TY - JOUR

T1 - Group actions on central simple algebras

T2 - A geometric approach

AU - Reichstein, Z.

AU - Vonessen, N.

N1 - Funding Information:
* Corresponding author. E-mail addresses: reichst@math.ubc.ca (Z. Reichstein), nikolaus.vonessen@umontana.edu (N. Vonessen). URLs: http://www.math.ubc.ca/~reichst (Z. Reichstein), http://www.math.umt.edu/vonessen (N. Vonessen). 1 The author was supported in part by an NSERC research grant. 2 The author gratefully acknowledges the support of the University of Montana and the hospitality of the University of British Columbia during his sabbatical in 2002/2003, when part of this research was done.

PY - 2006/10/15

Y1 - 2006/10/15

N2 - 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).

AB - 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).

KW - Central simple algebra

KW - Division algebra

KW - Group action

KW - Linear algebraic group

UR - http://www.scopus.com/inward/record.url?scp=33748057527&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2005.09.022

DO - 10.1016/j.jalgebra.2005.09.022

M3 - Article

AN - SCOPUS:33748057527

SN - 0021-8693

VL - 304

SP - 1160

EP - 1192

JO - Journal of Algebra

JF - Journal of Algebra

IS - 2

ER -