofD–categories

in terms of I(D) suggests that a morphism of D–categories C, C ′ is one of the

corresponding functors

I(D)

C''

C′

::

Cat

This yields the following definition. For simplicity, we shall assume throughout

that C∅ = C ′∅.

Definition. A strict morphism of D–categories C, C ′ is the datum of

• for any B ⊆ D, a functor HB : CB → C ′B• for any B ⊆ B′, a natural transformation

CB′HB′

//

FBB′

C ′B′

F ′BB′

γBB′

z

CBHB

// C ′B

(2.1)

such that

• H∅ = id

• γBB = idHB

• For any B ⊆ B′ ⊆ B′′,

γBB′′ = γBB′ γB′B′′


where is the composition of natural transformations defined by

CB′′ //

C ′B′′

y

CB′ //

C ′B′

y

CB // C ′B

(2.2)

The diagram (2.1), withB = ∅, induces an algebra homomorphism End(F ′B′)→

End(FB′) which, by (2.2) is compatible with the maps End(FB) → End(FB′)

and End(F ′B)→ End(F ′B′) for any B ⊂ B′. As pointed out in [TL4, 3.3], this

condition is too restrictive and will be weakened in the next paragraph.

2.4. Morphisms of D–categories.

Definition. A morphism of D–categories C, C ′, with C∅ = C ′∅, is the datum

of

• for any B ⊆ D a functor HB : CB → C ′B• for any B ⊆ B′ and F ∈ Mns(B,B′), a natural transformation

CB′HB′

//

FBB′

C ′B′

F ′BB′

γFBB′

z

CBHB

// C ′B

such that

• H∅ = id

• γFBB = idHB


• for any B ⊆ B′ ⊆ B′′, F ∈ Mns(B,B′), G ∈ Mns(B′, B′′),

γFBB′ γGB′B′′ = γF∨GBB′′

Remark. For any F ∈ Mns(B′), the natural transformation γFB′ induces

an algebra homomorphism ΨFB′ : End(F ′B′)→ End(FB′) such that the following

diagram commutes for any B ∈ F

End(F ′B′)ΨFB′// End(FB′)

End(F ′B)ΨFBB//

OO

End(FB)

OO

In particular, the collection of homomorphisms ΨFD defines a morphism of

D–algebras End(F ′D)→ End(FD) in the sense of [TL4, 3.4].

Remark. The above definition may be rephrased as follows. Let M(D) be

the category with objects the subdiagramsB ⊆ D and morphisms Hom(B′, B) =

Mns(B′, B), with composition given by union and lift (cf. 1.6). There is a

forgetful functor M(D) → I(D) which is the identity on objects and maps

F ∈ Mns(B′, B) to the inclusion B ⊆ B′. Given two D–categories C, C ′ :

I(D) → Cat a morphism C → C ′ as defined above coincides with a morphism

of the functors M(D)→ Cat given by the composition

M(D) // I(D)C//

C′// Cat

2.5. Quasi–Coxeter categories.


Definition. A labeling of the diagram D is the assignment of an integer

mij ∈ 2, 3, . . . ,∞ to any pair i, j of distinct vertices of D such that

mij = mji mij = 2

if and only if i ⊥ j.

Let D be a labeled diagram.

Definition. The generalized braid group of type D BD is the group gen-

erated by elements Si labeled by the vertices i ∈ D with relations

SiSj · · ·︸︷︷︸mij

= SjSi · · ·︸︷︷︸mij

for any i 6= j such that mij <∞. We shall also refer to BD as the braid group

corresponding to D.

Definition. A quasi–Coxeter category of type D

C = (CB, FBB′, ΦFG, Si)

is the datum of

• a D–category C = (CB, FBB′)

• for any F ,G in Mns(B,B′), a natural transformation

ΦFG ∈ Aut(FBB′)

• for any vertex i ∈ V(D), an element

Si ∈ Aut(Fi)


satisfying the following conditions

• Orientation. For any F ,G ∈ Mns(B,B′),

ΦGF = Φ−1FG

• Transitivity. For any F ,G,H ∈ Mns(B,B′),

ΦFG ΦGH = ΦFH

• Factorization. The assignment

Φ : Mns(B,B′)2 → Aut(FB′B)

is compatible with the embedding

∪ : Mns(B,B′)×Mns(B′, B′′)→ Mns(B,B′′)

for any B′′ ⊂ B′ ⊂ B, i.e., the diagram

Mns(B,B′)2 ×Mns(B′, B′′)2

∪

Φ×Φ// Aut(FB′′B′)× Aut(FB′B)

Mns(B,B′′)2Φ

// Aut(FB′′B)

is commutative.

• Braid relations. For any pairs i, j of distinct vertices of B, such

that 2 < mij < ∞, and F ,G in Mns(B) such that i ∈ F , j ∈ G, the

following relations hold in End(FB)

Ad(ΦGF)(Si) · Sj · · · = Sj · Ad(ΦGF)(Si) · · ·


where, by abuse of notation, we denote by Si its image in End(FB)

and the number of factors in each side equals mij.

The elements Si will be referred to as local monodromies.

Remark. The factorization property implies the support and forgetful

properties of [TL4, 3.12].

• Support. For any elementary pair (F ,G) in Mns(B,B′), let S =

supp(F ,G), Z = z supp(F ,G) ⊆ D. Then there are uniquely defined

maximal nested sets F , G ∈ Mns(Z, S) such that

ΦFG = idBZ ΦF G idB′S

where the expression above denotes the composition of natural trans-

formations

CB′

FBB′

FBB′

ΦGF+3

CB′

FSB′

CS

FZS

FZS

ΦGF+3=

CZFBZ

CB CB

• Forgetfulness. For any equivalent elementary pairs (F ,G), (F ′,G ′)

in Mns(B,B′)

ΦFG = ΦF ′G′


Remark. To rephrase the above definition, consider the 2–category qC(D)

obtained by adding to M(D) a unique 2–isomorphism ϕBB′

GF : F → G for any

pair of 1–morphisms F ,G ∈ Mns(B′, B), with the compositions

ϕBB′

HG ϕBB′

GF = ϕBB′

HF and ϕBB′

F2G2 ϕB′B′′F1G1

= ϕBB′′

F2∪F1 G2∪G1

where F ,G,H ∈ Mns(B′, B), B ⊂ B′ ⊆ B′′ and F1,G1 ∈ Mns(B′′, B′), F2,G2 ∈

Mns(B′, B). There is a unique functor qC(D) → I(D) extending M(D) →

I(D), and a quasi–Coxeter category is the same as a 2–functor qC(D)→ Cat

fitting in a diagram

qC(D) //

##

Cat

I(D)

<<

Note that, for any B ⊂ B′, the category HomqC(D)(B′, B) is the 1–groupoid of

the De Concini–Procesi associahedron on B′/B [TL4].

2.6. Morphisms of quasi–Coxeter categories.

Definition. A morphism of quasi–Coxeter categories C, C ′ of type D is a

morphism (H, γ) of the underlying D–categories such that

• For any i ∈ B, the corresponding morphism Ψi : End(F ′i )→ End(Fi)

satisfies

Ψi(S′i) = Si

• For any elementary pair (F ,G) in Mns(B,B′),

HB(ΦFG) γFBB′ (Φ′GF)HB′ = γGBB′


in Nat(F ′BB′ HB′ , HB FBB′), as in the diagram

C ′B′

Φ′FG//

55

HB′

CB′

ΦFG+3 γGrz

γF

C ′B55

HB

CB

Remark. Note that the above condition can be alternatively stated in

terms of morphisms ΨF as the identity

ΨG Ad(ΦGF) = Ad(Φ′GF) ΨF

2.7. Strict monoidal D–categories.

Definition. A strict monoidal D–category C = (CB, FBB′, JBB′)

is a D–category C = (CB, FBB′) where

• for any B ⊆ D, (CB,⊗B) is a strict monoidal category

• for any B ⊆ B′, the functor FBB′ is endowed with a tensor structure

JBB′

with the additional condition that, for every B ⊆ B′ ⊆ B′′, JBB′ JB′B′′ =

JBB′′ .

Remark. The tensor structure JB induces on End(FB) a coproduct ∆B :

End(FB)→ End(F 2B), where F 2

B := ⊗ (FB FB), given by

gV V ∈CB 7→ ∆B(g)VW := Ad(JBVW )(gV⊗W )V,W∈CB


Moreover, for any B ⊆ B′, End(FB) is a subbialgebra of End(FB′), i.e., the

following diagram is commutative

End(FB)

∆B// End(F 2

B)

End(FB′)∆B′

// End(F 2B′)

Remark. Note that a strict monoidal D–category can be thought of as

functor

C : I(D)→ Cat⊗0

where Cat⊗0 denotes the 2–category of strict monoidal category, with monoidal

functors and gauge transformations.

Definition. A morphism of strict monoidal D–categories is a natural

transformation of the corresponding 2–functors M(D) → Cat⊗0 , obtained by

composition with M(D)→ I(D).

2.8. Monoidal D–categories.

Definition. A monoidal D–category

C = ((CB,⊗B,ΦB), FBB′, JFBB′)

is the datum of

• A D–category ((CB, FBB′) such that each (CB,⊗B,ΦB) is a ten-

sor category, with C∅ a strict tensor category, i.e., Φ∅ = id.

• for any pair B ⊆ B′ and F ∈ Mns(B,B′), a tensor structure JBB′

F on

the functor FBB′ : CB′ → CB


with the additional condition that, for any B ⊆ B′ ⊆ B′′, F ∈ Mns(B′′, B′),

G ∈ Mns(B′, B),

JGBB′ JFB′B′′ = JF∪GBB′′

Remark. The usual comparison with the algebra of endomorphisms leads

to a collection of bialgebras (End(FB),∆F , ε) endowed with multiple coprod-

ucts, indexed by Mns(B).

Remark. A monoidal D–category can be thought of as a functor M(D)→

Cat⊗ fitting in a diagram

M(D)

// Cat⊗

w

I(D) // Cat

Accordingly, a morphism of monoidal D–categories is one of the corresponding

functors.

M(D)

C))

C′

99

Cat⊗

2.9. Fibered monoidal D–categories. We shall often be concerned

with monoidal D–categories such that the underlying categories (CB,⊗B) are

strict, and the functors FBB′ : (CB′ ,⊗B′)→ (CB,⊗B) are tensor functors. This

may be described in terms of the category M(D) as follows. Let DCat⊗ be

the 2–category of Drinfeld categories, that is strict tensor categories (C,⊗)

endowed with an additional associativity constraint Φ making (C,⊗,Φ) a

monoidal category. There is a canonical forgetful 2–functor DCat⊗ → Cat⊗0 .


We shall say that a monoidal D–category fibers over a strict monoidal D–

category if the corresponding functor M(D)→ Cat⊗ maps into DCat⊗ and fits

in a commutative diagram

M(D)

// DCat⊗

v~

I(D) // Cat⊗0

In this case, the coproduct ∆F on a bialgebra End(FB) is the twist of a reference

coassociative coproduct ∆0 on End(FD) such that ∆0 : End(FB)→ End(F 2B).

2.10. Braided monoidal D–categories.

Definition. A braided monoidal D–category

C = ((CB,⊗B,ΦB, βB), (FBB′ , JFBB′)

is the datum of

• a monoidal D–category ((CB,⊗B,ΦB), (FBB′ , JFBB′)

• for every B ⊆ D, a commutativity constraint βB in CB, defining a

braiding in (CB,⊗B,ΦB).

Remark. Note that the tensor functors (FBB′ , JFBB′) : CB′ → CB are not

assumed to map the commutativity constraint βB′ to βB.

Definition. A morphism of braided monoidal D–categories from C to C ′

is a morphism of the underlying monoidal D–categories such that the functors

HB : CB → C ′B are braided tensor functors.


Remark. The fact that HB are braided tensor functors automatically

implies that

Ψ⊗2F ((RB)JF ) = (R′B)J ′F

in analogy with [TL4], where RB = (12) βB. We are assuming that C∅ = C ′∅is a symmetric strict tensor category.

2.11. Quasi–Coxeter braided monoidal categories.

Definition. A quasi–Coxeter braided monoidal category of type D

C = ((CB,⊗B,ΦB, βB), (FBB′ , JFBB′), ΦFG, Si)

is the datum of

• a quasi–Coxeter category of type D,

C = (CB, FBB′, ΦFG, Si)

• a braided monoidal D–category

C = ((CB,⊗B,ΦB, βB), (FBB′ , JBB′

F ))

satisfying the following conditions

• for any B ⊆ B′, and G,F ∈ Mns(B,B′), the natural transformation

ΦFG ∈ Aut(FBB′) determines an isomorphism of tensor functors from

(FBB′ , JGBB′) to (FBB′ , J

FBB′), that is for any V,W ∈ CB′ ,

(ΦGF)V⊗W (JFBB′)V,W = (JGBB′)V,W ((ΦGF)V ⊗ (ΦGF)W )


• for any i ∈ D, the following holds:

∆i(Si) = (Ri)21Ji· (Si ⊗ Si)

A morphism of quasi–Coxeter braided monoidal categories of type D is a

morphism of the underlying quasi–Coxeter categories and braided monoidal

D–categories.

Remark. A quasi–Coxeter braided monoidal category of type D deter-

mines a 2–functor qC(D)→ Cat⊗ fitting in a diagram

qC(D)

// Cat⊗

w

I(D) // Cat

Note however that this functor does not entirely capture the quasi–Coxeter

braided monoidal category since it does not encode the commutativity con-

straints βB and automorphisms Si.

CHAPTER 3

An equivalence of quasi–Coxeter categories

In this chapter, we establish an equivalence of quasi–Coxeter braided monoidal

categories for Ug[[~]] and U~g, where g is a Kac–Moody algebra endowed with

a D–algebra structure. The construction is based on the equivalence, defined

by Etingof and Kazhdan in [EK6], between the categoryO for Ug[[~]] and U~g.

This amounts essentially to building a suitable tensor structure on the

restriction functors and natural transformations as described in 2.6. Our con-

struction works more generally for an inclusion (gD, gD,−, gD,+) ⊂ (g, g−, g+)

of Manin triples over a field k of characteristic zero, so we adopt this setting

until Section 6, where we apply the results to the Kac–Moody case.

In Section 1, we start with a short review the construction of the Etingof–

Kazhdan quantization functor and the associated equivalence of categories,

following [EK1, EK6]. In Section 2, we modify this construction by using

generalized Verma modules L−, N+, and we obtain a relative fiber functor

Γ : DΦ(g) → DΦD(gD). In Section 3, we define the quantum generalized

Verma modules L~− and N~+. Using suitably defined Props we then show, in

Section 4 that these are isomorphic to the EK quantization of their classical

counterparts. In Section 5, we use these results to show that, for any given

chain of Manin triples ending in a given g, there exists a natural transformation

63

64 3. AN EQUIVALENCE OF QUASI–COXETER CATEGORIES

that preserves the given chain. Finally, in Section 6, we apply these results

to the case of a Kac–Moody algebra g and obtain the desired tranport of its

quasi–Coxeter quasitriangular quasibialgebra structure to the completion of

Ug[[~]] with respect to category O, integrable modules.

1. Etingof-Kazhdan quantization

We review in this section the results obtained in [EK1, EK6]. More

specifically, we follow the quantization of Lie bialgebras given in [EK1, Part

II] and the case of generalized Kac–Moody algebras from [EK6].

1.1. Topological vector spaces. The use of topological vector spaces

is needed in order to deal with convergence issues related to duals of infinite

dimensional vector spaces and tensor product of such spaces.

Let k be a field of characteristic zero with the discrete topology and V a

topological vector space over k. The topology on V is linear if open subspaces

in V form a basis of neighborhoods of zero. Let V be endowed with a linear

topology and pV the natural map

pV : V → lim(V/U)

where the limit is taken over the open subspaces U ⊆ V . Then V is called

separated if pV is injective and complete if pV is surjective. Throughout this

section, we shall call topological vector space a linear, complete, separated

topological space.

1. ETINGOF-KAZHDAN QUANTIZATION 65

If U is an open subspace of a topological vector space V , then the quotient

V/U is discrete. It is then possible, given two topological vector spaces V and

W , to define the topological tensor product as

V ⊗W := limV/V ′ ⊗W/W ′

where the limit is take over open subspaces of V and W . We then denote by

Homk(V,W ) the topological vector space of continuous linear operators from

V to W equipped with the weak topology. Namely, a basis of neighborhoods

of zero in Homk(V,W ) is given by the collection of sets

Y (v1, . . . , vn,W1, . . . ,Wn) := f ∈ Homk(V,W ) | f(vi) ∈ Wi, i = 1, . . . , n

for any n ∈ N, vi ∈ V and Wi open subspace in W for all i = 1, . . . , n.In par-

ticular, if W = k with the discrete topology, the space V ∗ = Homk(V, k) has

a basis of neighborhoods of zero given by orthogonal complements of finite–

dimensional subspaces in V . When V is finite–dimensional, V ∗ coincides with

the linear dual and the weak topology coincides with the discrete topology.

The space of formal power series in ~ with coefficients in a topological

vector space V , V [[~]] = V ⊗k[[~]], is also a complete topological space with a

natural structure of a topological k[[~]]-module. A topological k[[~]]-module is

complete if it is isomorphic to V [[~]] for some complete V . The additive cat-

egory of complete k[[~]]-module, denoted A, where morphisms are continuous

k[[~]]-linear maps, has a natural symmetric monoidal structure. Namely, the

tensor product on A is defined to be the quotient of the tensor product V ⊗W

by the image of the operator ~ ⊗ 1 − 1 ⊗ ~. This tensor product will be still


denoted by ⊗. There is an extension of scalar functor from the category of

topological spaces to A, mapping V to V [[~]]. This functor respects the tensor

product, i.e., (V ⊗W )[[~]] is naturally isomorphic to V [[~]]⊗W [[~]].

1.2. Equicontinuous modules. Fix a topological Lie algebra g.

Definition. Let V be a topological vector space. We say that V is an

equicontinuous g-module if:

• the map πV : g → Endk V is a continuous homomorphism of topo-

logical Lie algebras;

• πV (g)g∈g is an equicontinuous family of linear operators,i.e., for

any open subspace U ⊆ V , there exists U ′ such that πV (g)U ′ ⊂ U

for all g ∈ g.

Clearly, a topological vector space with a trivial g-module structure is an

equicontinuous g-module. Moreover, given equicontinuous g-modules V,W,U ,

the tensor product V ⊗W has a natural structure of equicontinuous g-module

and (V ⊗W )⊗U is naturally identified with V ⊗(W ⊗U). The category of

equicontinuous g-modules is then a symmetric monoidal category, with braid-

ing defined by permutation of components. We denote this category by Repeq g.

1.3. Lie bialgebras and Manin triples. A Manin triple is the data of

a Lie algebra g with

• a nondegenerate invariant inner product 〈, 〉;

• isotropic Lie subalgebras g± ⊂ g;

such that

• g = g+ ⊕ g− as vector space;


• the inner product defines an isomorphism g+ ' g∗−;

• the commutator of g is continuous with respect to the topology ob-

tained by putting the discrete and the weak topology on g−, g+ re-

spectively.

Under these assumptions, the commutator on g+ ' g∗− induces a cobracket

on g−, satisfying the cocycle condition [D1]. Therefore, g− is canonically

endowed with a Lie bialgebra structure. Notice that, in absolute generality,

g+ is only a topological Lie bialgebra, i.e., δ(g+) ⊂ g+⊗g+. The inner product

also gives rise to an isomorphism of vector spaces g− ' g∗∗− ' g∗+, where the

latter is the continuous dual, though this isomorphism does not respect the

topology. Conversely, every Lie bialgebra a defines a Manin triple (a⊕a∗, a, a∗).

1.4. Verma modules. In [EK1], Etingof and Kazhdan constructed two

main examples of equicontinuous g-modules in the case when g belongs to a

Manin triple (g, g+, g−). The modules M±, defined as

M+ = Indgg− k M− = Indg

g+k

are freely generated over U(g±) by a vector 1± such that g∓1±. Therefore, they

are naturally identified, as vector spaces, to U(g±) via x1± → x. The modules

M− and M∗+, with appropriate topologies, are equicontinuous g-modules.

The module M− is an equicontinuous g-module with respect to the discrete

topology. The topology on M+ comes, instead, from the identification of vector

spaces

M+ ' U(g+) =⋃n≥0

U(g+)n


where U(g+)n is the set of elements of degree at most n. The topology on

U(g+)n is defined through the linear isomorphism

ξn :n⊕j=0

Sjg+ → U(g+)n

where Sjg+ is considered as a topological subspace of (g⊗j− )∗, embedded with

the weak topology. Finally, U(g+) is equipped with the topology of the col-

imit. Namely, a set U ⊆ U(g+) is open if and only if U ∩ U(g+)n is open for

all n. With respect to the topology just described, the action of g on M+ is

continuous.

Consider now the vector space of continuous linear functionals on M+

M∗+ = Homk(M+, k) ' colim Homk(U(g+)n, k)

It is natural to put the discrete topology on U(g+)∗n, since, as a vector space,

U(g+)∗n 'n⊕j=0

Sjg∗+ 'n⊕j=0

Sjg− ' U(g−)n

We then consider on M∗+ the topology of the limit. This defines, in particular,

a filtration by subspaces (M∗+)n satisfying

0→ (M∗+)n →M∗

+ → (U(g+)n)∗ → 0

and such that M∗+ = limM∗

+/(M∗+)n. The topology of the limit on M∗

+ is, in

general, stronger than the weak topology of the dual. Since the action of g on

M+ is continuous, M∗+ has a natural structure of g–module. In particular, this

is an equicontinuous g–action.


1.5. Drinfeld category. The natural embedding

g− ⊗ g∗− ⊂ Endk(g−)

induces a topology on g−⊗g∗− by restriction of the weak topology in Endk(g−).

With respect to this topology, the image of g− ⊗ g∗− is dense in Endk(g−) and

the topological completion g− ⊗ g∗− is identified with Endk(g−). Under this

identification, the identity operator defines an element r ∈ g−⊗g∗−.

Given two equicontinuous g–modules V,W , the map

πV ⊗ πW : g− ⊗ g∗− → Endk(V ⊗W )

naturally extends to a continuous map g−⊗g∗− → Endk(V ⊗W ). Therefore,

the Casimir operator

Ω = r + rop ∈ g−⊗g∗− ⊕ g∗−⊗g−

defines a continuous endomorphism of V ⊗W , ΩVW = (πV ⊗πW )(Ω), commut-

ing with the action of g.

Following [D2], it is possible to define a structure of braided monoidal

category on the category of deformed equicontinuous g–module, depending on

the choice of a Lie associator Φ, the bifunctor ⊗ and the Casimi operator Ω.

The commutativity constraint is explicitly defined by the formula

βVW = (12) e~2

ΩVW ∈ Homg(V ⊗W,W ⊗V )[[~]]


We denote this braided tensor category braided tensor category DΦ(Ug). The

category of equicontinuous g–modules is equivalent to the category of Yetter-

Drinfeld module over g−, YD(g−). The equivalence holds at the level of tensor

structure induced by the choice of an associator Φ,

DΦ(Ug) ' YDΦ(Ug−[[~]])

1.6. Verma modules. The modules M± are identified, as vector spaces,

with the enveloping universal algebras Ug±. Their comultiplications induce

the Ug–intertwiners i± : M± → M±⊗M±, mapping the vectors 1± to the g∓-

invariant vectors 1± ⊗ 1±.

For any f, g ∈ M∗+, consider the linear functional M+ → k defined by

v 7→ (f⊗g)(i+(v)). This functional defines a map i∗+ : M∗+⊗M∗

+ →M∗+, which

is continuous and extends to a morphism in Rep g[[~]], i∗+ : M∗+⊗M∗

+ → M∗+.

The pairs (M−, i−) and (M∗+, i∗+) form, respectively, a coalgebra and an algebra

object in DΦ(Ug).

