The Calabi-Yau property of Hopf algebras and braided Hopf algebras Xiaolan YU joint work with Yinhuo Zhang Hangzhou Normal University September 16th, 2011 1 / 35
Outline Motivation Braided Hopf algebras Calabi-Yau algebras The Calabi-Yau property of Hopf algebras The Calabi-Yau property of braided Hopf algebras 2 / 35
Outline Motivation Braided Hopf algebras Calabi-Yau algebras The Calabi-Yau property of Hopf algebras The Calabi-Yau property of braided Hopf algebras 3 / 35
❦ ❦ ❦ ❦ ❦ Motivation We fix an algebraically closed field ❦ of characteristic 0. 4 / 35
❦ ❦ ❦ ❦ ❦ Motivation We fix an algebraically closed field ❦ of characteristic 0. Let D be a generic datum of finite Cartan type, and λ a family of linking parameters for D . 4 / 35
❦ ❦ ❦ Motivation We fix an algebraically closed field ❦ of characteristic 0. Let D be a generic datum of finite Cartan type, and λ a family of linking parameters for D . Gr U ( D , λ ) ∼ = U ( D , 0) ∼ = R # ❦ Γ, where Γ is the group formed by group-like elements of U ( D , λ ) and R is a braided Hopf algebra in the category of Yetter-Drinfeld modules over the group algebra ❦ Γ. 4 / 35
Motivation We fix an algebraically closed field ❦ of characteristic 0. Let D be a generic datum of finite Cartan type, and λ a family of linking parameters for D . Gr U ( D , λ ) ∼ = U ( D , 0) ∼ = R # ❦ Γ, where Γ is the group formed by group-like elements of U ( D , λ ) and R is a braided Hopf algebra in the category of Yetter-Drinfeld modules over the group algebra ❦ Γ. Let A be a pointed Hopf algebra. We have Gr A ∼ = R # ❦ Γ, where ❦ Γ is the group algebra of the group formed by the group-like elements of A and R is a braided Hopf algebra in the category of Yetter-Drinfeld modules over ❦ Γ. 4 / 35
Motivation Question 1 Let H be a Hopf algebra, and R a braided Hopf algebra in the category of Yetter-Drinfeld modules over H. What is the relation between the Calabi-Yau property of R and that of R # H? 5 / 35
Outline Motivation Braided Hopf algebras Calabi-Yau algebras The Calabi-Yau property of Hopf algebras The Calabi-Yau property of braided Hopf algebras 6 / 35
Braided Hopf algebras All Hopf algebras mentioned are assumed to be Hopf algebras with bijective antipodes. 7 / 35
Braided Hopf algebras All Hopf algebras mentioned are assumed to be Hopf algebras with bijective antipodes. We consider braided Hopf algebras in the category of Yetter-Drinfeld modules. 7 / 35
Braided Hopf algebras All Hopf algebras mentioned are assumed to be Hopf algebras with bijective antipodes. We consider braided Hopf algebras in the category of Yetter-Drinfeld modules. Let H be a Hopf algebra. A Yetter-Drinfeld module V over H is simultaneously a left H -module and a left H -comodule satisfying the compatibility condition δ ( h · v ) = h 1 v ( − 1) S ( h 3 ) ⊗ h 2 · v (0) , for any v ∈ V , h ∈ H . 7 / 35
Braided Hopf algebras All Hopf algebras mentioned are assumed to be Hopf algebras with bijective antipodes. We consider braided Hopf algebras in the category of Yetter-Drinfeld modules. Let H be a Hopf algebra. A Yetter-Drinfeld module V over H is simultaneously a left H -module and a left H -comodule satisfying the compatibility condition δ ( h · v ) = h 1 v ( − 1) S ( h 3 ) ⊗ h 2 · v (0) , for any v ∈ V , h ∈ H . We denote by H H YD the category of Yetter-Drinfeld modules over H with morphisms given by H -linear and H -colinear maps. 7 / 35
❦ Braided Hopf algebras The category H H YD is a braided tensor category. For any two Yetter-Drinfeld modules M and N , the braiding c M , N : M ⊗ N → N ⊗ M is given by c M , N ( m ⊗ n ) = m ( − 1) · n ⊗ m (0) , for any m ∈ M and n ∈ N . 8 / 35
Braided Hopf algebras The category H H YD is a braided tensor category. For any two Yetter-Drinfeld modules M and N , the braiding c M , N : M ⊗ N → N ⊗ M is given by c M , N ( m ⊗ n ) = m ( − 1) · n ⊗ m (0) , for any m ∈ M and n ∈ N . A braided Hopf algebra in H H YD is a Hopf algebra in the category H H YD . ( R , m , u ) is an algebra in H H YD . ( R , ∆ , ε ) is a coalgebra in H H YD . ∆ : R → R ⊗ R and ε : R → ❦ are morphisms of algebras. The identity is convolution invertible in End( R ). The notation R ⊗ R denotes the Yetter-drinfeld module R ⊗ R in H H YD , whose algebra multiplication is defined as m R ⊗ R := ( m R ⊗ m R )(id ⊗ c ⊗ id). 8 / 35
Braided Hopf algebras Let H be a Hopf algebra and R a braided Hopf algebra in the category H H YD . Then R # H is an ordinary Hopf algebra. 9 / 35
