Gelfand-Kirillov dimension of cosemisimple Hopf algebras

Chirvăsitu, Alexandru; Walton, Chelsea; Wang, Xingting

doi:10.1090/proc/14616

Cited by 2 publications

(1 citation statement)

References 21 publications

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“…There are many instances of the situation, see for example [6,42] for a large review of examples, and [27,32,33,37] for more recent ones. (3) As just said, the Hopf algebras in Question 1 are non-isomorphic in general, but worst, some of their ring-theoretical properties, such as Gelfand-Kirillov dimension, can be very different, see [15]. The interest in the question is thus both theoretical, in the investigation of which properties of a Hopf algebra are preserved under monoidal equivalence of the category of comodules, and practical, in the determination of the global dimension of new Hopf algebras from known old ones.…”

Section: Question 1 If a And B Are Hopf Algebras Having Equivalent Li...mentioning

confidence: 99%

On the monoidal invariance of the cohomological dimension of Hopf algebras

Bichon

2022

Comptes Rendus. Mathématique

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We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We provide several positive new answers to this question, under various assumptions of smoothness, cosemisimplicity or finite dimension. We also discuss the comparison between the global dimension and the Gerstenhaber-Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. One of our main tools is the new concept of twisted separable functor.Résumé. Nous étudions la question de l'invariance monoïdale de la dimension globale des algèbres de Hopf : si deux algèbres de Hopf ont des catégories de comodules monoïdalement équivalentes, ont-elles même dimension globale ? Nous apportons plusieurs nouvelles réponses positives dans les cas d'algèbres de Hopf lisses, cosemisimples ou de dimension finie. Nous comparons également la dimension globale et la dimension cohomologique de Gerstenhaber-Schack dans le cas cosemisimple. Un outil important pour obtenir ces divers résultats est la nouvelle notion de foncteur séparable twisté.

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Section: Question 1 If a And B Are Hopf Algebras Having Equivalent Li...mentioning

confidence: 99%

On the monoidal invariance of the cohomological dimension of Hopf algebras

Bichon

2022

Comptes Rendus. Mathématique

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Balmer spectra and Drinfeld centers

Vashaw

2024

Alg. Number Th.

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The Balmer spectrum of a monoidal triangulated category is an important geometric construction which is closely related to the problem of classifying thick tensor ideals. We prove that the forgetful functor from the Drinfeld center of a finite tensor category C to C extends to a monoidal triangulated functor between their corresponding stable categories, and induces a continuous map between their Balmer spectra. We give conditions under which it is injective, surjective, or a homeomorphism. We apply this general theory to prove that Balmer spectra associated to finite-dimensional cosemisimple quasitriangular Hopf algebras (in particular, group algebras in characteristic dividing the order of the group) coincide with the Balmer spectra associated to their Drinfeld doubles, and that the thick ideals of both categories are in bijection. An analogous theorem is proven for certain Benson-Witherspoon smash coproduct Hopf algebras, which are not quasitriangular in general.

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Gelfand-Kirillov dimension of cosemisimple Hopf algebras

Abstract: In this note, we compute the Gelfand-Kirillov dimension of cosemisimple Hopf algebras that arise as deformations of a linearly reductive algebraic group. Our work lies in a purely algebraic setting and generalizes results of Goodearl

Cited by 2 publications

References 21 publications

On the monoidal invariance of the cohomological dimension of Hopf algebras

On the monoidal invariance of the cohomological dimension of Hopf algebras

Balmer spectra and Drinfeld centers

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