Finite generation of the cohomology rings of some pointed Hopf - PowerPoint PPT Presentation

Finite generation of the cohomology rings of some pointed Hopf algebras Van C. Nguyen Hood College, Frederick MD nguyen@hood.edu joint work with Xingting Wang and Sarah Witherspoon Maurice Auslander Distinguished Lectures and International Conference April 25 – 30, 2018 Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Setting & Motivation Let k be a field and H be a finite-dimensional Hopf algebra over k . � Ext n The cohomology of H is H ∗ ( H , k ) := H ( k , k ). n ≥ 0 Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Setting & Motivation Let k be a field and H be a finite-dimensional Hopf algebra over k . � Ext n The cohomology of H is H ∗ ( H , k ) := H ( k , k ). n ≥ 0 Conjecture (Etingof-Ostrik ’04) For any finite-dimensional Hopf algebra H , H ∗ ( H , k ) is finitely generated. Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Setting & Motivation Let k be a field and H be a finite-dimensional Hopf algebra over k . � Ext n The cohomology of H is H ∗ ( H , k ) := H ( k , k ). n ≥ 0 Conjecture (Etingof-Ostrik ’04) For any finite-dimensional Hopf algebra H , H ∗ ( H , k ) is finitely generated. GOAL: Study the finite generation of H ∗ ( H , k ), for some pointed Hopf algebras. Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Finite generation of cohomology ring F.g. Cohomology Conjecture Applications: Partial Results: Quillen’s stratification theorem, finite group algebras over pos. char., modular representation theory, finite group schemes over pos. char., support variety theory, Lusztig’s small quantum group over C , algebraic geometry, Drinfeld double of Frob. kernels of finite alg. groups, commutative algebra, certain pointed Hopf algebras some homological conjectures Remarks: H ∗ ( H , k ) is a graded-commutative ring. H ∗ ( H , k ) is a finitely generated k -algebra ⇐ ⇒ H ∗ ( H , k ) is left (or right) Noetherian ⇒ H ev ( H , k ) is Noetherian and H ∗ ( H , k ) is a f.g. module over H ev ( H , k ). ⇐ Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Finite generation of cohomology ring F.g. Cohomology Conjecture Applications: Partial Results: Quillen’s stratification theorem, finite group algebras over pos. char., modular representation theory, finite group schemes over pos. char., support variety theory, Lusztig’s small quantum group over C , algebraic geometry, Drinfeld double of Frob. kernels of finite alg. groups, commutative algebra, certain pointed Hopf algebras some homological conjectures Remarks: H ∗ ( H , k ) is a graded-commutative ring. H ∗ ( H , k ) is a finitely generated k -algebra ⇐ ⇒ H ∗ ( H , k ) is left (or right) Noetherian ⇒ H ev ( H , k ) is Noetherian and H ∗ ( H , k ) is a f.g. module over H ev ( H , k ). ⇐ Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Finite generation of cohomology ring F.g. Cohomology Conjecture Applications: Partial Results: Quillen’s stratification theorem, finite group algebras over pos. char., modular representation theory, finite group schemes over pos. char., support variety theory, Lusztig’s small quantum group over C , algebraic geometry, Drinfeld double of Frob. kernels of finite alg. groups, commutative algebra, certain pointed Hopf algebras some homological conjectures Remarks: H ∗ ( H , k ) is a graded-commutative ring. H ∗ ( H , k ) is a finitely generated k -algebra ⇐ ⇒ H ∗ ( H , k ) is left (or right) Noetherian ⇒ H ev ( H , k ) is Noetherian and H ∗ ( H , k ) is a f.g. module over H ev ( H , k ). ⇐ Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Finite generation of cohomology ring F.g. Cohomology Conjecture Applications: Partial Results: Quillen’s stratification theorem, finite group algebras over pos. char., modular representation theory, finite group schemes over pos. char., support variety theory, Lusztig’s small quantum group over C , algebraic geometry, Drinfeld