Manuel L. Reyes

Skew Calabi-Yau algebras and homological identities

Manuel L. Reyes

Bowdoin College

Joint international AMS-RMS meetingAlba Iulia, Romania

June 30, 2013

(joint work with Daniel Rogalski and James J. Zhang)

Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, 2013 1 / 35

Outline

1 Calabi-Yau algebras: basics and sources of examples

2 Calabi-Yau algebras from smash products

3 A curious example of a Calabi-Yau algebra

4 Skew Calabi-Yau algebras and homological identities

5 Problems and questions


The Calabi-Yau property in noncommutative algebra

The Calabi-Yau property has its origin in geometry:

But it has now made its way into noncommutative algebra!

Geometry Physics

(string theory)

triangulated

categories

noncomm.

algebras

There are (at least) two ways to think of these algebras:(1) Noncommutative version of coordinate ring of Calabi-Yau variety.

(2) Graded case: noncommutative version of polynomial ring k[x , y , z ].

I will emphasize (2).


Preliminaries: the enveloping algebra

Let A be an algebra over a field k . We write −⊗− for −⊗k −.

The enveloping algebra of A is Ae = A⊗ Aop. A left Ae-module M is thesame as a k-central (A,A)-bimodule, via:

(a⊗ bop) ·m = a ·m · b.

Provides a convenient way to discuss homological algebra for bimodules:

Projective/injective bimodules ! Projective/injective Ae-modules

Resolutions of (A,A)-bimodules ! resolutions of Ae-modules

Note: A bimodule AMA that is left or right A-projective need not beAe-projective. Most important example: AAA. (It’s only Ae-projective if Ais a “separable” k-algebra.)


Ginzburg’s definition

Here’s how Ginzburg generalized the Calabi-Yau condition tononcommutative algebras:

Definition

(i) A is homologically smooth if A has a projective resolution in Ae-Mod offinite length whose terms are finitely generated over Ae . (A is a “perfectobject” in Ae-Mod.)

(ii) [Ginzburg, 2006] A is Calabi-Yau of dimension d if it is homologicallysmooth and if

ExtiAe (A,Ae) ∼=

{0 if i 6= d ,

A if i = d ,

as Ae-modules.

This condition amounts to a “self-duality” under M∨ = RHomAe (M,Ae)in the derived bimodule category: A ∼= A∨[d ].


First examples of Calabi-Yau algebras

Commutative examples: coordinate rings of Calabi-Yau varieties[Ginzburg].

From now on, we will consider only graded Calabi-Yau algebras: take theprojective Ae-resolution and Ext isomorphism to be in the graded category.

Graded commutative examples: k[x1, . . . , xn].

So graded Calabi-Yau algebras are “noncommutative polynomial rings.”But so are the Artin-Schelter regular algebras. How do these compare?


AS-regular vs. Calabi-Yau properties

Definition

A connected graded k-algebra A is Artin-Schelter Gorenstein if

A has left and right injective dimension d <∞there is an integer l (“AS index”) such that, as both left and right

modules, ExtiA(k ,A) ∼=

{0, i 6= d ,

k(l) i = d .

An algebra A as above is AS-regular if it also has finite global dimension,which will equal d .

(This amounts to a “duality” between the one-sided resolutions of k underthe dual functor HomA(−,A).)

Note the similarity with the CY condition! However:

Theorem: An AS-regular algebra A (with finite GK dimension) ofdimension 3 is Calabi-Yau if and only if it is of “type A” in Artin &Schelter’s terminology [Berger & Taillefer].Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, 2013 7 / 35

Further examples

Here are some more ways to find noncommutative examples:

Enveloping algebras: U(g) is Calabi-Yau if g is finite-dimensional andN-graded.

E.g., g = {upper-triangular n × n matrices}, graded by “level of thediagonal.”

Morita invariance: The CY property is Morita invariant.In particular, A is Calabi-Yau if and only if Mn(A) is.

Polynomial rings: If A is Calabi-Yau then so is A[x ].


Calabi-Yau algebras from skew group algebras

Here is a well known way to produce Calabi Yau algebras.

