Every central simple algebra is Hopf Schur

TL;DR

This paper proves that every central simple algebra over a field is Brauer equivalent to a quotient of a finite dimensional Hopf algebra, with conditions for semisimplicity, and explores implications for algebraic closure assumptions.

Contribution

It establishes that all central simple algebras are Hopf Schur, extending the connection between central simple algebras and Hopf algebras, and discusses conditions for semisimplicity.

Findings

Every central simple algebra is Brauer equivalent to a Hopf algebra quotient.

If characteristic is zero or Galois splitting degree is prime to characteristic, the Hopf algebra can be semisimple.

Finite extensions of the base field are also quotients of finite dimensional Hopf algebras.

Abstract

We show that every central simple algebra A over a field k is Brauer equivalent to a quotient of a finite dimensional Hopf algebra over the same field (that is- A is Hopf Schur). If the characteristic of the field is zero, or if the algebra has a Galois splitting field of degree prime to the characteristic of k, we can take this Hopf algebra to be semisimple. We also show that if F is any finite extension of k, then F is a quotient of a finite dimensional Hopf algebra over k. We use it in order to show why the algebric closeness assumption is necessary in a weak form of Kaplansky's tenth conjecture, due to Stefan