Every central simple algebra is Brauer equivalent to a Hopf Schur algebra

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. This shows that the natural generalization of the Schur group for Hopf algebras (which we call the Hopf Schur group) is in fact the entire Brauer group of $k$. If the characteristic of the field is zero, or if the algebra has a Galois splitting field with certain properties, we can take this Hopf algebra to be semisimple. We also show that if $F$ is any finite separable extension of $k$, then $F$ is a quotient of a finite dimensional commutative semisimple and cosemisimple Hopf algebra over $k$.