Hopf automorphisms and twisted extensions

Montgomery, Susan; Vega, Maria D.; Witherspoon, Sarah

doi:10.1142/s0219498816501036

Cited by 12 publications

(6 citation statements)

References 22 publications

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“…Benson-Witherspoon smash coproduct Hopf algebras. We will now consider the Benson-Witherspoon smash coproducts which were originally studied in [8], with generalizations studied in [24] and [33]; their Balmer spectra and thick ideals were classified in [25]. We recall the general construction of these algebras.…”

Section: Since On the Basis {Cmentioning

confidence: 99%

Balmer spectra and Drinfeld centers

Vashaw¹

2020

Preprint

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The Balmer spectrum of a monoidal triangulated category is an important geometric construction which, in many cases, can be used to obtain a classification of thick tensor ideals. Many interesting examples of monoidal triangulated categories arise as stable categories of finite tensor categories. Given a finite tensor category, its Drinfeld center is a braided finite tensor category constructed via half-braidings; in the case that the original category is the category of representations of a Hopf algebra, then its Drinfeld center is the category of representations of its Drinfeld double. 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. We then prove that this functor induces a continuous map between the Balmer spectra of these two stable categories. In the finite-dimensional Hopf algebra setting, we give conditions under which the image of this continuous map can be realized as a certain intersection of open sets in the Balmer spectrum, and prove conditions under which it is injective, surjective, or a homeomorphism. We apply this general theory to the cases that the finite tensor category C is the category of modules for group algebras of finite groups G in characteristic p dividing the order of G, more generally to cosemisimple Hopf algebras, and for Benson-Witherspoon smash coproduct Hopf algebras HG,L. In the first case, we are able to prove that the Balmer spectrum for the stable module category of the Drinfeld double of any cosemisimple quasitriangular Hopf algebra H is homeomorphic to the Balmer spectrum for the stable module category of H itself, and that thick ideals of the two categories are in bijection.

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Section: Since On the Basis {Cmentioning

confidence: 99%

Balmer spectra and Drinfeld centers

Vashaw¹

2020

Preprint

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“…The importance of the FS-indicators is illustrated in their applications to semisimple Hopf algebras and spherical fusion categories (see for examples [3], [4], [16], [25], [28], [29] and [37]). The arithmetic properties of the values of the FS-indicators have played an integral role in all these applications, and remains the main interest of FS-indicators (see for example [13], [14], [24], [33], and [34]).…”

Section: Then We Havementioning

confidence: 99%

Gauge invariants from the powers of antipodes

Negron

2017

Pacific J. Math.

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Abstract. We prove that the trace of the nth power of the antipode of a Hopf algebra with the Chevalley property is a gauge invariant, for each integer n. As a consequence, the order of the antipode, and its square, are invariant under Drinfeld twists. The invariance of the order of the antipode is closely related to a question of Shimizu on the pivotal covers of finite tensor categories, which we affirmatively answer for representation categories of Hopf algebras with the Chevalley property.

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“…then H has dimension pn 2 and H * does not have the Chevalley property. This situation occurs in the classification of the Hopf algebras of dimension n ≤ 23, for n = pq 2 (12,18,20), n = 8 and n = 16.…”

Section: Remark 26mentioning

confidence: 99%

“…From this result arises the natural question about the existence of Hopf algebra automorphisms satisfying this property on arbitrary Hopf algebras. Knowledge of the existence of a such automorphism also provides information about the group of automorphisms of H , a topic of interest in which there has been several recent contributions (see [20] and references therein).…”

Section: Introductionmentioning

confidence: 99%