Calabi-Yau pointed Hopf algebras of finite Cartan type Yinhuo Zhang University of Hasselt Noncommutative Algebraic Geometry 2011 Shanghai Workshop 1 / 39
Outline Definition of Calabi-Yau algebras Pointed Hopf algebras The Calabi-Yau property of U ( D , λ ) The Calabi-Yau property of Nichols algebras of finite Cartan type joint work with Xiaolan Yu 2 / 39
❦ Notations ❦ is a fixed algebraically closes field with characteristic 0. 3 / 39
Notations ❦ is a fixed algebraically closes field with characteristic 0. All vector spaces and algebras are assumed to be over ❦ . 3 / 39
Notations ❦ is a fixed algebraically closes field with characteristic 0. All vector spaces and algebras are assumed to be over ❦ . All Hopf algebras mentioned are assumed to be Hopf algebras with bijective antipodes. 3 / 39
Notations ❦ is a fixed algebraically closes field with characteristic 0. All vector spaces and algebras are assumed to be over ❦ . All Hopf algebras mentioned are assumed to be Hopf algebras with bijective antipodes. Given an algebra A , we write A op for the opposite algebra of A and A e for the enveloping algebra A ⊗ A op of A . 3 / 39
Notations Let A be an algebra and M an A - A -bimodule. For algebra automorphisms σ and τ , the bimodule σ M τ is defined by a · m · b := σ ( a ) m τ ( b ) . 4 / 39
Notations Let A be an algebra and M an A - A -bimodule. For algebra automorphisms σ and τ , the bimodule σ M τ is defined by a · m · b := σ ( a ) m τ ( b ) . When σ or τ is the identity map, we shall simply omit it. For example, 1 M τ is denoted by M τ . 4 / 39
Notations Let A be an algebra and M an A - A -bimodule. For algebra automorphisms σ and τ , the bimodule σ M τ is defined by a · m · b := σ ( a ) m τ ( b ) . When σ or τ is the identity map, we shall simply omit it. For example, 1 M τ is denoted by M τ . We have A τ ∼ = τ − 1 A as A - A -bimodules. 4 / 39
Definition of Calabi-Yau algebras 5 / 39
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 = A , i = d . In the following, Calabi-Yau will be abbreviated to CY for short. 6 / 39
Examples of Calabi-Yau algebras The polynomial algebra ❦ [ x 1 , · · · , x n ] with n variables is a CY algebra of dimension n . (Berger) Any Sridharan enveloping algebra of an n -dimensional abelian Lie algebra is a CY algebra of dimension n . (Bocklandt) Let A be the algebra ❦ � x 0 , x 1 , x 2 , x 3 � / I , where the ideal I is generated by the following relations x 0 x 1 − x 1 x 0 − α ( x 2 x 3 + x 3 x 2 ) , x 0 x 1 + x 1 x 0 − ( x 2 x 3 − x 3 x 2 ), x 0 x 2 − x 2 x 0 − β ( x 3 x 1 + x 1 x 3 ) , x 0 x 2 + x 2 x 0 − ( x 3 x 1 − x 1 x 3 ), x 0 x 3 − x 3 x 0 − γ ( x 1 x 2 + x 2 x 1 ) , x 0 x 3 + x 3 x 0 − ( x 1 x 2 − x 2 x 1 ), α + β + γ + αβγ = 0 and ( α, β, γ ) / ∈ { ( α, − 1 , 1) , (1 , β, − 1) , ( − 1 , 1 , γ ) } . The algebra A is a 4-dimensional Sklyanian algebra and a CY algebra of dimension 4. 7 / 39
Dualizing complexes (Yekutieli) Let A be a Noetherian algebra. Roughly speaking, a complex R ∈ D b ( A e ) is called a dualizing complex 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. 8 / 39
Dualizing complexes (Yekutieli) Let A be a Noetherian algebra. Roughly speaking, a complex R ∈ D b ( A e ) is called a dualizing complex 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. Example: The complex R = A is a dualizing complex over A if and only if A is a Gorenstein ring (i.e. A has finite injective dimension as left and right module over itself). 8 / 39
Rigid Dualizing complexes Dualizing complexes are not unique. 9 / 39
Rigid Dualizing complexes Dualizing complexes are not unique. (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 ). 9 / 39
Rigid Dualizing complexes Theorem 1 (Van den Bergh, Brown-Zhang) Let A be a Noetherian algebra. Then the following two conditions are equivalent: (1) A has a rigid dualizing complex R = A ψ [ s ] , where ψ is an algebra automorphism and s ∈ Z . (2) A has finite injective dimension d and there is an algebra automorphism φ such that � 0 , i � = d ; A e ( A , A e ) ∼ Ext i = A φ i = d as A-A-bimodules. In this case, φ = ψ − 1 and s = d. 10 / 39
