Symmetry and pseudosymmetry of v-Yetter–Drinfeld categories for Hom–Hopf algebras

2017 ◽  
Vol 14 (09) ◽  
pp. 1750129 ◽  
Author(s):  
Xiao-Li Fang ◽  
Tae-Hwa Kim ◽  
Xiao-Hui Zhang
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Necessary Conditions ◽  
Tensor Category ◽  
Baxter Equation ◽  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Sufficient And Necessary Conditions

The purpose of this paper is to introduce the category of [Formula: see text]-Yetter–Drinfeld modules ([Formula: see text]) over a Hom–Hopf algebra. We first prove that every category of [Formula: see text]-Yetter–Drinfeld modules over a Hom–Hopf algebra with a bijective antipode [Formula: see text] is a braided tensor category and that every [Formula: see text]-Yetter–Drinfeld module can provide the solution of the Hom–Yang–Baxter equation. Secondly, we find sufficient and necessary conditions for [Formula: see text] to be symmetric and pseudosymmetric, respectively. Finally, we construct examples of [Formula: see text]-Yetter–Drinfeld modules by a quasitriangular Hom–Hopf algebra and study their relationship.

scholarly journals Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras

2021 ◽  
Vol 19 (1) ◽  
pp. 1231-1244
Author(s):  
Wei Liu ◽  
Xiaoli Fang
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Tensor Category ◽  
Necessary Condition ◽  
Baxter Equation ◽  
General Category ◽  
Drinfeld Modules ◽  
Sufficient And Necessary Condition

Abstract In this paper, we investigate a more general category of Θ \Theta -Yetter-Drinfeld modules ( Θ ∈ Aut H ( H ) \Theta \in {\rm{Aut}}\hspace{0.33em}H\left(H) ) over a Hom-Hopf algebra, which unifies two different definitions of Hom-Yetter-Drinfeld category introduced by Makhlouf and Panaite, Li and Ma, respectively. We show that the category of Θ \Theta -Yetter-Drinfeld modules with a bijective antipode S S is a braided tensor category and some solutions of the Hom-Yang-Baxter equation and the Yang-Baxter equation can be constructed by this category. Also by the method of symmetric pairs, we prove that if a Θ \Theta -Yetter-Drinfeld category over a Hom-Hopf algebra H H is symmetric, then H H is trivial. Finally, we find a sufficient and necessary condition for a Θ \Theta -Yetter-Drinfeld category to be pseudosymmetric.

scholarly journals Two-cocycles and cleft extensions in left braided categories

2020 ◽  
pp. 2140013
Author(s):  
István Heckenberger ◽  
Kevin Wolf
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Cleft Extensions ◽  
Braided Categories

We define two-cocycles and cleft extensions in categories that are not necessarily braided, but where specific objects braid from one direction, like for a Hopf algebra [Formula: see text] a Yetter–Drinfeld module braids from the left with [Formula: see text]-modules. We will generalize classical results to this context and give some application for the categories of Yetter–Drinfeld modules and [Formula: see text]-modules. In particular, we will describe liftings of coradically graded Hopf algebras in the category of Yetter–Drinfeld modules with these techniques.

scholarly journals Handlebody-knot invariants derived from unimodular Hopf algebras

2014 ◽  
Vol 23 (07) ◽  
pp. 1460001 ◽  
Author(s):  
Atsushi Ishii ◽  
Akira Masuoka
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Tensor Category ◽  
Symmetric Algebra ◽  
Computational Results ◽  
Drinfeld Modules ◽  
Generators And Relations ◽  
Knot Invariants ◽  
Finite Dimensional ◽  
Tensor Functor

To systematically construct invariants of handlebody-links, we give a new presentation of the braided tensor category [Formula: see text] of handlebody-tangles by generators and relations, and prove that given what we call a quantum-commutative quantum-symmetric algebra A in an arbitrary braided tensor category [Formula: see text], there arises a braided tensor functor [Formula: see text], which gives rise to a desired invariant. Some properties of the invariants and explicit computational results are shown especially when A is a finite-dimensional unimodular Hopf algebra, which is naturally regarded as a quantum-commutative quantum-symmetric algebra in the braided tensor category [Formula: see text] of Yetter–Drinfeld modules.

scholarly journals ON NICHOLS ALGEBRAS OVER PGL(2, q) AND PSL(2, q)

2010 ◽  
Vol 09 (02) ◽  
pp. 195-208 ◽  
Author(s):  
SEBASTIÁN FREYRE ◽  
MATÍAS GRAÑA ◽  
LEANDRO VENDRAMIN
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Necessary Conditions ◽  
Drinfeld Modules ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Mathieu Groups ◽  
Nichols Algebras

We compute necessary conditions on Yetter–Drinfeld modules over the groups PGL(2, q) = PGL(2, 𝔽q) and PSL(2, q) = PSL(2, 𝔽q) to generate finite-dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with group of group-likes isomorphic to one of these groups. As a by-product of the techniques developed in this work, we prove that any finite-dimensional pointed Hopf algebra over the Mathieu groups M20 or M21 = PSL(3, 4) is the group algebra.

