Cohomology rings of finite-dimensional pointed Hopf algebras over abelian groups

Andruskiewitsch, Nicolás; Angiono, Iván; Pevtsova, Julia; Witherspoon, Sarah

doi:10.1007/s40687-021-00287-y

Cited by 4 publications

(3 citation statements)

References 44 publications

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“…In the subsequent corollary, it is shown that, for a braided Hopf algebra H with bijective antipode, H is isomorphic to its H op,cop -version (see also [8]). For more information on braided Hopf algebras, we refer the reader to [7,16,25,30]. Proposition 3.1.…”

Section: Braided Products Coproducts and (Co)actionsmentioning

confidence: 99%

Duality for infinite-dimensional braided bialgebras and their (co)modules

Wagner

2023

Revista De La Unión Matemática Argentina

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The paper presents a detailed description of duality for braided algebras, coalgebras, bialgebras, Hopf algebras, and their modules and comodules in the infinite setting. Assuming that the dual objects exist, it is shown how a given braiding induces compatible braidings for the dual objects, and how actions (resp., coactions) can be turned into coactions (resp., actions) of the dual coalgebra (resp., algebra), with an emphasis on braided bialgebras and their braided (co)module algebras. Examples are provided by considering these structures in a graded (or filtered) setting, where each degree is finite-dimensional.

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Section: Braided Products Coproducts and (Co)actionsmentioning

confidence: 99%

Duality for infinite-dimensional braided bialgebras and their (co)modules

Wagner

2023

Revista De La Unión Matemática Argentina

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“…We also have a natural isomorphism Θ M : M * → * M, which can be described as follows, see [14] for details.…”

Section: The Canonical Isomorphisms In H H Y D F Dmentioning

confidence: 99%

The Canonical Isomorphisms in the Yetter-Drinfeld Categories for Dual Quasi-Hopf Algebras

Yan

Zhao

2022

Symmetry

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Hopf algebras, as a crucial generalization of groups, have a very symmetric structure and have been playing a prominent role in mathematical physics. In this paper, let H be a dual quasi-Hopf algebra which is a more general Hopf algebra structure. A. Balan firstly introduced the notion of right-right Yetter-Drinfeld modules over H and studied its Galois extension. As a continuation, the aim of this paper is to introduce more properties of Yetter-Drinfeld modules. First, we will describe all the other three kinds of Yetter-Drinfeld modules over H, and the monoidal and braided structure of the categories of Yetter-Drinfeld modules explicitly. Furthermore, we will prove that the category HHYDfd of finite dimensional left-left Yetter-Drinfeld modules is rigid. Then we will compute explicitly the canonical isomorphisms in HHYDfd. Finally, as an application, we will rewrite the isomorphisms in the case of coquasitriangular dual quasi-Hopf algebra.

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“…Thus, the algebra structure of DD m is completely determined by the equality ı t st 1 t D t ı s for all s; t 2 D m :In this case, the R-matrix reads R D P t 2D m ı t ˝t 2 DD m ˝DD m .4.1. The weights of DD mIt is well known that the simple modules over the Drinfeld double of a group algebra are classified by the conjugacy classes of the group and irreducible representations of their centralizers, cf [4]. and references therein.…”

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confidence: 99%

Simple modules of small quantum groups at dihedral groups

García,

Vay

2024

Doc. Math.

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Based on previous results on the classification of finite-dimensional Nichols algebras over dihedral groups and the characterization of simple modules of Drinfeld doubles, we compute the irreducible characters of the Drinfeld doubles of bosonizations of finite-dimensional Nichols algebras over the dihedral groups D 4t with t 3. To this end, we develop new techniques that can be applied to Nichols algebras over any Hopf algebra. Namely, we explain how to construct recursively irreducible representations when the Nichols algebra is generated by a decomposable module, and show that the highest-weight of minimum degree in a Verma module determines its socle. We also prove that tensoring a simple module by a rigid simple module gives a semisimple module.

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Cohomology rings of finite-dimensional pointed Hopf algebras over abelian groups

Cited by 4 publications

References 44 publications

Duality for infinite-dimensional braided bialgebras and their (co)modules

Duality for infinite-dimensional braided bialgebras and their (co)modules

The Canonical Isomorphisms in the Yetter-Drinfeld Categories for Dual Quasi-Hopf Algebras

Simple modules of small quantum groups at dihedral groups

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