Consider k[x1, . . . , xn] as the symmetric algebra on V =⊕

k · xi . LetG ≤ SL(V ) be a finite subgroup. Then the skew group algebrak[x1, . . . , xn] o G is Calabi-Yau. Recall, this has multiplication:

(a⊗ g)(b ⊗ h) = ag(b)⊗ gh.

Theorem: Given G as above, the algebra k[x1, . . . , xn] o G is Calabi-Yauof dimension n [“well-known,” see Bocklandt/Schedler/Wemyss].

One might think of this as a ”factory” for producing CY algebras that arenot commutative.


Calabi-Yau algebras from smash products

Thm: If G ≤ SL(V ) is a finite subgroup, then k[V ] o G is CY.

The idea of using an action on a CY algebra to produce a new CY algebrahas received much interest. Just a sample of those who’ve been workingon this: Bocklandt/Schedler/Wemyss, Farniati, LeMeur, Liu/Wu/Zhu,Yu/Zhang, etc.

Q: If A is CY and a finite group G acts on A, what condition ensures thatA o G is CY?

One of the most general results is due to Liu/Wu/Zhu.

What is their analogue of the condition that G ≤ SL(V )?

That a certain homological determinant is trivial.


Motivation: determinants the “coordinate-free” way

Suppose that a group G acts on V = Cd .

This induces a graded action on the exterior algebra

Λ(V ) = Λ0(V )⊕ Λ1(V )⊕ · · · ⊕ Λd(V ).

Here, Λ1(V ) ∼= V and Λ0(V ) ∼= Λd(V ) ∼= C.

As the action is graded, G acts in particular on Λd(V ). Fix any generatore ∈ Λd(V ). Then for each g ∈ G , ge is a scalar multiple of e. It turns outthat this is the determinant!

g(e) = det(g)e

And this can be (almost) recovered from homological algebra: over thesymmetric algebra A = k[V ], we have Ext•A(k , k) ∼= Λ(V ∗).


The homological determinant

There are a few ways to define the homological determinant, that agree inthe cases where they make sense.

The one that directly generalizes the above requires A to be AS-regular[Kirkman/Kuzmanovich/Zhang].

An action of G on A induces an action on Ext•A(k , k).

The action is graded, so the highest-degree part ExtdA(k, k) isG -invariant. It’s also one dimensional.

Fix generator e, then for he = η(h)e, we have hdet = η ◦ S : kG → k .

(Here S : kG → kG is the antipode:∑

ag g 7→∑

ag g−1.)

There is another formulation using local cohomology, which works for moregeneral AS-Gorenstein algebras. Uses top-degree local cohomology instead.


Examples of the homological determinant

In general, it’s not easy to compute the homological determinant. Here aresome special cases where we do know how to compute it:

Below, let g ∈ GrAut(A).

Examples

1 If A = k[x1, . . . , xn], then hdet(g) = det(g |A1).

2 If A = k−1[x , y ] = k〈x , y | xy = −yx〉, then hdet(g) = − det(g |A1).

3 If A = k〈x , y | x2y = yx2, xy2 = y2x〉, then hdet(g) = det(g |A1)2.

We could use more techniques for computing hdet!


Hopf algebra actions

At this level of generality, we might as well work with Hopf algebra actions(“quantized group actions”).Recall: A Hopf algebra H has a comultiplication ∆: H → H ⊗H, a counitε : H → k (respectively “dual” to multiplication m : H ⊗ H → H and unitη : k → H), and an antipode S : kG → kG . Assume S bijective.

Ex: for a group G , the group algebra H = kG is a Hopf algebra under thecomultiplication ∆(

∑ag g) =

∑ag g ⊗ g and counit ε(

∑ag g) =

∑ag .

Sweedler notation: ∆(h) =∑

h1 ⊗ h2.

An algebra A is a left H-module algebra if its a left H-module satisfying

h(ab) =∑

h1(a)h2(b).

Note: Group action of G on A ! left kG -module action on A.


