Triviality Theorems for Yetter-Drinfel'd Hopf Algebras

TL;DR

This paper proves triviality results for certain Yetter-Drinfel'd Hopf algebras, showing under specific conditions they reduce to ordinary Hopf algebras or contain trivial subalgebras, advancing understanding of their structure.

Contribution

It establishes new triviality theorems for finite-dimensional Yetter-Drinfel'd Hopf algebras over finite abelian groups, under semisimplicity and cocommutativity assumptions.

Findings

Commutative semisimple Yetter-Drinfel'd Hopf algebra is trivial if its dimension is coprime to the group order.

Finite-dimensional cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra contains a nontrivial trivial subalgebra.

Results depend on the algebra's dimension and the properties of the base field.

Abstract

Under suitable assumptions on the base field, we prove that a commutative semisimple Yetter-Drinfel'd Hopf algebra over a finite abelian group is trivial, i.e., is an ordinary Hopf algebra, if its dimension is relatively prime to the order of the finite abelian group. Furthermore, we prove that a finite-dimensional cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra contains a trivial Yetter-Drinfel'd Hopf subalgebra of dimension greater than one, at least if the Yetter-Drinfel'd Hopf algebra itself has dimension greater than one.