Contents Calabi-Yau algebra Deformation and extension Main results PBW deformation of Koszul Calabi-Yau algebras Can Zhu Shanghai Polytechnic University Noncommutative Algebraic Geometry 2011 Shanghai Workshop 9-12, 2011 Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents Calabi-Yau algebra Deformation and extension Main results Joint work with Quan Shui WU. Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents Calabi-Yau algebra Deformation and extension Main results Contents Calabi-Yau algebra 1 Calabi-Yau algebra and Calabi-Yau category Examples and properties Yonada Ext algebra Deformation and extension 2 PBW deformation Central regular extension Relations Main results 3 Central Regular Extension PBW deformation Application Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents Calabi-Yau algebra Deformation and extension Main results Notations k is field with characteristic 0 graded algebras are generated in degree 1 all modules are left graded ones A e := A ⊗ A o , ∗ := Hom k ( , k ) Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents Calabi-Yau algebra and Calabi-Yau category Calabi-Yau algebra Examples and properties Deformation and extension Yonada Ext algebra Main results Calabi-Yau algebra Definition (V. Ginzburg, 2007) A graded algebra A is d -Calabi-Yau ( d -CY, for short)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 � A ( l ) , if i = d ; Ext i A e ( A , A e ) ∼ = 0 , if i � = d as A e -modules. Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents Calabi-Yau algebra and Calabi-Yau category Calabi-Yau algebra Examples and properties Deformation and extension Yonada Ext algebra Main results Calabi-Yau category Suppose that ( T , Σ) is a triangulated k-category which is Hom-finite. Let d be a nonnegative integer. Definition (M. Kontsevich, 1998) The triangulated category ( T , Σ) is called a d -Calabi-Yau ( d -CY, for short) category if Σ d is a Serre functor, i.e., there is a bifunctorial isomorphism Hom T ( X , Y ) ∼ = Hom T ( Y , Σ d X ) ∗ . Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents Calabi-Yau algebra and Calabi-Yau category Calabi-Yau algebra Examples and properties Deformation and extension Yonada Ext algebra Main results Remarks (1) (Yekutieli-Zhang, Jørgensen, 1997) A is a d -CY algebra ⇒ D b ( qgrA ) is a ( d − 1 ) -CY category (2) (Keller, 2008) A is a d -CY algebra ⇒ D b fd ( Mod A ) is a d -CY category (3) (He-Oystaeyen-Zhang, 2010) Suppose A is p -Koszul. A is a d -CY algebra ⇔ D b fd ( Mod A ) is a d -CY category Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents Calabi-Yau algebra and Calabi-Yau category Calabi-Yau algebra Examples and properties Deformation and extension Yonada Ext algebra Main results Examples (1) k [ x 1 , x 2 , · · · , x n ]# G for a finite subgroup G of SL ( n , k ) (2) 3-dim. AS-regular algebras of type diag(1, 1), (1, 1, 1) (3) Yang-Mills algebras (4) Sklyanin algebras of dimension 4 (5) Weyl algebras A n (6) preprojective algebras of non-Dynkin quivers (7) quantum enveloping algebras (8) rational Cherednik algebras Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents Calabi-Yau algebra and Calabi-Yau category Calabi-Yau algebra Examples and properties Deformation and extension Yonada Ext algebra Main results Proposition (Berger-Taillefer, Iyama-Reiten) If A is a graded d -CY algebra, then (1) the global dimension of A is d ; (2) the Hochschild dimension of A is d ; (3) if, moreover, A is connected, then A is AS-regular; (4) d = 0 if and only if A is semisimple. Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents Calabi-Yau algebra and Calabi-Yau category Calabi-Yau algebra Examples and properties Deformation and extension Yonada Ext algebra Main results Remarks (1) For ungraded d -CY algebra, its global dimension is not necessary to be d . Eg.: Weyl algebra A n (2 n -CY, BUT gl.dim A n = n ). (2) Finite dim. CY algebras must be 0-CY. Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents Calabi-Yau algebra and Calabi-Yau category Calabi-Yau algebra Examples and properties Deformation and extension Yonada Ext algebra Main results Yonada Ext algebras of AS-regular algebras Theorem(Bondal-Polishchuk, Smith, Berger-Marconnet, LPWZ) A is an AS-regular algebra if and only if its Yoneda Ext algebra E ( A ) := � Ext ∗ A ( k , k ) is a Frobenius algebra. Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents Calabi-Yau algebra and Calabi-Yau category Calabi-Yau algebra Examples and properties Deformation and extension Yonada Ext algebra Main results Yonada Ext algebras of Calabi-Yau algebras Theorem (Van den Bergh, 2008) A is a d -CY algebra if and only if its Yoneda Ext algebra E ( A ) has a cyclic A ∞ -structure of degree d . Here, a cyclic structure on an A ∞ -algebra ( E , m 2 , m 3 , · · · ) is a symmetric bilinear form ( ., . ) on E such that for any n , ( m n ( a 1 , · · · , a n ) , a n + 1 ) = ± ( a 1 , m n ( a 2 , · · · , , a n + 1 )) . Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents PBW deformation Calabi-Yau algebra Central regular extension Deformation and extension Relations Main results Definitions Deformation A = T k ( V ) / � R � where R = { r 1 , · · · , r m } . U = T k ( V ) / � P � where P = { r 1 + l 1 , · · · , r m + l m } with | l i | < | r i | . U is called a deformation of A . ∃ A ։ gr U PBW deformation U is called a Poincaré-Birkhoff-Witt (PBW) deformation of A if A → gr U is an isomorphism as graded algebras. Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents PBW deformation Calabi-Yau algebra Central regular extension Deformation and extension Relations Main results Examples (1) universal enveloping algebras U ( g ) (2) Weyl algebras (3) Sridharan enveloping algebras (4) symplectic reflection algebras Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents PBW deformation Calabi-Yau algebra Central regular extension Deformation and extension Relations Main results PBW-theorem for Koszul algebras Let A be a quadratic algebra. Then ∃ α : R − → V , β : R − → k s. t. P = { r − α ( r ) − β ( r ) | r ∈ R } . Theorem (Braverman-Gaitsgory, 1996) Let A be a Koszul algebra. Then U is a PBW deformation if and only if the following are satisfied: (on R ⊗ V � V ⊗ R ); (1) Im ( α ⊗ id − id ⊗ α ) ⊂ R (2) α ( α ⊗ id − id ⊗ α ) = − ( β ⊗ id − id ⊗ β ) ; (3) β ( α ⊗ id − id ⊗ α ) = 0. Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents PBW deformation Calabi-Yau algebra Central regular extension Deformation and extension Relations Main results Remarks (1) Similar results due to Polishchuk-Positselski, Berger-Ginzburg. (2) Universal enveloping algebras U ( g ) α ( xy − yx ) = [ x , y ] , β = 0 PBW-theorem = “ Jacobi identity of Lie algebra g ” Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents PBW deformation Calabi-Yau algebra Central regular extension Deformation and extension Relations Main results PBW deformation of 3-Calabi-Yau algebras Let U be a PBW deformation of a 3-CY algebra A . Theorem (Berger-Taillefer, 2007) U is a 3-CY algebra if the following equivalent conditions are satisfied (1) U is derived from a potential, (2) id ⊗ α − α ⊗ id = 0. Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents PBW deformation Calabi-Yau algebra Central regular extension Deformation and extension Relations Main results Central regular extension Definition Let A and D be two graded algebras. If there is a central regular element t such that A ∼ = D / � t � , then D is called a central regular extension of A . Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents PBW deformation Calabi-Yau algebra Central regular extension Deformation and extension Relations Main results Rees algebra Suppose that U has an increasing filtration FU = { F n U } n ∈ Z . n ∈ Z F n U t n ⊂ U [ t , t − 1 ] Rees algebra Rees ( U ) = � Fact Rees ( U ) is a central regular extension of A . Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents PBW deformation Calabi-Yau algebra Central regular extension Deformation and extension Relations Main results Relations Let U be a deformation of A and H ( U ) be the central extension associated to U ( homogenization ). Theorem (Cassidy-Shelton, 2007) U is a PBW deformation ⇔ H ( U ) is a central regular extension. Theorem If U is a PBW deformation, then Rees ( U ) ∼ = H ( U ) . Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Contents Central Regular Extension Calabi-Yau algebra PBW deformation Deformation and extension Application Main results Central Regular Extension Theorem Suppose that A = T ( V ) / � R � is a Koszul algebra. Let D be a central regular extension of A . Then (1) if D is a CY algebra, then so is A ; (2) if A is a d -CY algebra, then, D is ( d + 1 ) -CY if and only if d − 2 ( − 1 ) i id ⊗ i ⊗ α ⊗ id ⊗ ( d − 2 − i ) ( x ) = 0 , � i = 0 for any x ∈ � V ⊗ i ⊗ R ⊗ V ⊗ ( d − 2 − i ) . Can Zhu PBW deformation of Koszul Calabi-Yau algebras
Recommend
More recommend
