The Grothendieck group of a Hopf algebra
type: Item
instance of: scholarly article
title: The Grothendieck group of a Hopf algebra
zbMATH Open document ID: 0848.16034
DOI: 10.1016/0022-4049(95)00023-2
author: Warren D. Nichols
author: M. Bettina Richmond
published in: Journal of Pure and Applied Algebra
publication date: 1996-10-24T00:00:00Z
review text: \textit{I. Kaplansky} conjectured [in Bialgebras (Univ. Chicago Lect. Notes 1975)] that if a finite-dimensional cosemisimple Hopf algebra \(H\) over an algebraically closed field contains a simple subcoalgebra of dimension \(n^2\), then \(n\) divides the dimension of \(H\). In this paper, the authors verify this conjecture when \(H\) contains a simple subcoalgebra of dimension 4. The main result is: if \(H\) is a cosimple Hopf algebra over an algebraically closed field and contains a simple subcoalgebra of dimension 4, then \(H\) contains either a Hopf subalgebra of dimension 2, 12 or 60, or a simple subcoalgebra of dimension \(n^2\) for each positive integer \(n\). To achieve this generality, the Grothendieck group of the category of finite-dimensional right \(H\)-comodules is defined and is studied. Some interesting properties are obtained. As an application, the main result is proved. As a corollary, if \(H\) is still finite dimensional, then it has even dimension.
reviewed by: Fang Li
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16E20
zbMATH DE Number: 858648
zbMATH Keywords: finite-dimensional cosemisimple Hopf algebras
zbMATH Keywords: simple subcoalgebras
zbMATH Keywords: cosimple Hopf algebras
zbMATH Keywords: Grothendieck groups
zbMATH Keywords: categories of finite-dimensional right comodules
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00023-2
OpenAlex ID: W1976080513
cites work: Q3922829
cites work: An Isomorphism for the Grothendieck Ring of a Hopf Algebra Order
cites work: Q5250139
cites work: Characters of Hopf algebras
cites work: Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple
cites work: Q4315520
cites work: A Hopf Algebra Freeness Theorem
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Wikidata QID: Q127203107
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description: scientific article; zbMATH DE number 858648