Lecture 16 - Stanford Universitysporadic.stanford.edu/quantum/lecture16.pdf · 2019-06-12 ·...

Lusztig’s quantum group Tilting modules The Fusion Category in Action

Lecture 16

Daniel Bump

June 12, 2019

v2 v1

v3 v0

u1 u0

[2]

[3]

[3][2]

[1]

[3]

[1]

[3][2][1]

[1]

[2]


Andersen and Paradowski

In Lecture 15 we described a certain modular tensor categoryC(g, k) associated with the following data: a complexsemisimple Lie algebra g and a “level” k. The simple objects arein bijection with weights λ in the fundamental alcove of level k.Thus λ dominant and satisfy 〈θ∨, λ〉 6 k, where θ is the highestroot and θ∨ is the associated coroot.

We claimed that this MTC could be constructed from thequantum group Uq(g) where q is a suitable root of unity, but thecategory of modules for Uq(g) is not semisimple (though it isribbon), as we will see very soon.

Andersen and Paradowski gave a description of the semisimplecategory C(g, k) from Uq(g) which discuss today.


Groups over Z: Chevalley and Kostant

Chevalley (1955) gave a construction of groups of Lie typeassociated with a semisimple Lie algebra g over C. A key stepwas to prove that the Lie algebra g has a basis with integerstructure constants, leading to an algebraic group over Z. Thisproduces a group scheme G over Z such that G(F) contains asimple group in its composition series for most finite fields F.

Since a group scheme is a commutative Hopf algebra, thereshould be a dual Hopf algebra that is a Z-form of U(g). Thiswas constructed in 1966 by Kostant.


Lusztig’s quantum group

There are different version of Uq(g). Lusztig’s version of Uq(g),which we will describe, is related Kostant’s enveloping algebraover Z. This version is defined over A = C[v, v−1], and givesrise to versions defined over other rings, particularly C(q)where q can be a root of unity, by extension of scalars. The keyidea is to include divided powers in the generating set.

For simplicity we will restrict ourselves to the simply-laced case,i.e. we assume all roots have the same length. Let αi

(i = 1, · · · , r) be the simple roots and aij = 〈αi,αj〉 with aW-invariant inner product on the weight lattice normalized soaii = 2. If αi,αj are not orthogonal then aij = −1.


The generic case

We review the construction of Uv(g) when v is an indeterminate.This is the C(v)-algebra with generators Ei, Fi and Ki

(i = 1, · · · , r) with Ki invertible and relations:

KiEjK−1i = vaijEj, KiFjK−1

i = v−aijFj, [Ei,Fj] = δijKi − K−1

i

v − v−1 ,

and the quantum Serre relations: if αi and αj are orthogonalthen Ei and Ej commute, otherwise

E2i Ej − [2]vEiEjEi + EjE2

i = 0

where [2]v = v + v−1, and similar relations for the Fi. Let Uv(g)be the algebra generated by the Ei, Fi and Ki.


The generic case (continued)

We then define the comultiplication by

∆(Ei) = Ei ⊗ 1 + K ⊗ Ei, ∆(Fi) = Fi ⊗ K−1i + 1⊗ Fi

∆(Ki) = Ki ⊗ Ki,

with a suitable counit and antipode.

This differs from our previous definition, the comultiplicationbeing conjugated by Ki ⊗ Ki.

We are following Lusztig, Modular Representations andQuantum Groups (1989). See also Chari and Pressley, A guideto Quantum groups Section 9.3.


The restricted quantum group

To define the quantum group at a root of unity, it is necessary tohave a version in which the indeterminate v can be specialized.We make use of the divided powers

E(k)i =

Eki

[k]v!, F(k)

i =Fk

i

[k]v!.

Let A = C[v, v−1].

Let Uresv (g) be the A-subalgebra of Uv(g) generated by the Ki,

E(k)i and F(k)

i for all k > 1. This is closed under thecomultiplication, so it is a Hopf algebra, defined over A.

It may be shown that the A-algebra Uresv (g) has a basis that is

also a C(v)-basis of Uv(g). Thus

Uv(g) = C(v)⊗A Uresv (g).


Braid group action

There is an action of the braid group on Uresv (g). This comes

through operators Ti (satisfying the braid relations) in which

Ti(Ej) =

−FjKj if i = j,Ej if αi,αj are orthogonal,−EiEj + v−1EjEi if aij = −1.

Ti(Fj) =

−K−1

j Ej if i = j,Fj if αi,αj are orthogonal,−FjFi + vFiFj if aij = −1.

