scholarly journals Finite Dimensional Hopf Algebras Over the Dual Group Algebra of the Symmetric Group in Three Letters

2011 ◽  
Vol 39 (12) ◽  
pp. 4507-4517 ◽  
Author(s):  
Nicolás Andruskiewitsch ◽  
Cristian Vay
Keyword(s):  
Symmetric Group ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Dual Group ◽  
Finite Dimensional

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 Structure Coefficients of the Hecke Algebra of $(\mathcal{S}_{2n},\mathcal{B}_n)$

10.37236/3592 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Omar Tout
Keyword(s):  
Symmetric Group ◽  
Group Algebra ◽  
Main Tool ◽  
Hecke Algebra ◽  
Natural Analogue ◽  
New Class ◽  
Finite Dimensional ◽  
Finite Dimensional Algebras ◽  
Combinatorial Algebra

The Hecke algebra of the pair $(\mathcal{S}_{2n},\mathcal{B}_n)$, where $\mathcal{B}_n$ is the hyperoctahedral subgroup of $\mathcal{S}_{2n}$, was introduced by James in 1961. It is a natural analogue of the center of the symmetric group algebra. In this paper, we give a polynomiality property of its structure coefficients. Our main tool is a combinatorial algebra which projects onto the Hecke algebra of $(\mathcal{S}_{2n},\mathcal{B}_n)$ for every $n$. To build it, by using partial bijections we introduce and study a new class of finite dimensional algebras.

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.

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.

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.

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 Pointed p3-Dimensional Hopf Algebras in Positive Characteristic

2018 ◽  
Vol 25 (03) ◽  
pp. 399-436
Author(s):  
Van C. Nguyen ◽  
Xingting Wang
Keyword(s):  
Group Algebra ◽  
Hopf Algebras ◽  
Positive Characteristic ◽  
Group Algebras ◽  
Closed Field ◽  
Graded Algebra ◽  
Prime Characteristic ◽  
Characteristic P ◽  
Finite Dimensional

We focus on the classification of pointed p3-dimensional Hopf algebras H over any algebraically closed field of prime characteristic p > 0. In particular, we consider certain cases when the group of grouplike elements is of order p or p2, that is, when H is pointed but is not connected nor a group algebra. The structures of the associated graded algebra gr H are completely described as bosonizations of graded braided Hopf algebras over group algebras, and most of the lifting structures of H are given. This work provides many new examples of (parametrized) non-commutative, non-cocommutative finite-dimensional Hopf algebras in positive characteristic.

scholarly journals EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

2020 ◽  
pp. 1-14
Author(s):  
NICOLÁS ANDRUSKIEWITSCH ◽  
DIRCEU BAGIO ◽  
SARADIA DELLA FLORA ◽  
DAIANA FLÔRES
Keyword(s):  
Hopf Algebras ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Group Algebras ◽  
Finite Abelian Groups ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Characteristic 2 ◽  
Nichols Algebras ◽  
Kirillov Dimension

Abstract We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

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