Rigid Dualizing complexes Theorem 1 (Van den Bergh, Brown-Zhang) Let A be a Noetherian algebra. Then the following two conditions are equivalent: (1) A has a rigid dualizing complex R = A ψ [ s ] , where ψ is an algebra automorphism and s ∈ Z . (2) A has finite injective dimension d and there is an algebra automorphism φ such that � 0 , i � = d ; A e ( A , A e ) ∼ Ext i = A φ i = d as A-A-bimodules. In this case, φ = ψ − 1 and s = d. Corollary 2 A Noetherian algebra A is CY of dimension d if and only if A is homologically smooth and has a rigid dualizing complex A [ d ] . 10 / 39
Artin-Schelter Gorenstein algebras Let A be a Noetherian augmented algebra with a fixed augmentation map ε : A → ❦ . The algebra A is said to be AS-Gorenstein, if (i) injdim A A = d < ∞ , where injdim stands for injective dimension. � 0 , i � = d ; (ii) dim Ext i A ( A ❦ , A A ) = 1 , i = d . (iii) the right version of the conditions (i) and (ii) hold. 11 / 39
Artin-Schelter Gorenstein algebras Let A be a Noetherian augmented algebra with a fixed augmentation map ε : A → ❦ . The algebra A is said to be AS-Gorenstein, if (i) injdim A A = d < ∞ , where injdim stands for injective dimension. � 0 , i � = d ; (ii) dim Ext i A ( A ❦ , A A ) = 1 , i = d . (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. 11 / 39
Examples A finite dimensional Hopf algebra H is AS-Gorenstein. Since a finite dimensional Hopf algebra H is Frobenius, injdim H H =injdim H H = 0, dim Hom H ( H ❦ , H H ) = dim Hom H ( ❦ H , H H ) = 1. 12 / 39
Examples A finite dimensional Hopf algebra H is AS-Gorenstein. Since a finite dimensional Hopf algebra H is Frobenius, injdim H H =injdim H H = 0, dim Hom H ( H ❦ , H H ) = dim Hom H ( ❦ H , H H ) = 1. Let g be a Lie algebra of dimension d . Then the universal enveloping algebra U ( g ) is AS-regular and of global dimension d . 12 / 39
Examples A finite dimensional Hopf algebra H is AS-Gorenstein. Since a finite dimensional Hopf algebra H is Frobenius, injdim H H =injdim H H = 0, dim Hom H ( H ❦ , H H ) = dim Hom H ( ❦ H , H H ) = 1. Let g be a Lie algebra of dimension d . Then the universal enveloping algebra U ( g ) is AS-regular and of global dimension d . (Wu-Zhang) Noetherian affine PI Hopf algebras are AS-Gorenstein. 12 / 39
❦ Homological integrals of AS-Gorenstein algebras (Lu, Wu and Zhang) Let A be an AS-Gorenstein algebra of injective dimension d . Then Ext d A ( A ❦ , A A ) is a 1-dimensional right A -module. Any non-zero element in Ext d A ( A ❦ , A A ) is called a left homological � l A for Ext d integral of A . We write A ( A ❦ , A A ). 13 / 39
Homological integrals of AS-Gorenstein algebras (Lu, Wu and Zhang) Let A be an AS-Gorenstein algebra of injective dimension d . Then Ext d A ( A ❦ , A A ) is a 1-dimensional right A -module. Any non-zero element in Ext d A ( A ❦ , A A ) is called a left homological � l A for Ext d integral of A . We write A ( A ❦ , A A ). Similarly, we have the right homological integrals. � r Ext d A ( ❦ A , A A ) is denoted by A . 13 / 39
Homological integrals of AS-Gorenstein algebras (Lu, Wu and Zhang) Let A be an AS-Gorenstein algebra of injective dimension d . Then Ext d A ( A ❦ , A A ) is a 1-dimensional right A -module. Any non-zero element in Ext d A ( A ❦ , A A ) is called a left homological � l A for Ext d integral of A . We write A ( A ❦ , A A ). Similarly, we have the right homological integrals. � r Ext d A ( ❦ A , A A ) is denoted by A . � l � r A and A are called left and right homological integral modules of A respectively. 13 / 39
❦ Examples If H is a finite dimensional Hopf algebra, then the homological � l and � r are just the classical integrals of H . integrals 14 / 39
❦ Examples If H is a finite dimensional Hopf algebra, then the homological � l and � r are just the classical integrals of H . integrals Let g be the 2-dimensional Lie algebra generated by x , y such that [ x , y ] = x . Let H = U ( g ). � l is given by: Then the right H -action on � l , for 0 � = t ∈ t · x = 0 and t · y = − t . 14 / 39
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