Transmutation theory and rank for quantum braided groups

1993 ◽  
Vol 113 (1) ◽  
pp. 45-70 ◽  
Author(s):  
Shahn Majid
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Braided Monoidal Category ◽  
Quasitriangular Hopf Algebra ◽  
Cocommutative Hopf Algebra

AbstractLet f: H1 → H2be any pair of quasitriangular Hopf algebras over k with a Hopf algebra map f between them. We construct in this situation a quasitriangular Hopf algebra B(H1, f, H2) in the braided monoidal category of H1-modules. It consists in the same algebra as H2 with a modified comultiplication and has a quasitriangular structure given by the ratio of those of H1 and H2. This transmutation procedure trades a non-cocommutative Hopf algebra in the category of k-modules for a more cocommutative object in a more non-commutative category. As an application, every Hopf algebra containing the group algebra of ℤ2 becomes transmuted to a super-Hopf algebra.

scholarly journals Comodule algebras and 2-cocycles over the (Braided) Drinfeld double

2019 ◽  
Vol 21 (04) ◽  
pp. 1850045 ◽  
Author(s):  
Robert Laugwitz
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Crossed Product ◽  
Braided Monoidal Category ◽  
Drinfeld Double ◽  
Comodule Algebra ◽  
Category Of Modules ◽  
Cohomology Spaces ◽  
Quasitriangular Hopf Algebra

We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces a 2-cocycle over their Drinfeld double. This construction also has an analogue for braided Drinfeld doubles.

scholarly journals The Hopf algebra structure of a crossed product in a braided monoidal category

2002 ◽  
Vol 26 (2) ◽  
pp. 299-311 ◽  
Author(s):  
J. N. Alonso Alvarez ◽  
J. M. Fernández Vilaboa ◽  
R. González Rodriguez
Keyword(s):  
Hopf Algebra ◽  
Monoidal Category ◽  
Crossed Product ◽  
Algebra Structure ◽  
Braided Monoidal Category ◽  
Hopf Algebra Structure

HOPF ALGEBRAS OF DIMENSION 30

2010 ◽  
Vol 09 (01) ◽  
pp. 11-15 ◽  
Author(s):  
DAIJIRO FUKUDA
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

scholarly journals Varieties of Null-Filiform Leibniz Algebras Under the Action of Hopf Algebras

2021 ◽  
Author(s):  
Lucio Centrone ◽  
Chia Zargeh
Keyword(s):  
Hopf Algebra ◽  
Free Algebra ◽  
Hopf Algebras ◽  
Leibniz Algebra ◽  
Module Structure ◽  
Leibniz Algebras ◽  
Finite Dimensional ◽  
Filiform Leibniz Algebra ◽  
Algebra Over A Field ◽  
Cocommutative Hopf Algebra

AbstractLet L be an n-dimensional null-filiform Leibniz algebra over a field K. We consider a finite dimensional cocommutative Hopf algebra or a Taft algebra H and we describe the H-actions on L. Moreover we provide the set of H-identities and the description of the Sn-module structure of the relatively free algebra of L.

scholarly journals INVOLUTORY HOPF ALGEBRAS AND 3-MANIFOLD INVARIANTS

1991 ◽  
Vol 02 (01) ◽  
pp. 41-66 ◽  
Author(s):  
GREG KUPERBERG
Keyword(s):  
Hopf Algebra ◽  
Fundamental Group ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Tensor Products ◽  
State Model ◽  
Formal Expression ◽  
Finite Dimensional ◽  
Heegaard Diagram ◽  
Structure Tensors

We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is a group algebra G, the invariant counts homomorphisms from the fundamental group of the manifold to G. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3-manifold modulo a well-understood equivalence. This raises the possibility of an algorithm which can determine whether two given 3-manifolds are homeomorphic.

