The set of types of \(n\)-dimensional semisimple and cosemisimple Hopf algebras is finite
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.
\(n\)-dimensional semisimple cosemisimple Hopf algebras
bialgebra structures on \(n\)-dimensional vector spaces
scientific article; zbMATH DE number 1051576