For any V ∈ DΦ(Ug), the vector space Homg(M−,M∗+⊗V ) is naturally

isomorphic to V , as topological vector space, through the isomorphism f 7→

(1+ ⊗ 1)f(1−).

1.7. The fiber functor and the EK quantization. We will now recall

the main results from [EK1, EK2]. Where no confusion is possible, we will

abusively denote ⊗ by ⊗. Let then F be the functor

F : DΦ(Ug)→ A F (V ) = HomDΦ(Ug)(M−,M∗+ ⊗ V )


There is a natural transformation

J ∈ Nat(⊗ (F F ), F ⊗)

defined, for any v ∈ F (V ), w ∈ F (W ), by

JVW (v ⊗ w) = (i∨+ ⊗ 1⊗ 1)A−1β−123 A(v ⊗ w)i−

where A is defined as a morphism

(V1 ⊗ V2)⊗ (V3 ⊗ V4)→ V1 ⊗ ((V2 ⊗ V3)⊗ V4)

by the action of (1⊗ Φ2,3,4)Φ1,2,34.

Theorem. The natural transformation J is invertible and defines a tensor

structure on the functor F .

The tensor functor (F, J) is called fiber functor. The algebra of endo-

morphisms of F is therefore naturally endowed with a topological bialgebra

structure, as described in the previous section.1

The object F (M−) ∈ A has a natural structure of Hopf algebra, defined

by the multiplication

m : F (M−)⊗ F (M−)→ F (M−) m(x, y) = (i∨+ ⊗ 1)Φ−1(1⊗ y)x

1By topological bialgebra we do not mean topological over k[[~]]. We are instead referringto the fact that the algebra End(F ) has a natural comultiplication ∆ : End(F )→ End(F 2),where End(F 2) can be interpreted as an appropriate completion of End(F )⊗2.


and the comultiplication

∆ : F (M−)→ F (M−)⊗ F (M−) ∆(x) = J−1(F (i−)(x))

The algebra F (M−) is naturally isomorphic as a vector space with M−[[~]] '

Ug−[[~]] and

Theorem. The algebra UEK

~ g− = F (M−) is a quantization of the algebra

Ug−.

In [EK2], it is shown that this construction defines a functor

QEK : LBA(k)→ QUE(K)

where LBA(k) denotes the category of Lie bialgebras over k and QUE(K) de-

notes the category of quantum universal enveloping algebras over K = k[[~]].

Another important result in [EK2] states the invertibility of the functor QEK.

The map

m− : UEK

~ g− → End(F ) m−(x)V (v) = (i∨+ ⊗ 1)Φ−1(1⊗ v)x

where V ∈ YDΦ(Ug−[[~]]) and v ∈ F (V ), is, indeed, an inclusion of Hopf

algebras. The map m− defines an action of UEK

~ g− on F (V ). Moreover, the

map

F (V )→ F (M−)⊗ F (V ) v 7→ RJ(1⊗ v)

where RJ denotes the twisted R–matrix, defines a coaction of UEK

~ g− on F (V )

compatible with the action, therefore


Theorem. The fiber functor F : YDΦ(Ug−[[~]]) → A lifts to an equiva-

lence of braided tensor categories

F : YDΦ(Ug−[[~]])→ YD(UEK

~ g−)

1.8. Generalized Kac-Moody algebras. Denote by k a field of char-

acteristic zero. We recall definitions from [Ka] and [EK6]. Let A = (aij)i,j∈I

be an n× n symmetrizable matrix with entries in k, i.e. there exists a (fixed)

collection of nonzero numbers dii∈I such that diaij = djaji for all i, j ∈ I.

Let (h,Π,Π∨) be a realization of A. It means that h is a vector space of di-

mension 2n− rank(A), Π = α1, . . . , αn ⊂ h∗ and Π∨ = h1, . . . , hn ⊂ h are

linerly independent, and (αi, hj) = aji.

Definition. The Lie algebra g = g(A) is generated by h, ei, fii∈I with

defining relations

[h, h′] = 0 h, h′ ∈ h; [h, ei] = (αi, h)ei

[h, fi] = −(αi, h)fi; [ei, fj] = δijhi

There exists a unique maximal ideal r in g that intersects h trivially. Let

g := g/r. The algebra g is called generalized Kac-Moody algebra. The Lie alge-

bra g is graded by principal gradation deg(ei) = 1, deg(fi) = −1, deg(h) = 0,

and the homogenous component are all finite-dimentional.

Let us now choose a non–degenerate bilinear symmetric form on h such

that 〈h, hi〉 = d−1i (αi, h). Following [Ka], there exists a unique extension of

the form 〈, 〉 to an invariant symmetric bilinear form on g. For this extension,


one gets 〈ei, fj〉 = δijd−1i . The kernel of this form on g is r, therefore it de-

scends to a non–degenerate bilinear form on g.

Let n±, b± be the nilpotent and the Borel subalgebras of g, i.e., n± are

generated by ei, fi, respectively, and b± := n±⊕h. Since [n±, h] ⊂ n±, we

get Lie algebra maps ¯ : b± → h and we can consider the embeddings of Lie

subalgebras η± : b± → g⊕ h given by

η±(x) = (x,±x)

Define the inner product on g⊕ h by 〈, 〉g⊕h = 〈, 〉g − 〈, 〉h.

Proposition. The triple (g⊕ h, b+, b−) with inner product 〈−,−〉g⊕h and

embeddings η± is a graded Manin triple.

Under the embeddings η±, the Lie subalgebras b± are isotropic with re-

spect to 〈, 〉g⊕h. Since 〈, 〉g and 〈, 〉h are invariant symmetric non–degenerate

bilinear form, so is 〈, 〉g⊕h.

The proposition implies that g ⊕ h, b+, b− are Lie bialgebras. Moreover,

b∗+ ' bcop− as Lie bialgebras (where b∗+ :=⊕

(b+)∗n denotes the restricted dual

space and by cop we mean the opposite cocommutator). The Lie bialgebra

structures on b± are then described by the following formulas:

δ(h) = 0, h ∈ h ⊂ b±;

δ(ei) =di2

(ei ⊗ hi − hi ⊗ ei) =di2ei ∧ hi; δ(fi) =

di2fi ∧ hi


The Lie subalgebra (0, h) | h ∈ h is therefore an ideal and a coideal in g⊕h.

Thus, the quotient g = (g⊕h)/h is also a Lie bialgebra with Lie subbialgebras

b± and the same cocommutator formulas.

1.9. Quantization of Kac–Moody algebras and category O. In

[EK6], Etingof and Kazhdan proved that, for any symmetrizable irreducible

Kac-Moody algebra g, the quantization UEK

~ g is isomorphic with the Drinfeld–

Jimbo quantum group U~g.

In particular, they construct an isomorphism of Hopf algebras U~b+ '

UEK

~ b+, inducing the identity on Uh[[~]], where b+ is the Borel subalgebra and

h is the Cartan subalgebra of g. Thanks to the compatibility with the doubling

operations

DUEK

~ b+ ' UEK

~ Db+

proved by Enriquez and Geer in [EG], the isomorphism for the Borel subalge-

bra induces an isomorphism U~g ' UEK

~ g.

Recall that the category O for g, denoted Og is defined to be the category

of all h–diagonalizable g–modules V , whose set of weights P(V ) belong to a

union of finitely many cones

D(λs) = λs +∑i

Z≥0αi λs ∈ h∗, s = 1, ..., r


and the weight subspaces are finite–dimensional. We denote by Og[[~]] the

category of deformation g–representations,i.e., representations of g on topo-

logically free k[[~]]–modules with the above properties (with weights in h∗[[~]]).

In a similar way, one defines the category OU~g: it is the category of U~g–

modules which are topologically free over k[[~]] and satisfy the same conditions

as in the classical case.

The morphism of Lie bialgebras

Db+ → g ' Db+/(h ' h∗)

gives rise to a pullback functors

Og → YD(Ub+) Og,Φ[[~]]→ YDΦ(Ub+[[~]])

where Og,Φ denotes the category Og with the tensor structure of the Drinfeld

category. Similarly, the morphism of Hopf algebras

DUEK

~ b+ → UEK

~ g ' U~g

gives rise to a pullback functor

OU~g → YD(UEK

~ b+)

Theorem. The equivalence F reduces to an equivalence of braided tensor

categories

FO : Og,Φ[[~]]→ OU~g


which is isomorphic to the identity functor at the level of h–graded k[[~]]–

modules.

1.10. The isomorphism ΨEK. In [EK6], Etingof–Kazhdan showed that

the equivalence F induces an isomorphism of algebras

ΨEK : Ug[[~]]→ U~g

where

Ug = limUβ Uβ = Ug/Iβ, β ∈ NI

Iβ being the left ideal generated by elements of weight less or equal β (analo-

gously for U~g, cf. [EK6, Sec. 4] and [D5, 8.1]).

Proposition. The isomorphism ΨEK coincides with the isomorphism in-

duced by the equivalence FO, as explained in Section 2.1.

Proof. The identification of the two isomorphism is constructed in the

following way:

(a) First, we show that there is a canonical map

End(fO)→ CEnd(U)(Endg(U))

(b) There is a canonical multiplication in U , so that

(i) There is a canonical map

CEnd(U)(Endg(U))→ U


(ii) For every V ∈ O the action of Ug lifts to an action of U

Ug //

End(V )

U

<<

(c) It defines a map U → End(fO) and we have an isomorphism of alge-

bras

U ' End(fO)

A detailed proof of the equivalence is given in Appendix A Section 1.

If g is a semisimple Lie algebra, the equivalence of categories F leads to an

isomorphism of algebras

U(Db+)[[~]] ' DUEK

~ b+ =⇒ Ug[[~]] ' U~g

which is the identity modulo h. Toledano Laredo proved in [TL4, Prop. 3.5]

that such an isomorphism cannot be compatible with all the isomorphisms

Uslαi2 [[~]] ' U~slαi2 ∀i

where αi are the simple roots of g. This amounts to a simple proof that the

isomorphism ΨEK cannot be, in general, an isomorphism of D–algebras.

2. A relative Etingof–Kazhdan functor

2.1. In this section, we consider a split inclusion of Manin triples

iD : (gD, gD,+, gD,−) → (g, g+, g−)

2. A RELATIVE ETINGOF–KAZHDAN FUNCTOR 79

We then define a relative version of the Verma modules M±, and use them to

prove the following

Theorem. There is a tensor functor

Γ : DΦ(Ug)→ DΦD(UgD)

canonically isomorphic, as abelian functor, to the restriction functor i∗D.

2.2. Split inclusions of Manin triples.

Definition. An embedding of Manin triples

i : (gD, gD,−, gD,+) −→ (g, g−, g+)

is a Lie algebra homomorphism i : gD → g preserving inner products, and

such that i(gD,±) ⊂ g±.

Denote the restriction of i to gD,± by i±. i± give rise to maps p± = i∗∓ :

g± → gD,±, defined via the identifications g± ' g∗∓ and gD,± ' g∗D,∓ by

〈p±(x), y〉D = 〈x, i∓(y)〉

for any x ∈ g± and y ∈ gD,∓. These map satisfy p± i± = idgD,± since, for any

x ∈ gD,±, y ∈ gD,∓,

〈p± i±(x)− x, y〉D = 〈i±(x), i∓(y)〉 − 〈x, y〉D = 0

This yields in particular a a direct sum decomposition g± = i(gD,±) ⊕ m±,

where

m± = Ker(p±) = g± ∩ i(gD)⊥


Definition. The embedding i : (gD, gD,−, gD,+) −→ (g, g−, g+) is called

split if the subspaces m± ⊂ g± are Lie subalgebras.

2.3. Split pairs of Lie bialgebras. For later use, we reformulate the

above notion in terms of bialgebras via the double construction.

Definition. A split pair of Lie bialgebras is the data of

• Lie bialgebras (a, [, ]a, δa) and (b, [, ]b, δb).

• Lie bialgebra morphisms i : a→ b and p : b→ a such that pi = ida.

Proposition. There is a one–to–one correspondence between split inclu-

sions of Manin triples and split pairs of Lie bialgebras. Specifically,

(i) If i : (gD, gD,−, gD,+) −→ (g, g−, g+) is a split inclusion of Manin

triples, then (gD,−, g−, i−, i∗+) is a split pair of Lie bialgebras.

(ii) Conversely, if (a, b, i, p) is a split pair of Lie bialgebras, then i⊕ p∗ :

(Da, a, a∗) −→ (Db, b, b∗) is a split inclusion of Manin triples.

2.4. Proof of (i) of Proposition 2.3. Given a split inclusion

i = i− ⊕ i+ : (gD, gD,−, gD,+) −→ (g, g−, g+)

we need to show that i− and i∗+ are Lie bialgebra morphisms. By assumption,

i− is a morphism of Lie algebras, and i∗+ one of coalgebras. Since i− = (i∗−)∗,

it suffices to show that p± = i∗∓ preserve Lie brackets.

We claim to this end that m± are ideals in g±. Since [m±,m±] ⊆ m± by

assumption, this amounts to showing that [i(gD,±),m±] ⊆ m±. This follows


from the fact that [i(gD,±),m±] ⊆ g±, and from

〈[i(gD,±),m±], i(gD,∓)〉 = 〈m±, [i(gD,±), i(gD,∓)]〉 ⊂ 〈m±, i(gD,±)〉+〈m±, i(gD,∓)〉

where the first term is zero since g± is isotropic, and the second one is zero by

definition of m±.

Let now X1, X2 ∈ g±, and write Xj = i±(xj) + yj, where xj ∈ gD,± and

yj ∈ m±. Since m± = Ker(p±) and p± i± = id, we have [p±(X1), p±(X2)] =

[x1, x2], while

p±[X1, X2] = p± (i±[x1, x2] + [i±x1, y2] + [y1, i±x2] + [y1, y2]) = [x1, x2]

where the last equality follows from the fact that m± is an ideal.

2.5. Proof of (ii) of Proposition 2.3. The bracket on Da is defined by

[a, φ] = ad∗(a)(φ)− ad∗(φ)(a) = −〈φ, [a,−]a〉+ 〈φ⊗ id, δa(a)〉

for any a ∈ a, φ ∈ a∗. Analogously for Db. Therefore, the equalities

〈p∗(φ)⊗ id, δb(i(a))〉 =〈φ⊗ id, (p⊗ id)(i⊗ i)δa(a)〉

=〈φ⊗ id, (id⊗i)δa(a)〉 = i(〈φ⊗ id, δa(a)〉)

and

〈p∗(φ), [i(a), b]b〉 = 〈φ, p([i(a), b]b)〉 = 〈φ, [a, p(b)]a〉


for all a ∈ a and b ∈ b, imply that the map i⊕ p∗ is a Lie algebra map. It also

respects the inner product, since for any a ∈ a, φ ∈ a∗,

〈p∗(φ), i(a)〉 = 〈φ, p i(a)〉 = 〈φ, a〉

Finally, m− = Ker(p) and m+ = Ker i∗ are clearly subalgebras.

2.6. Parabolic Lie subalgebras. Let

iD = i− ⊕ i+ : (gD, gD,−, gD,+)→ (g, g−, g+)

be a split embedding of Manin triples. We henceforth identify gD as a Lie

subalgebra of g with its induced inner product, and gD,± as subalgebras of g±

noting that, by Proposition 2.3, gD,− is a sub Lie bialgebra of g−.

The following summarizes the properties of the subspaces m± = g± ∩ g⊥D

and p± = m± ⊕ gD.

Proposition.

(i) m± is an ideal in g±, so that g± = m± o gD,±.

(ii) [gD,m±] ⊂ m±, so that p± = m± o gD are Lie subalgebras of g.

(iii) δ(m−) ⊂ m− ⊗ gD,− + gD,− ⊗m−, so that m− ⊆ g− is a coideal.

Proof. (i) was proved in 2.4. (ii) Since

〈[gD,m±], gD〉 = 〈m±, [gD, gD]〉 = 0


we have [gD,m±] ⊂ g⊥D = m− ⊕m+. Moreover,

〈[gD,m±],m±〉 = 〈gD, [m±,m±]〉 = 〈gD,m±〉 = 0

since m± is a subalgebra, and it follows that [gD,m±] ⊂ m±. (iii) is clear since

m− is the kernel of a Lie coalgebra map.

Remark. If the inclusion iD is compatible with a finite type N–grading,

then m+ ⊂ g+ is a coideal. Moreover, p± are Lie subbialgebras of g such

that the projection p± → gD is a morphism of bialgebras. Namely, a finite

type N–grading allows to define a Lie bialgebra structure on g, g+. We then

get a vector space decomposition g± = m± ⊕ gD,± and a Lie bialgebra map

g± → gD,±. It is also possible to define the Lie subalgebras

p± = m± ⊕ gD ⊂ g

If we assume the existence of a compatible grading on g and gD,i.e., preserved

by iD, then the natural maps

p± ⊂ g p± → gD

are morphisms of Lie bialgebras.

2.7. The relative Verma Modules.

Definition. Given a split embedding of Manin triples gD ⊂ g, and the

corresponding decomposition g = m− ⊕ p+, let L−, N+ be the relative Verma

modules defined by

L− = Indgp+

k and N+ = Indgm− k


Proposition. The g–modules L− and N∗+ are equicontinuous.

The description of the appropriate topologies on L− and N∗+, and the proof of

their equicontinuity will be carried out in 2.8–2.11.

2.8. Equicontinuity of L−. As vector spaces,

L− ' Um− ⊂ Ug−

so it is natural to equip L− with the discrete topology. The set of opera-

tors πL−(x)x∈g is then an equicontinuous family, and the continuity of πL−

reduces to checking that, for every element v ∈ L−, the set

Yv = b ∈ g+| b.v = 0

is a neighborhood of zero in g+. Since Um− embeds naturally in Ug− the proof

is identical to [EK1, Lemma 7.2]. We proceed by induction on the length of

v = ai1 . . . ain1−. If n = 0, then v = 1− and Yv = g+. If n > 1, then assume

v = ajw, with w = ai1 . . . ain−11− and Yw open in g+. For every x ∈ g+

x.v = x.(ajw) = [x, aj].w + (ajx).w

Call Z the subset of g+

Z = x ∈ g+| [x, aj] ∈ Yw

Z is open in g+, by continuity of bracket [, ], and clearly Z ∩ Yw ⊂ Yv.

2.9. Topology of N+. As vector spaces,

N+ = Indgm−k ' Up+ ' colimUnp+


where Unp+ denotes the standard filtration of Up+, so that

Unp+ 'n⊕

m=0

Smp+ =⊕i+j≤n

(Sig+ ⊗ SjgD,−

)We turn this isomorphism into an isomorphism of topological vector spaces,

by taking on Sig+ and SjgD,− the topologies induced by the embeddings

Sig+ → (g⊗i− )∗ and SjgD,− → g⊗jD,−

With respect to these topologies, Ump+ is closed inside Unp+ for m < n, and

we equip N+ with the direct limit topology. We shall need the following

Lemma. For any x ∈ g, the map πN+(x) : N+ → N+ is continuous.

Proof. We need to show that for any neighborhood of the origin U ⊂ N+,

there exists a neighborhood of zero U ′ ⊂ N+ such that πN+(x)U ′ ⊂ U . The

topology on N+ comes from the decomposition Up+ ' Ug+ ⊗ UgD,−, so that

an open neighborhood of zero in N+ has the form U ⊗UgD,−+Ug+⊗V , with

U open in Ug+ and V open in UgD,−. We apply the same procedure used in

[EK1, Lemma 7.3] to construct a set U ′ ⊗ UgD,−, with U ′ open in Ug+, such

that

πN+(x)(U ′ ⊗ UgD,−) ⊂ U ⊗ UgD,− ⊂ U ⊗ UgD,− + Ug+ ⊗ V

Since the topology on UgD,− is discrete, the set U ′⊗UgD,− is open in N+ and

the lemma is proved.

2.10. Topology of N∗+. As vector spaces,

N∗+ ' (Up+)∗ ' lim(Unp+)∗


Define a filtration (N∗+)n on N∗+ by

0→ (N∗+)n → (Up+)∗ → (Unp+)∗ → 0

so that N∗+ ⊃ (N∗+)0 ⊃ (N∗+)1 ⊃ · · · , and we get an isomorphism of vector

spaces

N∗+ ' limN∗+/(N∗+)n

Finally, we use the isomorphism to endow N∗+ with the inverse limit topology.

Lemma. πN∗+(x)x∈g is an equicontinuous family of operators.

Proof. Since p+ acts on N+ by multiplication,

p+(N∗+)n ⊂ (N∗+)n−1

If x ∈ m− and xi ∈ Up+ for i = 1, . . . , n, then in Ug,

xx1 · · · xn = x1 · · ·xnx−n∑i=0

x1 · · ·xi−1[xi, x]xi+1 · · ·xn

where [xi, x] ∈ g. Iterating shows that (x.f)(x1 · · ·xn) = 0 if f ∈ (N∗+)n,

so that x(N∗+)n ⊂ (N∗+)n. Then, for any neighborhood of zero of the form

U = (N∗+)n, it is enough to take U ′ = (N∗+)n+1 to get g(N∗+)n+1 ⊂ (N∗+)n.

2.11. Equicontinuity of N∗+.

Lemma. The map πN∗+ : g→ End(N∗+) is a continuous map.

Proof. Since g− is discrete, it is enough to check that, for any f ∈ N∗+

and n ∈ N, the subset

Y (f, n) = b ∈ g+| b.f ∈ (N∗+)n


is open in g+, i.e.

bi.f ∈ (N∗+)n for a.a. i ∈ I

Since f ∈ N∗+ ' limN∗+/(N∗+)n, we have f = fn where fn is the class of f

modulo (N∗+)n. Therefore bi.f ∈ (N∗+)n iff

(bi.f)n = bi.fn+1 = 0

Now, for any x1 · · ·xn ∈ Unp+, we have

bi.fn+1(x1 · · ·xn) = −fn+1(bix1 · · ·xn) = 0

for a.a. i ∈ I and the lemma is proved (it is enough to exclude the indices

corresponding to the generators involved in the expression of fn+1).

As a vector spaces, we can identify

p∗+ = g∗+ ⊕ g∗D,− ' g− ⊕ gD,+ = p−

We can give as a basis for p+ and p−

p+ ⊃ bii∈I , arr∈I(D) p− ⊃ aii∈I , brr∈I(D)

and obvious relations

(bi, aj) = δij (bi, br) = 0

(ar, aj) = 0 (ar, bs) = δrs


with i, j ∈ I, r, s ∈ I(D). We can then identify fn+1 with an element in

Un+1p−. Call Tn+1(f) the set of indices of all ai involved in the expression of

fn+1. Excluding these finite set of indices we get the result.

2.12. Coalgebra structure on L−, N+. Define g–module maps

i− : L− → L−⊗L− and i+ : N+ → N+⊗N+

by mapping 1∓ to 1∓⊗1∓. Note that, under the identification L− ' Um− and

N+ ' Up+, i∓ correspond to the coproduct on Um− and Up+ respectively.

Following [D3, Prop. 1.2], we consider the invertible element T ∈ (Ug⊗Ug)[[~]]

satisfying relations:

S⊗3(Φ321) · (T ⊗ 1) · (∆⊗ 1)(T ) = (1⊗ T )(1⊗∆)(T ) · Φ

T∆(S(a)) = (S ⊗ S)(∆(a))T

LetN∗+ be as before and f, g ∈ N∗+. Consider the linear functional in Homk(N+, k)

defined by

v 7→ (f ⊗ g)(T · i+(v))

This functional is continuous, so it belongs to N∗+ and allow us to define the

map

i∨+ ∈ Homk(N∗+ ⊗N∗+, N∗+)[[~]] , i∨+(f ⊗ g)(v) = (g ⊗ f)(T · i+(v))


This map is continuous and extends to a map from N∗+⊗N∗+ to N∗+. For any

a ∈ g, we have

i∨+(a(f ⊗ g))(v) = (f ⊗ g)((S ⊗ S)(∆(a))T · i+(v)) =

= (f ⊗ g)(T∆(S(a)) · i+(v)) =

= i∨+(f ⊗ g)(S(a).v) = (a.i∨+(f ⊗ g))(v)

and then i∨+ ∈ Homg(N∗+⊗N∗+, N∗+)[[~]].

