Z. Reichstein, N. Vonessen, Group actions on central simple algebras: a geometric approach, J. Algebra 304 (2006), 1160--1192.
We study algebraic group actions on central simple algebras.
Suppose and algebraic group G acts on a central simple algebra A
of degree n. We are interested in questions of the following type:
(a) Do the G-fixed elements form a central simple subalgebra of A of degree n? (b) Does A have a G-invariant maximal subfield? (c) Does A have a splitting field with a G-action, extending the G-action on the center of A?
Somewhat surprisingly, we find that under mild assumptions on A and the actions, one can answer these questions by using techniques from birational invariant theory (i.e., the study of group actions on algebraic varieties, up to an equivariant birational isomorphism). In fact, group actions on central simple algebras turn out to be related to some of the central problems in birational invariant theory, such as the existence of sections, stabilizers in general position, affine models, etc. In this paper we explain these connections and explore them to give partial answers to questions (a) - (c).