Homological determinants and smash products

Homological determinant: The definition of the homologicaldeterminant extends to actions of Hopf algebras in a straightforward way:if A is an AS-regular left H-module algebra, we get an induced algebrahomomorphism hdet : H → k, via H-action on ExtdA(k, k).

Smash products: Generalizing the skew group algebra A o G , we canform the smash product A#H when A is a left H-module algebra.

This is defined to be the vector space A⊗ H with multiplication given by

(a⊗ g)(b ⊗ h) =∑

ag1(b)⊗ g2h.

When H = kG is a group algebra, we have A#H ∼= A o G .


Smash products of Calabi-Yau algebras

Basic example: Given G a group of graded automorphisms ofA = k[x1, . . . , xn], the resulting map hdet : kG → k gives the usualdeterminant of each group element.In particular: G ≤ SLn(k) ⇐⇒ hdet is trivial.

For our context, this is the appropriate generalization of the conditionG ≤ SL(V )!

Theorem: [Liu/Wu/Zhu] Let A be an N-Koszul Calabi-Yau algebra andH an involutory Calabi-Yau Hopf algebra, where A is a left H-modulealgebra. Then A#H is Calabi-Yau if and only if the homologicaldeterminant of the H-action on A is trivial.

Related result: Yu/Zhang have a related result when A is also a Hopfalgebra.


Calabi-Yau algebras from quivers

Next I want to present an example that complicates this nice picture. Itarises as a quotient of a path algebra, and gives us an excuse to discussmore examples of CY algebras that are not connected.

A quiver Q is a directed graph (multiple arrows and loops are allowed).

The path algebra kQ is the k-vector space spanned by (possibly trivial)paths in Q (read right-to-left), where the product of two paths is eithertheir concatenation (if defined) or zero (if concatenation is undefined).

Quick examples:

Q = a1 << a2bb Q ′ =b //

kQ ∼= k〈x , y〉 kQ ′ ∼=(

k 0k k

)


Calabi-Yau algebras and superpotentials

A popular theme in the study of CY algebras (coming from physics),especially in dimension 3, is that they tend to have relations defined by“superpotentials.”

Superpotential W : a linear combination of cycles in Q. (May beidentified, up to permutation, with an element of kQ/[kQ, kQ].)

If a ∈ ar(Q) is an arrow, then the “cyclic partial derivative” ∂aW is thepartial derivative with respect to a of all cyclic permutations of W .

The Jacobi algebra of W is kQ/(∂aW : a ∈ Q1).

CY-3 Ex: Q = x ,y ,zYh with W = xyz − xzy .

kQ

(∂aW : a ∈ ar(Q))∼=

k〈x , y , z〉(yz − zy , zx − xz , xy − yx)

∼= k[x , y , z ].


Calabi-Yau algebras from superpotentials

(Note: not every Jacobi algebra is CY; only for “good” superpotentials.)

It’s been shown that many CY algebras come from superpotentials[Bocklandt, Bocklandt/Schedler/Wemyss, Van den Bergh] but not allCY-3 algebras come from superpotentials [Davidson].

Example (Bocklandt)

For Q =a1,a2

!)

a3,a4

ai and the superpotential W = a1a3a2a4 + a3a1a4a2, the

resulting Jacobi algebra B = CQ/(∂aW ) is Calabi-Yau of dimension 3.The explicit relations are:

∂a1W = 0 a3a2a4 = −a4a2a3

∂a2W = 0 a4a1a3 = −a2a3a1

∂a3W = 0 a2a4a1 = −a1a4a2

∂a4W = 0 a1a3a2 = −a2a3a1


Why Bocklandt’s example is “strange”

Ex [Bocklandt]: For the quiver Q =a1,a2

!)

a3,a4

ai and the superpotential

W = a1a3a2a4 + a3a1a4a2, the resulting Jacobi algebraB = CQ/(∂ai W : i = 1, . . . , 4) is Calabi-Yau of dimension 3.

Fact: We noticed that this can be realized as a skew group algebra: Let

A = C〈x , y | x2y = yx2, y2x = xy2〉.

Let µ : A→ A send x 7→ −x and y 7→ −y . Then for G = 〈µ〉 = {1, µ}, itturns out that B ∼= A o G .

