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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.

##### MSC:
 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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##### References:
 [1] Bourbaki, N., Algebra II, chapters 4-7, (1990), Springer Berlin, Translated from the French by P.M. Cohn and J. Howie · Zbl 0719.12001 [2] Humphreys, J., Linear algebraic groups, (1981), Springer New York [3] Jacobson, N., Basic algebra II, (1980), Freeman New York · Zbl 0441.16001 [4] Le Bruyn, L., The artin – schofield theorem and some applications, Comm. algebra, 14, 8, 1439-1455, (1986) · Zbl 0601.16014 [5] Luna, D.; Richardson, R.W., A generalization of the Chevalley restriction theorem, Duke math. J., 46, 3, 487-496, (1979) · Zbl 0444.14010 [6] Montgomery, S., Fixed rings of finite automorphism groups of associative rings, Lecture notes in math., vol. 818, (1980), Springer Berlin · Zbl 0449.16001 [7] Popov, V.L., Sections in invariant theory, (), 315-361 [8] Popov, V.L.; Vinberg, E.B., Invariant theory, (), 123-284 · Zbl 0783.14028 [9] Procesi, C., Non-commutative affine rings, Atti accad. naz. lincei mem. cl. sci. fis. mat. natur. sez. I (8), 8, 237-255, (1967) · Zbl 0204.04802 [10] Procesi, C., The invariant theory of n×n matrices, Adv. math., 19, 3, 306-381, (1976) · Zbl 0331.15021 [11] Reichstein, Z., On the notion of essential dimension for algebraic groups, Transform. groups, 5, 3, 265-304, (2000) · Zbl 0981.20033 [12] Reichstein, Z.; Vonessen, N., Torus actions on rings, J. algebra, 170, 781-804, (1994) · Zbl 0832.16022 [13] Reichstein, Z.; Vonessen, N., Rational central simple algebras, Israel J. math., 95, 253-280, (1996) · Zbl 0869.16014 [14] Reichstein, Z.; Vonessen, N., Stable affine models for algebraic group actions, J. Lie theory, 14, 2, 563-568, (2004) · Zbl 1060.14067 [15] Z. Reichstein, N. Vonessen, Polynomial identity rings as rings of functions, J. Algebra, in press. Preprint available at http://arxiv.org/math.RA/0507152 · Zbl 1117.16013 [16] Z. Reichstein, N. Vonessen, Group actions and invariants in algebras of generic matrices, Adv. Appl. Math., in press. Preprint available at http://arxiv.org/math.RA/0507548 · Zbl 1117.16014 [17] Reichstein, Z.; Youssin, B., Conditions satisfied by characteristic polynomials in fields and division algebras, J. pure appl. algebra, 166, 1-2, 165-189, (2002) · Zbl 0998.12003 [18] Richardson, R.W., Deformations of Lie subgroups and the variation of isotropy subgroups, Acta math., 129, 35-73, (1972) · Zbl 0242.22020 [19] Rowen, L.H., Polynomial identities in ring theory, (1980), Academic Press New York · Zbl 0461.16001 [20] Rowen, L.H., Ring theory, vol. II, (1988), Academic Press New York [21] Sage, D.S., Group actions on central simple algebras, J. algebra, 250, 18-43, (2002) · Zbl 1017.16010 [22] Scott, W.R., Group theory, (1987), Dover New York · Zbl 0641.20001 [23] Springer, T.A.; Steinberg, R., Conjugacy classes, (), 167-266 · Zbl 0249.20024 [24] Steinberg, R., Torsion in reductive groups, Adv. math., 15, 63-92, (1975) · Zbl 0312.20026 [25] Vonessen, N., Actions of linearly reductive groups on affine PI-algebras, Mem. amer. math. soc., vol. 414, (1989) · Zbl 0691.16025 [26] Vonessen, N., Actions of linearly reductive groups on PI-algebras, Trans. amer. math. soc., 335, 425-442, (1993) · Zbl 0789.16025 [27] N. Vonessen, Actions of solvable algebraic groups on central simple algebras, Algebr. Represent. Theory, in press. Preprint available at http://www.math.umt.edu/vonessen/publications.html · Zbl 1132.16030 [28] Voskresenskiĭ, V.E., Algebraic groups and their birational invariants, (1998), Amer. Math. Soc. Providence, RI · Zbl 0974.14034
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