The following shows that L− and N+ are coalgebra objects in the Drinfeld

categories of g–modules and (g, gD)–bimodules respectively.

Proposition. The following relations hold

(i) Φ(i− ⊗ 1)i− = (1⊗ i−)i−.

(ii) i∨+(1⊗ i∨+)Φ = i∨+(i∨+ ⊗ 1)S⊗3(Φ−1D )ρ

where (−)ρ denotes the right gD–action on N∗+.

Proof. We begin by showing that

Φ(1⊗3− ) = 1⊗3

− and Φ(1⊗3+ ) = ΦD(1⊗3

+ ) (2.1)

To prove the first identity, it is enough to notice that, since g+1− = 0 and

Ω =∑

(ai ⊗ bi + bi ⊗ ai),

Ωij(1⊗3− ) = 0


Then Φ(1⊗3− ) = 1⊗3

− . To prove the second one, we notice that m−1+ = 0 and

that we can rewrite

Ω =∑j∈ID

(aj⊗bj+bj⊗aj)+∑i∈I\ID

(ai⊗bi+bi⊗ai) = ΩD+∑i∈I\ID

(ai⊗bi+bi⊗ai)

where ajj∈ID is a basis of gD,− and bjj∈ID is the dual basis of gD,+. Then

Ωij(1⊗3+ ) = ΩD,ij(1

⊗3+ )

and, since for any element x ∈ gD, the right and the left gD-action coincide

on 1+, i.e. x.1+ = 1+.x, we have

Ωij(1⊗3+ ) = (1⊗3

+ )ΩD,ij

and consequently Φ(1⊗3+ ) = ΦD(1⊗3

+ ).

To prove (i), note that since the comultiplication in Um− is coassociative,

we have (i−⊗1)i− = (1⊗ i−)i−. We therefore have to show that Φ(i−⊗1)i− =

(1 ⊗ i−)i−. This is an obvious consequence of (2.1) and the fact that m− is

generated by 1−.


To prove (ii), consider v ∈ N+,

i∨+(1⊗ i∨+)(Φ(f ⊗ g ⊗ h))(v) =

= (h⊗ g ⊗ f)((S⊗3(Φ321) · (T ⊗ 1) · (∆⊗ 1)(T )) · (i+ ⊗ 1)i+(v)) =

= (h⊗ g ⊗ f)((1⊗ T )(1⊗∆)(T ) · Φ(i+ ⊗ 1)i+(v)) =

= (h⊗ g ⊗ f)((1⊗ T )(1⊗∆)(T )(1⊗ i+)i+(v)ΦD) =

= (S⊗3(ΦD)ρ(h⊗ g ⊗ f))((1⊗ T )(1⊗∆)(T )(1⊗ i+)i+(v)) =

= i∨+(i∨+ ⊗ 1)(S⊗3(Φ321D )ρ(f ⊗ g ⊗ h))(v) =

= i∨+(i∨+ ⊗ 1)S⊗3(Φ−1D )ρ(f ⊗ g ⊗ h)(v)

and (ii) is proved.

2.13. The fiber functor over gD. To any representation V [[~]] ∈ RepUg[[~]],

we can associate the k[[~]]–module

Γ(V ) = Homg(L−, N∗+⊗V )[[~]]

where Homg is the set of continuous homomorphisms, equipped with the weak

topology. The right gD–action on N∗+ endows Γ(V ) with the structure of a left

gD–module.

Proposition. The complete vector space Homg(L−, N∗+⊗V ) is isomorphic

to V as equicontinous gD–module. The isomorphism is given by

αV : f 7→ (1+ ⊗ 1)f(1−)

for any f ∈ Homg(L−, N∗+⊗V ).


Proof. By Frobenius reciprocity, we get an isomorphism

Homg(L−, N∗+⊗V ) ' Homp+(k, N∗+⊗V ) ' Homk(k, V ) ' V

given by the map

f 7→ (1+ ⊗ 1)f(1−)

For f ∈ Γ(V ) and x ∈ UgD, x.f ∈ Γ(V ) is defined by

x.f = (S(x)ρ ⊗ id) f

For any x ∈ UgD, we have

∑i,j

x(1)i fj ⊗ x(2)

i vj = ε(x)f(1−)

where ∆(x) =∑

i x(1)i ⊗ x

(2)i and f(1−) =

∑j fj ⊗ vj. Using the identity

1⊗ x =∑i

(S(x(1)i )⊗ 1) ·∆(x

(2)i )

holding in any Hopf algebra, we obtain

(1⊗ x)f(1−) =∑i

(S(x(1)i ε(x

(2)i ))⊗ 1)f(1−) = (S(x)⊗ 1)f(1−)

Finally, we have

x.αV (f) = 〈1+ ⊗ id, (1⊗ x)f(1−)〉 =

= 〈1+ ⊗ id, (S(x)⊗ 1)f(1−)〉 =

= 〈1+ ⊗ id, (S(x)ρ ⊗ 1)f(1−)〉 = αV (x.f)

Therefore, Γ(V ) is isomorphic to V [[~]] as equicontinuous gD-module.


2.14. For any continuous ϕ ∈ Homg(V, V′), define a map Γ(ϕ) : Γ(V )→

Γ(V ′) by

Γ(ϕ) : f 7→ (id⊗ϕ) f

This map is clearly continuous and for all x ∈ gD

Γ(ϕ)(x.f) = (S(x)ρ ⊗ ϕ) f = x.Γ(ϕ)(f)

then Γ(ϕ) ∈ HomgD(Γ(V ),Γ(V ′)).

Since the diagram

Γ(V )

αV

Γ(ϕ)// Γ(V ′)

αV ′

V [[~]]ϕ// V ′[[~]]

is commutative for all ϕ ∈ Homg(V, V′), we have a well–defined functor

Γ : Repeq Ug[[~]]→ Repeq UgD[[~]]

which is naturally isomorphic to the pullback functor induced by the inclusion

iD : gD → g via the natural transformation

αV : Γ(V ) ' i∗DV [[~]]

2.15. Tensor structure on Γ. Denote the tensor product in the cate-

gories DΦ(Ug), DΦD(UgD) by ⊗, and let B1234 and B′1234 be the associativity

constraints

B1234 : (V1 ⊗ V2)⊗ (V3 ⊗ V4)→ V1 ⊗ ((V2 ⊗ V3)⊗ V4)


and

B′1234 : (V1 ⊗ V2)⊗ (V3 ⊗ V4)→ (V1 ⊗ (V2 ⊗ V3))⊗ V4

For any v ∈ Γ(V ), w ∈ Γ(W ), define JVW (v ⊗ w) to be the composition

L−i−−→ L− ⊗ L−

v⊗w−−→ (N∗+ ⊗ V )⊗ (N∗+ ⊗W )A−→ N∗+ ⊗ ((V ⊗N∗+)⊗W )

β−132−−→ N∗+⊗ ((N∗+⊗ V )⊗W )

A′−→ (N∗+⊗N∗+)⊗ (V ⊗W )i∨+⊗1−−−→ N∗+⊗ (V ⊗W )

where the pair (A,A′) can be chosen to be (BN∗+,V,N∗+,W

, B−1N∗+,N

∗+,V,W

) or (B′N∗+,V,N∗+,W , B′−1N∗+,N

∗+,V,W

).

The map JVW (v ⊗ w) is clearly a continuous g-morphism from L− to N∗+ ⊗

(V ⊗W ), so we have a well-defined map

JVW : Γ(V )⊗ Γ(W )→ Γ(V ⊗W )

Proposition. The maps JVW are isomorphisms of gD–modules and define

a tensor structure on the functor Γ.

The proof of Proposition 2.15 is given in 2.16–2.19.

2.16. We easily observe that the maps JVW are compatible with the gD–

action. Indeed, the map i∨+ is compatible with the right gD–action on N∗+ and,

for any x ∈ gD,

x.JVW (v ⊗ w) = (Sρ(x)⊗ id)(i∨+ ⊗ id⊗ id)A(v ⊗ w)i− =

= (i∨+ ⊗ id⊗ id)(∆(S(x))ρ)12A(v ⊗ w)i− =

= (i∨+ ⊗ id⊗ id)A((S ⊗ S)(∆(x)))ρ)13(v ⊗ w)i− = JVW (x.(v ⊗ w))


where A = A′β−132 A.

JVW is an isomorphism, since it is an isomorphism modulo ~. Indeed,

JVW (v ⊗ w) ≡ (i∗+ ⊗ 1)(1⊗ s⊗ 1)(v ⊗ w)i− mod ~

Finally, to prove that JVW define a tensor structure on Γ, we need to show

that, for any V1, V2, V3 ∈ DΦ(Ug) the following diagram is commutative

(Γ(V1)⊗ Γ(V2))⊗ Γ(V3)

ΦD

J12⊗1// Γ(V1 ⊗ V2)⊗ Γ(V3)

J12,3// Γ((V1 ⊗ V2)⊗ V3)

Γ(Φ)

Γ(V1)⊗ (Γ(V2)⊗ Γ(V3))1⊗J23

// Γ(V1)⊗ Γ(V2 ⊗ V3)J1,23

// Γ(V1 ⊗ (V2 ⊗ V3))

where Jij denotes the map JVi,Vj and Jij,k the map JVi⊗Vj ,Vk .

2.17. For any vi ∈ Γ(Vi), i = 1, 2, 3, the map Γ(Φ)J12,3J12⊗1(v1⊗v2⊗v3)

is given by the composition

(1⊗ Φ)(i∗+ ⊗ 1⊗3)A4(1⊗ β1⊗2,N∗+⊗ 1)A3((i∗+ ⊗ 1)⊗ 1⊗3)(A2 ⊗ 1⊗ 1)

· (1⊗ βN∗+,1 ⊗ 1⊗3)(A1 ⊗ 1⊗ 1)(v1 ⊗ v2 ⊗ v3)(i− ⊗ 1)i−

where

A1 = BN∗+,1,N∗+,2

A3 = BN∗+,1⊗2,N∗+,3

A2 = B−1N∗+,N

∗+,1,2

A4 = B−1N∗+,N

∗+,1⊗2,3


illustrated by the diagram

L−

i−// L− ⊗ L−

i−⊗1

// (L− ⊗ L−)⊗ L−

v1⊗v2⊗v3// ((N∗+ ⊗ V1)⊗ (N∗+ ⊗ V2))⊗ (N∗+ ⊗ V3)

A1⊗1⊗1// (N∗+ ⊗ ((V1 ⊗N∗+)⊗ V2))⊗ (N∗+ ⊗ V3)

1⊗βN∗+

,1⊗1⊗3

// (N∗+ ⊗ ((N∗+ ⊗ V1)⊗ V2))⊗ (N∗+ ⊗ V3)A2⊗1⊗1

// ((N∗+ ⊗N∗+)⊗ (V1 ⊗ V2))⊗ (N∗+ ⊗ V3)

(i∗+⊗1)⊗1⊗3

// (N∗+ ⊗ (V1 ⊗ V2))⊗ (N∗+ ⊗ V3)A3

// N∗+ ⊗ (((V1 ⊗ V2)⊗N∗+)⊗ V3)

1⊗β1⊗2,N∗+⊗1

// N∗+ ⊗ ((N∗+ ⊗ (V1 ⊗ V2))⊗ V3)A4

// (N∗+ ⊗N∗+)⊗ ((V1 ⊗ V2)⊗ V3)

i∗+⊗1⊗3

// N∗+ ⊗ ((V1 ⊗ V2)⊗ V3)1⊗Φ

// N∗+ ⊗ (V1 ⊗ (V2 ⊗ V3))

By functoriality of associativity and commutativity isomorphisms, we have

A3(i∗+ ⊗ 1⊗4) = (i∗+ ⊗ 1⊗4)A5

where A5 = BN∗+⊗N∗+,12,N∗+,3,

(1⊗ β12,N∗+⊗ 1)(i∗+ ⊗ 1⊗4) = (i∗+ ⊗ 1⊗4)(1⊗2 ⊗ β12,N∗+

⊗ 1⊗2)

and

A4(i∗+ ⊗ 1⊗4) = (i∗+ ⊗ 1⊗4)A6

where A6 = B−1N∗+⊗N∗+,N∗+,1⊗2,3. Finally, we have

Γ(Φ)J12,3(J12 ⊗ 1)(v1 ⊗ v2 ⊗ v3)

= (1⊗3 ⊗ Φ123)((i∗+(i∗+ ⊗ 1))⊗ 1⊗3)A (v1 ⊗ v2 ⊗ v3)(i− ⊗ 1)i− (2.2)


where

A = A6(1⊗2 ⊗ β1⊗2,N∗+⊗ 1⊗2)A5(A2 ⊗ 1⊗2)(1⊗ βN∗+,1 ⊗ 1⊗3)(A1 ⊗ 1⊗ 1)

2.18. On the other hand, J1,2⊗3(1⊗ J23)ΦD(v1 ⊗ v2 ⊗ v3) corresponds to

the composition

(i∗+ ⊗ 1⊗3)A′4(1⊗ βN∗+,1 ⊗ 1⊗2)A′3(1⊗2 ⊗ i∗+ ⊗ 1⊗2)(1⊗ 1⊗ A′2)

(1⊗3 ⊗ β2,N∗+⊗ 1)(1⊗ 1⊗ A′1)ΦD(v1 ⊗ v2 ⊗ v3)(1⊗ i−)i−

where

A′1 = BN∗+,2,N∗+,3

A′3 = BN∗+,1,N∗+,2⊗3

A′2 = B−1N∗+,N

∗+,2,3

A′4 = B−1N∗+,N

∗+,1,2⊗3

illustrated by the diagram

L−

i−// L− ⊗ L−

1⊗i−// L− ⊗ (L− ⊗ L−)

ΦD(v1⊗v2⊗v3)// (N∗+ ⊗ V1)⊗ ((N∗+ ⊗ V2)⊗ (N∗+ ⊗ V3))

1⊗1⊗A′1// (N∗+ ⊗ V1)⊗ (N∗+ ⊗ ((V2 ⊗N∗+)⊗ V3))

1⊗3⊗β2,N∗+⊗1

// (N∗+ ⊗ V1)⊗ (N∗+ ⊗ ((N∗+ ⊗ V2)⊗ V3))1⊗1⊗A′2

// (N∗+ ⊗ V1)⊗ ((N∗+ ⊗N∗+)⊗ (V2 ⊗ V3))

1⊗2⊗i∗+⊗1⊗2

// (N∗+ ⊗ V1)⊗ (N∗+ ⊗ (V2 ⊗ V3))A′3

// N∗+ ⊗ ((V1 ⊗N∗+)⊗ (V2 ⊗ V3))

1⊗β1,N∗+⊗1

// N∗+ ⊗ ((N∗+ ⊗ V1)⊗ (V2 ⊗ V3))A′4

// (N∗+ ⊗N∗+)⊗ (V1 ⊗ (V2 ⊗ V3))

i∗+⊗1⊗3

// N∗+ ⊗ (V1 ⊗ (V2 ⊗ V3))

By functoriality of associativity and commutativity isomorphisms, we have

A′3(1⊗2 ⊗ i∗+ ⊗ 1⊗2) = (1⊗2 ⊗ i∗+ ⊗ 1⊗2)A′5


where A′5 = BN∗+,1,N∗+⊗N∗+,2⊗3,

(1⊗ β1,N∗+⊗ 1⊗2)(1⊗2 ⊗ i∗+ ⊗ 1⊗2) = (1⊗ i∗+ ⊗ 1⊗3)(1⊗ β1,N∗+⊗N∗+ ⊗ 1⊗2)

and

A′4(1⊗ i∗+ ⊗ 1⊗3) = (1⊗ i∗+ ⊗ 1⊗3)A′6

where A′6 = B−1N∗+,N

∗+⊗N∗+,1,2⊗3. Thus,

J1,23(1⊗ J23)ΦD(v1 ⊗ v2 ⊗ v3)

= (i∗+ ⊗ 1⊗3)((1⊗ i∗+)⊗ 1⊗3)B ΦD(v1 ⊗ v2 ⊗ v3)(1⊗ i−)i− (2.3)

where

B = A′6(1⊗ β1,N∗+⊗N∗+ ⊗ 1⊗2)A′5(1⊗2 ⊗ A′2)(1⊗3 ⊗ β2,N∗+⊗ 1)(1⊗ 1⊗ A′1)


2.19. Comparing (2.2) and (2.3), we see that it suffices to show that the

outer arrows of the following form a commutative diagram.

L−

(i−⊗1)i−

xx

(1⊗i−)i−

&&

(L− ⊗ L−)⊗ L−

v1⊗v2⊗v3

L− ⊗ (L− ⊗ L−)

ΦD(v1⊗v2⊗v3)

((N∗+ ⊗ V1)⊗ (N∗+ ⊗ V2))⊗ (N∗+ ⊗ V3)

A

Φ// (N∗+ ⊗ V1)⊗ ((N∗+ ⊗ V2)⊗ (N∗+ ⊗ V3))

B

((N∗+ ⊗N∗+)⊗N∗+)⊗ ((V1 ⊗ V2)⊗ V3)

(i∗+(i∗+⊗1))⊗1⊗3

Φ⊗Φ// (N∗+ ⊗ (N∗+ ⊗N∗+))⊗ (V1 ⊗ (V2 ⊗ V3))

(i∗+(1⊗i∗+))⊗1⊗3

N∗+ ⊗ ((V1 ⊗ V2)⊗ V3)1⊗Φ

// N∗+ ⊗ (V1 ⊗ (V2 ⊗ V3))

Using the pentagon and the hexagon axiom, we can show that

(Φ⊗ Φ)A = BΦ

We have to show that

Γ(Φ)J12,3(J12 ⊗ 1)(v1 ⊗ v2 ⊗ v3) = J1,23(1⊗ J23)ΦD(v1 ⊗ v2 ⊗ v3)

in Homg(L−, N∗+ ⊗ (V1 ⊗ (V2 ⊗ V3))):


J1,23(id⊗J23)ΦD(v1 ⊗ v2 ⊗ v3) =

= (i∨+(id⊗i∨+)⊗ id⊗3)BΦD(v1 ⊗ v2 ⊗ v3)(id⊗i−)i−

= (i∨+(id⊗i∨+)⊗ id⊗3)BΦD(v1 ⊗ v2 ⊗ v3)Φ(i− ⊗ id)i−

= (i∨+(id⊗i∨+)⊗ id⊗3)BΦΦD(v1 ⊗ v2 ⊗ v3)(i− ⊗ id)i−

= (i∨+(id⊗i∨+)Φ⊗ Φ)AΦD(v1 ⊗ v2 ⊗ v3)(i− ⊗ id)i−

= (i∨+(id⊗i∨+)Φ⊗ Φ)(S⊗3(ΦD)ρ ⊗ id⊗3)A(v1 ⊗ v2 ⊗ v3)(i− ⊗ id)i−

= (i∨+(id⊗i∨+)ΦS⊗3(ΦD)ρ ⊗ Φ)A(v1 ⊗ v2 ⊗ v3)(i− ⊗ id)i−

= (i∨+(i∨+ ⊗ id)⊗ Φ)A(v1 ⊗ v2 ⊗ v3)(i− ⊗ id)i−

= Γ(Φ)J12,3(J12 ⊗ id)(v1 ⊗ v2 ⊗ v3)

where the second and seventh equalities follow from Proposition 2.12, the fifth

one from the definition of the gD–action on the modules Γ(Vi) and the others

from functoriality of the associator Φ. This complete the proof of Theorem

2.1.

2.20. 1–Jets of relative twists. The following is a straightforward ex-

tension of the computation of the 1–jet of the Etingof–Kazhdan twist given in

[EK1].

Proposition. Under the natural identification

αV : Γ(V )→ V [[~]]

3. QUANTIZATION OF VERMA MODULES 101

the relative twist JΓ satisfyies

αV⊗W JΓ (α−1V ⊗ α

−1W ) ≡ 1 +

~2

(r + r21D ) mod ~2

in End(V ⊗W )[[~]].

Proof. For v ∈ V,w ∈ W , let

α−1V (v)(1−) =

∑fi ⊗ vi α−1

W (w)(1−) =∑

gj ⊗ wj

in (N∗+ ⊗ V )p+ and (N∗+ ⊗W )p+ respectively. Then using

〈(1+ ⊗ 1)⊗2,Ω23

∑i,j

fi ⊗ vi ⊗ gj ⊗ wj〉 = −r(v ⊗ w)

and

〈(1+ ⊗ 1)⊗2,ΩD,23

∑i,j

fi ⊗ vi ⊗ gj ⊗ wj〉 = −ΩD(v ⊗ w)

where Ω = Ω + ΩD, we get

αV⊗W JΓ (α−1V ⊗ α

−1W )(v ⊗ w) ≡ v ⊗ w +

~2

(r + r21D )(v ⊗ w) mod ~2

because the definition of JΓ involves the braiding β′XY = β−1Y X .

Corollary. The relative twist JΓ satisfies

Alt2JΓ ≡~2

(r − r21

2− rD − r21

D

2

)mod ~2

3. Quantization of Verma modules

This section and the next contain results about the quantization of classical

Verma modules, which are required to construct the morphism of D–categories


between the representation theory of Ug[[~]] and that of U~g. In particular,

from now on, we will assume the existence of a finite N–grading on g−, which

induces on g = g−⊕g+ a Lie bialgebra structure and allows us to consider the

quantization of g through the Etingof–Kazhdan functor, UEK

~ g.

3.1. Quantum Verma Modules. Let (g, g−, g+) be a graded Manin

triple. This means that we have an isomorphism between g+ and the graded

dual of g− and that g+ is naturally endowed with a Lie bialgebra structure.

Because of the functoriality of the quantization defined by Etingof and Kazh-

dan in [EK2], in the category of Drinfeld-Yetter modules over UEK

~ g− we can

similarly define quantum Verma modules.

The standard inclusions of Lie bialgebras g± ⊂ g ' Dg− lift to UEK

~ g± ⊂

UEK

~ g ' DUEK

~ g−, and we can define the induced modules of the trivial repre-

sentation over UEK

~ g±

M~± = Ind

UEK

~ g

UEK~ g±

k[[~]]

Similarly, we have Hopf algebra maps UEK

~ p± ⊂ UEK

~ g and UEK

~ p± → UEK

~ gD,

and we can define induced modules

L~− = IndU

EK

~ g

UEK~ p+

k[[~]] N~+ = Ind

UEK

~ g

UEK~ p−

UEK

~ gD

We want to show that the equivalence F : YDΦ(Ug−)[[~]]→ YDU

EK~ g−

matches

these modules. We start proving the statement for M−,M∗+.

3.2. Quantization of M±. We denote by (M~+)∗ the U

EK

~ g–module

Homk(IndU

EK

~ g

UEK~ g−

k[[~]], k[[~]])


Theorem. In the category of left UEK

~ g–modules,

(a) F (M−) 'M~−

(b) F (M∗+) ' (M~

+)∗

Proof. The Hopf algebra UEK

~ g− is constructed on the space F (M−) with

unit element u ∈ F (M−) defined by u(1−) = ε+⊗1−, where ε+ ∈M∗+ is defined

as ε+(x1+) = ε(x) for any x ∈ Ug+. Consequently, the action of UEK

~ g− on

u ∈ F (M−) is free, as multiplication with the unit element. The coaction

of UEK

~ g− on F (M−) is defined using the R-matrix associated to the braided

tensor functor F ,i.e.,

π∗M− : F (M−)→ F (M−)⊗ F (M−), π∗(x) = R(u⊗ x)

where x ∈ F (M−) and RVW ∈ EndU

EK~ g

(F (V ) ⊗ F (W )) is given by RVW =

σJ−1WV F (βVW )JVW , JV,WV,W∈YDUg−

being the tensor structure on F . It is

easy to show that J(u⊗ u)|1− = ε+ ⊗ 1− ⊗ 1−, and, since Ω(1− ⊗ 1−) = 0, we

have

R(u⊗ u) = u⊗ u

For a generic V ∈ YDUg− [[~]], the action of UEK

~ g∗− is defined as

F (M−)∗ ⊗ F (V )→ F (M−)∗ ⊗ F (M−)⊗ F (V )→ F (V )

This means, in particular that, for every φ ∈ I ⊂ UEK

~ g∗−, where I is the max-

imal ideal corresponding to u⊥, we have φ.u = 0. This proves (a).


