Grothendieck rings of universal quantum groups.
type: Item
instance of: scholarly article
title: Grothendieck rings of universal quantum groups.
zbMATH Open document ID: 1261.16029
author: Alexandru Chirvasitu
published in: Journal of Algebra
publication date: 2012-05-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1006.3464
reviewed by: Sorin Dascalescu
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 16T15
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 6037567
zbMATH Keywords: free Hopf algebras
zbMATH Keywords: matrix coalgebras
zbMATH Keywords: cosovereign Hopf algebras
zbMATH Keywords: corepresentations
zbMATH Keywords: Grothendieck rings of comodules
MaRDI profile type: Publication
OpenAlex ID: W2963038304
arXiv ID: 1006.3464
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DOI: 10.1016/J.JALGEBRA.2011.09.020
review text: Let \(k\) be an algebraically closed field. The author considers Hopf algebras of the following types: 1) Images of matrix coalgebras \(M_n(k)^*\) through the left adjoint of the forgetful functor from Hopf algebras to coalgebras; these were introduced by \textit{M. Takeuchi} [J. Math. Soc. Japan 23, 561-582 (1971; Zbl 0217.05902)]; 2) Images of matrix coalgebras \(M_n(k)^*\) through the left adjoint of the forgetful functor from Hopf algebras with bijective antipode to coalgebras; these were introduced by \textit{P. Schauenburg} [Lect. Notes Pure Appl. Math. 210, 331-344 (2000; Zbl 0977.16019)]; 3) If \(d\) is a positive integer and \(F\) is an invertible \(n\times n\) matrix over \(k\), a new Hopf algebra \(H_d(F)\) is constructed as a free Hopf algebra generated by a matrix coalgebra \(u=(u_{ij})\) such that the \(2d\)-th power of the antipode acts on \(u\) as conjugation by \(F\) (this generalizes the universal cosovereign Hopf algebras defined by \textit{J. Bichon} [J. Pure Appl. Algebra 157, No. 2-3, 121-133 (2001; Zbl 0976.16027)]).NEWLINENEWLINE The Grothendieck ring of finite dimensional comodules over a Hopf algebra of one of these 3 types is determined, and it is shown to be isomorphic to the non-commutative polynomials over a certain set. An explicit multiplication table is given for these Grothendieck rings.
description: scientific article; zbMATH DE number 6037567