The set of types of \(n\)-dimensional semisimple and cosemisimple Hopf algebras is finite
type: Item
instance of: scholarly article
title: The set of types of \(n\)-dimensional semisimple and cosemisimple Hopf algebras is finite
zbMATH Open document ID: 0882.16029
author: Dragoş Ştefan
published in: Journal of Algebra
publication date: 1998-03-04T00:00:00Z
review text: Let \(k\) be an algebraically closed field. The author shows that for any positive integer \(n\), the set of isomorphism classes of \(n\)-dimensional semisimple cosemisimple Hopf algebras over \(k\) is finite. The method of proof involves studying the affine algebraic set \(B(V)\) of bialgebra structures on an \(n\)-dimensional vector space \(V\). \(\text{GL}(V)\) acts on \(B(V)\) so that the orbits are the isomorphism classes of \(n\)-dimensional bialgebras. For a given \(A\) in \(B(V)\), the authors studies cohomology theory of \(A\) involving the tangent spaces at \(A\) of \(B(V)\) and at the closure of the orbit through \(A\). If the cohomology is zero, then the orbit is open, so that the number of orbits corresponding to bialgebras with trivial cohomology is finite. The main result is obtained by showing that the cohomology of a semisimple cosemisimple Hopf algebra is zero. This involves using the category equivalence of Yetter-Drinfeld modules, 2-sided and 2-cosided Hopf modules and left \(D(A)\)-modules, where \(D(A)\) is the Drinfeld double.
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 1051576
zbMATH Keywords: \(n\)-dimensional semisimple cosemisimple Hopf algebras
zbMATH Keywords: \(n\)-dimensional bialgebras
zbMATH Keywords: bialgebra structures on \(n\)-dimensional vector spaces
zbMATH Keywords: cohomology
zbMATH Keywords: numbers of orbits
zbMATH Keywords: Yetter-Drinfeld modules
zbMATH Keywords: Hopf modules
zbMATH Keywords: Drinfeld double
reviewed by: Earl J. Taft
MaRDI profile type: Publication
full work available at URL: jabr.1996.6991
OpenAlex ID: W2014098065
cites work: Bialgebra cohomology, deformations, and quantum groups.
cites work: Q5250139
cites work: Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple
cites work: Q4274976
cites work: Algebras and their automorphism groups
cites work: The Group of Automorphisms of a Semisimple Hopf Algebra Over a Field of Characteristic 0 is Finite
cites work: Hopf modules and Yetter-Drinfel'd modules
cites work: Q4135685
DOI: 10.1006/JABR.1996.6991
description: scientific article; zbMATH DE number 1051576