Graded Calabi-Yau Algebras actions and PBW deformations

Transcription

1 Actions on Graded Calabi-Yau Algebras actions and PBW deformations Q. -S. Wu Joint with L. -Y. Liu and C. Zhu School of Mathematical Sciences, Fudan University International Conference at SJTU, Shanghai Oct. 6, 2011

2 Actions on Definitions Examples Notations k is a field of characteristic 0 all algebras are k-algebras all graded algebras are generated in degree 1 all modules are left graded modules; A e := A A op ; ( ) := Hom k (, k).

3 Actions on Definitions Examples Definition (Ginzburg, 2006) A graded algebra A is called d-calabi-yau if (1) A is homologically smooth, i.e., A has a finitely generated A e -projective resolution of finite length; (2) there exists an integer l such that, as A e -modules { Ext i A e(a, Ae ) A(l), if i = d; = 0, if i d.

4 Actions on Definitions Examples Remarks Condition (1) A D per (A e ) A is a compact object in D(A e ) Hom D(A e )(A, ) commutes with arbitrary direct sum. (Ungraded) are defined similarly, but with no degree shift in condition (2).

5 Actions on Definitions Examples Calabi-Yau Categories Definition (Kontsevich, 1998) A hom-finite triangulated category C is called d-calabi-yau for some integer d if Hom C (A, B) iso = Hom C (B, A(d)) for any A, B ob C, and the "iso" is natural. This really means that C has a Serre duality with the Serre functor being trivial.

6 Actions on Definitions Examples Calabi-Yau Derived Calabi-Yau Definition (Bocklandt, Iyama-Reiten, 2008) An algebra A is called a derived d-calabi-yau algebra if Df b.dim(a-mod) is a d-calabi-yau triangulated category. Theorem (Keller, VdB, 2009) If A is a d-calabi-yau algebra, then Df b.dim(a-mod) is a d-calabi-yau category, i.e., A is derived d-calabi-yau.

7 Actions on Definitions Examples Examples of CY-algebras I (1) k[x 1, x 2,, x n ]: n-calabi-yau (2) 3-dim. AS-regular algebras of type diag(1, 1), (1, 1, 1) (3) Sklyanin algebras of dimension 4 (4) Weyl algebras A n (k): 2n-Calabi-Yau (JLMS, 2009) (5) preprojective algebras of non-dynkin quivers (6) Yang-Mills algebras (7) rational Cherednik algebras

8 Actions on Definitions Examples Examples of CY-algebras (II) (8) Let X be a smooth Calabi-Yau variety (i.e., the canonical sheaf is trivial), then C[X] is a Calabi-Yau algebra of dimension dim X. (9) Let G be a finite group acting on X = C n, X/G be the quotient variety, and Y f X/G be the crepant resolution (f ω X/G = ω Y ), then D b (C[x 1, x 2,, x n ]#G) = D b (Coh Y ). Theorem (Iyama-Reiten, AMJ 2008, Farinati, JA 2005) k[x 1, x 2,, x n ]#G with G GL(n, k) is Calabi-Yau if and only if that G SL(n, k).

9 Actions on Definition Twisted Definition (Artin-Schelter) A connected graded algebra A is called left Artin-Schelter(AS, for short) regular (resp. Gorenstein) if the following conditions hold. (1) A has finite global (resp. A A has finite injective) dim. d; (2) there exists an integer l such that { Ext i A (k, A) k(l), if i = d, = 0, if i d. (1)

10 Actions on Definition Twisted Definition (Twisted ) A k-algebra is called a (left) σ-twisted Calabi-Yau algebra of dimension d, for σ Aut(A), if A D per (A e ) and as A e -modules { Ext i A e(a, Ae ) 1A σ, i = d, = 0, i d.

11 Actions on Definition Twisted "Twisted" graded Calabi-Yau is AS-regular A connected graded algebra is twisted Calabi-Yau if and only if it is AS-regular.

12 Actions on Definition Twisted Twisted AS-regular Hopf algebra is twisted Calabi-Yau (Brown-Zhang) Let H be a noetherian Hopf algebra with bijective antipode S. If H is AS-regular with global dimension gldim(h) = d, and if l H = ε k π and ξ is the winding automorphism determined by π, then Ext i H e(h, He ) = { 1H σ, i = d, 0, i d, where σ = S 2 ξ.