Ti(Kj) =

K−1

i if i = j,Kj if αi,αj are orthogonal,KiKj if aij = −1.


Certain important elements

Lusztig defines certain elements that are analogous to theGaussian binomial coefficients[

ct

]=

[c]v![t]q![c − t]q!

=

t∏s=1

vc−s+1 − v−c+s−1

vs − v−s .

If t > 0 and c are integers let[Ki; ct

]=

t∏s=1

Kivc−s+1 − K−1i v−c+s−1

vs − v−s .

Proposition (Lusztig)[Ki; ct

]∈ Ures

v (g).


Proof

The proof of this fact appears in another paper, Quantumdeformations of certain modules over enveloping algebras.By induction assume true for smaller values of t. Use[

Ki; ct

]− vt

[Ki; c + 1

t

]= K−1

i v−(c+1)[

Ki; ct − 1

].

to reduce to the case c = 0. Then we may use

E(r)i F(r)

i =

r∑t=0

F(r−t)i

[Ki; 2t − 2r

t

]E(r−t)

i

and another induction.


Quantum groups at roots of unity

For simplicity let ` be an odd integer > 3 and let q = eiπ/`. Thenthe specialization v→ q gives a ring homomorphism A→ C,and we may define

Uq(g) = C⊗A Uresv (g)

by extension of scalars.

We denote by Ki, E(n)i , F(n)

i the images of these elements underthe natural map from Ures

v (g). Lusztig proves that K`i is centraland K2`

i = 1.

We will restrict ourselves to finite-dimensional modules in whichK`i acts as the identity. Such modules are called Type 1.


Why we need tilting modules

All modules ofUq(g) := C⊗A Ures

v (g)

will be assumed finite-dimensional and Type 1. Such modulesform a ribbon category Mod(Uq(g))) that is not semisimple.

Andersen and Paradowski gave a description of a semisimplecategory derived from Mod(Uq(g))). The objects will be tiltingmodules, and the morphisms will have to be described. Wealso recommend Chari and Pressley, A guide to QuantumGroups Chapter 11 and Sawin, Quantum Groups andModularity for this subject.


Weight space decomposition

Let V be a Uq(g)-module (always Type 1). We will assume thatV has a decomposition

V =⊕λ∈Λ

Vλ,

where the sum is over the weight lattice Λ, where Ki acts on Vλby the scalar q〈α

∨i ,λ〉.

Unfortunately this is not a good description of Vλ becauseq〈α

∨i ,λ〉 does not determine λ, so we can’t just define Vλ to be

the eigenspace of the Ki. Instead we define Vλ to be the set ofvectors such that

Kiv = q〈α∨i ,λ〉v,

[Ki; 0`

]v =

[〈α∨

i , λ〉`

]v .


Weyl modules: the generic case

With this definition,V =

⊕λ∈Λ

Vλ,

andE(k)

i Vµ ⊂ Vµ+kαi , F(k)i Vµ ⊂ Vµ−kαi .

Note that without the Lusztig quantum group, we would not beable to decompose over Λ, only Λ/`Λ.

If λ is a dominant weight, there is a unique irreducible Type 1Uv(g)-module Wλ

v with highest weight λ. Its character is givenby the Weyl character formula, that is, its weight spaces havethe same dimensions as its classical counterpart.


Weyl modules: root of unity case

We may extend scalars to obtain Weyl modules for quantumgroups at roots of unity.

If x is a highest weight vector then WλA = Ures

v (g) · x is a A-formof Wλ

v in thatWλ

v∼= C(v)⊗A Wλ

A.

Now let q be an arbitrary nonzero complex number. We make Cinto an A-module Cq via the homomorphism v→ q. ThenWλ

q = Cq ⊗A WλA is called a Weyl module for Uq(g).


Example: sl2

Let α = α1 be the unique positive root, so ρ = 12α. If λ = nρ is a

positive weight we describe Wλq . Let v be a highest weight

vector and vk = F(k)v. It has weight λ− iα. We have

Kvk = qn−2k, Evk = [n − k + 1]vk−1, Fvk = [k + 1]vk+1.

For example if n = 3, using solid lines to denote E and dashedlines to denote F:

v3 v2 v1 v0

[1] [2]

[3]

[3]

[2] [1]

Note that if ` = 3 then [3] = 0 so the module is reducible. Thatis, v1 and v2 span an irreducible submodule L, andM = W3ρ

q /〈v1, v2〉 is the direct sum of two one-dimensionalsubmodules Cv0 and Cv3.