However, A is not Calabi-Yau! It’s AS-regular of dimension 3, but of the“wrong” type (type A).

What’s going on here? To better understand this example, we must stepoutside of the world of CY algebras.


Skew Calabi-Yau algebras

Let M be an (A,A)-bimodule and µ, ν : A→ A two automorphisms. Thetwist µMν is equal to M as a k-vector space, with

a ·m · b = µ(a)mν(b).

Definition

A (graded) k-algebra A is (graded) skew Calabi-Yau of dimension d if it ishomologically smooth and there is a (graded) automorphism µ : A→ Aand isomorphisms of (graded) bimodules

ExtiAe (A,Ae) ∼=

{0, i 6= d ,1Aµ, i = d .

(Also called twisted Calabi-Yau algebras.)

This µ = µA is called the Nakayama automorphism (generalized fromFrobenius algebras). It’s only unique up to an inner automorphism. So Ais CY ⇐⇒ µ is inner.Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, 2013 24 / 35

Examples of skew Calabi-Yau algebras

Where do we find examples of skew CY algebras? There are two majorclasses.

(1) A connected graded algebra A is graded skew Calabi-Yau if and only ifit is AS-regular [Yekutieli/Zhang, R./Rogalski/Zhang].

(Note: no assumption of finite GK dimension here!)

In this sense, graded skew CY algebras are a non-connected generalizationof AS-regular algebras.

Some particular examples:

1 Skew polynomial rings kqij [x1, . . . , xn]

2 Coordinate ring of quantum matrices

3 Sklyanin algebras


Examples of skew Calabi-Yau algebras

(2) A noetherian Hopf algebra H is skew CY if and only if it is AS-regularin the sense of Brown/Zhang [Brown/Zhang, Lu/Wu/Zhu]. ExplicitNakayama automorphism for such H: S−2 ◦ Ξr∫ l .

(For an algebra map α : H → k , the right winding automorphism of α is

defined by Ξrα(h) =

∑h1α(h2). Also,

∫ l: H → k is a map induced by the

“left homological integral.”)

Some examples:

1 U(g) for finite-dimensional g. (It’s CY if g is N-graded.)

2 O(G )

3 Lots of quantum groups (Uq(g), Oq(G ))

4 Affine noetherian PI Hopf algebras with finite global dimension

Question of Brown: are all noetherian Hopf algebras “AS-Gorenstein”?Positive answer ⇒ all noetherian H with gl .dim(H) <∞ are skew CY.


Three homological identities

So what happens when we take smash products with skew CY algebras?This is our first “homological identity.”

Theorem (R./Rogalski/Zhang)

Let H be a finite dimensional semisimple Hopf algebra acting on aconnected graded skew Calabi-Yau algebra A (compatible with grading).Then A#H is skew Calabi-Yau as well, with Nakayama automorphism

(HI1) µA#H = µA#(µH ◦ Ξlhdet),

where hdet is the homological determinant of the action of H on A.

In particular, if hdet is trivial, then the winding automorphism of hdet isthe identity and

hdet trivial =⇒ µA#H = µA#µH .


Constructing Calabi-Yau algebras

This formula indicates a way to construct new Calabi-Yau algebras.


Let A be a noetherian AS-regular algebra with Nakayama automorphismµ. Suppose that hdet(µ) = 1. Then B = A o 〈µ〉 is Calabi-Yau.

Proof.

For simplicity, assume µ has finite order. Then H = k〈µ〉 has µH = idH .Also, hdet(µ) = 1 means that hdetH is trivial, so Ξl

hdet = idH . So

µB = µA#(µH ◦ Ξlhdet) = µA# idH = µ#1.

But this is an inner automorphism of B = A#H, implemented by1⊗ µ ∈ A#H. So µB inner ⇒ B is CY!


Bocklandt’s example revisited

Recall Bocklandt’s quiver algebra B = kQ/(∂aW ) ∼= A o {1, µ} for

A = C〈x , y | x2y = yx2, y2x = xy2〉

and µ : A→ A given by µ(x) = −x , µ(y) = −y .Set H = k{1, µ}. Then B ∼= A#H is Calabi-Yau, while A is notCalabi-Yau.

