On the Hopf-Schur group of a field

TL;DR

This paper explores the Hopf-Schur group of a field, showing it can be significantly larger than the Schur group by representing tensor products of cyclic algebras as quotients of Hopf algebras.

Contribution

It demonstrates that twisted group algebras and abelian extensions are quotients of cocommutative and commutative Hopf algebras, respectively, expanding understanding of the Hopf-Schur group's scope.

Findings

Twisted group algebras are quotients of cocommutative Hopf algebras.

Abelian extensions are quotients of commutative Hopf algebras.

Tensor products of cyclic algebras are quotients of Hopf algebras.

Abstract

Let k be any field. We consider the Hopf-Schur group of k, defined as the subgroup of the Brauer group of k consisting of classes that may be represented by homomorphic images of Hopf algebras over k. We show here that twisted group algebras and abelian extensions of k are quotients of cocommutative and commutative Hopf algebras over k, respectively. As a consequence we prove that any tensor product of cyclic algebras over k is a quotient of a Hopf algebra over k, revealing so that the Hopf-Schur group can be much larger than the Schur group of k.