Mathematics – Rings and Algebras
Scientific paper
2004-08-31
Journal of Algebra 304 (2006), no. 2, 1160--1192.
Mathematics
Rings and Algebras
33 pages. Final version, to appear in Journal of Algebra. Includes a short new section on Brauer-Severi varieties
Scientific paper
10.1016/j.jalgebra.2005.09.022
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).
Reichstein Zinovy
Vonessen Nikolaus
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