13 Actions on Definition Twisted When AS-regular is Calabi-Yau? Let H be a noetherian Hopf algebra with bijective antipode S. Then H is Calabi-Yau of dimension d if and only if (i) H is AS-regular with global dimension gldim(h) = d and l H = ε k ε, (ii) S 2 is an inner automorphism of H.

14 Actions on Questions Homological determinant Two results Questions-I Poly. algebras AS-regular algebras G k[x 1, x 2,, x n ]; k[x 1,, x n ]#G is Calabi-Yau G SL(n, k), G Calabi-Yau algebra A A#G is Calabi-Yau G???

15 Actions on Questions Homological determinant Two results Question-II Let H be a (Calabi-Yau) Hopf algebra, and A be a Calabi-Yau algebra. If H A, when is A#H Calabi-Yau?

16 Actions on Questions Homological determinant Two results Homological determinant for group actions Let A be an AS-Gorenstein algebra, and σ GrAut(A). (1) H i m(m) = lim Ext i A (A/A n, M) is the i-th local cohomology of M for any graded module M; { (2) H i m(a) 0, i d, = AA (l), i = d; (3) there exists c k so that the following is commutative. H d H d m m(a) (σ) H d m(a) = = AA (l) c(σ 1 ) AA (l).

17 Actions on Questions Homological determinant Two results Homological determinant for group actions Definition (Jørgensen & Zhang, Adv. Math. 1998) hdet(σ) := c 1 is called the homological determinant of σ. If A = k[x 1, x 2,, x n ] and σ GrAut(A). Then σ V GL(n, k) where V = kx 1 kx 2 kx n and hdet(σ) = det(σ V ).

18 Actions on Questions Homological determinant Two results Remarks Remarks ( ) 0 1 (1) A := k x, y / xy + yx, σ A1 =, 1 0 then hdet(σ) = 1, but det(σ A1 ) = 1. (2) Let A be a p-koszul AS-regular algebra of dimension d and σ GrAut(A). If σ A1 = (c ij ), then σ τ GrAut(A! ) where σ τ A 1 = (c ij ) Tr and σ τ (u) = hdet(σ)u for u Ext d A (k, k);

19 Actions on Questions Homological determinant Two results Homological determinant for Hopf actions Hypothesis: (1) H is a Hopf algebra, (2) A is a connected graded AS-Gorenstein algebra, and (3) A is a graded left H-module algebra, or, A is a left H-module algebra and each A i is a left H-submodule for each i.

20 Actions on Questions Homological determinant Two results Homological determinant for Hopf actions Definition (Kirkman, Kuzmanovich & Zhang, J. Alg. 2009) Let A be an AS-regular algebra of global dimension d. There is a left H-action on Ext d A (k, A) induced by the left A#H-action on A. Let e be a nonzero element in Ext d A (k, A). Then there is an algebra morphism η : H k satisfying h e = η(h)e for all h H. 1 The composite map η S : H k is called the homological determinant of the H-action on A, and is denoted by hdet (or more precisely hdet A ) 2 The homological determinant hdet A is said to be trivial if hdet A = ɛ, the counit of the Hopf algbera H.

21 Actions on Questions Homological determinant Two results Theorem (W-Zhu, J. Alg. 2011) Let A be a Koszul Calabi-Yau algebra, G be a finite subgroup of GrAut(A). Then A#G is Calabi-Yau G SL(A) where SL(A) := {σ GrAut(A) hdet(σ) = 1}. Theorem (Liu-W-Zhu, Contemp. Math., AMS, 2011) Let A be a Koszul Calabi-Yau algebra, H be a Calabi-Yau involutory Hopf algebra, and A be a left graded H-module algebra. Then A#H is Calabi-Yau The homological determinant hdet H is trivial.

22 Motivation Actions on Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application Theorem (Berger-Taillefer, 2007) Let U be a PBW deformation of a 3-Calabi-Yau algebra A. If U is derived from a superpotential, then U is 3-Calabi-Yau. Some PBW deformations of polynomial algebras, say, Weyl algebras A n (k) (U(g)) are (sometimes) Calabi-Yau. Symplectic refection algebra is a PBW deformation of a skew group algebra of a symmetric algebra.

