Group actions on central simple algebras: a geometric approach

We study actions of linear algebraic groups on central simple algebras using algebro-geometric techniques. Suppose an 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 equivariant birational isomorphisms). 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).Comment: 33 pages. Final version, to appear in Journal of Algebra. Includes a short new section on Brauer-Severi varietie

An Embedding Property of Universal Division Algebras

AbstractLet A be a central simple algebra of degree n and let k be a subfield of its center. We show that A contains a copy of the universal division algebra Dm, n(k) generated by m generic n × n matrices if and only if trdegkA ≥ trdegkDm, n(k) = (m − 1)n2 + 1. Moreover, if in addition the center of A is finitely and separately generated over k then "almost all" division subalgebras of A generated by m elements are isomorphic to Dm, n(k). In the last section we give an application of our main result to the question of embedding free groups in division algebras

Tame group actions on central simple algebras

We study actions of linear algebraic groups on finite-dimensional central simple algebras. We describe the fixed algebra for a broad class of such actions.Comment: 19 pages, LaTeX; slightly revised; final version will appear in Journal of Algebr

Polynomial identity rings as rings of functions

We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where "varieties" carry a PGL_n-action, regular and rational "functions" on them are matrix-valued, "coordinate rings" are prime polynomial identity algebras, and "function fields" are central simple algebras of degree n. In particular, a prime polynomial identity algebra of degree n is finitely generated if and only if it arises as the "coordinate ring" of a "variety" in this setting. For n = 1 our definitions and results reduce to those of classical affine algebraic geometry.Comment: 24 pages. This is the final version of the article, to appear in J. Algebra. Several proofs have been streamlined, and a new section on Brauer-Severi varieties has been adde