The module M∗+ satisfies the following universal property: for any V in the

Drinfeld category of equicontinuous Ug-modules, we have

HomUg(V,M∗+) ' HomUg−(V, k)

Indeed, to any map of Ug-modules f : V → M∗+, we can associate f : V → k,

f(v) = 〈f(v), 1+〉. It is clear that f factors through V/g−.V . The equicon-

tinuity property is necessary to show the continuity of f with respect to the

topology on V .

Since F defines an equivalence of categories, we have

HomU

EK~ g

(F (V ), F (M∗+)) ' HomUg(V,M

∗+)[[~]] ' HomUg−(V, k)[[~]]

Using the natural isomorphism αV : F (V )→ V [[~]], defined by

αV (f) = 〈f(1−), 1+ ⊗ id〉

we obtain a map HomUg−(V, k)[[~]] → Homk(F (V ), k[[~]]). Consider now the

linear isomorphism α : UEK

~ g− → Ug−[[~]] and for any x ∈ Ug− consider the

g-intertwiner ψx : M− →M∗+ ⊗M− defined by ψx(1−) = ε+ ⊗ x1−. It is clear

that, if f(1−) = f(1) ⊗ f(2) in Swedler’s notation,

αV (ψx.f) = 〈(i∨+ ⊗ id)Φ−1(id⊗f)(ε+ ⊗ x.1−), 1+ ⊗ id〉

= 〈Φ−1(ε+ ⊗ id⊗ id)(id⊗∆(x))(id⊗f(1) ⊗ f(2)), (T ⊗ id)(1+ ⊗ 1+ id)〉

= 〈∆(x)(f(1) ⊗ f(2)), 1+ ⊗ id〉

= 〈f(1), 1+〉x.f(2)

= x.αV (f)


using the fact that (ε⊗ 1⊗ 1)(Φ) = 1⊗2 and (ε⊗ 1)(T ) = 1. So, clearly, if φ ∈

HomUg−(V, k), then φ αV ∈ HomU

EK~ g−

(F (V ), k[[~]]). Then F (M∗+) satisfies

the universal property of Homk(IndU

EK

~ g

UEK~ g−

k[[~]], k[[~]]) and (b) is proved.

3.3. Quantization of relative Verma modules. The proof of (b) shows

that the linear functional F (M∗+) → k[[~]] is, in fact, the trivial deformation

of the functional M∗+ → k. These results extend to the relative case and hold

for the right gD–action on L−, N∗+.

Theorem. In the category YDU

EK~ g−

(a) F (L−) ' L~−

(b) F (N∗+) ' (N~+)∗

Moreover, as right UEK

~ gD–module

(c) FD(L−) ' L~−

(d) FD(N∗+) ' (N~+)∗

The proof of (a) and (b) amounts to constructing the morphisms

k[[~]]→ F (L−) F (N∗+)→ UEK

~ g∗D

equivariant under the action of UEK

~ p+ and UEK

~ p− respectively.

A direct construction along the lines of the proof of Theorem 3.2 is however

not straightforward. We prove this theorem in the next section by using a

description of the modules L−, N∗+ and their images through F via Prop


categories. These descriptions show that the classical intertwiners

k→ L− N∗+ → Ug∗D

satisfy the required properties and yield canonical identifications

F (L−) ' L~− F (N∗+) ' (N~+)∗

4. Universal relative Verma modules

In this section, we prove Theorem 3.3 by using suitable Prop (product-

permutation) categories compatible with the EK universal quantization functor

[EK2, EG].

4.1. Prop description of the EK quantization functor. We will

briefly review the construction of Etingof–Kazhdan in the setting of Prop

categories [EK2].

A Prop is a symmetric tensor category generated by one object. More

precisely, a cyclic category over S is the datum of

• a symmetric monoidal k–linear category (C,⊗) whose objects are

non–negative integers, such that [n] = [1]⊗n and the unit object is

[0]

• a bigraded set S =⋃m,n∈Z≥0

Snm of morphism of C, with

Snm ⊂ HomC([m], [n])

such that any morphism of C can be obtained from the morphisms in S and

permutation maps in HomC([m], [m]) by compositions, tensor products or lin-

ear combinations over k. We denote by FS the free cyclic category over S.

4. UNIVERSAL RELATIVE VERMA MODULES 107

Then there exists a unique symmetric tensor functor FS → C, and the follow-

ing holds (cf. [EK2])

Proposition. Let C be any cyclic category generated by a set S of mor-

phisms. Then C has the form FS/I, where I is a tensor ideal in FS.

Let N be a symmetric monoidal k–linear category, and X an object in

N . A linear algebraic structure of type C on X is a symmetric tensor functor

GX : C → N such that GX([1]) = X. A linear algebraic structure of type C on

X is a collection of morphisms between tensor powers of X satisfying certain

consistency relations.

We mainly consider the case of non–degenerate cyclic categories, i.e., sym-

metric tensor categories with injective maps k[Sn]→ HomC([n], [n]). We first

consider the Karoubian envelope of C obtained by formal addition to C of the

kernel of the idempotents in k[Sn] acting on [n]. Furthermore, we consider the

closure under inductive limits. In this category, denoted S(C), every object is

isomorphic to a direct sum of indecomposables, corresponding to irreducible

representations of Sn (cf. [EK2, EG]). In particular, in S(C), we can consider

the symmetric algebra

S[1] =⊕n≥0

Sn[1]

If N is closed under inductive limits, then any linear algebraic structure of

type C extends to an additive symmetric tensor functor

GX : S(C)→ N

We introduce the following fundamental Props .


• Lie bialgebras. In this case the set S consists of two elements of

bidegrees (2, 1), (1, 2), the universal commutator and cocommutator.

The category C = LBA is FS/I, where I is generated by the classi-

cal five relations.

• Hopf algebras. In this case, the set S consists of six elements of

bidegrees (2, 1), (1, 2), (0, 1), (1, 0), (1, 1, ), (1, 1), the universal prod-

uct, coproduct, unit, count, antipode, inverse antipode. The cate-

gory C = HA is FS/I, where I is generated by the classical four

relations.

The quantization functor described in Section 1 can be described in this

generality, as stated by the following (cf. [EK2, Thm.1.2])

Theorem. There exists a universal quantization functor Q : HA→ S(LBA).

Let g− be the canonical Lie dialgebra [1] in LBA with commutator µ and

cocommutator δ. Let Ug− := Sg− ∈ S(LBA) be the universal enveloping

algebra of g−. The construction of the Etingof–Kazhdan quantization functor

amounts to the introduction of a Hopf algebra structure on Ug−, which coin-

cides with the standard one modulo 〈δ〉, and yields the Lie bialgebra structure

on g− when considerd modulo 〈δ2〉. This Hopf algebra defines the object Q[1],

where [1] is the generating object in HA. The formulae used to defined the

Hopf structure coincide with those defined in [EK1, Part II] and described in

Section 1. In particular, they rely on the construction of the Verma modules

M− := Sg− M∗+ = Sg−


realized in the category of Drinfeld–Yetter modules over g− as object of LBA

.

4.2. Props for split pairs of Lie bialgebras. Let (g−, gD,−) be a split

pair of Lie bialgebras, i.e., there are Lie bialgebra maps

gD,−i−→ g−

p−→ gD,−

such that p i = id. These maps induce an inclusion DgD,− ⊂ Dg− and

consequently an inclusion of Manin triple (gD, gD,−, gD,+) ⊂ (g, g−, g+), as

described in Section 2.6.

Definition. We denote by PLBA the Karoubian envelope of the multi-

colored Prop, whose class of objects is generated by the Lie bialgebra objects

[g−], [gD,−], related by the maps i : [gD,−]→ [g−] , p : [g−]→ [gD,−], such that

p i = id[gD,−].

The Karoubian envelope implies that [m−] := ker(p) ∈ PLBA.

Proposition. The multicolored Prop PLBA is endowed with a pair of

functors U,L

U,L : LBA→ PLBA U [1] := [g−], L[1] := [gD,−]

and natural transformations i, p, induced by the maps i, p in PLBA,

LBA

U

**

L

44PLBAp

i

KS


such that p i = id. Moreover, it satisfies the following universal property:

for any tensor category C, closed under kernels of projections, with the same

property as PLBA , there exists a unique tensor functor PLBA→ C such that

the following diagram commutes

LBA

U

**

L

44

$$

::PLBAp

i

KS

// C

4.3. Props for split pairs of Hopf algebras. We can analogously de-

fine suitable Prop categories corresponding to split pairs of Hopf algebras. In

particular, we consider the Prop PHA characterized by functors U~, L~ and

natural transformations p~, i~ satisfying

HA

U~

))

L~

55

$$

::PHAp~

i~

KS∃!

// C

where HA denotes the Prop category of Hopf algebras. These also satisfy

HAQEK

//

S(LBA)

PHAQPLBA

// S(PLBA)

where QPLBA is the extension of the Etingof–Kazhdan quantization func-

tor to PLBA, obtaine by the universal property described above with C =

S(PLBA).


4.4. Props for parabolic Lie subalgebras. In order to describe the

module N∗+ it is necessary to deal with the Lie bialgebra object p− or, in other

words to introduce the double of gD,− and the Prop D⊕(LBA) [EG]. We then

introduce the multicolored Prop as a cofiber product of PLBA and D⊕(LBA)

over LBA .

Proposition. The multicolored Prop PLBAD is endowed with canonical

functors

D⊕(LBA)→ PLBAD ← PLBA

and satisfies the following universal property:

LBAdouble

//

D⊕(LBA)

PLBA

00

// PLBAD∃!

(( C

where double is the Prop map introduced in [EG].

In PLBAD we can consider the Lie bialgebra object [p−].

4.5. Props for parabolic Hopf subalgebras. Similarly, we introduce

the multicolored Prop PHAD , endowed with canonical functors (cf. [EG])

D⊗(HA)→ PHAD ← PHA


and satisfying an analogous universal property:

HAdouble

//

D⊗(HA)

PHA

00

// PHAD∃!

'' C

Moreover, we then have a canonical functor

QPLBAD : PHAD → S(PLBAD)

obtained applying such universal property with C = S(PLBAD) and satisfying

HA

LHAxx

double//

QEK

D⊗(HA)

Q2

vv

PHA

QPLBA

// PHAD

QPLBAD

S(LBA)

xx

S(double)// S(D⊕(LBA))

vv

S(PLBA) // S(PLBAD)

The commutativity of the square on the back is given by the compatibility of

the quantization functor with the doubling operations, proved in [EG].

4.6. Prop description of L−, N∗+. The modules L−, N

∗+ can be real-

ized in S(PLBAD). The module L− is constructed over the object Sm− ∈


S(PLBA). The structure of Drinfeld-Yetter module over g− is determined in

the following way:

• the free action of the Lie algebra object m− is defined by the map

Sm− ⊗ Sm− → Sm−

given by Campbell-Hausdorff series, describing on Sm− the multipli-

cation in Um−.

• we define the action of gD,− to be trivial on 1→ Sm−.

• The actions of m−, gD,−, the relation

π ([, ]⊗ 1) = π (1⊗ π)− π (1⊗ π) σ12

and the map [, ] : gD,− ⊗m− → m− define the action of g−.

• We then impose the trivial coaction on 1 → Sm− and the compati-

bility condition between action and coaction

π∗ π = (1⊗ π)σ12(1⊗ π∗)− (1⊗ π)(δ ⊗ 1) + (µ⊗ 1)(1⊗ π∗)

determines the coaction for Sm−. The action defined is compatible

with [, ] : gD,− ⊗m− → m−

Similarly, the module N∗+ can be realized on the object Sp−, formally added

to S(PLBAD).


We determine the formulae for the action and the coaction of g− by direct

inspection of the action of g = g− ⊕ g+ on N+ in the category Vect. Namely,

the identification N∗+ = Sp− is clearly obtained through the invariant bilinear

form 〈−,−〉 and there are topological formulae expressing the action of g on

N+. Therefore we determine action and coaction on N∗+ in the following way:

• the g+–action on N+ = Sp+ = Sg+ ⊗ SgD,− is given by the free ac-

tion on the first factor Sg+ expressed by Campbell–Hausdorff series.

• the action of g− = m− ⊕ gD,− on the subspace SgD,− ⊂ Sp+ is given

by the trivial action of m− and the usual free action of gD,− by mul-

tiplication.

• The action of g− is then interpreted as a topological coaction of g+

and the aforementioned compatibility condition between action and

coaction allows to extend the formula for the topological g+ coaction

on the entire space Sp+.

• Through the invariant bilinear form 〈−,−〉, these formulae are car-

ried over N∗+ = Sp−, by switching, in particular, the bracket and the

topological cobracket on g+ with the cobracket and the bracket in

g−, respectively.

• The obtained formulae, describing the action and the coaction of

g− on N∗+, are well–defined in the category PLBAD and define the

requested structure of Drinfeld-Yetter module over g−.


4.7. Proof of Theorem 3.3. The relative Verma module

N+ = Indgm− k ' Indg

p− UgD

satisfies

HomUg(N+, V ) ' HomUp−(UgD, V )

for every Ug-module V . We have a canonical map of p−-modules ρD : UgD →

N+ corresponding to the identity in the case V = N+. We get a map of

p−-modules ρ∗D : N∗+ → Ug∗D inducing an isomorphism

HomUg(V,N∗+) ' HomUp−(V, Ug∗D)

The morphism ρ∗D can indeed be thought as

Up− ⊗N∗+

// N∗+

ρ∗D

UgD ⊗ Ug∗D // Ug∗D

Assuming the existence of a suitable finite N–grading, a split pair of Lie

bialgebras (g−, gD,−), gives rise to a functor

PLBAD → Vect

Consider now the trivial split pair given by (gD,−, gD,−). We have a natural

transformation

PLBAD

(g−,gD,−)

))

(gD,−,gD,−)

55Vectp


where p naturally extends to the projection p− → gD.

The module U(gD)∗ is indeed the module N∗+ with respect to the trivial pair

(gD,−, gD,−). Consequently, the existence of the p−–intertwiner ρ∗D can be in-

terpreted as a simple consequence of the existence of natural transformation p.

The quantization functor QPLBAD extends the natural transformation p to

PHAD$$

::

// S(PLBAD)

(g−,gD,−)

))

(gD,−,gD,−)

55VectS(p)

and shows that

F (N∗+) ' (N~+)∗

Similarly, we can consider the natural transformation S(i) and the diagram

PHAD$$

::

// S(PLBAD)

(g−,gD,−)

))

(gD,−,gD,−)

55VectS(i)

KS

implying

F (L−) ' L~−

5. CHAINS OF MANIN TRIPLES 117

We can make analogous consideration for the right gD–action on L−, N∗+. This

leads to isomorphisms of right UEK

~ gD–modules

FD(N∗+) ' (N~+)∗ FD(L−) ' L~−

5. Chains of Manin triples

5.1. Chains of length 2. In Section 3, given an inclusion of Manin triples

iD : gD ⊆ g, we introduced the relative quantum Verma modules

L~− = IndU

EK

~ g

UEK~ p+

k[[h]] N~+ = Ind

UEK

~ g

UEK~ p−

UEK

~ gD

These modules allow to define the functor

Γ~ : D(UEK

~ g)→ D(UEK

~ gD)

by

Γ~(V) = HomU

EK~ g

(L~−, (N~+)∗ ⊗ V)

Lemma. The functor Γ~ is naturally tensor isomorphic to the restriction

functor (UEK

~ (i~D))∗.

Proof. The proof of the existence of the natural isomorphism as UEK

~ gD–

module is identical to that of Prop. 2.13. The isomorphism respects the tensor

structures, because there are only trivial associators involved.

We now prove the following

Theorem. Let g, gD be Manin triples with a finite Z–grading and iD :

gD ⊆ g an inclusion of Manin triples compatible with the grading. Then, there


exists an algebra isomorphism

Ψ : UEK

~ g→ Ug[[~]]

restricting to ΨEKD on U

EK

~ gD, where the completion is given with respect to

Drinfeld–Yetter modules.

Proof. In the previous section, we showed that the quantization of the

(UEK

~ g, UEK

~ gD)–modules N∗+, L− gives

F (N∗+)U

EK

~ g−−−→ (N~

+)∗U

EK

~ gD←−−−− F EK

D (N∗+)

F (L−)U

EK

~ g−−−→ L~−

UEK

~ gD←−−−− F EK

D (L−)

Recall that the standard natural transformations αV : F (V ) ' V [[~]], (αD)V :

FD(V ) ' V [[~]] give isomorphisms of right UgD[[~]]–modules

F (N∗+) ' N∗+[[~]] F (L−) ' L−[[~]]

and isomorphisms of Ug[[~]]–modules

F EK

D (N∗+) ' N∗+[[~]] F EK

D (L−) ' L−[[~]]

In particular, we get isomorphisms of right UEK

~ gD–modules

F EK

D F (N∗+) ' F EK

D (N∗+) ' (N~+)∗ F EK

D F (L−) ' F EK

D (L−) ' L~−

and isomorphisms of UEK

~ g–modules

F EK

D F (N∗+) ' F (N∗+) ' (N~+)∗ F EK

D F (L−) ' F (L−) ' L~−


We have a natural isomorphism through J :

HomU

EK~ g

(F (L−), F (N∗+)⊗ F (V )) ' Homg(L−, N∗+ ⊗ V )[[~]]

This is indeed an isomorphism of UgD[[~]]–modules, since, for x ∈ UgD, φ ∈

HomU

EK~ g

(F (L−), F (N∗+)⊗ F (V )), we have

x.φ := (F (x)⊗ id) φ J (F (x)⊗ id)) = F (x⊗ id) J

Quantizing both side and using the isomorphism F EKD F (N∗+) ' (N~

+)∗, we

obtain a natural transformation

γD : Γ~ F ' FD Γ

making the following diagram commutative

DΦ(Ug)F

//

Γ

D(U~g)

Γ~

γD

s

DΦD(UgD[[~]])FD

// D(U~gD)

Applying the construction above to the algebra of endomorphisms of the

fiber functor, we get the result.

5.2. Chains of arbitrary length. For any chain

C : 0 = g0 ⊆ g1 ⊆ · · · ⊆ gn−1 ⊆ gn = g

of inclusions of Manin triples, the natural transformations

γi,i+1 ∈ Nat⊗(Γ~i,i+1 Fi+1, Fi Γi,i+1)


where 0 ≤ i ≤ n − 1, Γ0,1 : DΦ(g1) → Vectk[[~]] is the EK fiber functor, and

F EK0 = id, yield a natural transformation

γC = γ0,1 · · · γn−1,n

∈ Nat⊗(Γ~0,1 · · · Γ~n−1,n Fn,Γ0,1 · · · Γn−1,n)

∼= Nat⊗((i∗0,n)~ Fn,Γ0,1 · · · Γn−1,n)

= Nat⊗(Fn,Γ0,1 · · · Γn−1,n)

where we used Γ~i,i+1∼= (i∗i,i+1)~, and the fact that the composition (i∗0,n)~ Fn

is the EK fiber functor for gn, which we denote by the same symbol as Fn.

This proves the following

Theorem.

(i) For any chain of Manin triples

C : g0 ⊆ g1 ⊆ · · · ⊆ gn ⊆ g

there exists an isomorphism of algebras

ΨC : UEK

~ g→ Ug[[~]]

such that ΨC(UEK

~ gi) = Ugi[[~]] for any gi ∈ C.

(ii) Given two chains C,C′, the natural transformation

ΦCC′ := γ−1C γC′ ∈ Aut(F EK)

satisfies

Ad(ΦCC′)ΨC′ = ΨC


The following proposition is clear.

Proposition. The natural transformations ΦCC′C,C′ satisfy the follow-

ing properties

(i) Orientation. Given two chains C,C′

ΦCC′ = Φ−1C′C

(ii) Transitivity. Given the chains C,C′,C′′

ΦCC′ ΦC′C′′ = ΦCC′′

(iii) Factorization. Given the chains

C,C′ : g0 ⊆ g0 ⊆ · · · ⊆ gn D,D′ : gn ⊆ · · · ⊆ gn+n′

Φ(C∪D)(C′∪D′) = ΦCC′ ΦDD′

5.3. Abelian Manin triples and central extensions. We will now

consider the following special case, that generalizes the role of Levi subalgebras

for Kac–Moody algebras.

Proposition. If g admits a Manin subtriple lD, obtained by a central

extension of gD, then the relative twists and the gauge transformations are

invariant under lD. In particular, the Etingof–Kazhdan constructions are in-

variant under abelian Manin subtriples.

Proof. For gD = 0, the statement reduces to prove that the Etingof–

Kazhdan functor preserves the action of an abelian Manin subtriple a ⊂ g


(cf. [EK6, Thm. 4.3], with a− = h). Under this assumption, the natural map

Ua− // UEK

~ g− := F (M−)

a // ψa : 1− 7→ ε⊗ a1−

defines an inclusion of bialgebras. For any V [[~]] ∈ DΦ(Ug), the natural

identification

αV : F (V )→ V [[~]]

is then an isomorphism of Ua–modules. This gives the following commutative

diagram

DΦ(Ug)F

//

&&

F

''

D(UEK

~ g)

ww

xx

D(Ua[[~]])

A

We can observe that the tensor restriction functor fits in an analogous dia-

gram. It is easy to show that the object ΓD(L−) is naturally a pointed Hopf

algebra in the category DΦD(UgD). We denote by DgD(ΓD(L−)) the category

of Drinfeld–Yetter modules over ΓD(L−) in the category DΦD(UgD). This cat-

egory is naturally equivalent to the category of Drinfeld–Yetter module over

the Radford’s product UgD,−[[~]]#ΓD(L−) and there is a natural identification

DΦ(Ug)

ΓD &&

// DgD(ΓD(L−))

ww

DΦD(UgD)


Moreover, there is a natural inclusion of bialgebras

U lD ⊂ UgD[[~]]#ΓD(L−)

and a natural U lD–module identification ΓD(V ) → V [[~]]. This originates

natural identifications

DΦ(Ug) //

&&

ΓD

%%

DgD(ΓD(L−))

xx

ww

D(U lD)

DΦD(UgD)

This proves that relative twists are invariant under lD. It is clear that the

Casimir operator ΩD ∈ (gD ⊗ gD)lD defines a braided tensor structure on

D(U lD[[~]]) which is preserved by the restriction functor induced by the inclu-

sion jD : gD ⊂ lD. Given the decomposition lD = gD o cD, the natural map

UcD → EndgD(j∗DV ) induces an action of UcD on FD(j∗DV ), commuting with

the action of UEK

~ gD. Therefore, we obtain a naturally commutative diagram

DΦD(U lD)˜FD//

j∗D

D(UEK

~ lD)

DΦD(UgD)FD// D(U

EK

~ gD)

where˜FD is the tensor functor induced by the composition FD j∗D. The

natural transformation γ automatically lifts to the level of lD, as showed in


the following diagram

DΦ(Ug)F

//

&&

Γ

D(UEK

~ g)

xx

DΦD(U lD)˜FD//

j∗D

xx

D(UEK

~ lD)

&&

DΦD(UgD)FD

// D(UEK

~ gD)

6. The equivalence of quasi–Coxeter categories

The following is the main result of this Chapter.


structure. Then the completion U~g is isomorphic to a quasi-Coxeter quasitri-

angular quasibialgebra of type Dg on the quasitriangular Dg–quasibialgebra

(Ug[[~]], UgD[[~]],∆0, ΦKZD , RKZ

D )

where the completion is taken with respect to the integrable modules in category

O.

