Endomorphism algebras over weak (α, β)-Yetter–Drinfeld modules

2015 ◽  
Vol 14 (07) ◽  
pp. 1550097
Author(s):  
Ling Liu ◽  
Bingliang Shen
Keyword(s):  
Hopf Algebra ◽  
Weak Hopf Algebra ◽  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Azumaya Algebra ◽  
Finite Dimensional ◽  
Endomorphism Algebras

Let H be a weak Hopf algebra with a bijective antipode, α, β ∈ Aut weak Hopf (H) and M a finite-dimensional weak (α, β)-Yetter–Drinfeld module. Then in this paper we prove that the endomorphism algebras End Hs(M) and End Ht(M) op endowed with certain structures become algebras in H𝒲𝒴𝒟H and we also study the isomorphic relations between different endomorphism algebras. We prove that End Hs(M) endowed with certain structures becomes an H-Azumaya algebra.

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 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 Finitistic dimension of weak Hopf-Galois extensions

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2825-2828
Author(s):  
Xiao-Yan Zhou ◽  
Qiang Li
Keyword(s):  
Hopf Algebra ◽  
Galois Extension ◽  
Duality Theorem ◽  
Weak Hopf Algebra ◽  
Crossed Products ◽  
Finitistic Dimension ◽  
Galois Extensions ◽  
Finite Dimensional ◽  
Dimension Conjecture

Let H be a finite dimensional weak Hopf algebra and A/B be a right faithfully flat weak H-Galois extension. Then in this note, we first show that if H is semisimple, then the finitistic dimension of A is less than or equal to that of B. Furthermore, using duality theorem, we obtain that if H and its dual H* are both semisimple, then the finitistic dimension of A is equal to that of B, which means the finitistic dimension conjecture holds for A if and only if it holds for B. Finally, as applications, we obtain these relations for the weak crossed products and weak smash products.

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.

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 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.

Invariants of 3-manifolds derived from universal invariants of framed links

1995 ◽  
Vol 117 (2) ◽  
pp. 259-273 ◽  
Author(s):  
Tomotada Ohtsuki
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Independent Method ◽  
Topological Invariant ◽  
Finite Dimensional

Reshetikhin and Turaev [10] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group Uq(sl2)) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For Uq(sl2) these two methods give different invariants of 3-manifolds.

scholarly journals Rota-Baxter Leibniz Algebras and Their Constructions

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Liangyun Zhang ◽  
Linhan Li ◽  
Huihui Zheng
Keyword(s):  
Hopf Algebra ◽  
Weak Hopf Algebra ◽  
Associative Algebras ◽  
Leibniz Algebras

In this paper, we introduce the concept of Rota-Baxter Leibniz algebras and explore two characterizations of Rota-Baxter Leibniz algebras. And we construct a number of Rota-Baxter Leibniz algebras from Leibniz algebras and associative algebras and discover some Rota-Baxter Leibniz algebras from augmented algebra, bialgebra, and weak Hopf algebra. In the end, we give all Rota-Baxter operators of weight 0 and -1 on solvable and nilpotent Leibniz algebras of dimension ≤3, respectively.

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