Skew Calabi-Yau algebras and homological identities

Transcription

1 Skew Calabi-Yau algebras and homological identities Manuel L. Reyes Bowdoin College Joint international AMS-RMS meeting Alba Iulia, Romania June 30, 2013 (joint work with Daniel Rogalski and James J. Zhang) Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

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

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

4 Preliminaries: the enveloping algebra Let A be an algebra over a field k. We write for k. The enveloping algebra of A is A e = A A op. A left A e -module M is the same as a k-central (A, A)-bimodule, via: (a b op ) m = a m b. Provides a convenient way to discuss homological algebra for bimodules: Projective/injective bimodules Projective/injective A e -modules Resolutions of (A, A)-bimodules resolutions of A e -modules Note: A bimodule A M A that is left or right A-projective need not be A e -projective. Most important example: A A A. (It s only A e -projective if A is a separable k-algebra.) Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

5 Ginzburg s definition Here s how Ginzburg generalized the Calabi-Yau condition to noncommutative algebras: Definition (i) A is homologically smooth if A has a projective resolution in A e -Mod of finite length whose terms are finitely generated over A e. (A is a perfect object in A e -Mod.) (ii) [Ginzburg, 2006] A is Calabi-Yau of dimension d if it is homologically smooth and if { Ext i A e (A, Ae ) 0 if i d, = A if i = d, as A e -modules. This condition amounts to a self-duality under M = RHom A e (M, A e ) in the derived bimodule category: A = A [d]. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

6 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 the projective A e -resolution and Ext isomorphism to be in the graded category. Graded commutative examples: k[x 1,..., x n ]. So graded Calabi-Yau algebras are noncommutative polynomial rings. But so are the Artin-Schelter regular algebras. How do these compare? Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

7 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, Ext i A (k, A) 0, i 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 under the dual functor Hom A (, A).) Note the similarity with the CY condition! However: Theorem: An AS-regular algebra A (with finite GK dimension) of dimension 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, / 35

8 Further examples Here are some more ways to find noncommutative examples: Enveloping algebras: U(g) is Calabi-Yau if g is finite-dimensional and N-graded. E.g., g = {upper-triangular n n matrices}, graded by level of the diagonal. Morita invariance: The CY property is Morita invariant. In particular, A is Calabi-Yau if and only if M n (A) is. Polynomial rings: If A is Calabi-Yau then so is A[x]. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

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

10 Calabi-Yau algebras from skew group algebras Here is a well known way to produce Calabi Yau algebras. Consider k[x 1,..., x n ] as the symmetric algebra on V = k x i. Let G SL(V ) be a finite subgroup. Then the skew group algebra k[x 1,..., x n ] G is Calabi-Yau. Recall, this has multiplication: (a g)(b h) = ag(b) gh. Theorem: Given G as above, the algebra k[x 1,..., x n ] G is Calabi-Yau of dimension n [ well-known, see Bocklandt/Schedler/Wemyss]. One might think of this as a factory for producing CY algebras that are not commutative. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

11 Calabi-Yau algebras from smash products Thm: If G SL(V ) is a finite subgroup, then k[v ] G is CY. The idea of using an action on a CY algebra to produce a new CY algebra has received much interest. Just a sample of those who ve been working on 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 that A 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. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

12 Motivation: determinants the coordinate-free way Suppose that a group G acts on V = C d. This induces a graded action on the exterior algebra Λ(V ) = Λ 0 (V ) Λ 1 (V ) Λ d (V ). Here, Λ 1 (V ) = V and Λ 0 (V ) = Λ d (V ) = C. As the action is graded, G acts in particular on Λ d (V ). Fix any generator e Λ d (V ). Then for each g G, ge is a scalar multiple of e. It turns out that this is the determinant! g(e) = det(g)e And this can be (almost) recovered from homological algebra: over the symmetric algebra A = k[v ], we have Ext A (k, k) = Λ(V ). Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

13 The homological determinant There are a few ways to define the homological determinant, that agree in the 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 Ext d A (k, k) is G-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: a g g a g g 1.) There is another formulation using local cohomology, which works for more general AS-Gorenstein algebras. Uses top-degree local cohomology instead. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