6.1. D–structures on Kac–Moody algebras. Let A = (aij)i,j∈I be a

complex n×n matrix and g = g(A) the corresponding generalized Kac–Moody

algebra defined in Section 1. Let J be a nonempty subset of I. Consider the

submatrix of A defined by

AJ = (aij)i,j∈J

We recall the following proposition from [Ka, Ex.1.2]

6. THE EQUIVALENCE OF QUASI–COXETER CATEGORIES 125

Proposition. Let

ΠJ := αj | j ∈ J Π∨J := hj | j ∈ J

Let h′J be the subspace of h generated by Π∨J and

tJ =⋂j∈J

Kerαj = h ∈ h | 〈αj, h〉 = 0 ∀j ∈ J

Let h′′J be a supplementary subspace of h′J + tJ in h and let

hJ = h′J ⊕ h′′

J

Then,

(i) (hJ,ΠJ,Π∨J) is a realization of the generalized Cartan matrix AJ.

(ii) The subalgebra gJ ⊂ g, generated by ej, fjj∈J and hJ, is the Kac–

Moody algebra associated to the realization (hJ,ΠJ,Π∨J) of AJ.

Set

QJ =∑j∈J

Zαj ⊂ Q g = g(A) =⊕α∈Q

gα

Then,

(iii)

gJ = hJ ⊕⊕

α∈QJ\0

gα

Let A be a symmetrizable matrix with a fixed decomposition and (−|−) be the

standard normalized non–degenerate bilinear form on h. Then,

(iv) The restriction of (−|−) to hJ is non–degenerate.


Proof. Since dim(h′J ∩ tJ) = dim(z(gJ) = nJ − lJ, where nJ = |J| and

lJ = rank(AJ), it follows that

dim h′′J = nJ − lJ dim hJ = 2nJ − lJ

Moreover, by construction, the restriction of αjj∈J to hJ are linearly inde-

pendent. Indeed, since 〈∑cjαj, tJ〉 = 0 for all cj ∈ C,

〈∑j∈J

cjαj, hJ〉 = 0 =⇒ 〈∑j∈J

cjαj, h〉 = 0 =⇒ cj = 0

This proves (i). The proof of (ii) and (iii) is clear.

Assume now that A is irreducible and symmetrizable and there exists h ∈

hJ such that

(h|h′) = 0 ∀h′ ∈ hJ

In particular, (h|α∨j ) = 0 and h ∈ h′J ∩ tJ ⊂ h′J. Therefore, h =∑cjα

∨j and

(∑

cjα∨j |h′) =

∑cj(α

∨j |h′) = 〈

∑cjdjαj, h

′〉 = 0

Since the operators αj are linearly independent over hJ and dj 6= 0, we have

cj = 0 and h = 0. We conclude that (|) is non–degenerate on hJ and (iv) is

proved.

Remark. The derived algebra g′J = [gJ, gJ] is generated by ej, fj, hjj∈J,

where hj = [ej, fj]. Therefore, it does not depend of the choice of the subspace

h′′J. The assignment J 7→ g′J defines a structure that coincides with the one

provided in [TL4, 3.2.2].


Let now A be an irreducible, generalized Cartan matrix. Let Dg = D(A)

be the Dynkin diagram of g, that is, the connected graph having I as vertex

set and an edge between i and j if aij 6= 0. For any i ∈ I, let sli2 ⊂ g be the

three–dimensional subalgebra spanned by ei, fi, hi.

Any connected subdiagram D ⊆ Dg defines a subset JD ⊂ I. We would like

to use the assignement J 7→ gJ to define a Dg–algebra structure on g = g(A).

Remark. For any subset J of finite type, dim h′′J = nJ − lJ = 0 and

hJ = h′J. Therefore, if A is a generalized Cartan matrix of finite type, h′′J = 0

for any subset J ⊂ I. The Dg–algebra structure on g = g(A) is then uniquely

defined by the subalgebras sli2i∈I and the Cartan subalgebra is defined for

any subdiagram D ⊂ Dg by

hD = hi | i ∈ V(D)

If A is a generalized Cartan matrix of affine type, we obtain diagrammatic

Cartan subalgebras hD, where

hD =

hi | i ∈ V(D) if D ⊂ Dg

h if D = Dg

If A is an irreducible generalized Cartan matrix of hyperbolic type, i.e., every

submatrix is of finite or affine type, it is still possible to define a Dg–algebra

structure, depending upon the choice of the subspaces h′′J for |I \ J| = 1.

It is not always possible to define a Dg–algebra structure for a generic

matrix of order ≥ 3. In order to obtain a Dg–algebra structure on g = g(A),


we have to satisfy the following condition:

hJ ⊂ tJ⊥ ∩⋂J⊂J′

hJ′

Since tJ⊥ + tJ = h, we can always choose hJ ⊆ tJ⊥ .

Lemma. Assume given a Dg–algebra structure on g = g(A). Then for any

two subsets J′,J′′ ⊂ I,

corank(AJ′∩J′′) ≤ corank(AJ′) + corank(AJ′′)

In particular, if corank(AJ′) = corank(AJ′′) = 0, then corank(AJ′∩J′′) = 0.

Proof. The result is an immediate consequence of the estimate, given by

the construction,

dim(hJ′ ∩ hJ′′) ≤ |J′ ∩ J′′|+ (corank(AJ′) + corank(AJ′′))

and the constraint

hJ′∩J′′ ⊆ hJ′ ∩ hJ′′

Remark. Indeed, it is easy to show that the symmetric irreducible Cartan

matrix

A =

2 −1 0 0

−1 2 −2 0

0 −2 2 −1

0 0 −1 2


does not admit any Dg–algebra structure on g(A), since dim h23 = 3 and

dim h123 ∩ h234 = 2.

The previous condition on the corank is not sufficient to obtain a Dg–

algebra structure on g(A). Consider the symmetric Cartan matrix

A =

2 −2 0 0

−2 2 −1 0

0 −1 2 −1

0 0 −1 2

A clearly satisfies the above condition. Nonetheless, a suitable h′′12, complement

in h of (h′12 + t12), should satisfies:

h′′12 ⊂ h123 = h′123 and h′′12 ⊆ t4 = 〈h′12,−2α∨3 + α∨4 〉

that are clearly not compatible conditions. Therefore, there is no suitable

structure for A.

In the following, we will consider only symmetrizable Kac–Moody algebras

g that admit such a structure. It automatically defines an analogue structure

on UEK

~ g.

6.2. qCqtqba structure on U~g. Given a fixed Dg–structure on the

Kac–Moody algebra g, the quantum enveloping algebra U~g is naturally en-

dowed with a quasi–Coxeter quasitriangular quasibialgebra structure of type

Dg defined by


(i) Dg-algebra: for any D ∈ SD(Dg), let gD ⊂ g be the corresponding

Kac–Moody subalgebra. The Dg-algebra structure is given by the

subalgebras U~gD.

(ii) Quasitriangular quasibialgebra: the universalR-matrices R~,D,

with trivial associators ΦD = 1⊗3 and structural twists FF = 1⊗2.

(iii) Quasi-Coxeter: the local monodromies are the quantum Weyl group

elements S~i i∈I. The Casimir associators ΦGF are trivial.

We transfer this qCqtqba structure on Ug[[~]]. More precisely, we define

an equivalence of quasi–Coxeter categories between the representation theories

of U~g and Ug[[~]].

6.3. Gauge transformations for g(A). For any D ⊂ Dg, the inclusion

gD ⊂ g, defined in the previous section, lifts to an inclusion of Manin triples

gD ⊕ hD ⊂ g⊕ h

We denote by gD = (gD ⊕ hD, bD,+, bD,−) the Manin triple attached to gD, for

any D ⊆ Dg.

Theorem. There exists an equivalence of braided Dg–monoidal categories

from

((DΦB(U gB[[~]]),⊗B,ΦB, σRB), (ΓBB′ , JBB′

F ))

to

((D(U~gB),⊗B, id, σR~B), (Γ~BB′ , id))

given by (FB, γFBB′).


Proof. The natural transformations γBB′ , B ⊆ B′ ⊆ Dg constructed in

Section 5, define, by vertical composition, a natural transformation

γCBB′ ∈ Nat⊗(Γ~BB′ FB′ , FB ΓBB′)

for any chain of maximal length

C : B = C0 ⊂ C1 ⊂ · · · ⊂ Cr = B′

Any chain of maximal length defines uniquely a maximal nested set FC ∈

Mns(B,B′), but this is not a one to one correspondence. For example, for

D = A3, the maximal nested set

F = α1, α3, α1, α2, α3

corresponds to two different chains of maximal length

C1 : α1 ⊂ α1 t α3 ⊂ A3 C2 : α3 ⊂ α1 t α3 ⊂ A3

In order to prove that the natural transformations γ define a morphism of

braided Dg–monoidal categories, we need to prove that the transformation

γCBB′ depend only on the maximal nested set corresponding to C.


In particular, we have to prove that, for any B1 ⊥ B2 in I(D), the con-

struction of the fiber functor

CB1tB2

FB1,B1tB2

##

FB2,B1tB2

CB1

FB1 ##

CB2

FB2

C∅

is independent of the choice of the chain. In our case,

CB1tB2 = D(U gB1 [[~]]⊗ U gB2 [[~]])

and the braided tensor structure is given by product of the braided tensor

structures on

CB1 = DΦB1(U gB1 [[~]]) CB2 = DΦB2

(U gB2 [[~]])

Similarly, the tensor structure on the forgetful functor

CB1tB2 → CBi i = 1, 2

is obtained killing the tensor structure on CBi , i = 1, 2, i.e., applying the tensor

structure on CBi → C∅. In particular, the tensor structure on FB1 FB1,B1tB2

and FB2 FB2,B1tB2 coincide, since [gB1 , gB2 ] = 0.

Analogously we have an equality of natural transformation

γB1 γB1,B1tB2 = γB2 γB2,B1tB2


Therefore, for any maximal nested set F ∈ Mns(B,B′), it is well defined a

natural transformation

γFBB′ ∈ Nat⊗(Γ~BB′ FB′ , FB ΓBB′)

so that the data (FB, γFBB′) define an isomorphism of D–categories from

DΦB(U gB[[~]]) to D(U~gB).

6.4. Extension to Levi subalgebras. In analogy with [TL4, Thm. 9.1],

we want to show that the relative twists and the Casimir associators are weight

zero elements. This corresponds to show that the corresponding tensor func-

tors Γ and the natural transformations γ lift to the level of Levi subalgebras:

gD ⊂ lD = nD,+ ⊕ h⊕ nD,− ⊂ g

Proposition. The relative twists and the Casimir associators are weight

zero elements.

Proof. For D = ∅, the statement reduces to prove that the Etingof–

Kazhdan functor preserves the h–action [EK6, Thm. 4.3]. The result is a

consequence of Proposition 5.3 applied to Levi subalgebras.

6.5. Reduction to category Oint. The Etingof–Kazhdan functor gives

rise, by restriction, to an equivalence of categories

F : Og[[~]]→ OU~g


We will show now that this equivalence can be further restricted to integrable

modules in category O, i.e., modules in category O with a locally nilpotent

action of the elements ei, fii∈I (respectively Ei, Fi).

Proposition. The Etingof–Kazhdan functor restricts to an equivalence of

braided tensor categories

F : Ointg [[~]]→ Oint

U~g

which is isomorphic to the identity functor at the level of h–graded k[[~]]–

modules.

Proof. Let V ∈ Ointg . Then, the elements ei, fi for i ∈ I act nilpotently

on V . Then, by [Ka], for all λ ∈ P(V ), there exist p, q ∈ Z≥0 such that

t ∈ Z | λ+ tαi ∈ P(V ) = [−p, q]

Since the Cartan subalgebra h is not deformed by the quantization, the functor

F preserves the weight decomposition. In U~g, for any h ∈ h and i ∈ I, we

have

[h,Ei] = αi(h)Ei

Therefore the action of the Ei’s on V is locally nilpotent. The action of the

Fi’s is always locally nilpotent, since

P(V ) ⊂r⋃s=1

D(λs)

The result follows.


Corollary. .

(i) There exists an equivalence of braided Dg–monoidal categories be-

tween


′

F ))

and

O~ := ((OintU~gB,⊗B, id, σR~B), (Γ~BB′ , id))

(ii) There exists an isomorphism of Dg–algebras

ΨF : U~g→ Ug[[~]]

such that ΨF(U~gDi) = UgDi [[~]] for any Di ∈ F , where the comple-

tion is taken with respect to the integrable modules in category O.

6.6. Quasi–Coxeter structure. The previous equivalence of braided

Dg–monoidal categories induces on

O = ((OintgB

[[~]],⊗B,ΦB, σRB), (ΓBB′ , JBB′

F ))

a structure of quasi–Coxeter category of tipe Dg, given by the Casimir asso-

ciators ΦGF ∈ Nat⊗(ΓF ,ΓG) and the local monodromies Si ∈ End(Γi) defined

for any G,F ∈ Mns(B,B′) and i ∈ I(D) by

FB(ΦGF) = (γFBB′)−1 γGBB′ Si = ΨEK

i (S~i )


where ΨEKi : U~sl

i2 → Usli2[[~]] is the isomorphism induced at the sli2 level by

the Etingof–Kazhdan functor.

Proposition. The equivalence of braided Dg–monoidal categories O →

O~ induces a structure of quasi–Coxeter category on O.

Proof. In order to prove the proposition, we have to prove the compatibil-

ity relations of the elements ΦGF , Si with the underlying structure of braided

Dg– monoidal category on O.

The element Si’s satisfy the relation

∆F(Si) = (Ri)21F · (Si ⊗ Si)

since ΨF is given by an isomorphism of braided monoidal D–categories and

therefore

ΨF((R~i )F) = (Ri)F

Similarly, the braid relations are easily satisfied, since

Ad(ΦGF)ΨF = ΨG

The elements ΦFG defined above satisfy all the required properties:

(i) Orientation For any elementary pair (F ,G) in Mns(B,B′)

FB(ΦFG) = (γFBB′)−1 γGBB′ =

(FB(ΦGF)

)−1


(ii) Coherence For any F ,G,H ∈ Mns(B,B′)

FB(ΦFG) = (γFBB′)−1γHBB′ (γHBB′)

−1 γGBB′ =

=FB(ΦFH) HB(ΦHG)

This property implies the coherence.

(iii) Factorization. Clear by construction.

Finally, the elements ΦGF satisfy

∆(ΦGF) JF = JG Φ⊗2GF

because they are given by composition of invertible natural tensor transforma-

tions.

6.7. Normalized isomorphisms. In the completion Usli2[[~]] with re-

spect to category O integrable modules, there are preferred element Si,C

Si,C = si exp(~2Ci)

where

si = exp(ei) exp(−fi) exp(ei) Ci =(αi, αi)

2(eifi + fiei +

1

2h2i )

Proposition. There exists an equivalence of quasi–Coxeter categories of

type Dg between


′

F ), ΦGF, Si,C)


and

O~ := ((OintU~gB,⊗B, id, σR~B), (Γ~BB′ , id), id, S~i )

Proof. Using the result of Proposition 6.6, it is enough to prove that the

natural transformation γi

Ointi

Fi//

f

Ointi,~

f~~~

γipx

A

can be modified in such a way that the induced isomorphism at the level of

endomorphism algebras U~sli2 → Usli2[[~]] maps S~i to Si,C . The natural trans-

formation used in Corollary 6.5 induces the Etingof–Kazhdan isomorphism

ΨEK

i : U~sli2 → Usli2[[~]]

which is the identity mod ~ and the identity on the Cartan subalgebra. As

above, we denote by Si the element ΨEKi (S~i ). Then Si ≡ si mod ~ and, by

[TL4, Proposition 8.1, Lemma 8.4], we have

S2i = S2

i,C Si = Ad(x)(Si,C)

on the integrable modules in category O, for x = (Si,C · S−1i )

12 . Therefore, the

modified isomorphism

Ψi := Ad(x) ΨEK

i


maps S~i to Si,C . Moreover, Ψi correspond with the natural transformation

given by the composition of γi with x ∈ Usli2[[~]] = End(f)

Ointi

f

Ointi

F//

f

Ointi,~

f~

γiks

xks

A A A

The result follows substituting γi with x γi in Proposition 6.6.

6.8. The main theorem. We now state in more details the main theorem

of this Chapter and summarize the proof outlined in the previous results.


structure and U~g the corresponding Drinfeld–Jimbo quantum group with the

analogous Dg–structure. For any choice of a Lie associator Φ, there exists an

equivalence of quasi–Coxeter categories between


′

F ), ΦGF, Si,C)

and

O~ := ((OintU~gB

,⊗B, id, σR~B), (Γ~BB′ , id), id, S~i )

where ⊗B denotes the standard tensor product in OintgB

and

Si,C = si exp(h

2· Ci)

ΦB = 1 mod ~2


RB = exp(~2

ΩD)

Alt2JBB′

F =~2

(rB′ − r21B′

2− rB − r21

B

2

)and ΦGF , JBB

′F are weight zero elements.

Proof. The existence of an equivalence is a consequence of the construc-

tions of Section 5 and proved in Theorem 6.3 and Proposition 6.6, 6.7, con-

cerning the local monodromies Si,C .

The properties of associators ΦB and R–matrices RB are direct conse-

quences of the construction in Section 1,2. The relation satisfied by the relative

twists JBB′

F is proven by a simple application of Proposition 2.20 and Corollary

2.20. It is easy to check that the 1–jet of the twist JBB′

F differs from the 1–jet

of the twist JBB′(as defined in Section 2) by a symmetric element that cancels

out computing the alternator. Therefore, Corollary 2.20 holds for JBB′

F as well.

Finally, as previously explained, the weight zero property of the relative

twists JBB′

F and the Casimir associators ΦGF is proved in Proposition 5.3, 6.4.

This complete the proof of Theorem 6.8.

CHAPTER 4

A rigidity theorem for quasi–Coxeter categories

The goal of this chapter is to show that, under appropriate hypotheses,

there is at most one quasi–Coxeter braided monoidal category on the classi-

cal enveloping algebra Ug[[~]] with prescribed local monodromies (Thm 4.2).

This result is needed to show that the quasi–Coxeter braided monoidal cate-

gory naturally attached to the quantum affine algebra U~g is equivalent to the

one constructed for Ug[[~]], which underlies the monodromy of the rational

KZ and Casimir connections of g.

In Section 1, we review a number of basic notions from [TL4], in particular

that of the Dynkin–Hochschild bicomplex of a D–bialgebra A which controls

the deformations of quasi–Coxeter quasitriangular quasibialgebra structures

on A. In Section 2, we extend the notion of the bicomplex to the case of com-

pletion of algebras, using the language of endomorphisms algebra of monoidal

functors and the corresponding cohomology theory. In Section 3 we study in-

variant properties of the relative twist under the Chevalley involution of g and

we prove a uniqueness property up to gauge transformation. Section 4 contains

the rigidity result, consisting in the construction of a natural transformation

matching the Casimir associators and preserving the local monodromies.

141

142 4. A RIGIDITY THEOREM FOR QUASI–COXETER CATEGORIES

1. The Dynkin-Hochschild bicomplex

In this section, we briefly recall the definition of D–bialgebras and the con-

struction of the Dynkin–Hochschild complex introduced in [TL4].

1.1. The Dynkin complex of a D–algebra. We begin by defining the

category of coefficients of the Dynkin complex of D–algebra A (2.2).

Definition. A D–bimodule over A is an A–bimodule M , with left and

right actions denoted by am and ma respectively, endowed with a family of

subspaces MD1 indexed by the connected subdiagrams D1 ⊆ D of D such that

the following properties hold

• for any D1 ⊆ D,

AD1 MD1 ⊆MD1 and MD1 AD1 ⊆MD1

• For any pair D2 ⊆ D1 ⊆ D,

MD2 ⊆MD1

• For any pair of orthogonal subdiagrams D1, D2 of D, aD1 ∈ AD1 and

mD2 ∈MD2 ,

aD1mD2 = mD2aD1

A morphism of D–bimodules M,N over A is an A–bimodule map T : M → N

such that T (MD) ⊆ ND for all D ⊆ D.

Clearly, A is a D–bimodule over itself. We denote by BimodD(A) the

abelian subcategory of Bimod(A) consisting of D–bimodules over A. If M ∈

1. THE DYNKIN-HOCHSCHILD BICOMPLEX 143

BimodD(A), and D1 ⊆ D is a subdiagram, we set

MD1 = m ∈M | am = ma for any a ∈ ADi1

where Di1 are the connected components of D1. In particular, if D1, D2 ⊆ D

are orthogonal, and D1 is connected, then MD1 ⊆MD2 .

Let M ∈ BimodD(A). For any integer 0 ≤ p ≤ n = |D|, set

Cp(A;M) =⊕

α⊆D1⊆D,|α|=p

MD1\αD1

where the sum ranges over all connected subdiagrams D1 of D and ordered

subsets α = α1, . . . , αp of cardinality p of D1 and MD1\αD1

= (MD1)D1\α. We

denote the component of m ∈ Cp(A;M) along MD1\αD1

by m(D1;α).

Definition. The group of Dynkin p–cochains on A with coefficients in M

is the subspace CDp(A;M) ⊂ Cp(A;M) of elements m such that

m(D1;σα) = (−1)σm(D1;α)

where, for any σ ∈ Sp, σα1, . . . , αp = ασ(1), . . . , ασ(p).

Note that

CD0(A;A) =⊕D1⊆D

Z(AD1) and CDn(A;M) ∼= MD

For 1 ≤ p ≤ n− 1, define a map dpD : Cp(A;M)→ Cp+1(A;M) by

dDm(D1;α) =

p+1∑i=1

(−1)i−1

(m(D1;α\αi) −m(

D1\αiα\αi

;α\αi)

)


where α = α1, . . . , αp+1, D1\αiα\αij

is the connected component of D1 \ αi con-

taining α \ αi if one such exists and the empty set otherwise, and we set

m(∅;−) = 0. For p = 0, define d0D : C0(A;M)→ C1(A;M) by

d0Dm(D1;αi) = mD1 −mD1\αi

where mD1\αi is the sum of mD2 with D2 ranging over the connected com-

ponents of D1 \ αi. Finally, set dnD = 0. It is easy to see that the Dynkin

differential dD is well–defined and that it leaves CD∗(A;M) invariant.

Theorem. (CD∗(A;M), dD) is a complex. The cohomology groups

HDp(A;M) = Ker(dpD)/ Im(dp−1D )

for p = 0, . . . , |D| are called the Dynkin diagram cohomology groups of A with

coefficients in M .

1.2. The Dynkin–Hochschild bicomplex of a D–bialgebra. Let D

be a connected diagram and A a D–bialgebra (the algebraic counterpart of

strict D–monoidal categories 2.7, cf. [TL4, 6.1]). By combining the Dynkin

complex of A with the cobar complexes of its subalgebras AB, we review the

definition of a bicomplex which controls the deformations of quasi–Coxeter

quasibialgebra structures on A introduced in [TL4, 7.1].

Let A be a bialgebra and C∗(A) the cobar complex of A, regarded as a

coalgebra. If C ⊆ A is a sub–bialgebra, let

Cn(A)C = a ∈ Cn(A)| [a,∆(n)(c)] = 0 for any c ∈ C

2. DYNKIN–HOCHSCHILD COMPLEX OF MONOIDAL FUNCTORS 145

where ∆(n) : C → C⊗n is the nth iterated coproduct, be the submodule of

C–invariants. It is easy to check that C∗(A)C is a subcomplex of C∗(A).

Assume now that A is a D–bialgebra. For p ∈ N and 0 ≤ q ≤ |D|, let

CDq(A;A⊗p) ⊂⊕

α⊆B⊆D,|α|=q

(A⊗pB )B\α

be the group of Dynkin q–cochains with values in the D–bimodule A⊗p over

A. The Dynkin differential dDD defines a vertical differential CDq(A;A⊗p)→

CDq+1(A;A⊗p) while the Hochschild differential dH defines a horizontal dif-

ferential CDq(A;A⊗p) → CDq(A;A⊗(p+1)). A straightforward computation

yields the following.