Example (continued)

v3 v2 v1 v0

[1] [2]

[3]

[3]

[2] [1]

Assuming as before that l = 3 so [3] = 0 we have remarked thatthere is a submodule L ∼= Wρ

q spanned by v1 and v2. It mightseem that the quotient Wλ

q /L is the direct sum of twoone-dimensional submodules, since E and F both annihilate v0and v3 in Wλ

q /L.

However E(3)v3 = v0 and F(3)v0 = v3, so taking the dividedpowers into account, the two-dimensional module Wλ/L isirreducible.


Example (continued)

v3 v2 v1 v0

[1] [2]

[3]

[3]

[2] [1]

Above: W3ρq . Below: its contragredient representation.

v∨3 v∨2 v∨1 v∨0

[3] [2]

[1]

[1]

[2] [3]

Assuming as before that l = 3 so [3] = 0, W3ρq is not self-dual.

The dual has the two-dimensional module spanned by v0 and v3as a submodule and Wρ

q as a quotient.


Simple modules

TheoremFor every dominant weight λ, there is a unique irreduciblefinite-dimensional representation L(λ) with highest weight λ. Itis the unique irreducible quotient of Wλ

q .

Proof: It follows from standard category O arguments that forevery weight λ there is a unique irreducible highest weightmodule L(λ) with highest weight λ. This module is a quotient ofany highest weight module with highest weight λ, in particularof Wλ

q , if λ is dominant. Therefore L(λ) is finite-dimensional if λis dominant.


Irreducible Weyl modules

Proposition (Andersen, Polo and Wen)

The module Wλq is irreducible if and only if either λ is in the

fundamental alcove F consisting of dominant weights such that

〈λ+ ρ, θ〉 < `,

or λ+ ρ ∈ `Λ.

Here θ is the highest root. There are two types of irreducibleWeyl modules. The second type, Wλ

q where λ+ ρ is divisible by`, are different because their quantum dimensions are zero.The quantum dimension is given by the Weyl dimensionformula:

dλ =∏α∈Φ+

[〈λ+ ρ,α〉]q[〈ρ,α〉]q

.


Tilting modules

We define a Weyl filtration of a module V to be a sequence ofsubmodules

0 = V0 ⊆ V1 ⊆ · · · ⊆ Vr = V,

where the successive quotients Vi+1/Vi are Weyl modules.If both V and its contragredient V∨ have Weyl filtrations, V iscalled a tilting module.

As an example of a module that is not tilting, consider with ` = 3the four-dimensional module W3ρ

q . This is a Weyl module, sotrivially it has a Weyl filtration. But its dual, we have seen, is notdual, and a subquotient L(3ρ) occurs in its (unique)composition series

0→ L(3ρ)→ W3ρq → Wρ

q → 0.

It has no Weyl filtration, so W3ρq is not a tilting module.


History

Tilting theory has a long history going back to Brenner andButler (1980).

Tilting modules exist in different categories, for examplecategory O, where a tilting module is defined to be one in whichboth V and V∨ have filtrations whose quotients are Vermamodules.

Tilting modules reached quantum groups after prior work in thealgebraic category by many, including Auslander and Reites,Ringel and Donkin. In quantum groups they were introducedindependently by Andersen and by S. Gelfand and Kazhdan in1992.


Quotation from Brenner and Butler (1980)

It turns out that there are applications of our functors whichmake use of the analogous transformations which we like tothink of as a change of basis for a fixed root-system—a tilting ofthe axes relative to the roots which results in a different subsetof roots lying in the positive cone .... For this reason, andbecause the word ’tilt’ inflects easily, we call our functors tiltingfunctors or tilts.


Example ` = 3

Since with ` = 3 the Uq(sl2) module W3ρq is not tilting, let us

describe a unique tilting module T3ρq with highest weight 3ρ.

v2 v1

v3 v0

u1 u0

[2]

[3]

[3][2][1]

[3]

[1]

[3][2][1]

[1]

[2]


Example ` = 3 (continued)

v2 v1

v3 v0

u1 u0

[2]

[3]

[3][2][1]

[3]

[1]

[3][2][1]

[1]

[2]

This module is self-dual, so to verify that it is tilting, v1 andd v2span a submodule L ∼= Wρ

q , and the quotient T3ρq /Wρ

q ∼= W3ρq .

