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

Let F be a -field (differential field) of characteristic zero with an algebraically closed field of constants F^, A be a -F-central simple algebra, K be a Picard-Vessiot extension for the -F-module A and G (K|F) be the -Galois group of K over F. We prove that a -field extension L of F, having F^ as its field of constants, splits the -F-central simple algebra A if and only if the -field K embeds in L. We then extend the theory of -F-matrix algebras over a -field F, put forward by Magid & Juan (2008), to arbitrary -F-central simple algebras. In particular, we establish a natural bijective correspondence between the isomorphism classes of -F-central simple algebras of dimension n² over F that are split by the -field K and the classes of inequivalent representations of the algebraic group G (K|F) in PGLₙ (F^). We show that G (K|F) is a reductive or a solvable algebraic group if and only if A has certain kinds of -right ideals.