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

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