Thus it has a Weyl filtration. Since [4] = [1] = −[2]:

dim T3ρq = dim W3ρ

q + dim Wρq = [4] + [2] = 0.


Properties of tilting modules

Andersen (1992) and Paradowski (1992) proved:

Any direct sum or tensor product of tilting modules is tilting.The dual of a tilting module is tilting.Any direct summand in a tilting module is tilting.

The indecomposable tilting modules are in bijection withdominant weights. If λ is a dominant weight, then there is aunique indecomposable tilting module Tλq that contains a vectorvλ of weight λ, unique up to scalar multiple. The weight λ ismaximal in the sense that λ < µ for every weight of Tλq . Theweights of Tλq are contained in the convex hull of the W-orbit ofλ.


Negligible morphisms and objects

If f : X → Y are morphisms in a ribbon category we say that f isnegligible if tr(fg) = 0 for every g : Y → X. An object X isnegligible if 1X is negligible. Obviously this implies thatdim(X) = 0.

With ` = 3, the Weyl module W2ρq is an irreducible Weyl module

of quantum dimension [3] = 0. It is a tilting module. It isnegligible.

The tilting module T3ρq is negligible even though in its Weyl

filtration:0→ Wρ

q → T3ρq → W3ρ

q → 0

dim(Wρq ) and dim(W3ρ

q ) are both not negligible.


Negligible tilting modules

Theorem (Andersen 1992)

The irreducible Weyl modules that are not negligible are the Wλq

such that λ lies in the fundamental alcove F, that is,〈λ+ ρ, θ〉 < `.

Now the plan is to consider the category of tilting modules andsomehow “quotient out” the negligible objects. One concreteway of doing this is consider the category C of tilting modulesall of whose indecomposable direct summands are Tλq whereλ ∈ F.


Monoidal structure

This category is semisimple, generated by the modulesWλ

q = Tλq , with λ ∈ F. The tensor product rule is to discardnegligible tilting modules in the decomposition. That is, if A andB are tilting modules, the A⊗ B is a tilting module, so we mayuniquely write A⊗ B = C ⊕ Z with C in C and Z negligible. Wediscard the Z and C becomes a monoidal category. It inheritsthe braiding and twist from the category of all finite-dimensionalType 1 Uq(g)-modules, and it is a ribbon category.


Example

Let us consider the case where ` = 5 and g = sl2. We need toknow some indecomposable tilting modules. First, we have theWeyl modules Wλ with

λ ∈ F = {λ|〈α+ ρ, θ〉 < `}.

These are the simple objects in the fusion category.

W0q , Wρ

q , ,W2ρq , W3ρ

q .

There are other irreducible Weyl modules Wkρ where k ≡ 4 mod5, that is, λ+ ρ ∈ `Λ. These are tilting modules, but they arenegligible.

Now some other indecomposable tilting modules may be found.



v4 v3 v2 v1

v5 v0

u3 u2 u1 u0

[2]

[5]

[3]

[4]

[4]

[3] [5][2][1]

[5]

[1]

[5][4][1] [3]

[2]

[2]

[3]

[1]

[4]

With ` = 5, the quantum dimension is [4] + [6] = 0.

0→ W3ρq → T5ρ

q → W5ρq → 0



Similarly:

0→ W2ρq → T6ρ

q → W6ρq → 0

0→ W1ρq → T7ρ

q → W7ρq → 0

0→ W0q → T8ρ

q → W8ρq → 0

These tilting modules all have quantum dimension 0, andordinary dimension 10. After which we have another irreducibleWeyl module T9ρ

q = W9ρq .



If λ ∈ {0, ρ, 2ρ, 3ρ} let xλ denote the class of the irreducible Weylmodule Wλ

0 , which is also a non-negligible indecomposabletilting module. We wish to compute some products in two ways,using the Kac-Walton formula from the last lecture, and usingtilting modules.

You can do compute these products in Sage as follows:

sage: A13=FusionRing("A1",3)sage: A13.fusion_labels([’x0’,’x1’,’x2’,’x3’])sage: x1*x3x2


Racah Speiser method

We review the analogous computation for a Lie group G. Themethod Racah-Speiser method decomposes πλπν intoirreducibles. We need the character of one factor, say

χλ(z) =∑

K(λ,µ)zµ.

Then provided all µ+ ν are in the positive Weyl chamber C:

χλχν =∑µ

K(λ,µ)χµ+ν(z).

If some of the µ+ ν are not dominant, we may still try to movethem into the positive Weyl chamber using the dot actionw · µ = w(µ+ ρ) − ρ. We have to take into account a sign, andthere will be cancellations.