Theorem. One has dDD dH = dH dDD. The corresponding cohomology

of the bicomplex CDq(A;A⊗p) is called the Dynkin–Hochschild cohomology of

the D–bialgebra A.

2. Dynkin–Hochschild complex of monoidal functors

We use the language of monoidal functor and cohomology of monoidal

functor to reformulate the definition of the Dynkin–Hochschild bicomplex in a

more general setting. This approach allows to consider the Dynkin–Hochschild

cohomology for completions of bialgebras with respect to a fixed tensor sub-

category.

2.1. Hochschild complex of monoidal functors. Let F : C → C ′ be

a monoidal functor of associative monoidal k–linear categories with associated


tensor structure J . The nth tensor power of F is the functor

F⊗n : Cn → C ′ F⊗n(X1, . . . , Xn) =n⊗i=1

F (Xi)

Through repeated applications of J , we obtain a canonical isomorphism

F⊗n(X1, · · · , Xn) ' F (n⊗i=1

Xi)

The k–algebra of endomorphisms End(F⊗n) form the cosimplicial complex of

algebras

EndC′(1) //// End(F ) ////// End(F⊗2) //

//////

End(F⊗3) · · ·

with face homomorphisms din : End(F⊗n) → End(F⊗n+1), i = 0, . . . , n + 1,

defined as follows:

(d00f)X : F (X)→ F (X)⊗ 1→ F (X)⊗ 1→ F (X)

(d10f)X : F (X)→ 1⊗ F (X)→ 1⊗ F (X)→ F (X)

for X ∈ C, f ∈ EndC′ 1, and

(dinf)X1,...,Xn+1 =

id⊗fX2,...,Xn+1 i = 0

Ad(JXiXi+1)fX1,...,Xi⊗Xi+1,...Xn+1 1 ≤ i ≤ n

fX1,...,Xn ⊗ id i = n+ 1

for f ∈ End(F⊗n), Xi ∈ C, i = 1, . . . , n + 1. The Hochschild differential is

defined in the usual way:

dn =n+1∑i=0

(−1)idin : End(F⊗n)→ End(F⊗n+1)


The degeneration homomorphisms sin : End(F⊗n) → End(F⊗n−1) are de-

fined, for i = 1, . . . , n, by

(sinf)X1,...,Xn−1 := fX1,...,Xi−1,1,Xi,...,Xn−1

The morphisms sin, din satisfy the standard relations

djn+1din = din+1d

j−1n i < j

sjnsin+1 = sins

j+1n+1 i ≤ j

sjn+1din =

din−1s

j−1n i < j

id i = j, j + 1

di−1n−1s

jn i > j + 1

2.2. Composition of monoidal functor and Hochschild subcom-

plexes. Let F : C → C ′ and G : C ′ → C ′′ be monoidal functors of associative

monoidal categories with associated tensor structures J and K, respectively.

For any n ∈ N there are well– defined homomorphisms

End(F⊗n)⊗ End(G⊗n)→ End((G F )⊗n)

mapping the tensor α ⊗ β into α β ∈ End((G F )⊗n) defined to be the

composition α β = Ad(K)(G(α)) β

⊗ni=1 GF (Xi)

βF//⊗n

i=1GF (Xi)

//⊗n

i=1 GF (Xi)

G(⊗n

i=1 F (Xi))G(α)// G(

⊗ni=1 F (Xi))

OO

The functor G F is naturally endowed with a tensor structure induced by J

and K. A quick computation proves the following


Proposition.

(i) The composition maps are compatible with the Hochschild differential

maps, i.e., the diagram

End(F⊗n)⊗ End(G⊗n)

dFn⊗dGn

n// End((G F )n)

dGFn

End(F⊗n+1)⊗ End(G⊗n+1)n+1// End((G F )⊗n+1)

is commutative for any n ∈ N.

(ii) The natural maps

End(F⊗n)→ End((G F )⊗n) End(G⊗n)→ End((G F )⊗n)

are well–defined morphisms of the associated cochain complexes.

Remark. If F = G = id, then the composition law described above

coincides with the multiplication in End(id⊗n).

2.3. Cohomology of monoidal functors. Let C be a k–linear abelian

monoidal category and F : C → Vect a faithful, exact, associative monoidal

functor. Then we can consider on End(F ) the initial topology with basis of

neighborhood of zero the kernels of the maps End(F )→ End(F (X)). We get

canonical isomorphisms

End(F ) ' limX∈C

End(F (X))


and

End(F⊗n) ' End(F )⊗n

where ⊗ denotes the completion of the tensor product with respect to the

topology of the inverse limit. This identification allows in particular to look

at the canonical map

∆ : End(F )→ End(F⊗2) ' End(F )⊗End(F )

as a topological coproduct on End(F ).

Definition. The cohomology of the cochain complex (End(F⊗•), d) is

called additive cohomology of the monoidal functor F and denoted H•(F ).

The external multiplication

End(F⊗n)⊗ End(F⊗m)→ End(F⊗n+m)

defined in the obvious way on the complex End(F⊗•) induces a structure of

graded ring on H•(F ). This follows from the relations

dn+mi =

dni (α) · β i ≤ n

α · dmn−i(β) i > ndnn+1(α) · β = α · dm0 (β)

The differential graded algebra (End(F⊗•),⊗, d) is homotopically commutative

[D]. Thus the cohomology Hn(F ) is a skew–commutative graded algebra.

Moreover, the homotopy of the commutativity defines on H•(F ) a structure of

Lie superalgebra J·, ·K : Hn(F )⊗Hm(F )→ Hn+m−1(F ), which coincides on the


subalgebra∧• H1(F ) with the Schouten bracket

Ju1∧· · ·∧un, v1∧· · ·∧vmK =∑i,j

(−1)i+j[ui, vj]∧u1∧· · ·∧ui∧. . . un∧v1∧· · ·∧vj∧· · ·∧vm

2.4. Cohomology of the forgetful functor in category O. Let g be

a Lie algebra over a field k, with char k = 0, and F : RepUg → Vect be the

forgetful functor. Tannaka–Krein duality gives a canonical isomorphism Ug '

End(F ). The cochain complex End(F⊗•) coincides with the co–Hochschild

complex of Ug and H1(F ) = Prim(Ug) = g. Since dH(g⊗n) = 0, we obtain

a map from T •g → H•(F ) and we denote by µ the restriction to the exterior

algebra

µ :•∧g→ H•(F )

It is well–known, in this case, that the map µ is an isomorphism [D3].

As mentioned above, for an arbitrary monoidal functor F : C → C ′ we have

a canonical inclusion

µ :•∧H1(F )→ H•(F )

observing that dH(H1(F )⊗n) = 0. For an arbitrary monoidal functor, the mor-

phism µ is not surjective. Easy counterexamples are given by the identity func-

tor on RepUg (cocommutative case) or the forgetful functor RepH → Vect

where H is the Sweedler’s Hopf algebra (non cocommutative). In both cases

we have H1(F ) = 0 and H•(F ) 6= 0.

Our main interest lies in the forgetful functor F : Og → Vect, where g is

a Kac–Moody algebra and Og is the corresponding category O. In this case,


we assume that there exists an appropriate completion of the exterior algebra

that extends µ to an isomorphism.

Conjecture. Let g be a Kac–Moody algebra and F : Og → Vect be the

forgetful functor from the category O, then

H•(F ) '∧•

g H1(F )h ' h

2.5. Dynkin complex of D–categories. Let C = (CB, FBB′)B,B′∈SD(D)

be a D–category. We recall that, by definition, for any B ⊂ B′ ⊂ B′′, we are

given maps

End(FBB′)→ End(FBB′′) End(FB′B′′)→ End(FBB′′)

and, for any B′ ⊥ B′′, B′, B′′ ⊂ B′′′, and B ⊂ B′,

End(FBB′)→ End(F(B∪B′′)B′′′)

For any integer 0 ≤ p ≤ n = |D|, we set

Cp(C) =⊕

α⊆B∈Csd(D),|α|=p

End(F(B,α))

where F(B;α) := F(B\α)B.

Definition. The group of Dynkin p–cochains on C is the subspace CDp(C) ⊂

Cp(C) of elements m such that

f(B;σα) = (−1)σf(B;α)

where, for any σSp, σα1, . . . , αp = ασ(1), . . . , ασ(p).


Note that

CD0(C) =⊕B⊂D

End(idCB) CDn(C) = End(FD)

For 1 ≤ p ≤ n− 1, we define the map dpD : CDp(C)→ CDp+1(C) by

(dDf)(B;α) =

p+1∑i=1

(−1)i−1(f(B;α\αi) − f(

B\αiα\αi

;α\αi)

)as in Section 1.1. By abuse of notation, by f(B;α\αi) we mean its image under

the map

End(F(B;α\αi))→ End(F(B;α))

and by f(B\αiα\αi

;α\αi)its image under the map

End(F(B\αiα\αi

;α\αi))→ End(F(B\αi;α\αi))→ End(F(B;α))

For p = 0, define d0D : CD0(C)→ CD1(C) by

d0Df(B;α) = fB − fB\α

where fB\α is the sum over B′, the connected components of B \ α, of fB′ .

Finally, we set dnD = 0. The Dynkin differential is well–defined and preserves

CD∗(C).

Proposition. (CD∗(C); dD) is a cochain complex.

2.6. Dynkin-Hochschild bicomplex for associative D–monoidal cat-

egories. We now combine the Dynkin complex for a D–category and the

3. UNIQUENESS OF THE RELATIVE TWIST 153

Hochschild complex for monoidal functors to define a suitable bicomplex for as-

sociative D–monoidal categories, which controls the deformation of the quasi–

Coxeter monoidal structure.

For p ∈ N and 0 ≤ q ≤ |D|, we denote by

CDp(Cq) ⊂⊕

α⊆B∈Csd(D),|α|=p

End(F⊗q(B,α))

the group of Dynkin p–cochains for Cq. The Dynkin differential dD defines a

vertical differential

dpD : CDp(Cq)→ CDp+1(Cq)

while the Hochschild differential dH defines a horizontal differential

dqH : CDp(Cq)→ CDp(Cq+1)

A simple computation yields the following.

Proposition. One has dD dH = dH dD.

The corresponding cohomology of the bicomplex CDq(Cp) is called the

Dynkin–Hochschild cohomology of the monoidal D–category C. As in [TL4,

7.2], the DynkinHochschild bicomplex of C controls the formal, one- parameter

deformations of the trivial quasi-Coxeter monoidal structure on C.

3. Uniqueness of the relative twist

3.1. Chevalley invariance. We denote by θ the Chevalley involution on

g and the induced morphism on the Manin triple attached to g. In this section


we will study the following invariant property of a twist under θ:

Jθ = J21 (3.1)

Proposition. The Etingof–Kazhdan twist JEKfd , defined in [EK1, Part I],

satisfies

(JEK

fd )θ = (JEK

fd )21

Proof. The twist JEKfd defines a tensor structure on the functor hM+⊗M−

and it can be identified with the action on V ⊗W of the element J ∈ Ug⊗2[[~]]

J := (ϕ⊗ ϕ)−1(Φ−11,2,34 · Φ234 · e

~2

Ω23 · Φ−1324 · Φ1,3,24)(1+ ⊗ 1− ⊗ 1+ ⊗ 1−)

where ϕ : Ug∼→ M+ ⊗M−. The involution θ defines an autoequivalence of

DΦ(g) and a natural identification

hM+⊗M− θ∗ = hMθ+⊗Mθ

−

Using the isomorphism M θ± ' M∓ and the relations Ωθ = Ω, Φθ = Φ, we

observe that the twist induced on the functor hM+⊗M− θ∗, denoted (JEK)θ, is

given by the same formula defining JEK on the functor hM−⊗M+ . It corresponds

to the action on V ⊗W of the element

Jθ := (ψ ⊗ ψ)−1(Φ−11,2,34 · Φ234 · e

~2

Ω23 · Φ−1324 · Φ1,3,24)(1− ⊗ 1+ ⊗ 1− ⊗ 1+)


where ψ : Ug∼→ M− ⊗M+. It is easy to check that ϕ ψ−1 : M− ⊗M+ →

M+ ⊗M− corresponds to the permutation σ12. We then obtain

J21 =(ψ−1 ⊗ ψ−1)σ14σ23(Φ−11,2,34 · Φ234 · e

~2

Ω23 · Φ−1324 · Φ1,3,24)(1+ ⊗ 1− ⊗ 1+ ⊗ 1−) =

=(ψ−1 ⊗ ψ−1)(Φ−14,3,21 · Φ321 · e

~2

Ω23 · Φ−1231 · Φ4,2,31)(1− ⊗ 1+ ⊗ 1− ⊗ 1+) =

=(ψ−1 ⊗ ψ−1)(Φ12,3,4 · Φ−1123 · e

~2

Ω23 · Φ132 · Φ13,2,4)(1− ⊗ 1+ ⊗ 1− ⊗ 1+) = Jθ

where the last equivalence follows by the pentagon axiom. Alternatively, we

can also notice that J21 can be thought as the twist of the functor F EK with

respect to the opposite tensor product, naturally identified with hM−⊗M+ .

The Etingof–Kazhdan functor F EK can be naturally identified with hM+⊗M− ,

in the finite dimensional setting, via the natural transformation χ : F EK →

hM+⊗M−

χV (v) := (ev ⊗ id) Φ−1 (id⊗v)

The identification preserves the tensor structure, but it does not satisfy χθ = χ.

We therefore conclude that the twist JEK does not satisfy 3.1. The following

lemma shows that we are allowed to weaken the requirement.

Lemma. Let u ∈ End(F )[[~]] an invertible natural transformation of the

tensor functor (F, J) satisfying

u ? Jθ = J21

where u ? J := ∆(u−1) · J · u ⊗ u. If uθ = u−1, J is gauge equivalent to a

solution of 3.1.


Proof. We first prove that the condition uθ = u−1 is equivalent to the

existence of an invertible element v ∈ End(F )[[~]] such that u = v−1vθ. One

implication is trivial.

We have u ≡ 1 mod ~ and uθ = u−1. Denote u = 1 +∑

n≥0 un~n, so that

uθ1 = −u1. The element v1 := −u1/2 satisfies

vθ1 − v1 = u1

Consider the natural transformation

u(2) := (1 + v1~) · u · (1 + vθ1~)−1

Then

u(2) ≡ 1 mod ~2 (u(2))θ = (u(2))−1

Assume we constructed u(n) satisfying u(n) ≡ 1 mod ~n and (u(n))θ = (u(n))−1.

Similarly, consider the element vn = −u(n)n /2 and

u(n+1) := (1 + vn~n) · u(n) · (1 + vθn~n)−1

Then u(n+1) satisfies u(n+1) ≡ 1 mod ~n+1 and (u(n+1))θ = (u(n+1))−1. By

induction, we obtain

u =∏n≥0

(1 + vn~n)−1 ·∏n≥0

(1 + vn~n)θ

and we denote by v the natural transformation∏

n≥0(1 + vn~n). The factor-

ization is uniquely defined up to a natural transformation w ∈ End(F )[[~]]


invariant under θ.

Finally, the relation u ? Jθ = J21 and the factorization u = v−1vθ imply

(v ? J)θ = (v ? J)21

An easy computation on the 2–jet of JΓ (cf. 2.15) shows that the relative

twist does not satisfy, in the finite dimensional setting, the condition 3.1. We

observe that the natural transformation constructed in 5.1 gives rise to a gauge

transformation relating JΓ and the composition of JEK and (JEKD )−1. We get

the following

Corollary. In the finite–dimensional setting, there exists a natural trans-

formation u ∈ End(Γ) such that

(u ? JΓ)θ = (u ? JΓ)21

where (u ? J)V⊗W := u−1V⊗WJVWuV ⊗ uW for any V,W ∈ DΦ(g).

3.2. Uniqueness property. We now prove a uniqueness property of the

relative twist. By 2.4 it is easy to check that

H2S(Γ) := H2((

∧H1(F ))lD , Jr − rD, ·K) = (

∧2lD)lD =

∧2cD

Theorem. Let Ji, i = 1, 2 be two tensor structures on the functor Γ :

Oint(g)→ Oint(lD) such that Ji ≡ 1⊗2 + ~fi mod ~ and

Alt2Ji ≡ 1⊗2 +~2

(r − r21

2− rD − r21

D

2

)mod ~2 i = 1, 2


Then there exist

u ∈ End(Γ) λ ∈ H2S(Γ)

such that

expλ · J2 = u ? J1

In the finite dimensional case, under the additional assumption that

Jθi = J21i

we have λ = 0 and u is uniquely determined by the property uθ = u.

Proof. Denote

r =r − r21

2rD =

rD − r21D

2

and assume

Alt2fi =1

2(r − rD) + νi νi ∈ H2

S(Γ)

Then there exist gi ∈ End(Γ) such that

fi − Alt2fi = dHgi

Therefore, we can replace Ji with

exp(−~νi)(1− ~gi) ? Ji

and assume that fi = 1/2(r − rD). In the particular case of the relative twist

JΓ we have

f − Alt2f =1

4(Ω− ΩD) =

1

4dH(m(rD)−m(r))


where m(r) is the element obtained by multiplication of the components of

the r–matrix.

The additional assumption Jθi = J21i implies

dHgθi + νθi = dHgi + ν21

i ⇒ dHgθi = dHgi, νθi = ν21

i

since dHgi is symmetric and (r− rD)θ = −(r− rD) = (r− rD)21. Therefore, we

have νi = 0, since θ acts as the identity on∧2cD, and we may assume gθi = gi,

replacing gi with 12(gi + gθi ).

We will now construct inductively two sequences

vn ∈ End(Γ) µn ∈ H2S(Γ)

such that

J2 ≡ exp(λn)(un ? J1) mod ~n+1

where

un =n∏i=1

(1 + ~nvn) λn =n∑i=1

~iµi

Since we reduced both twists to the form

Ji ≡ 1⊗2 +~2

(r − rD) mod ~2

we may assume v1 = 0, µ1 = 0. Therefore, we assume elements vk, µk defined

for k = 1, . . . , n, n ≥ 1. Denote J ′1 the element exp(λn)(un ? J1) so that

J2 = J ′1 + ~n+1η mod ~n+2 η ∈ End(Γ⊗2)


We clearly have ΦJ ′1= ΦD = ΦJ2 . Computing mod ~n+2 we obtain

1⊗ η − (∆⊗ id)(η) + (id⊗∆)(η)− η ⊗ 1 = 0

i.e., dHη = 0. Moreover, if Jθi = J21i , we get ηθ = η21. Then, there exist

v ∈ End(Γ) and µ ∈ H2(Γ)= (∧2H1(F ))gD such that η = dHv + µ and

vθ = v µθ = µ21= −µ

Therefore, we set vn+1 = −v and

J ′′1 =(1 + ~n+1vn+1) ? J ′1 =

=(1 + ~n+1vn+1) ? (exp(λn) · (un ? J1)) =

= exp(λn) ·((1 + ~n+1vn+1) · un

)? J1

since clearly vn+1 and λn commute. We now have

J2 ≡ exp(−~n+1µ) · J ′′1 mod ~n+2

So we set µn+1 = −µ and it remains to show that µ lies in H2S(Γ).

Consider the truncation of J2 mod ~n+2

J(n+2)2 = 1⊗2 +

~2

(r − rD) +n+1∑i=2

~ifi

Then we have

ΦJ

(n+2)2

≡ ΦD + ~n+2ξ mod ~n+3


for some ξ ∈ End(Γ⊗3). Since J(n+2)2 is the truncation of a relative twist mod-

ulo ~n+3, we have dHξ = 0 and Alt3ξ = 0. Similarly, we can consider the

truncation of J ′′1 modulo ~n+2, J ′′1(n+2), and denote by ξ′′ the corresponding

error. Then it satisfies dHξ′′ = 0 and Alt3ξ

′′ = 0 (cf. [TL3, 5.4]).

Now, since J ′′1(n+2) = J

(n+2)2 + ~n+1µ mod ~n+2, denote by f = 1

2(r − rD)

and we obtain

ξ′′ − ξ = f 23(µ12 + µ13) + µ23(f 12 + f 13)− f 12(µ13 + µ23)− µ12(f 13 + f 23)

then we get Jf, µK = 0. Then we have

µ = [[f, x]] + y for some x ∈ H1(F )lD = cD ⊂ h, y ∈ H2S(Γ)

Since f is a weight zero element, we have Jf, xK = − ad(x)f = 0. We conclude

that

µ ∈ H2S(Γ) = H2((

∧H1(F ))lD , Jr − rD, ·K)'

∧2cD[[~]]

Since we are assuming Jθi = J21i , we get µθ = −µ and µ = 0.

Let u ∈ End(Γ) be such that

u ? J1 = J1

with uθ = u. Assume u ≡ 1 mod ~n and write u ≡ 1 + ~nun mod ~n+1, with

uθn = un. Taking the coefficient of ~n+1 we get dHun = 0, implying un ∈ h,

since it is of weight zero. Then we get un = 0, θ acting on h by −1.


4. The main theorem

4.1. The D–bialgebra structure on Ug. Let A = (aij)i,j∈I be an irre-

ducible, symmetrizable, generalised Cartan matrix of affine type and g = g(A)

the corresponding affine Kac–Moody algebra [Ka]. Let Dg = D(A) be the

Dynkin diagram of g, that is the connected graph having I as its vertex set

and an edge between i and j if aij 6= 0. For any i ∈ I, let sli2 ⊆ g be the three–

dimensional subalgebra spanned by ei, fi, hi. In this case, any proper connected

subdiagram of Dg is a Dynkin diagram of finite type. Therefore, for any proper

subdiagram D ⊂ Dg, we consider the subalgebra gD spanned by ei, fi, hii∈D.

For D = Dg, we consider the entire algebra g. Then (Ug, UgD) is a Dg-

algebra over C.

4.2. The rigidity theorem. Extend the bilinear form (·, ·) on h to a

non–degenerate, symmetric, bilinear, ad–invariant form on g. For any proper

connected subdiagram D ⊂ Dg, let gD ⊂ lD ⊂ g be the corresponding simple

and Levi subalgebras. Denote by

ΩD = xa ⊗ xa, CD = xa · xa and rgD =∑α0:

supp(α)⊆D

(α, α)

2· eα ∧ fα

where xaa, xaa are dual basis of gD with respect to (·, ·), the correspond-

ing invariant tensor, Casimir operator and standard solution of the modified

classical Yang–Baxter equation for gD respectively. Abbreviate slαi2 , Ωαi and

Cαi to sli2, Ωi and Ci respectively and let si be the triple exponentials

si = exp(ei) exp(−fi) exp(ei)

4. THE MAIN THEOREM 163

where ei = eαi , fi = fαi are a fixed choice of simple root vectors.

In the next sections 4.3-4.5, we outline the proof of the following

Theorem. Up to equivalence, there exists a unique quasi-Coxeter braided

monoidal D–category of type Dg of the form

(Oint

g [[~]], OintgD

[[~]],⊗, RD,ΦD, Si,c, Φ(D;αi,αj), F(D;αi), J(D;αi))

where ⊗ is the standard tensor product in Og, (F(D;αi), J(D;αi)) are fiber func-

tors,

Si,c = si · exp (~/2 · Ci) (4.1)

RD = exp(~/2 · ΩD) (4.2)

Alt2 J(D;αi) =~2

(rgD − rgD\αi

)mod ~2 (4.3)

and Φ(D;αi,αj), J(D;αi) are of weight 0, i.e., they properly extend to the corre-

sponding Levi’s subalgebras.

4.3. Proof of Theorem 4.2. Let

(Oint

g [[~]], OintgD

[[~]],⊗, RD,ΦkD, Si,c, Φk

(D;αi,αj), F(D;αi), J

k(D;αi)

)

k = 1, 2 be two quasi–Coxeter braided monoidal categories.

By Drinfeld’s uniqueness, [TL3, TL4] and the uniqueness of the twist, we

may assume that

Φ2D = Φ1

D J2(D;αi)

= J1(D;αi)


and that

Φ1(D;αi,αj)

= Φ2(D;αi,αj)

+ ~nϕ(D;αi,αj) mod ~n+1

for some ϕ(D;αi,αj) ∈ End(FD\αi,αj).