14 Examples of the homological determinant In general, it s not easy to compute the homological determinant. Here are some special cases where we do know how to compute it: Below, let g GrAut(A). Examples 1 If A = k[x 1,..., x n ], 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 x 2 y = yx 2, xy 2 = y 2 x, then hdet(g) = det(g A1 ) 2. We could use more techniques for computing hdet! Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

15 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 counit ɛ: H k (respectively dual to multiplication m : H H H and unit η : 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 the comultiplication ( a g g) = a g g g and counit ɛ( a g g) = a g. Sweedler notation: (h) = h 1 h 2. An algebra A is a left H-module algebra if its a left H-module satisfying h(ab) = h 1 (a)h 2 (b). Note: Group action of G on A left kg-module action on A. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

16 Homological determinants and smash products Homological determinant: The definition of the homological determinant extends to actions of Hopf algebras in a straightforward way: if A is an AS-regular left H-module algebra, we get an induced algebra homomorphism hdet: H k, via H-action on Ext d A (k, k). Smash products: Generalizing the skew group algebra A G, we can form 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) = ag 1 (b) g 2 h. When H = kg is a group algebra, we have A#H = A G. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

17 Smash products of Calabi-Yau algebras Basic example: Given G a group of graded automorphisms of A = k[x 1,..., x n ], the resulting map hdet: kg k gives the usual determinant of each group element. In particular: G SL n (k) hdet is trivial. For our context, this is the appropriate generalization of the condition G SL(V )! Theorem: [Liu/Wu/Zhu] Let A be an N-Koszul Calabi-Yau algebra and H an involutory Calabi-Yau Hopf algebra, where A is a left H-module algebra. Then A#H is Calabi-Yau if and only if the homological determinant of the H-action on A is trivial. Related result: Yu/Zhang have a related result when A is also a Hopf algebra. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

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

19 Calabi-Yau algebras from quivers Next I want to present an example that complicates this nice picture. It arises as a quotient of a path algebra, and gives us an excuse to discuss more 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 either their concatenation (if defined) or zero (if concatenation is undefined). Quick examples: Q = a 1 a 2 Q = b ) ( kq = k x, y kq k 0 = k k Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

20 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 be identified, up to permutation, with an element of kq/[kq, kq].) If a ar(q) is an arrow, then the cyclic partial derivative a W is the partial derivative with respect to a of all cyclic permutations of W. The Jacobi algebra of W is kq/( a W : a Q 1 ). CY-3 Ex: Q = with x,y,z W = xyz xzy. kq ( a W : a ar(q)) = k x, y, z (yz zy, zx xz, xy yx) = k[x, y, z]. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

21 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 all CY-3 algebras come from superpotentials [Davidson]. Example (Bocklandt) For Q = a 1,a 2 and the superpotential W = a 1 a 3 a 2 a 4 + a 3 a 1 a 4 a 2, the a 3,a 4 resulting Jacobi algebra B = CQ/( a W ) is Calabi-Yau of dimension 3. The explicit relations are: a1 W = 0 a 3 a 2 a 4 = a 4 a 2 a 3 a2 W = 0 a 4 a 1 a 3 = a 2 a 3 a 1 a3 W = 0 a 2 a 4 a 1 = a 1 a 4 a 2 a4 W = 0 a 1 a 3 a 2 = a 2 a 3 a 1 Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

22 Why Bocklandt s example is strange Ex [Bocklandt]: For the quiver Q = a 1,a 2 and the superpotential a 3,a 4 W = a 1 a 3 a 2 a 4 + a 3 a 1 a 4 a 2, the resulting Jacobi algebra B = 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 x 2 y = yx 2, y 2 x = xy 2. Let µ: A A send x x and y y. Then for G = µ = {1, µ}, it turns out that B = A 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 step outside of the world of CY algebras. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

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

24 Skew Calabi-Yau algebras Let M be an (A, A)-bimodule and µ, ν : A A two automorphisms. The twist µ M ν is equal to M as a k-vector space, with Definition a m b = µ(a)mν(b). A (graded) k-algebra A is (graded) skew Calabi-Yau of dimension d if it is homologically smooth and there is a (graded) automorphism µ: A A and isomorphisms of (graded) bimodules { Ext i A e (A, Ae ) 0, i d, = 1 A µ, i = d. (Also called twisted Calabi-Yau algebras.) This µ = µ A is called the Nakayama automorphism (generalized from Frobenius algebras). It s only unique up to an inner automorphism. So A is CY µ is inner. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

