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 the adjoint representation of a hopf algebra

2020 ◽  
Vol 63 (4) ◽  
pp. 1092-1099
Author(s):  
Stefan Kolb ◽  
Martin Lorenz ◽  
Bach Nguyen ◽  
Ramy Yammine
Keyword(s):  
Group Theory ◽  
Hopf Algebra ◽  
Hopf Algebras ◽  
Adjoint Representation ◽  
Finite Part ◽  
Finitely Generated ◽  
Locally Finite ◽  
Hopf Subalgebra ◽  
Finite Dimensional ◽  
Tensor Functor

AbstractWe consider the adjoint representation of a Hopf algebra $H$ focusing on the locally finite part, $H_{{\textrm ad\,fin}}$, defined as the sum of all finite-dimensional subrepresentations. For virtually cocommutative $H$ (i.e., $H$ is finitely generated as module over a cocommutative Hopf subalgebra), we show that $H_{{\textrm ad\,fin}}$ is a Hopf subalgebra of $H$. This is a consequence of the fact, proved here, that locally finite parts yield a tensor functor on the module category of any virtually pointed Hopf algebra. For general Hopf algebras, $H_{{\textrm ad\,fin}}$ is shown to be a left coideal subalgebra. We also prove a version of Dietzmann's Lemma from group theory for Hopf algebras.

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.

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.

scholarly journals Finite-dimensional Nichols algebras over dual Radford algebras

2020 ◽  
pp. 2140001
Author(s):  
O. Márquez ◽  
D. Bagio ◽  
J. M. J. Giraldi ◽  
G. A. García
Keyword(s):  
Jorge Luis Borges ◽  
Drinfeld Modules ◽  
Indecomposable Modules ◽  
Simple Modules ◽  
Generators And Relations ◽  
Dimension Formula ◽  
Finite Dimensional ◽  
Nichols Algebras ◽  
The Way

For [Formula: see text], let [Formula: see text] be the dual of the Radford algebra of dimension [Formula: see text]. We present new finite-dimensional Nichols algebras arising from the study of simple Yetter–Drinfeld modules over [Formula: see text]. Along the way, we describe the simple objects in [Formula: see text] and their projective envelopes. Then we determine those simple modules that give rise to finite-dimensional Nichols algebras for the case [Formula: see text]. There are 18 possible cases. We present by generators and relations, the corresponding Nichols algebras on five of these eighteen cases. As an application, we characterize finite-dimensional Nichols algebras over indecomposable modules for [Formula: see text] and [Formula: see text], [Formula: see text], which recovers some results of the second and third author in the former case, and of Xiong in the latter. Cualquier destino, por largo y complicado que sea, consta en realidad de un solo momento: el momento en que el hombre sabe para siempre quién es. Jorge Luis Borges

scholarly journals A CLASS OF QUASITRIANGULAR GROUP-COGRADED MULTIPLIER HOPF ALGEBRAS

2018 ◽  
Vol 62 (1) ◽  
pp. 43-57
Author(s):  
TAO YANG ◽  
XUAN ZHOU ◽  
HAIXING ZHU
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Finite Dimensional ◽  
Special Cases ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Classical Construction

AbstractFor a multiplier Hopf algebra pairing 〈A,B〉, we construct a class of group-cograded multiplier Hopf algebras D(A,B), generalizing the classical construction of finite dimensional Hopf algebras introduced by Panaite and Staic Mihai [Isr. J. Math. 158 (2007), 349–365]. Furthermore, if the multiplier Hopf algebra pairing admits a canonical multiplier in M(B⊗A) we show the existence of quasitriangular structure on D(A,B). As an application, some special cases and examples are provided.

scholarly journals The Drinfel’d double versus the Heisenberg double for Hom-Hopf algebras

2016 ◽  
Vol 15 (04) ◽  
pp. 1650059 ◽  
Author(s):  
Daowei Lu ◽  
Shuanhong Wang
Keyword(s):  
Group Theory ◽  
Hopf Algebra ◽  
Quantum Group ◽  
Hopf Algebras ◽  
Duke Math ◽  
Dual Pairs ◽  
Cambridge University ◽  
Drinfel’D Double ◽  
Finite Dimensional ◽  
Heisenberg Double

Let ([Formula: see text], [Formula: see text]) be a finite-dimensional Hom-Hopf algebra. In this paper we mainly construct the Drinfel’d double [Formula: see text] in the setting of Hom-Hopf algebras by two ways, one of which generalizes Majid’s bicrossproduct for Hopf algebras (see [S. Majid, Foundations of Quantum Group Theory (Cambridge University Press, 1995)]) and another one is to introduce the notion of dual pairs of Hom-Hopf algebras. Then we study the relation between the Drinfel’d double [Formula: see text] and Heisenberg double [Formula: see text], generalizing the main result in [J. H. Lu, On the Drinfel’d double and the Heisenberg double of a Hopf algebra, Duke Math. J. 74 (1994) 763–776]. The examples given in the paper are especially, not obtained from the usual Hopf algebras.

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.

On the Hopf algebra dual of an enveloping algebra

1982 ◽  
Vol 91 (2) ◽  
pp. 215-224 ◽  
Author(s):  
Stephen Donkin
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Algebraic Groups ◽  
Field Extension ◽  
Smash Product ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Polycyclic Groups ◽  
Algebra Over A Field

In (1) it is claimed that the main results of that paper have applications to the representation theory of algebraic groups, of polycyclic groups and of Lie algebras. An application to algebraic groups is given in Corollary 6·4 of (1), the applications to polycyclic groups are given in (2), the purpose of this work is to deal with the outstanding case of enveloping algebras. To make use of the results of (1), in this context, we show that the Hopf algebra dual of the enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero is quasi-affine (see § 1·5). This is done by an easy field extension argument and a generalization, to the Hopf algebra dual of the smash product of Hopf algebras, of Proposition 1·6·3 of (2) on the dual of the group algebra of a semidirect product of groups. Since this paper is aimed at those readers interested in enveloping algebras, the Hopf theoretic aspects are dealt with at a fairly leisurely pace.

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).

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