Hopf algebras having a dense big cell

Bichon, Julien; Riche, Simon

doi:10.1090/tran/6335

Cited by 3 publications

(10 citation statements)

References 37 publications

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“…The category of comodules over H(F ) has been studied in [2,6,16,11]. In order to recall the characterization of the cosemisimplicity of H(F ), we need some vocabulary.…”

Section: Universal Cosovereign Hopf Algebrasmentioning

confidence: 99%

“…[12,18,19,45,48]. However, so far, the algebraic properties of the general H(F ) have only been analyzed through the study of its category of comodules [2,6,16,11]. We provide here a full computation of the cohomological dimensions of H(F ), when the matrix F is a generic asymmetry (see Section 2 for the definition of these notions): in that case we show that cd(H(F )) = 3 = cd GS (H(F )).…”

Section: Introductionmentioning

confidence: 99%

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Cohomological dimensions of universal cosovereign Hopf algebras

Bichon

2018

Publicacions Matemàtiques

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We compute the Hochschild and Gerstenhaber-Schack cohomological dimensions of the universal cosovereign Hopf algebras, when the matrix of parameters is a generic asymmetry. Our main tools are considerations on the cohomologies of free product of Hopf algebras, and on the invariance of the cohomological dimensions under graded twisting by a finite abelian group.2010 Mathematics Subject Classification. 16T05, 16E40, 16E10.

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“…The category of comodules over H(F ) has been studied in [2,6,16,11]. In order to recall the characterization of the cosemisimplicity of H(F ), we need some vocabulary.…”

Section: Universal Cosovereign Hopf Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Cohomological dimensions of universal cosovereign Hopf algebras

Bichon

2018

Publicacions Matemàtiques

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“…Suppose that is infinitesimally flat, that is, A/(A + ) n is finitely presented and flat as k-module (or equivalently, finite free) for any n ≥ 1. Then one sees that hy( ) has a structure of a cocommutative Hopf superalgebra such that the restriction For a left -supermodule V , we regard V as a left hy( )-supermodule by letting (1) is the right A-supercomodule structure of V . Suppose that V is finite free.…”

Section: Algebraicmentioning

confidence: 99%

“…are defined over the field k. In this section, we will construct all simple -supermodules, which extends Serganova's construction [27, §9] to arbitrary characteristic. The main idea is based on Brundan and Kleshchev's argument [3, §6], see also Parshall and Wang [24], Bichon and Riche [1] for the non-super situation.…”

Section: Borel-weil Theorem Formentioning

confidence: 99%

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Borel-Weil theorem for algebraic supergroups

Shibata

2020

Journal of Algebra

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We study the structure of an algebraic supergroup and establish the Borel-Weil theorem for to give a systematic construction of all simple supermodules over an arbitrary field. Especially when has a distinguished parabolic super-subgroup, we show that the set of all simple supermodules of is parameterized by the set of all dominant weights for the even part of , prove a super-analogue of the Kempf vanishing theorem, and give a description of Euler characteristics. Date

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Graded twisting of categories and quantum groups by group actions

Bichon¹,

Neshveyev²,

Yamashita³

2016

Ann. inst. Fourier

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Abstract. Given a Hopf algebra A graded by a discrete group together with an action of the same group preserving the grading, we define a new Hopf algebra, which we call the graded twisting of A. If the action is by adjoint maps, this new Hopf algebra is a twist of A by a pseudo-2-cocycle. Analogous construction can be carried out for monoidal categories. As examples we consider graded twistings of the Hopf algebras of nondegenerate bilinear forms, their free products, hyperoctahedral quantum groups and q-deformations of compact semisimple Lie groups. As applications, we show that the analogues of the Kazhdan-Wenzl categories in the general semisimple case cannot be always realized as representation categories of compact quantum groups, and for genuine compact groups, we analyze quantum subgroups of the new twisted compact quantum groups, providing a full description when the twisting group is cyclic of prime order.

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Hopf algebras having a dense big cell

Cited by 3 publications

References 37 publications

Cohomological dimensions of universal cosovereign Hopf algebras

Cohomological dimensions of universal cosovereign Hopf algebras

Borel-Weil theorem for algebraic supergroups

Graded twisting of categories and quantum groups by group actions

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