23 Question Actions on Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application Question If U is a PBW deformation of a graded Calabi-Yau algebra, when is U Calabi-Yau?

24 Deformation Actions on Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application Definition Let A = T k (V )/ R where R = {r 1,, r m }, with all the r j s homogeneous and U = T k (V )/ P where P = {r 1 + l 1,, r m + l m } with deg l i < deg r i. Then U is called a deformation of A. In this case, there exists a natural filtration on U, and a canonical homomorphism A gr U of graded algebras.

25 Actions on PBW Deformation Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application Definition The deformation U of the graded algebra A is said to be a Poincaré-Birkhoff-Witt deformation (PBW deformation) if A gr U is an isomorphism of graded algebras.

26 Actions on Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application Some examples of PBW-deformation (1) universal enveloping algebras U(g) (2) Weyl algebras A n (k) (3) Sridharan enveloping algebras U f (g) (4) symplectic reflection algebras

27 Actions on Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application More assumption Let A = T (V )/ R be a quadratic algebra and P (k V ) = 0. Then there exist two linear maps α : R V, β : R k such that P = {r α(r) β(r) r R}. Theorem (Braverman-Gaitsgory, 1996) Let A be a Koszul algebra. Then U is a PBW deformation of A if and only if the following are satisfied: (1) Im(α id id α) R ; (2) α(α id id α) = (β id id β); (3) β(α id id α) = 0.

28 Actions on Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application Universal enveloping algebras U(g) Universal enveloping algebras U(g) α(xy yx) = [x, y]; β = 0 Condition (2) the Jacobian identity on g

29 Actions on Central regular extension Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application Definition Let A and D be two graded algebras. If there is a central element t D 1 such that A = D/ t, then D is called a central extension of A. If further t D is regular, then D is called a central regular extension of A.

30 Rees algebra Actions on Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application An easy fact: Suppose that U has an increasing filtration FU = {F n U} n Z. Let Rees(U) = n Z F nu t n U[t, t 1 ] be the Rees algebra. Then Rees(U) is a central regular extension of gr U.

31 Homogenization Actions on Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application Let U = T (V )/ P be a deformation of A = T (V )/ R. Let H(U) = T (V )[t]/ h(r 1 + l 1 ), h(r 2 + l 2 ),, h(r m + l m ) be the central extension associated to U (homogenization). Theorem (Cassidy-Shelton, 2007) U is a PBW deformation of A H(U) is a central regular extension of A.

32 A connection Actions on Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application Theorem (W-Zhu, 2011) If U is a PBW deformation, then Rees(U) = H(U).

33 Actions on Central Regular Extension Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application Theorem (W-Zhu, 2011) Let D be a central regular extension of a Koszul algebra A. (1) if D is a Calabi-Yau algebra, so is A; (2) if A is a Calabi-Yau algebra, then D is Calabi-Yau if and only if d 2 ( 1) i id i α id (d 2 i) = 0. i=0

34 Actions on PBW deformation Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application Theorem (W-Zhu, 2011) Suppose that A is a noetherian Koszul Calabi-Yau algebra of dimension d (d 2) and U is a PBW deformation of A. Then the following are equivalent: (1) U is d-calabi-yau; (2) Rees(U) is (d + 1)-Calabi-Yau; (3) d 2 ( 1) i id i α id (d 2 i) = 0. i=0

35 Actions on PBW deformation Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application Remark (1) independent of β; (2) U is derived from a superpotential id α α id = 0; (3) Chevalley-Eilenberg resolution

36 Application Actions on Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application Let g be a k-lie algebra, f Z 2 (g, k) be a 2-cocycle. The Sridharan enveloping algebra of g is defined to be U f (g) = T k (g)/ x y y x [x, y] f (x, y). Theorem (He-Oystaeyen-Zhang, 2010) The following statements are equivalent. (1) dim g = d and tr(ad(x)) = 0 for all x g. (2) U(g) is d-calabi-yau. (3) U f (g) is d-calabi-yau.

37 Actions on Motivation and Question Definition of PBW deformation Central regular extension Connections Central Regular Extension PBW deformation Application Thank you for your attention!