Abstract:

Let k be an algebraically closed base field of arbitrary characteristic.  In this paper, we study actions of a connected solvable linear algebraic group G on a central simple algebra Q.  The main result is the following: Q can be split G-equivariantly by a finite-dimensional splitting field, provided that G acts "algebraically'', i.e., provided that Q contains a G-stable order on which the action is rational.  As an application, it is shown that rational torus actions on prime PI-algebras are induced by actions on commutative domains.