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Group actions on central simple algebras: a geometric approach. (English) Zbl 1112.16021
Let \(k\) be a field of characteristic zero. The paper under review deals with the study of actions of (linear) algebraic groups \(G\) on simple \(k\)-algebras \(A\) by \(k\)-automorphisms. The algebras \(A\) are always supposed to be finite-dimensional over their centres \(Z(A)\), which in turn are assumed to be finitely generated extensions of \(k\). Let \(A^G=\{a\in A:g(a)=a\}\) be the subalgebra of \(G\)-fixed elements of \(A\).
The authors are interested in questions of the following type: (i) whether \(A^G\) is a simple algebra of dimension \([A:Z(A)]\) over its centre; (ii) whether \(A\) possesses a \(G\)-invariant maximal subfield; (iii) whether the \(G\)-action on \(Z(A)\) can be extended to a splitting field \(L\) of \(A\), and if so, what is the minimal possible value of the transcendence degree \(\text{tr}(L/Z(A))\).
Their purpose is to introduce a geometric approach to this topic by relating it to what they call birational invariant theory, i.e., to the study of group actions on algebraic varieties, up-to birational isomorphisms; throughout, they use Popov-Vinberg’s “Invariant Theory” [see V. L. Popov and Eh. B. Vinberg, Encycl. Math. Sci. 55, 123-278 (1994); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 55, 137-309 (1989; Zbl 0735.14010)] as a reference to standard notions from this area.
Their starting point is the well-known fact that \(Z(A)\) can be identified with the field of rational functions on some irreducible algebraic variety \(X\) which is uniquely determined by \(Z(A)\), up-to a birational isomorphism. Similarly, if \([A:Z(A)]=n^2\), where \(n\in\mathbb{N}\), then \(A\) is \(k\)-isomorphic to the algebra \(k_n(X)\) of \(\text{PGL}_n\)-equivariant rational functions \(X\to M_n(k)\), where \(X\) is an irreducible variety with a generically free action. This leads the authors to the notion of a geometric action of \(G\) on \(A\). The main results of the paper give answers to the stated questions, for geometric actions and show that the questions are related to some of the central problems in birational invariant theory, such as the existence of affine models, quotients, stabilizers in general positions, sections, etc.

16K20 Finite-dimensional division rings
14L30 Group actions on varieties or schemes (quotients)
16R30 Trace rings and invariant theory (associative rings and algebras)
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