Let αi 6= αj ∈ D ⊆ Dg and F ,G ∈ Mns(Dg) such that

F \ G = D\αiαjG \ F = D\αjαi

Subtracting the equations

JF ·∆(Φ1(D;αi,αj)

) = Φ2(D;αi,αj)

⊗2 · JG

JF ·∆(Φ2(D;αi,αj)

) = Φ1(D;αi,αj)

⊗2 · JG

and equating the coefficients of ~n+1, we find

∆(ϕ(D;αi,αj))− ϕ(D;αi,αj) ⊗ 1− 1⊗ ϕ(D;αi,αj) = 0

so that ϕ(D;αi,αj) is a primitive element of End(FD). Since ϕ(D;αi,αj) is also of

weight 0, we find that

ϕ(D;αi,αj) ∈ hD

Moreover, since Φa(D;αi,αj)

satisfy the generalized pentagon identities corre-

sponding to the 2-faces of the De Concini–Procesi associahedron, we also find

dDϕ(D;αi,αj) = 0

We are then reduced to prove the following


Proposition. Let ϕ = ϕ(D;αi,αj) be a 2–cocycle in the Dynkin complex

of End(F ) such that


for any αi 6= αj ∈ D ⊆ Dg. Then, there exists a Dynkin 1–cochain a =

a(D;αi) such that

a(D;αi) ∈ hD and dDa = ϕ

The element a may be chosen such that a(αi;αi) = 0 for all αi and is then

unique with this additional property.

The element a = 1−~na(D;αi)αi∈D⊆Dg defines the twist matching the two

qCqtqba structures. Notice that, since a(αi;αi) = 0, then (Si,c)a = Si,c for all i.

The theorem is proved.

4.4. Proof of Prop. 4.3: simple case. Let us first give a proof of

Prop.4.3 for the simple case.1

Let then g be a simple complex Lie algebra of rank l, Dg be the Dynkin di-

agram associated to g with vertices α1, . . . , αl. For any i ∈ [1, l], let slαi2 ⊂ g

be the three dimensional subalgebra spanned by ei, fi, hi. Then the collec-

tion Uslαi2 defines a Dg–algebra structure on Ug.

Let hD ⊂ h be the subalgebra generated by α∨i | αi ∈ D, where D ⊂ Dg,

and λ∨i,D be the fundamental coweights associated to hD, for any D ⊂ Dg.

1the original proof in [TL4] is incomplete, since it is implicitly assumed that λ∨i,D does

not depends upon D or, in other words, that λ∨i,D′ ∈ Cλ∨i,D for D′ ⊂ D.


We wish to solve the equation ϕ = dDa. In components, this reads

ϕ(D;αi,αj) = a(D;αj) − a(D\αiαj

;αj)− a(D;αi) + a

(D\αjαi

;αi)(4.4)

for any connected subdiagram D ⊆ Dg and αi 6= αj, αi, αj ∈ D. The assump-

tions ϕ(D;αi,αj), a(D;αi) ∈ hD and the fact that ϕ, a lie in the Dynkin complex

of g imply that

ϕ(D;αi,αj) ∈ Cλ∨i,D ⊕ Cλ∨j,D and a(D;αk) ∈ Cλ∨k,D for k = i, j

respectively, where λ∨k,D is the fundamental coweight in hD dual to αk|hD .

Projecting (4.4) on λ∨i,D and λ∨j,D we therefore find that it is equivalent to

a(D;αi) = ai(D\αjαi

;αi)− ai

(D\αiαj

;αj)− ϕi(D;αi,αj)

a(D;αj) = aj(D\αiαj

;αj)− aj

(D\αjαi

;αi)+ ϕj(D;αi,αj)

where

• by (−)i we denote the component along λ∨i,D of the projection

pi,D : hD =⊕αj∈D

Cλ∨j,D −→ Cλ∨i,D

• for any subdiagram D′ ⊂ D, we look at

a(D′;αi) ∈ hD′ ⊂ hD

as an element of the bigger subspace, since in general

λ∨i,D′ 6∈ Cλ∨i,D λ∨i,D′ ∈⊕αk∈D

λ∨k,D


• we are identifying a(D;αi), a(D,αj) with their components along λ∨i,D, λ∨j,D

respectively.

Thus, since ϕ(D;αi,αj) = −ϕ(D;αj ,αi), ϕ = dDa iff

a(D;αi) = ai(D\αjαi

;αi)− ai

(D\αiαj


(4.5)

for all D and 1 ≤ i 6= j ≤ n with αi, αj ∈ D. Induction on the cardinality of

D readily shows that the above equations possess at most one solution once

the values of a(αi;αi) are fixed. To prove that one solution exists, assume that

a(D;αi) have been constructed for all D with at most m vertices in such a way

that the equations (4.5) hold for all such D. We claim that (4.5) may be

used to define a(D;αi) for all D with |D| = m + 1 in a consistent way, i.e.,

independently of j 6= i such that αj ∈ D. This amounts to showing that for

all such D, and distinct vertices αi, αj, αk ∈ D, one has

ai(D\αjαi

;αi)− ai

(D\αiαj


= ai(D\αkαi

;αi)− ai

(D\αiαk

;αk)− ϕi(D;αi,αk)

(4.6)

To see this, consider the (D;αi, αj, αk) component of dDϕ i.e., the sum(ϕ(D;αj ,αk) − ϕ(

D\αiαj,αk

;αj ,αk)

)−(ϕ(D;αi,αk) − ϕ

(D\αjαi,αk

;αi,αk)

)+

(ϕ(D;αi,αj) − ϕ(

D\αkαi,αj

;αi,αj)

)


Since dDϕ = 0 we get, by projecting on λ∨i,D and using ϕi(D;αj ,αk) = 0,

ϕi(D;αi,αj)= ϕi(D;αi,αk) − ϕi

(D\αjαi,αk

;αi,αk)+ ϕi

(D\αiαj,αk

;αj ,αk)+ ϕi

(D\αkαi,αj

;αi,αj)

so that (4.6) holds iff

ai(D\αjαi

;αi)− ai

(D\αiαj

;αj)− ai

(D\αkαi

;αi)+ ai

(D\αiαk

;αk)=

= −ϕi(D\αjαi,αk

;αi,αk)+ ϕi

(D\αiαj,αk

;αj ,αk)+ ϕi

(D\αkαi,αj

;αi,αj)

(4.7)

ai(D\αiαk

;αk)− ai

(D\αiαj

;αj)− ϕi

(D\αiαj,αk

;αj ,αk)=

= ai(D\αkαi

;αi)+ ϕi

(D\αkαi,αj

;αi,αj)− ai

(D\αjαi

;αi)− ϕi

(D\αjαi,αk

;αi,αk)

(4.8)

The inductive assumption yields the following relation for any distinct αi, αj, αk ∈

D

ϕ(D\αjαi,αk

;αi,αk)= a

(D\αjαi,αk

;αk)− aD\αjαi,αk

\αiαk

;αk

− a

(D\αjαi,αk

;αi)+ aD\αjαi,αk

\αkαi

;αi

(4.9)

We consider five separate cases.

4.4.1. At least two connected component among D\αjαi,αk , D\αkαi,αj ,D\αiαj ,αk

are empty. We may assume D\αjαi,αk = ∅, D\αkαi,αj = ∅. This case cannot arise

since the first condition implies that any path in D from αi to αk must pass

through αj before it reaches αk while the second one implies that the portion

of this path linking αi to αj must first pass through αk.


4.4.2. D\αjαi,αk = ∅, D\αkαi,αj 6= ∅, D\αiαj ,αk 6= ∅. In this case,

ϕ(D\αjαi,αk

;αi,αk)= 0

and (4.7) reads, using (4.9),

ai(D\αiαk

;αk)− ai

(D\αiαj

;αj)− ai

(D\αiαj,αk

;αk)

+ aiD\αiαj,αk\αj

αk;αk

+ ai(D\αiαj,αk

;αj)− aiD\αiαj,αk

\αkαj

;αj

=

= ai(D\αkαi

;αi)+ ai

(D\αkαi,αj

;αj)− aiD\αkαi,αj

\αiαj

;αj

− ai

(D\αkαi,αj

;αi)+ aiD\αkαi,αj

\αjαi

;αi

− ai

(D\αjαi

;αi)

This equation holds since:

• D\αkαi = D\αkαi,αj , so a(D\αkαi

;αi)= a

(D\αkαi,αj

;αi)

• D\αiαk = D\αiαj ,αk , so a(D\αiαk

;αk)= a

(D\αiαj,αk

;αk)

• D\αiαj = D\αiαj ,αk , so a(D\αiαj

;αj)= a

(D\αiαj,αk

;αj)

• Since D\αjαi,αk = ∅, then

D\αjαi= D\αj ,αkαi

= D\αkαi,αj

\αjαi =⇒ a

(D\αjαi

;αi)= aD\αkαi,αj

\αjαi

;αi

• Similarly,

D\αkαi,αj

\αiαj = D\αi,αkαj

= D\αiαj,αk

\αkαj =⇒ aD\αkαi,αj

\αiαj

;αj

= aD\αiαj,αk\αk

αj


Condition (4.7) reduces to:

ai(D\αkαi,αj

;αj)= aiD\αiαj,αk

\αjαk

;αk

that is

ai(D\αkαj

;αj)= ai

(D\αjαk

;αk)

We claim that for all distinct i, j, k ∈ I,

ai(D\αkαj

;αj)= 0

that is equivalent to:

λ∨j,D\αkαj

∈ Cλ∨j,D ⊕ Cλ∨k,D (4.10)

Indeed, if i ∈ Ikj := r ∈ I | αr ∈ D\αkαj , i 6= j, then

〈αi, λ∨j,D\αkαj

〉 = 0

by definition of λ∨j,D\αkαj

. On the other hand, if i 6∈ Ikj , then, by orthogonality,

we have:

∀s ∈ Ikj 〈αi, α∨s 〉 = 0

Consequently, since λ∨j,D\αkαj

∈ hD\αkαj

, we have

〈αi, λ∨j,D\αkαj

〉 = 0

and (4.10) is proved.

4.4.3. D\αjαi,αk 6= ∅, D\αkαi,αj = ∅, D\αiαj ,αk 6= ∅. This case reduces to the

previous one under the interchange αj ↔ αk.


4.4.4. D\αjαi,αk 6= ∅, D\αkαi,αj 6= ∅, D\αiαj ,αk = ∅. In this case,

ϕ(D\αkαi,αj

;αi,αk)= 0

and (4.7) reads, using (4.9),

ai(D\αiαk

;αk)− ai

(D\αiαj

;αj)=

= ai(D\αkαi

;αi)+ ai

(D\αkαi,αj


\αiαj

;αj

− ai

(D\αkαi,αj

;αi)

+aiD\αkαi,αj\αj

αi;αi

− ai

(D\αjαi

;αi)− ai

(D\αjαi,αk

;αk)− aiD\αjαi,αk

\αiαk

;αk

+ai

(D\αjαi,αk

;αi)− aiD\αjαi,αk

\αkαi

;αi

This equation holds, since:

• D\αjαi = D\αjαi,αk , so a(D\αjαi

;αi)= a

(D\αjαi,αk

;αi)


;αi)= a

(D\αkαi,αj

;αi)

• Since D\αiαj ,αk = ∅, then

D\αiαk= D\αi,αjαk

= D\αjαi,αk

\αiαk =⇒ a

(D\αiαk

;αk)= aD\αjαi,αk

\αiαk

;αk

D\αiαj= D\αi,αkαj

= D\αkαi,αj

\αiαj =⇒ a

(D\αiαj

;αj)= aD\αkαi,αj

\αiαj

;αj

• Similarly,

D\αjαi,αk

\αkαi = D\αj ,αkαi

= D\αkαi,αj

\αjαi


implies

aD\αjαi,αk\αk

αi;αi

= aD\αkαi,αj\αj

αi;αi

As before, it remains to check that:

ai(D\αkαi,αj

;αj)= ai

(D\αkαj

;αj)= 0

ai(D\αjαi,αk

;αk)= ai

(D\αjαk

;αk)= 0

that it is true by (4.10).

4.4.5. D\αjαi,αk 6= ∅, D\αkαi,αj 6= ∅, D\αiαj ,αk 6= ∅. It is interesting to observe

that this case can occur only if there is a cycle in D passing through αi, αj, αk

that is not our case, since g is simple. Instead, in the affine case g, it can

happen only if g = An and D = Dg.

Even in this case, anyway, the consistency relation holds. Indeed, (4.7) reads,


using (4.9),

ai(D\αiαk

;αk)− ai

(D\αiαj

;αj)− ai

(D\αiαj,αk

;αk)+ aiD\αiαj,αk

\αjαk

;αk

+ai

(D\αiαj,αk

;αj)− aiD\αiαj,αk

\αkαj

;αj

=

= ai(D\αkαi

;αi)+ ai

(D\αkαi,αj


\αiαj

;αj

− ai

(D\αkαi,αj

;αi)

+aiD\αkαi,αj\αj

αi;αi

− ai

(D\αjαi

;αi)− ai

(D\αjαi,αk

;αk)+ aiD\αjαi,αk

\αiαk

;αk

+ai

(D\αjαi,αk

;αi)− aiD\αjαi,αk

\αkαi

;αi

This equation holds, since:

• D\αjαi = D\αjαi,αk , so a(D\αjαi

;αi)= a

(D\αjαi,αk

;αi)

• D\αiαj = D\αiαj ,αk , so a(D\αiαj

;αj)= a

(D\αiαj,αk

;αj)


;αi)= a

(D\αkαi,αj

;αi)

• D\αiαk = D\αiαj ,αk , so a(D\αiαk

;αk)= a

(D\αiαj,αk

;αk)

• Similarly,

D\αjαi,αk

\αkαi = D\αj ,αkαi

= D\αkαi,αj

\αjαi

implies

aD\αjαi,αk\αk

αi;αi

= aD\αkαi,αj\αj

αi;αi


then

D\αiαj,αk

\αkαj = D\αi,αkαj

= D\αkαi,αj

\αiαj

implies

aD\αiαj,αk\αk

αj;αj

= aD\αkαi,αj\αi

αj;αj

and finally

D\αjαi,αk

\αiαk = D\αi,αjαk

= D\αiαj,αk

\αjαk

implies

aD\αjαi,αk\αi

αk;αk

= aD\αiαj,αk\αj

αk;αk

As before it remains to check that:

ai(D\αkαi,αj

;αj)= ai

(D\αkαj

;αj)= 0

ai(D\αjαi,αk

;αk)= ai

(D\αjαk

;αk)= 0

that is true, by (4.10).

This concludes the proof of Proposition 4.3

4.5. Proof of Prop. 4.3: Affine case. We will now discuss the previ-

ous proof using the combinatorics of the Dynkin diagrams of affine type.

Let g be a simple complex Lie algebra of rank l and g be the affine Lie

algebra of type g. We denote by Dg the Dynkin diagram associated to g with


vertices α0, α1, . . . , αl and by h = h ⊕ Cc ⊕ Cd ⊂ g the Cartan subalgebra

generated by α∨0 , α∨1 , . . . , α∨l , α∨l+1 where

α∨0 =c

(θ|θ)− θ∨ ⊗ 1 α∨l+1 = d

Let B = α0, α1, . . . , αl, αl+1 be a basis for h∗, where αl+1 = c∗ is defined by

∀i ∈ 0, 1, . . . , l, l + 1 〈αl+1, α∨i 〉 = δ0i

We denote by λ∨0 , λ∨1 , . . . , λ∨l , λ∨l+1 the fundamental coweights of h. for any

subdiagram D ⊆ Dg is given a subalgebra

hD =

〈α∨i | αi ∈ D〉 if D 6= Dg

h if D = Dg

As in the previous section, we aim at proving the following

Proposition. Let ϕ = ϕ(D;αi,αj) be a 2–cocycle in the Dynkin complex

of U g such that


for any αi 6= αj ∈ D ⊆ Dg. Then, there exists a Dynkin 1–cochain a =

a(D;αi) such that

a(D;αi) ∈ hD and dDa = ϕ

The element a may be chosen such that a(αi;αi) = 0 for all αi and is then

unique (up to a coboundary) with this additional property.


Proof. We will follow the same outline of Prop. 4.3. Indeed, as before,

the equation ϕ = dDa, in components, reads

ϕ(D;αi,αj) = a(D;αj) − a(D\αiαj

;αj)− a(D;αi) + a

(D\αjαi

;αi)(4.11)

for any connected subdiagram D ⊆ Dg and i < j such that αi, αj ∈ D. If D is

a proper subdiagram, then gD is of finite type and the discussion is the same

that before.

If D = Dg is the Dynkin diagram of g, then ϕ(D;αi,αj) ∈ Cλ∨i ⊕ Cλ∨j ⊕ Cλ∨l+1

a(D;αi) ∈ Cλ∨i ⊕ Cλ∨l+1

since λ∨l+1 = c. Componentwise (4.11) reads

ϕi(D;αi,αj)= ai

(D\αjαi

;αi)− ai

(D\αiαj

;αj)− ai(D;αi)

(4.12)

ϕj(D;αi,αj)= aj

(D\αjαi

;αi)− ai

(D\αiαj

;αj)+ aj(D;αj)

(4.13)

ϕl+1(D;αi,αj)

= al+1(D;αj)

− al+1

(D\αiαj

;αj)− al+1

(D;αi)+ al+1

(D\αjαi

;αi)(4.14)

It is useful to notice that, if D = Dg, (4.10) becomes

λ∨j,D\αkαj

∈ Cλ∨j,D ⊕ Cλ∨k,D ⊕ Cλ∨l+1,D (4.15)

then the consistency of (4.11) along λ∨i,D, λ∨j,D is granted by the proof of (4.3).

Moreover, it is easy to see that, if α0 6∈ D\αkαj , then

λ∨j,D\αkαj

∈ Cλ∨j,D ⊕ Cλ∨k,D (4.16)


or, in other words,

al+1

(D\α0αj

;αj)= 0 (4.17)

Indeed, if

λ∨j,D\αkαj

=l+1∑i=0

yij,kα∨i =

l+1∑i=0

xij,kλ∨i,D

we have xl+1j,k = y0

j,k.

It remains to prove the consistency of (4.11) along λ∨l+1,D. Up to a coboundary,

we may assume al+1(D;α0) = 0. It is enough to observe that the element a′ =

a′(D′;αi) defined by

a′(D′;αi) =

0 if D′ 6= Dg

δ0ic if D′ = Dg

is indeed a coboundary. Assume then al+1(D;α0) = 0. By (4.17), for each j 6= 0,

ϕl+1(D;α0,αj)

= al+1(D;αj)

+ al+1

(D\αjα0

;α0)(4.18)

Using this relation in (4.14), we get

ϕl+1(D;αi,αj)

= ϕl+1(D;α0,αj)

− al+1

(D\αjα0

;α0)− al+1

(D\αiαj

;αj)

−ϕl+1(D;α0,αi)

+ al+1

(D\αiα0

;α0)+ al+1

(D\αjαi

;αi)

(4.19)

To see this, consider the (D;α0, αi, αj) component of dDϕ i.e., the sum(ϕ(D;αi,αj) − ϕ(

D\α0αi,αj

;αi,αj)

)−(ϕ(D;α0,αj) − ϕ(

D\αkα0,αj

;α0,αj)

)+

(ϕ(D;α0,αi) − ϕ(

D\αjα0,α1

;α0,α1)

)


Since dDϕ = 0 and ϕl+1

(D\α0αi,αj

;αi,αj)= 0 by (4.16), we get, by projecting on

λ∨l+1,D,

ϕl+1

(D\αjα0,αi

;α0,αi)− ϕl+1

(D\αiα0,αj

;α0,αj)= al+1

(D\αiα0

;α0)+ al+1

(D\αjαi

;αi)

−al+1

(D\αjα0

;α0)− al+1

(D\αiαj

;αj)

(4.20)

Recall that a = a(D′;αi) is determined by induction on D and the relation

al+1(D;αi)

= ϕl+1(D;α0,αj)

− al+1

(D\αjα0

;α0)(4.21)

gives the recursion formula for the component along λ∨l+1. In particular, in-

duction on D gives us the following relation:

ϕ(D\αjα0,αi

;α0,αi)= a

(D\αjα0,αi

;αi)− aD\αjα0,αi

\α0αi

;αi

−a

(D\αjα0,αi

;α0)+ aD\αjα0,αi

\αiα0

;αi

(4.22)

Projecting along λ∨l+1,D, by (4.16),

ϕl+1

(D\αjα0,αi

;α0,αi)= al+1

(D\αjα0,αi

;αi)− al+1

(D\αjα0,αi

;α0)+ al+1D\αjα0,αi

\αiα0

;αi

(4.23)

We consider four separate cases.

4.5.1. D\αjα0,αi = ∅ and D\αiα0,αj = ∅. This case cannot arise since the first

condition implies that any path in D from α0 to αi must pass through αj

before it reaches αi while the second one implies that the portion of this path

linking α0 to αj must first pass through αi.


4.5.2. D\αjα0,αi 6= ∅ and D\αiα0,αj = ∅. In this case,

ϕ(D\αiα0,αj

;α0,αj)= 0

and (4.20) reads

al+1

(D\αjα0,αi

;αi)− al+1

(D\αjα0,αi


\αiα0

;α0

=

= al+1

(D\αiα0

;α0)+ al+1

(D\αjαi

;αi)− al+1

(D\αjα0

;α0)− al+1

(D\αiαj

;αj)

(4.24)

This equation holds since

• D\αjαi = D\αjα0,αi , so al+1

(D\αjαi

;αi)= al+1

(D\αjα0,αi

;αi)

• D\αjα0 = D\αjα0,αi , so al+1

(D\αjα0

;α0)= al+1

(D\αjα0,αi

;α0)

• Since D\αiα0,αj = ∅,

D\αiα0= D\αi,αjα0

= D\αjα0,αi

\αiα0 =⇒ al+1

(D\αiα0

;α0)= al+1

(D\αjα0,αi

\αiα0

;α0)

So, we need to verify

al+1

(D\αiαj

;αj)= 0

Since D\αiα0,αj = ∅, then α0 6∈ D\αiαj and, by (4.17), we can conclude.

4.5.3. D\αjα0,αi = ∅ and D\αiα0,αj 6= ∅. This case reduces to the previous one

under the interchange αj ↔ αk.


4.5.4. D\αjα0,αi = ∅ and D\αiα0,αj 6= ∅. This case is possible when α0 is a

vertex in D of degree at least 2 i.e., only for g = sln. Equation (4.20) reads:

al+1

(D\αiα0

;α0)+ al+1

(D\αjαi

;αi)− al+1

(D\αjα0

;α0)− al+1

(D\αiαj

;αj)=

= al+1

(D\αjα0,αi

;αi)− al+1

(D\αjα0,αi


\αiα0

;α0

− al+1

(D\αiα0,αj

;αj)+ al+1

(D\αiα0,αj

;α0)− al+1D\αiα0,αj

\αjα0

;α0

(4.25)

This equation holds since

• D\αjαi = D\αjα0,αi , so al+1

(D\αjαi

;αi)= al+1

(D\αjα0,αi

;αi)

• D\αjα0 = D\αjα0,αi , so al+1

(D\αjα0

;α0)= al+1

(D\αjα0,αi

;α0)

• D\αiαj = D\αiα0,αj , so al+1

(D\αiαj

;αj)= al+1

(D\αiα0,αj

;αj)

• D\αiα0 = D\αiα0,αj , so al+1

(D\αiα0

;α0)= al+1

(D\αiα0,αj

;α0)

• we have

D\αjα0,αi

\αiα0 = D\αi,αjα0

= D\αiα0,αj

\αjα0 =⇒ aD\αjα0,αi

\αiα0

;α0

= aD\αiα0,αj\αj

α0;α0

This concludes the proof of Proposition 4.5 and the proof of the main

theorem for the affine case.

