Differential Galois Groups of Differential Central Simple Algebras and their Projective Representations

Abstract

Let $F$ be a $\delta-$field (differential field) of characteristic zero with an algebraically closed field of constants $F^\delta$, $A$ be a $\delta-F-$central simple algebra, $K$ be a Picard-Vessiot extension for the $\delta-F-$module $A$ and $\mathscr G(K|F)$ be the $\delta-$Galois group of $K$ over $F.$ We prove that a $\delta-$field extension $L$ of $F,$ having $F^\delta$ as its field of constants, splits the $\delta-F-$central simple algebra $A$ if and only if the $\delta-$field $K$ embeds in $L.$ We then extend the theory of $\delta-F-$matrix algebras over a $\delta-$field $F,$ put forward by Magid & Juan (2008), to arbitrary $\delta-F-$central simple algebras. In particular, we establish a natural bijective correspondence between the isomorphism classes of $\delta-F-$central simple algebras of dimension $n^2$ over $F$ that are split by the $\delta-$field $K$ and the classes of inequivalent representations of the algebraic group $\mathscr G(K|F)$ in $\mathrm{PGL}_n(F^\delta).$ We show that $\mathscr G(K|F)$ is a reductive or a solvable algebraic group if and only if $A$ has certain kinds of $\delta-$right ideals.