On Semisimple Hopf Algebras of Dimension pqn

2014 ◽  
Vol 57 (2) ◽  
pp. 264-269
Author(s):  
Li Dai ◽  
Jingcheng Dong
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Prime Numbers ◽  
Closed Field ◽  
Semisimple Hopf Algebra ◽  
Algebraically Closed Field

AbstractLet p, q be prime numbers with p2 < q, n ∊ ℕ, and H a semisimple Hopf algebra of dimension pqn over an algebraically closed field of characteristic 0. This paper proves that H must possess one of the following two structures: (1) H is semisolvable; (2) H is a Radford biproduct R#kG, where kG is the group algebra of group G of order p and R is a semisimple Yetter–Drinfeld Hopf algebra in of dimension qn.

MULTIPLIER BI- AND HOPF ALGEBRAS

2010 ◽  
Vol 09 (02) ◽  
pp. 275-303 ◽  
Author(s):  
K. JANSSEN ◽  
J. VERCRUYSSE
Keyword(s):  
Hopf Algebra ◽  
Commutative Ring ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Left And Right ◽  
Projective Algebra

We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele [Multiplier Hopf algebras, Trans. Amer. Math. Soc.342(2) (1994) 917–932] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly nonunital, idempotent, nondegenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.

scholarly journals Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups

2020 ◽  
Author(s):  
Nicolás Andruskiewitsch ◽  
Giovanna Carnovale ◽  
Gastón Andrés García
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Mixed Classes ◽  
Type V ◽  
Groups Of Lie Type

Abstract We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from $$\mathbf {PSL}_n(q)$$ PSL n ( q ) collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we prove that the only finite-dimensional pointed Hopf algebra whose group of group-like elements is $$\mathbf {PSp}_{2n}(q)$$ PSp 2 n ( q ) , $$\mathbf {P}{\varvec{\Omega }}^+_{4n}(q)$$ P Ω 4 n + ( q ) , $$\mathbf {P}{\varvec{\Omega }}^-_{4n}(q)$$ P Ω 4 n - ( q ) , $$^3D_4(q)$$ 3 D 4 ( q ) , $$E_7(q)$$ E 7 ( q ) , $$E_8(q)$$ E 8 ( q ) , $$F_4(q)$$ F 4 ( q ) , or $$G_2(q)$$ G 2 ( q ) with q even is the group algebra.

Symmetries and the u-condition in weak monoidal Hom–Yetter–Drinfeld categories

2020 ◽  
pp. 2150194
Author(s):  
Linlin Liu ◽  
Shuanhong Wang
Keyword(s):  
Hopf Algebra ◽  
Monoidal Category ◽  
Drinfeld Modules ◽  
Braided Monoidal Category

The aim of this paper is to introduce and study Yetter–Drinfeld category over a weak monoidal Hom–Hopf algebra [Formula: see text]. We first show that the category [Formula: see text] of Yetter–Drinfeld modules over [Formula: see text] with a bijective antipode is a braided monoidal category. Secondly, we discuss some properties on the symmetries of the category [Formula: see text]. Finally, we prove that the representation category of triangular weak monoidal Hom–Hopf algebra is a symmetric braided monoidal subcategory of [Formula: see text]. Furthermore, a class of weak monoidal Hom–Yetter–Drinfeld modules are constructed by a quasitriangular weak monoidal Hom–Hopf algebra.

Induced Modules of Semisimple Hopf Algebras

2007 ◽  
Vol 14 (04) ◽  
pp. 571-584 ◽  
Author(s):  
Jun Hu ◽  
Yinhuo Zhang
Keyword(s):  
Finite Group ◽  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Direct Sum ◽  
Drinfel’D Double ◽  
Finite Dimensional ◽  
Induced Module ◽  
A Finite Group

Let K be a field. Let H be a finite-dimensional K-Hopf algebra and D(H) be the Drinfel'd double of H. In this paper, we study Radford's induced module Hβ, where β is a group-like element in H∗. Using the commuting pair established in [7], we obtain an analogue of the class equation for [Formula: see text] when H is semisimple and cosemisimple. In case H is a finite group algebra or a factorizable semisimple cosemisimple Hopf algebra, we give an explicit decomposition of each Hβ into a direct sum of simple D(H)-modules.

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