Braided Hopf algebras Let H be a Hopf algebra and R a braided Hopf algebra in the category H H YD . Then R # H is an ordinary Hopf algebra. Let A and H be two Hopf algebras and π : A → H , ι : H → A Hopf algebra homomorphisms such that πι = id H . In this case the algebra of right coinvariants with respect to π R = A co π := { a ∈ A | (id ⊗ π )∆( a ) = a ⊗ 1 } , is a braided Hopf algebra in H H YD . 9 / 35
Outline Motivation Braided Hopf algebras Calabi-Yau algebras The Calabi-Yau property of Hopf algebras The Calabi-Yau property of braided Hopf algebras 10 / 35
Calabi-Yau algebras 11 / 35
Calabi-Yau algebras (Ginzburg) An algebra A is called a Calabi-Yau algebra of dimension d if (i) A is homologically smooth. That is, A has a bounded resolution of finitely generated projective A - A -bimodules. (ii) There are A - A -bimodule isomorphisms � 0 , i � = d ; A e ( A , A e ) ∼ Ext i = i = d . A , In the following, Calabi-Yau will be abbreviated to CY for short. 11 / 35
Rigid Dualizing complexes (Yekutieli) Let A be a Noetherian algebra. Roughly speaking, a complex R ∈ D b ( A e ) is called dualizing if the functor RHom A ( − , R ) : D b fg ( A ) → D b fg ( A op ) is a duality, with adjoint RHom A op ( − , R ). Here D b fg ( A ) is the full triangulated subcategory of the derive category D ( A ) of A consisting of bounded complexes with finitely generated cohomology modules. (Van den Bergh) Let A be a Noetherian algebra. A dualizing complex R over A is called rigid if RHom A e ( A , A R ⊗ R A ) ∼ = R in D ( A e ). Rigid dualizing complexes are unique up to isomorphism. 12 / 35
Outline Motivation Braided Hopf algebras Calabi-Yau algebras The Calabi-Yau property of Hopf algebras The Calabi-Yau property of braided Hopf algebras 13 / 35
Question : Let H be a Hopf algebra, and R a braided Hopf algebra in H H YD . What is the relation between the CY property of R and that of R # H ? If R is a CY algebra, when is R # H a CY algebra? 14 / 35
❦ ❦ Homological determinants Let R be a p -Koszul CY algebra (not necessarily a braided Hopf algebra) and H an involutory CY Hopf algebra. Liu, Wu and Zhu showed that the smash product R # H is CY if and only if the homological determinant of the H -action is trivial. 15 / 35
Homological determinants Let R be a p -Koszul CY algebra (not necessarily a braided Hopf algebra) and H an involutory CY Hopf algebra. Liu, Wu and Zhu showed that the smash product R # H is CY if and only if the homological determinant of the H -action is trivial. Let A be a Noetherian augmented algebra with a fixed augmentation map ε : A → ❦ . A is said to be AS-Gorenstein (Artin-Schelter Gorenstein), if (i) injdim A A = d < ∞ , � 0 , i � = d ; (ii) dim Ext i A ( A ❦ , A A ) = i = d , where injdim stands for 1 , injective dimension. (iii) the right version of the conditions (i) and (ii) hold. 15 / 35
Homological determinants Let R be a p -Koszul CY algebra (not necessarily a braided Hopf algebra) and H an involutory CY Hopf algebra. Liu, Wu and Zhu showed that the smash product R # H is CY if and only if the homological determinant of the H -action is trivial. Let A be a Noetherian augmented algebra with a fixed augmentation map ε : A → ❦ . A is said to be AS-Gorenstein (Artin-Schelter Gorenstein), if (i) injdim A A = d < ∞ , � 0 , i � = d ; (ii) dim Ext i A ( A ❦ , A A ) = i = d , where injdim stands for 1 , injective dimension. (iii) the right version of the conditions (i) and (ii) hold. An AS-Gorenstein algebra A is said to be regular if in addition, the global dimension of A is finite. 15 / 35
Homological determinants (Jørgensen-J. Zhang) Let R be an AS-Gorenstein algebra of injective dimension d . If R is an H -module algebra, then there is a left H -action on Ext d R ( ❦ , R ) induced by the left H -action on R . Let e be a non-zero element in Ext d R ( ❦ , R ). Then there is an algebra homomorphism η : H → ❦ satisfying h · e = η ( h ) e for all h ∈ H . (i) The composite map η S H : H → ❦ is called the homological determinant of the H -action on R , and it is denoted by hdet (or more precisely hdet R ). (ii) The homological determinant hdet R is said to be trivial if hdet R = ε H , where ε H is the counit of the Hopf algebra H . 16 / 35
❦ ❦ Example Let V be an n -dimensional vector space with basis x 1 , x 2 , · · · , x n ( n � 2) and Γ a finite subgroup of GL n ( ❦ ). 17 / 35
❦ ❦ Example Let V be an n -dimensional vector space with basis x 1 , x 2 , · · · , x n ( n � 2) and Γ a finite subgroup of GL n ( ❦ ). The symmetric algebra A = S ( V ) is a Koszul CY algebra of dimension n . 17 / 35
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