But A is skew Calabi-Yau! (It’s AS-regular.)

It turns out that µ = µA.

Kirkman and Kuzmanovich: for any σ ∈ GrAut(A),hdet(σ) = det(σ|A1)2. Thus, hdet(µ) = 1 and hdet : H → k is trivial.

Thus the previous theorem recovers the fact that B is Calabi-Yau!

Again, the key idea is that while µA was not an inner automorphism of A,it “becomes an inner automorphism” when we pass to B = A o 〈µ〉.


hdet of the Nakayama automorphism

Here is a second “homological identity” that we established.


Let A be a noetherian connected graded Koszul skew Calabi-Yau algebra.Then

hdet(µA) = 1.

We strongly believe that it should be possible to omit the Koszul property.Here’s a little evidence:

Recall that, if hdet(µA) = 1, then A o 〈µA〉 is Calabi-Yau.

But more recently, J. Goodman and U. Krahmer have shown that if A isskew CY, not necessarily graded, then A o 〈µA〉 is CY. This is consistentwith hdet(µA) = 1 in the connected graded case.


Food for thought

Problems1 If A and H are skew CY with A a left H-module algebra, is A#H

skew Calabi-Yau? If so, what is µA#H (in terms of µA and µH)?

2 If A is connected graded skew Calabi-Yau (or even AS-Gorenstein), ishdet(µA) = 1?

3 What are some good, general techniques to compute the Nakayamaautomorphism of an algebra or the homological determinant of anaction?

4 Big, big question, even for regular algebras: How can we understandthe ring-theoretic structure of these algebras?

There is much that is not yet understood. Skew CY algebras are verynatural in noncommutative algebra/geometry and deserve more attention!


Selected references

R. Berger and R. Taillefer, Poincare-Birkhoff-Witt deformations ofCalabi-Yau algebras, J. Noncommut. Geom. (2007).

R. Bocklandt, Graded Calabi Yau algebras of dimension 3, J. Pure Appl.Algebra (2008).

R. Bocklandt, T. Schedler, M. Wemyss, Superpotentials and higher orderderivations, J. Pure Appl. Algebra (2010).

K.A. Brown and J.J. Zhang, Dualizing complexes and twisted Hochschild(co)homology for Noetherian Hopf algebras, J. Algebra (2008).

B. Davidson, Superpotential algebras and manifolds, Adv. Math. (2012).

V. Ginzburg, Calabi-Yau algebras, arXiv:math/0612139 (2006).

J. Goodman and U. Krahmer, Untwisting a twisted Calabi-Yau algebra,arXiv:1304.0749 (2013).Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, 2013 33 / 35

More selected references

P. Jørgensen and J.J. Zhang, Gourmet’s guide to Gorensteinness, Adv.Math. (2000).

E. Kirkman, J. Kuzmanovich, J.J. Zhang, Gorenstein subrings of invariantsunder Hopf algebra actions, J. Algebra (2009).

P. LeMeur, Crossed products of Calabi-Yau algebras by finite groups,arXiv:1006.1082 (2010).

L.-Y. Liu, Q.-S. Wu, and C. Zhu, Hopf action on Calabi-Yau algebras,Contemp. Math. 562 (2012).

M. Reyes, D. Rogalski, J.J. Zhang, Skew Calabi-Yau algebras andhomological identities, arXiv:1302.0437 (2013).

M. Van den Bergh, Calabi-Yau algebras and superpotentials,arXiv:1008.0599 (2010).


Even more selected references

Q.-S. Wu and C. Zhu, Skew group algebras of Calabi-Yau algebras, J.Algebra (2011).

A. Yekutieli and J.J. Zhang, Homological transcendence degree, Proc.London Math. Soc. (2006).

X. Yu and Y. Zhang, The Calabi-Yau property of smash products, J.Algebra (2012).

Thank you!


Manuel L. Reyes

Documents

Transcript of Manuel L. Reyes