A family of braided cosemisimple Hopf algebras of finite dimension
type: Item
instance of: scholarly article
title: A family of braided cosemisimple Hopf algebras of finite dimension
zbMATH Open document ID: 0939.16024
author: Satoshi Suzuki
published in: Tsukuba Journal of Mathematics
publication date: 2000-02-01T00:00:00Z
review text: The author constructs a family of Hopf algebras as described in the title over an algebraically closed field of characteristic not two. This family \(\mathcal F\) has 4 types called \(A^{ij}_{NL}\), where \(i,j\in\{+1,-1\}\) and \(N\geq 1\), \(L\geq 2\). They are all quotients of a quadratic bialgebra \(B\). It is too technical to describe \(B\) here, but its construction starts with the two-by-two matrix coalgebra. Two other quadratic bialgebras \(B^+\) and \(B^-\) are constructed, which lead to braidings on a subfamily of \(\mathcal F\). \(B\), \(B^+\) and \(B^-\) are all cosemisimple, and their braidings are determined. The main results are: (1) \(A^{ij}_{NL}\) is non-cocommutative involutory cosemisimple of dimension \(4NL\). It is non-commutative if \((L,j)\neq(2,1)\), and is semisimple if \(NL\neq 0\) in the base field. (2) Any non-commutative sub-Hopf algebra of \(A^{ij}_{NL}\) generated by a simple subcoalgebra is in \(\mathcal F\). (3) All braidings of \(A^{ij}_{NL}\) are determined. (4) It is determined when two members of \(\mathcal F\) are isomorphic.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1383394
zbMATH Keywords: cosemisimple Hopf algebras
zbMATH Keywords: matrix coalgebras
zbMATH Keywords: quadratic bialgebras
zbMATH Keywords: braidings
zbMATH Keywords: simple subcoalgebras
reviewed by: Earl J. Taft
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496163467
OpenAlex ID: W349525578
DOI: 10.21099/TKBJM/1496163467
description: scientific article; zbMATH DE number 1383394