APPENDIX A

In this Appendix, we provide a detailed proof of two results used in Chapter

3. Namely, in Section 1, we prove that the algebra Ug, described by Drinfeld

in [D5, 8.1] and by Etingof–Kazhdan in [EK6], is isomorphic to the algebra of

endomorphisms of the forgetful functor with respect to the category O of g. In

Section 2, we prove that the isomorphism ΨEK between Ug[[~]] and its quantum

counterpart given in [EK6] corresponds, under the previous identification,

with the natural isomorphism induced by the equivalence of categoriesO[[~]] '

O~.

Finally, in Section 3, we prove, in analogy with the results of [EK1, Part

II], that the relative fiber functor Γ : DΦ(g) → DΦD(gD) gives rise naturally

to a braided Hopf algebra object in DΦD(gD). This result is used in Chapter

3 Section 5 to study the behavior of the functor and its tensor structure with

respect to abelian extensions of gD.

1. The completion of Ug in category O

Let Q be the root lattice of g. Consider the modules

Uβ := Ug/Iβ

181

182 A

where Iβ ⊂ Ug is the left ideal generated by all elements of weight ≤ β. Define

Ug := limβ∈Q

Uβ

It is easy to prove that there is an injection

limβ

colimγ HomUg(Uβ, Uγ) −→ HomUg(Ug, Ug)

and we want to prove that, in this case, this is an isomorphism obtained by

the identification

colimγ HomUg(Uγ, Uβ) ' HomUg(Ug, Uβ)

1.1. The Ug-module structure of Uβ. The module Uβ is associated to

the short exact sequence

0 // Iβiβ// Ug

pβ// Uβ // 0

In particular, Uβ is a cyclic module generated by an element 1β such that

Iβ.1β = 0.

Lemma. For each v ∈ Uβ, there exists a weight β′ = β′(v) ∈ Q, β′ ≤ β,

such that Iβ′ .v = 0

Proof. We can assume v = a1...an.1β for some ai ∈ Ug. Proceed by

induction. If n = 0, then v = 1β and β′ = β. Now assume v = a.w, where

w = a1...an.1β and there exists β′ = β′(w) such that Iβ′ .w = 0.

1. THE COMPLETION OF Ug IN CATEGORY O 183

(i) if a ∈ (Ug)µ, µ ≥ 0, then for each element x of weight λ ≤ β′ (i.e.

for each generator of Iβ′) we have

x.v = [x, a].w + a.x.w = [x, a].w

and [x, a] ∈ (Ug)λ+µ. Now, if we choose λ ≤ β′ − µ, we have [x, a] ∈

Iβ′ , so [x, a].w = 0. We conclude that, for β′′ := β′ − µ ≤ β′,

Iβ′′ .v = 0.

(ii) if a ∈ (Ug)µ, µ ≤ 0, for each generator x of Iβ′ of weight λ, the

commutator [x, a] has weight λ + µ ≤ β′, since λ ≤ β′ and µ ≤ 0.

Then Iβ′ .v = 0.

For general a ∈ Ug, use the weight decomposition

a ∈ Ug =⊕β∈Q

(Ug)β ⇒ v = a.w =m∑i=1

aµi .w =m∑i=1

vi

where aµi ∈ (Ug)µi . Then there exist β1...βm such that Iβi .vi = 0. Choosing

β′′ := minβi, i = 1..m (i.e. β′′ =∑

α∈∆ min(βi, α)α), we have

Iβ′′ .v =( m⋂i=1

Iβi).v = 0

As a consequence we can say that the action of Ug on Uβ, for every element

v, factors through Uβ′(v), β′(v) ≤ β, i.e. for every element u ∈ Ug u.v = uβ′ .v,

in the sense that the action of u on v is the same that the action of any

representative of the class of u in Uβ′ , i.e. uβ′ = pβ′(u).

1.2. The Ug-module structure of Ug. Define an action a : Ug⊗ Ug→

Ug by u.xβ = u.xβ. Since the maps πβ′β are maps of Ug-modules, the

184 A

map Ug⊗ Ug→ Ug is well-defined, because for every β′ ≤ β,

u.xβ = u.πβ′β(xβ′) = πβ′β(u.xβ′)

and it’s obviously an action of Ug-modules.

1.3. The multiplication map in Ug. Using the action

Ug⊗ Uβ → Uβ

for every β ∈ Q, we can define a bilinear map:

U ⊗ Uβ // Uβ

x⊗ v // xω(v).v

This map lifts up to a multiplication in Ug

Ug⊗ Ugm

// Ug

x⊗ y // xω(yβ).yβ

where ω(yβ) is a weight such that Iω(yβ).yβ = 0. In order to show that this

map is well-defined, we need to prove that, for every β′ ≤ β,

xω(yβ).yβ = πβ′β(xω(yβ′ ).yβ′)

Indeed, we have yβ′ ∈ Uβ′ and Iω(yβ′ ).yβ′ = 0. Then

0 = πβ′β(Iω(yβ′ ).yβ′) = Iω(yβ′ )

.πβ′β(yβ′) = Iω(yβ′ ).yβ


So, ω(yβ′) ≤ ω(yβ). In particular,

xω(yβ) = πω(yβ′ )ω(πβ)(xω(yβ))

Consequently, their action on yβ is the same, i.e.

xω(yβ).yβ = xω(yβ′ ).yβ = xω(yβ′ )

.πβ′β(yβ′) = πβ′β(xω(yβ′ ).yβ′)

So, we have an algebra structure on Ug.

1.4. Compatibility with the Ug-action. We have a map i : Ug→ Ug,

sending each element u in the collection of its image in Uβ for all β ∈ Q,

i(u) := uββ∈Q. Since

Ker(i) =⋂β∈Q

Iβ = 0

i is an injection. It’s very easy to show that the multiplication map is com-

patible with the Ug-action in the following way:

(a) By definition, the action of Ug on Ug is given by the formula

a(u,x) = u.x = uβ.x = m(i(u),x)

so that the following diagram commutes:

Ug⊗ Uga

//

i⊗1 %%

Ug

Ug⊗ Ug

m

OO

186 A

(b) Since the injection map is a map of Ug-modules, we have a commu-

tative diagram

Ug⊗ Ugm

//

1⊗i

Ug

i

Ug⊗ Uga

// Ug

Then

Ug⊗ Ug

m

Ug⊗ Ug

a

i⊗1oo Ug⊗ Ug

1⊗ioo

m

Ug Ug Ugi

oo

is commutative and Ug ⊂ Ug is a subalgebra.

1.5. Topology on Ug. We consider the modules Uβ with the discrete

topology and we equip Ug = limUβ with the topology induced by limit. In

particular, a basis for the open sets is given by the sets U(β1... βn, u1... un) for

n ≥ 0, βi ∈ Q, ui ∈ Uβi , where

U(β1..βn, u1...un) := x ∈ Ug | xβi = ui ∀i


Let β := minβi, i = 1... n. Then, by surjectivity of pβ, there exists an

element u ∈ Ug such that uβi = ui.

Ug

pβ

Uβπββ1

~~

πββn

!!

Uβ1 · · · Uβn

Then we have proved that Ug is dense in Ug.

For any element u ∈ Ug, we can find ui ∈ Ug such that

u = limi→∞

ui

In particular, if φ ∈ EndUg(Ug), then, for any u, x ∈ Ug, we have

φ(u.x) = φ((limiui).x) = lim

iφ(ui.x) = (lim

iui).φ(x) = u.φ(x)

and

EndUg(Ug) ' EndUg(Ug) ' Ug

1.6. HomUg(Ug, Uβ). The map

Ug⊗ Uβ // Uβ

is in fact an action of the algebra Ug on the module Uβ. Using the same

density argument as before, we have

HomUg(Ug, Uβ) ' HomUg(Ug, Uβ)

188 A

Consequently, for any map φ ∈ HomUg(Ug, Uβ), we have

φ(u) = u.φ(1)

for any u ∈ Ug. In particular, φ is completely determined by φ(1) ∈ Uβ. By

the previous discussion, we know that there exists a weight ω = ω(φ(1)) such

that Iω.φ(1) = 0. Then the action of u on φ(1) factors through Uω, i.e.

φ(u) = u.φ(1) = uω.φ(1) =: φ(uω)

and

Ugφ//

πω

Uβ

Uω

φ

>>

where φ ∈ HomUg(Uω, Uβ). It’s clear that we can associate to φ a uniquely

determined class [φ] ∈ colimγ HomUg(Uγ, Uβ) and consequently

colimγ HomUg(Uγ, Uβ) ' HomUg(Ug, Uβ)

and

limβ

colimγ HomUg(Uγ, Uβ) ' limβ

HomUg(Ug, Uβ) ' HomUg(Ug, Ug)

Notice that the same argument can be used to show that, for any module

V ∈ O, we have

HomUg(Ug, V ) ' HomUg(Ug, V )

2. THE ISOMORPHISM BETWEEN Ug[[h]] AND U~g 189

and consequently for any map f ∈ HomUg(Ug, V ) there exists β ∈ Q and

fβ ∈ HomUg(Uβ, V ) such that

Ugf

//

πβ

V

Uβ

fβ

??

is commutative.

2. The isomorphism between Ug[[h]] and U~g

The equivalence of categories between O[[h]] ' Oh, established by the

Etingof–Kazhdan functor, carries an isomorphism

HomUg(Uγ, Uβ)[[h]] ' HomUhg(Uhγ , U

hβ )

and then

limβ colimγ HomUg(Uγ, Uβ)[[h]]'//

o

limβ colimγ HomUhg(Uhγ , U

hβ )

o

EndUg(Ug)[[h]]

o

EndUhg(Uhg)

o

Ug[[h]]'

// Uhg

On the other hand, we can consider the forgetful functors

F : O[[h]]→ A Fh : Oh → A

190 A

and the associated completions End(F ), End(Fh). Using the equivalence be-

tween |Co[[h]] and Oh we have a commutative diagram

O[[h]]Φ//

F

""

OhFh

A

that induces a natural isomorphism

End(F ) ' End(Fh)

We want to show that this construction is equivalent to the previous one,

in particular we want to show that there exists an algebra isomorphism

Ug[[h]]'−→ End(F )

2.1. The map φ. First of all, let us construct a map φ : Ug[[h]] →

End(F).

Lemma. For any v ∈ V there exists β ∈ Q such that Iβ.v = 0

Proof. Assume first v ∈ Vλ. Set µ := minλi|λ ∈ D(λi). For any

x ∈ (Ug)λ′ where λ′ ≤ µ(v) := µ− λ, we have x.v = 0. Then Iµ(v).v = 0.

In general, v =∑vλ, where vλ ∈ Vλ. Set µ(v) := minλ∈P (V )µ(vλ). Then

Iµ(v).v = 0

For each V ∈ O we have a Ug-action given by, for any u ∈ Ug,

u.v = uµ(v).v


where by uµ(v) we mean any representative of the class uµ(v) ∈ Uµ(v) (to be

precise, any representative of the classes uβ ∈ Uβ with β ≤ µ(v)).

In particular, for any V ∈ O we have a map

φV : Ug→ End(V )

Define φ(u) := φV (u).

2.2. φ is well-defined. We need to show that for any V,W ∈ O and

f ∈ HomUg(V,W ) we have

Vf//

φV (u)

W

φW (u)

Vf// W

i.e. for any v ∈ V

f(uµ(v).v) = uµ(f(v)).f(v)

Since f ∈ HomUg(V,W ), then

0 = f(Iµ(v).v) = Iµ(v).f(v)

so µ(v) ≤ µ(f(v)) and the action of uµ(v) is equivalent to that one of uµ(f(v)).

So,

uµ(f(v)).f(v) = uµ(v).f(v) = f(uµ(v).v)

Then we have a well-defined map

φ : Ug[[h]]→ End(F )

192 A

2.3. φ is injective. Recall that for any β ∈ Q Uβ ∈ O. In particular, for

any v ∈ Uβ

φβ(u)(v) = uβ.v

Then, if φ(u) = 0, we have uβ = 0 for all β ∈ Q, and u = 0.

2.4. φ is surjective. For any f ∈ End(F ), β′ ≤ β, we have a commuta-

tive diagram

Uβ′πβ′β

//

fβ′

Uβ

fβ

Uβ′πβ′β

// Uβ

By universal property of limits in the category Vec(k), there exists a map

f ∈ Homk(Ug, Ug) such that the diagram

Ugπβ

//

πβ′

f

Uβ

fβ

Uβ′

fβ′

πβ′β

??

Ugπβ

//

πβ′

Uβ

Uβ′

πβ′β

??

is commutative.

2.5. Commutativity in O. Recall that HomUg(Ug, V ) ' HomUg(Ug, V ).

Then for any g ∈ HomUg(Ug, V ), there exists µ := µ(g, V ) ∈ Q and gµ ∈


HomUg(Uµ, V ) such that

Ugg

//

πµ

V

Uµ

gµ

??

Consequently, the following diagram

Ugg

//

πµ

f

V

fV

Uµ

fµ

gµ

??

Ugg

//

πµ

V

Uµ

gµ

??

is commutative and then, for any V ∈ O and g ∈ HomUg(Ug, V ), the diagram

Ugg//

f

V

fV

Ugg// V

is commutative. This fact, in particular, is true for Uβ, β ∈ Q.

2.6. Commutativity for EndUg(Ug). Suppose now g ∈ HomUg(U , U) '

limβ HomUg(U , Uβ). It means that for any β, there exists gβ ∈ HomUg(Ug, Uβ)

194 A

such that the diagram

Ugg//

gβ

Ug

πβ

Uβ

By density, there exists β′ ≤ β and gβ′β ∈ HomUg(Uβ′ , Uβ) such that the

diagram

Ugg//

πβ′

gβ

Ug

πβ

Uβ′gβ′β

// Uβ

is commutative. Then we have a box

Ugg

//

f

πβ

Ugπβ′

f

Uβ′gβ′β

//

fβ′

Uβ

fβ

Ugg//

πβ′

Ugπβ

Uβ′gβ′β

// Uβ

where every face is commutative except one behind, i.e. we have

πβ g f = πβ f g

It means that the maps f g and g f define the same endomorphism of Ug

as vector space. Then, for any g ∈ HomUgg(Ug, Ug) we have a commutative


diagram

Ugg//

f

Ug

f

Ugg// Ug

2.7. The inverse map φ−1. Notice that, for each element y ∈ Ug, the

map

my : Ug→ Ug|my(x) = x · y

is in HomUg(Ug, Ug). Indeed, for any u ∈ Ug,

my(u.x) = my(i(u) · x) = i(u) · x · y = u.my(x)

and, for any u ∈ Ug,

f(u) = f(mu(1)) = mu(f(1)) = u · f(1)

Then we have defined a map

φ−1 : End(F )→ Ug | φ−1(f) = f(1)

and we have also proved the surjectivity of φ. indeed, for any v ∈ V , take the

map g ∈ HomUg(Ug, V ) defined by g(1) = v. We have a commutative diagram

Ugg//

f

V

fV

Ugg// V

i.e.

fV (v) = fV (g(1)) = g(f(1)) = f(1).g(1) = f(1).v

196 A

It is equivalent to say that

f = φ(f(1))

We can conclude that the diagram

Ug[[h]]'

//

o

Uhg

o

End(F )'// End(Fh)

is commutative.

3. A BRAIDED HOPF ALGEBRA IN DΦD (gD) 197

3. A braided Hopf algebra in DΦD(gD)

3.1. This section is entirely dedicated to the proof of the following

Theorem.

(i) The object Γ(L−) has a structure of braided Hopf algebra in DΦD(gD).

(ii) On every object Γ(V ) there are well-defined actions and coalitions of

Γ(L−), so that the functor Γ naturally factors as

DΦ(g)

Γ

Γ

**

DΦD,UgD(Γ(L−)) ' DΦD(Γ(L−)]UgD,−)

fDtt

DΦD(gD)

where Γ(L−)]UgD,− denotes the Radford algebra attached to Γ(L−).

Theorem 3.1 is used in Chapter 3 Section 5 to study the behavior of the

relative twists with respect to abelian extensions of gD.

3.2. Let (Γ, JΓ) : DΦ(g)→ DΦD(gD) be the monoidal functor introduced

in Chapter 3. Following [EK1], for every V ∈ DΦ(g) we can define a map

µV : Γ(L−)⊗ Γ(V )→ Γ(V ) µV (x⊗ v) = (i∨+ ⊗ id)Φ−1(id⊗v)x

Proposition. For every V ∈ DΦ(g), the map µV is a morphism of gD–

modules. Moreover, it satisfies

µV (id⊗µV )ΦD = µV (µL− ⊗ id)

198 A

Therefore, (Γ(L−), µ) is an associative algebra object in DΦD(gD) and acts on

the image of the functor Γ.

Proof. Let ∆(a) = a1 ⊗ a2 for a ∈ UgD. Then

µV (a1.x⊗ a2.v) = (i∨+ ⊗ id)Φ−1(id⊗S(a2)ρ ⊗ id)(id⊗v)(S(a1)ρ ⊗ id)x

= (i∨+ ⊗ id)Φ−1(S(a1)ρ ⊗ S(a2)ρ ⊗ id)(id⊗v)x

= (i∨+ ⊗ id)(S(a1)ρ ⊗ S(a2)ρ ⊗ id)Φ−1(id⊗v)x

= (S(a)ρ ⊗ id)(i∨+ ⊗ id)Φ−1(id⊗v)x

= a.µV (x⊗ v)

The second statement is also easily proved. Let x, y ∈ Γ(L−), v ∈ Γ(V ) and

ΦD =∑Pk ⊗Qk ⊗Rk. Then

µV (id⊗µV )ΦD(x⊗ y ⊗ v) =

= (i∨+ ⊗ id)Φ−1(id⊗i∨+ ⊗ id)(id⊗Φ−1)(id⊗ id⊗Rk.v)(id⊗Qk.y)(Pk.x)

= (i∨+ ⊗ id)(id⊗i∨+ ⊗ id)Φ−11,23,4Φ−1

234ΦρD,123(id⊗ id⊗v)(id⊗y)x

= (i∨+ ⊗ id)(id⊗i∨+ ⊗ id)Φ123ΦρD,123Φ−1

12,3,4Φ−11,2,34(id⊗ id⊗v)(id⊗y)x

= (i∨+ ⊗ id)(i∨+ ⊗ id⊗ id)Φ−112,3,4Φ−1

1,2,34(id⊗ id⊗v)(id⊗y)x

= (i∨+ ⊗ id)Φ−1(id⊗ id⊗v)(i∨+ ⊗ id)Φ−1(id⊗y)x

= µV (µL− ⊗ id)(x⊗ y ⊗ v)

where the equalities are obtained by applications of the pentagon axiom and

the associativity of i∨+.


3.3. The tensor structure on Γ and the canonical coalgebra structure on

L− define the map

δ : Γ(L−)→ Γ(L−)⊗ Γ(L−) δ = J−1Γ Γ(i−)

Proposition. The map δ is a morphism of gD–modules and it satisfies

ΦD(δ ⊗ id)δ = (id⊗δ)δ

Therefore, (Γ(L−), δ) is a coassociative algebra object in DΦD(gD).

Proof. Let a ∈ gD. The maps JΓ and Γ(i−) are both morphisms of

gD–modules, therefore we have

δ(a.x) = ∆(a).δ(x)

Moreover,

ΦD(δ ⊗ id) = ΦDJ−1Γ,1,2(Γ(i−)⊗ id)J−1

Γ Γ(i−)

= ΦDJ−1Γ,1,2J

−1Γ,12,3Γ(i− ⊗ id)Γ(i−)

= J−1Γ,2,3J

−1Γ,1,23Γ(Φ)Γ(i− ⊗ id)Γ(i−)

= J−1Γ,2,3(id⊗Γ(i−))J−1

Γ Γ(i−)

= (id⊗δ)δ

3.4. We now want to study the action of Γ(L−) on the tensor product.

200 A

Proposition. The object (Γ(L−), µ, δ) is a bialgebra object in DΦD(gD).

The action on the tensor product is compatible with the coproduct and the

tensor structure JΓ, i.e.,

µV⊗W (id⊗JΓ) = JΓ(µV ⊗ µW )βD,23(δ ⊗ id⊗ id)

as morphisms from Γ(L−)⊗ Γ(V )⊗ Γ(W ) to Γ(V ⊗W )

Proof. We ignored the action of the associator ΦD since we proved that it

is compatible with µV and JΓ. The equality above corresponds to the identity

in Γ(V ⊗W )

(i∨+ ⊗ id⊗ id)(id⊗i∨+ ⊗ id⊗ id)β34(id⊗v ⊗ w)(id⊗i−)x =

= (i∨+ ⊗ id⊗ id)β23(i∨+ ⊗ id⊗i∨+ ⊗ id)(id⊗b.v ⊗ id⊗w)(x1 ⊗ ai.x2)i−

where v ∈ Γ(V ), w ∈ Γ(W ), x ∈ Γ(L−), δ(x) = x1 ⊗ x2 and exp(−~ΩD/2) =

ai⊗ bi. It is more convenient, in this case, to follow a pictorial proof. We have

to show that the following diagrams are equivalent:

v w

x

x1 ai.x2

bi.v w=


The relation satisfied by δ is represented by

δ(x)x

=

Since the action of gD on the objects Γ(V ) is given by right action on N∗+, we

represent the braiding β−1D as

ai.x2 = x2

•bi.v v

=

We notice that we are allowed to move the black and the white bullet along

the lines, since the commute with the left action of g. Clearly, the RHS corre-

sponds to

x1 x2

•v w

x1 x2

•v w

x1 x2

wv• = =

202 A

We now use the fact that the map i∨+ satisfies

i∨+ β (β−1D )ρ = i∨+

i.e.,

•

=

•

=

Finally we get

x1 x2

v w=

v w

x

3.5. The tensor functor (Γ, JΓ) induces a natural braiding on the sub-

category generated in DΦD(gD) b the objects Γ(V ), for any V ∈ DΦ(g). The

braiding is given by the usual formula

βJΓ:= J−1

Γ,21 Γ(β) JΓ


where β = σ · R is the braiding in DΦ(g). Clearly, βJΓis a morphism of gD–

module and so is the relative R–matrix RD := σβJΓ.

Let now u ∈ Γ(L−) be the unit element. For every V ∈ DΦ(g) we define

the trivial comodule structure on Γ(V ) by

ηV : Γ(V )→ Γ(L−)⊗ Γ(V ) ηV (v) = u⊗ v

The maps η are clearly morphisms of gD–modules and they satisfy

ΦD(η ⊗ id)ηV = (id⊗ηV )ηV

where η := ηL− and η(u) = u⊗ u = δ(u). Consider

RD,V ∈ EndUgD(Γ(L−),Γ(V ))

and define the following map:

δV : Γ(V )→ Γ(L−)⊗ Γ(V ) δV = RD,V ηV

Proposition. The maps δV define on any Γ(V ), V ∈ Dg(Φ), a structure

of comodule over Γ(L−).

Proof. Clearly we have that the relative R–matrix satisfies

RD,1,23 = Φ−1D,231RD,13ΦD,213RD,12Φ−1

D,123

RD,12,3 = ΦD,312RD,13Φ−1D,132RD,23ΦD,123

204 A

and therefore

RD,12ΦD,312RD,13Φ−1D,132RD,23ΦD,123 = ΦD,321RD,23Φ−1

D,231RD,13ΦD,213RD,12

Moreover, we have the obvious relations

RD,1,23(id⊗ηV ) = (id⊗ηV )RD RD,12,3(ηV ⊗ id) = (ηV ⊗ id)RD

for V ∈ DΦ(g). Then

(id⊗δV )δV = RD,23(id⊗ηV )RDηV

= RD,23RD,1,23(id⊗ηV )ηV

= ΦDRD,12RD,12,3Φ−1D (id⊗ηV )ηV

= ΦDRD,12RD,12,3(η ⊗ id)ηV

= ΦDRD,12(η ⊗ id)RDηV

= ΦD(δ ⊗ id)δ

This concludes the proof of the Theorem 3.1.

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