double of Frob. kernels of finite alg. groups, commutative algebra, certain pointed Hopf algebras some homological conjectures Remarks: H ∗ ( H , k ) is a graded-commutative ring. H ∗ ( H , k ) is a finitely generated k -algebra ⇐ ⇒ H ∗ ( H , k ) is left (or right) Noetherian ⇒ H ev ( H , k ) is Noetherian and H ∗ ( H , k ) is a f.g. module over H ev ( H , k ). ⇐ Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Preliminary Ingredients Definition A Hopf algebra H over a field k is a k -vector space which is an algebra ( m , u ) ♥ a coalgebra (∆ , ε ) ♥ together with an antipode map S : H → H . Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Preliminary Ingredients Definition A Hopf algebra H over a field k is a k -vector space which is an algebra ( m , u ) ♥ a coalgebra (∆ , ε ) ♥ together with an antipode map S : H → H . Example group algebra k G , polynomial rings k [ x 1 , x 2 , . . . , x n ], universal enveloping algebra U ( g ) of a Lie algebra g , etc. Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Today’s Object: p 3 -dim pointed Hopf algebras Let k = k with char( k ) = p > 2 and H be a p 3 -dim pointed Hopf algebra. The classification of H is done by (V.C. Nguyen-X. Wang ’16). Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Today’s Object: p 3 -dim pointed Hopf algebras Let k = k with char( k ) = p > 2 and H be a p 3 -dim pointed Hopf algebra. The classification of H is done by (V.C. Nguyen-X. Wang ’16). We are interested in the case when H 0 = k C q = � g � , q is divisible by p (more general) , gr H ∼ = B ( V )# k C q , where V = k x ⊕ k y is k C q -module . Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Today’s Object: p 3 -dim pointed Hopf algebras Let k = k with char( k ) = p > 2 and H be a p 3 -dim pointed Hopf algebra. The classification of H is done by (V.C. Nguyen-X. Wang ’16). We are interested in the case when H 0 = k C q = � g � , q is divisible by p (more general) , gr H ∼ = B ( V )# k C q , where V = k x ⊕ k y is k C q -module . B ( V ) is a rank two Nichols algebra of Jordan type over C q . B ( V ) = k � x , y � / ( x p , y p , yx − xy − 1 2 x 2 ) . with action g x = x and g y = x + y . Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Today’s Object: p 3 -dim pointed Hopf algebras Let k = k with char( k ) = p > 2 and H be a p 3 -dim pointed Hopf algebra. The classification of H is done by (V.C. Nguyen-X. Wang ’16). We are interested in the case when H 0 = k C q = � g � , q is divisible by p (more general) , gr H ∼ = B ( V )# k C q , where V = k x ⊕ k y is k C q -module . B ( V ) is a rank two Nichols algebra of Jordan type over C q . B ( V ) = k � x , y � / ( x p , y p , yx − xy − 1 2 x 2 ) . with action g x = x and g y = x + y . B ( V ) lifting bosonization pointed Hopf gr H ∼ = B ( V )# k C q C q Hopf alg in YD algebras H C q Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Today’s Object: two Hopf algebras Let k = k with char( k ) = p > 2 and w = g − 1. Consider the following Hopf algebras over k : The p 2 q -dim bosonization gr H ∼ = B ( V )# k C q is isomorphic to k � w , x , y � 1 subject to w q , x p , y p , yx − xy − 1 2 x 2 , xw − wx , yw − wy − wx − x . Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018

Today’s Object: two Hopf algebras Let k = k with char( k ) = p > 2 and w = g − 1. Consider the following Hopf algebras over k : The p 2 q -dim bosonization gr H ∼ = B ( V )# k C q is isomorphic to k � w , x , y � 1 subject to w q , x p , y p , yx − xy − 1 2 x 2 , xw − wx , yw − wy − wx − x . The 27-dim liftings in p = q = 3 are H = H ( ǫ, µ, τ ) ∼ = k � w , x , y � subject to 2 w 3 = 0 , x 3 = ǫ x , y 3 = − ǫ y 2 − ( µǫ − τ − µ 2 ) y , yw − wy = wx + x − ( µ − ǫ )( w 2 + w ) , xw − wx = ǫ ( w 2 + w ) , yx − xy = − x 2 + ( µ + ǫ ) x + ǫ y − τ ( w 2 − w ) , with ǫ ∈ { 0 , 1 } and τ, µ ∈ k . Nguyen*–Wang–Witherspoon Finite generation of cohomology Auslander Conference - April 29, 2018