25 Examples of skew Calabi-Yau algebras Where do we find examples of skew CY algebras? There are two major classes. (1) A connected graded algebra A is graded skew Calabi-Yau if and only if it 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 generalization of AS-regular algebras. Some particular examples: 1 Skew polynomial rings k qij [x 1,..., x n ] 2 Coordinate ring of quantum matrices 3 Sklyanin algebras Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

26 Examples of skew Calabi-Yau algebras (2) A noetherian Hopf algebra H is skew CY if and only if it is AS-regular in the sense of Brown/Zhang [Brown/Zhang, Lu/Wu/Zhu]. Explicit Nakayama automorphism for such H: S 2 Ξ r l. (For an algebra map α: H k, the right winding automorphism of α is defined by Ξ r α(h) = h 1 α(h 2 ). 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 (U q (g), O q (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. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

27 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 a connected graded skew Calabi-Yau algebra A (compatible with grading). Then A#H is skew Calabi-Yau as well, with Nakayama automorphism (HI1) µ A#H = µ A #(µ H Ξ l hdet ), 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 is the identity and hdet trivial = µ A#H = µ A #µ H. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

28 Constructing Calabi-Yau algebras This formula indicates a way to construct new Calabi-Yau algebras. Theorem (R./Rogalski/Zhang) Let A be a noetherian AS-regular algebra with Nakayama automorphism µ. Suppose that hdet(µ) = 1. Then B = A µ is Calabi-Yau. Proof. For simplicity, assume µ has finite order. Then H = k µ has µ H = id H. Also, hdet(µ) = 1 means that hdet H is trivial, so Ξ l hdet = id H. So µ B = µ A #(µ H Ξ l hdet ) = µ A# id H = µ#1. But this is an inner automorphism of B = A#H, implemented by 1 µ A#H. So µ B inner B is CY! Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

29 Bocklandt s example revisited Recall Bocklandt s quiver algebra B = kq/( a W ) = A {1, µ} for A = C x, y x 2 y = yx 2, y 2 x = xy 2 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 not Calabi-Yau. But A is skew Calabi-Yau! (It s AS-regular.) It turns out that µ = µ A. 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 µ A was not an inner automorphism of A, it becomes an inner automorphism when we pass to B = A µ. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

30 hdet of the Nakayama automorphism Here is a second homological identity that we established. Theorem (R./Rogalski/Zhang) Let A be a noetherian connected graded Koszul skew Calabi-Yau algebra. Then hdet(µ A ) = 1. We strongly believe that it should be possible to omit the Koszul property. Here s a little evidence: Recall that, if hdet(µ A ) = 1, then A µ A is Calabi-Yau. But more recently, J. Goodman and U. Krähmer have shown that if A is skew CY, not necessarily graded, then A µ A is CY. This is consistent with hdet(µ A ) = 1 in the connected graded case. Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

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

32 Food for thought Problems 1 If A and H are skew CY with A a left H-module algebra, is A#H skew Calabi-Yau? If so, what is µ A#H (in terms of µ A and µ H )? 2 If A is connected graded skew Calabi-Yau (or even AS-Gorenstein), is hdet(µ A ) = 1? 3 What are some good, general techniques to compute the Nakayama automorphism of an algebra or the homological determinant of an action? 4 Big, big question, even for regular algebras: How can we understand the ring-theoretic structure of these algebras? There is much that is not yet understood. Skew CY algebras are very natural in noncommutative algebra/geometry and deserve more attention! Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

33 Selected references R. Berger and R. Taillefer, Poincaré-Birkhoff-Witt deformations of Calabi-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 order derivations, 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/ (2006). J. Goodman and U. Krähmer, Untwisting a twisted Calabi-Yau algebra, arxiv: (2013). Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

34 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 invariants under Hopf algebra actions, J. Algebra (2009). P. LeMeur, Crossed products of Calabi-Yau algebras by finite groups, arxiv: (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 and homological identities, arxiv: (2013). M. Van den Bergh, Calabi-Yau algebras and superpotentials, arxiv: (2010). Manuel Reyes (Bowdoin College) Skew Calabi-Yau algebras June 30, / 35

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