Dual Quasi-Hopf Algebras and Antipodes

2006 ◽  
Vol 13 (01) ◽  
pp. 111-118 ◽  
Author(s):  
Jinqi Li
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Tensor Category

Let H be a coalgebra. In this paper, we show that H is a dual quasi-bialgebra if and only if the category [Formula: see text] of comodules is a tensor category; and H is a braided dual quasi-bialgebra if and only if [Formula: see text] is a braided tensor category. If H is a braided dual quasi-Hopf algebra, it is shown that the antipode of H is inner, i.e., s2(h) = ∑ τ (h1)h2τ-1(h3).

The crossed structure of Hopf bimodules

2018 ◽  
Vol 17 (09) ◽  
pp. 1850172 ◽  
Author(s):  
Haixing Zhu
Keyword(s):  
Hopf Algebra ◽  
Tensor Category ◽  
Drinfeld Modules

Let [Formula: see text] be a Hopf algebra with bijective antipode. We first define some generalized Hopf bimodules. Next, we show that these Hopf bimodules form a new tensor category with a crossed structure, which is equivalent to the category of some generalized Yetter–Drinfeld modules introduced by Panaite and Staic. Finally, based on this equivalence, we verify that the category of Hopf bimodules admits the structure of a braided [Formula: see text]-category in the sense of Turaev.

Quantum Cocommutative Coalgebras in $^{H}_{H}\mathcal{YD}$ and the Solutions of the Quasi-Yang-Baxter Equation

2013 ◽  
Vol 20 (02) ◽  
pp. 227-242
Author(s):  
Xiaoli Fang ◽  
Jinqi Li
Keyword(s):  
Hopf Algebra ◽  
Baxter Equation ◽  
Drinfeld Modules

In this paper, we construct a quantum cocommutative coalgebra in the category of Yetter-Drinfeld modules over a coquasi-Hopf algebra, and give some solutions of the quasi-Yang-Baxter equation.

scholarly journals ON POINTED HOPF ALGEBRAS ASSOCIATED WITH THE MATHIEU SIMPLE GROUPS

2009 ◽  
Vol 08 (05) ◽  
pp. 633-672 ◽  
Author(s):  
FERNANDO FANTINO
Keyword(s):  
Irreducible Representation ◽  
Hopf Algebra ◽  
Conjugacy Class ◽  
Simple Group ◽  
Hopf Algebras ◽  
Simple Groups ◽  
Drinfeld Module ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras

Let G be a Mathieu simple group, s ∈ G, [Formula: see text] the conjugacy class of s and ρ an irreducible representation of the centralizer of s. We prove that either the Nichols algebra [Formula: see text] is infinite-dimensional or the braiding of the Yetter–Drinfeld module [Formula: see text] is negative. We also show that if G = M22 or M24, then the group algebra of G is the only (up to isomorphisms) finite-dimensional complex pointed Hopf algebra with group-likes isomorphic to G.

scholarly journals On Cyclic Cohomology of ×-Hopf algebras

2014 ◽  
Vol 13 (1) ◽  
pp. 147-170
Author(s):  
Mohammad Hassanzadeh
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Cyclic Cohomology ◽  
Drinfeld Modules ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Algebraic Tori ◽  
Characteristic Map ◽  
Hopf Algebroids

AbstractIn this paper we study the cyclic cohomology of certain ×-Hopf algebras: universal enveloping algebras, quantum algebraic tori, the Connes-Moscovici ×-Hopf algebroids and the Kadison bialgebroids. Introducing their stable anti Yetter-Drinfeld modules and cocyclic modules, we compute their cyclic cohomology. Furthermore, we provide a pairing for the cyclic cohomology of ×-Hopf algebras which generalizes the Connes-Moscovici characteristic map to ×-Hopf algebras. This enables us to transfer the ×-Hopf algebra cyclic cocycles to algebra cyclic cocycles.

scholarly journals Hopf algebras arising from partial (co)actions

2020 ◽  
pp. 2140006
Author(s):  
Danielle Azevedo ◽  
Grasiela Martini ◽  
Antonio Paques ◽  
Leonardo Silva
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Product Formula ◽  
Matched Pair ◽  
Matched Pairs ◽  
Partial Action

In this paper, extending the idea presented by Takeuchi in [M. Takeuchi, Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra 9 (1981) 841–882.] and more generally by Majid in [S. Majid, Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra 130(1) (1990) 17–64.], we introduce the notion of partial matched pair [Formula: see text] involving the concepts of partial action and partial coaction between two bialgebras [Formula: see text] and [Formula: see text]. Furthermore, we present sufficient conditions for the corresponding bismash product [Formula: see text] to generate a new Hopf algebra and, as illustration, a family of examples is provided. Moreover, under some hypotheses such sufficient conditions are also necessary conditions.

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