Finite Splittings of Differential Matrix Algebras

TL;DR

This paper investigates conditions under which differential matrix algebras can be split by finite extensions, extending classical results on central simple algebras to the differential setting and exploring the relation to tensor powers.

Contribution

It provides new examples of differential matrix algebras split by finite extensions and links their splitting properties to the triviality of tensor powers, including divisibility results.

Findings

Finite splitting extensions exist for certain differential matrix algebras.

The triviality of tensor powers relates to the existence of finite splitting extensions.

Orders of differential matrix algebras divide their degrees in some cases.

Abstract

It is well known that central simple algebras are split by suitable finite Galois extensions of their centers. A counterpart of this result was studied by Juan and Magid in the set up of differential matrix algebras, wherein Picard-Vessiot extensions that split matrix differential algebras were constructed. In this article, we exhibit instances of differential matrix algebras which are split by finite extensions. In some cases, we relate the existence of finite splitting extensions of a differential matrix algebra to the triviality of its tensor powers, and show in these cases, that orders of differential matrix algebras divide their degrees.