Review: the Racah-Speiser method (SL(3,C))

−ρ

The green weights lie on a hyperplane through −ρ so they arediscarded. Three other red weights to the left of the hyperplaneare reflected and subtracted.


Review: after cancellation

−ρ

χ(3,1,0)χ(3,3,0) = χ(4,3,3)+χ(4,4,2)+χ(5,3,2)+χ(5,4,1)+χ(6,3,1)+χ(6,4,0)


The Kac Walton Formula (SL(3))

The Kac Walton formula proceeds similarly but we use thefundamental alcove:

−ρ


The Kac Walton Formula (` = 5)

For SL2 the fundamental alcove of level `− h∨ = `− 2 is a linesegment:

0 ρ 2ρ 3ρ

Now we are allowed to reflect on either line dashed lines, at −ρand 4ρ. When we have reflected µ+ ν if it lies on one of thedashed lines it is zero and we may discard it. Otherwise itreflects in and we may try to make cancellations..


Example (` = 5) continued

Let

z =

(z

z−1

).

We will try to compute Wρq ⊗W3ρ

q in the fusion category. Wewrite

χρ(z) = z + z−1 = z(1/2,−1/2) + z(−1/2,1/2).

If we are considering these as characters of SL(2,C) then theRacah-Speiser method gives

χρχ3ρ = χ4ρ + χ2ρ

For the Kac-Walton formula there is the complication that 4ρlies outside the fundamental alcove; in fact, it lies on the line ofreflection, so its contribution is zero. Thus in the fusion ring:

χρ ∗ χ3ρ = χ2ρ.


Using Tilting Modules

Now let us do the same computation using tilting modules. Weknow that Wρ

q ⊗W3ρq is a tilting module, so it decomposes into a

direct sum of indecomposable tilting modules. But thecharacters of these are linearly independent and so it is easy tocompute the decomposition.

Wρq ⊗W3ρ

q = W2ρq ⊕W4ρ

q .

We remember that W4ρ is negligible and so it is to bediscarded. We obtain the same formula:

Wρq ⊗W3ρ

q ∼ W2ρq

meaning that the two differ by a negligible tilting module.


Example (` = 5) continued

Similarly if we compute χ2ρ ⊗ χ3ρ we write

χ2ρ(z) = z2 + 1 + z−2 = z2ρ + 1 + z−2ρ

So we have three contributions corresponding to weights

µ+ ν ∈ {ρ, 3ρ, 5ρ}.

This time when we reflect 5ρ we get a term that cancels the 3row term and so the Kac-Walton formula predicts

χ2ρ ∗ χ3ρ = χρ.

We wish to understand this in terms of tilting modules. Todecompose W2ρ

q ⊗W3ρq it is enough to identify a tilting module

with the same character.


Tilting (` = 5)

Remember that the negligible tilting module T5ρq has character

z5/2 + 2z3/2 + 2z1/2 + 2z−1/2 + 2z−3/2 + z5/2.

v4 v3 v2 v1

v5 v0

u3 u2 u1 u0

[2]

[5]

[3]

[4]

[4]

[3] [5][2][1]

[5]

[1]

[5][4][1] [3]

[2]

[2]

[3]

[1]

[4]


More Tilting

Because W2ρ ⊗W3ρ and T5ρ + Wρ have the same character

z5/2 + 2z3/2 + 2z1/2 + 2z−1/2 + 2z−3/2 + z5/2,

and both are tilting modules, they are equal. Of course in thefusion ring, we discard the negligible tilting module T5ρ andobtain the same answer as with the Kac-Walton fomrula. Let usdenote by xi the class of W iρ in the fusion ring, for i = 0, 1, 2, 3.We have computed in two different ways

x1x3 = x2, x2x3 = x1.


Origin in Physics

Chari and Pressley cite the following to show how this kind offusion multiplication for quantum groups was understood byphysicists.

Pasquier and Saleur (1990). Common structures betweenfinite systems and conformal field theories throughquantum groups, Nucl. Phys. B. 330, 523-56.Fröhlich and Kerber (1993), Quantum Groups, QuantumCategories and Quantum Field Theory, Springer LNM1542.

Lecture 16 - Stanford Universitysporadic.stanford.edu/quantum/lecture16.pdf · 2019-06-12 ·...

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Transcript of Lecture 16 - Stanford Universitysporadic.stanford.edu/quantum/lecture16.pdf · 2019-06-12 ·...