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Invariant fields of linear groups and division algebras. (English) Zbl 0688.16020

Perspectives in ring theory, Proc. NATO Adv. Res. Workshop, Antwerp/Belg. 1987, NATO ASI Ser., Ser. C 233, 279-297 (1988).
[For the entire collection see Zbl 0676.00006.]
Let F be an algebraically closed field of characteristic 0, G a reductive linear algebraic group defined over F, V a finite dimensional representation over F which has an element with trivial G-stabilizer, F[V] the symmetric algebra on V and F(V) its field of fractions. Let \(F(V)^ G\) be the field of G-invariant elements of F(V). According to Bogomolov’s result \(F(V)^ G\) is up to stable isomorphism independent of the choice of V. This allows to talk about the invariant field of G without specifying V, denote it simply by F(G). In the paper under review some connections between invariant fields of reductive linear groups and division algebras are discussed.
The main result of the paper is the following Theorem 1. Suppose \(\phi\) : G\({}'\to G\) is a surjective homomorphism of linear connected reductive algebraic groups with central kernel cyclic of order n. Then \(F(G')\) is stably isomorphic to F(G)(A) for some central simple algebra \(A| F(G)\) of exponent n, where F(G)(A) is the function field of the Brauer- Severi variety defined by A. Furthermore some results are presented for some specific groups: \(SL_ n\), Spin groups of odd degree and \(SO_ n\). It should be noted the importance of the result about \(SL_ n\) and its images. Theorem 2. Let \(D=UD(F,n,2)\) be the generic division algebra over F of degree n in two variables, Z the center of D and A a division algebra in the class r[D]\(\in Br(Z)\). Set \(G_ r\) to be \(SL_ n(F)/C_ r\) where \(C_ r\) is the central cyclic subgroup of order r, let V be a \(G_ r\)-representation over F with an element with trivial stabilizer and \(F(G_ r)\) the invariant field of \(G_ r\) on F(V). Then \(F(G_ r)\) is stably isomorphic to Z(A), where Z(A) is the function field of the Brauer-Severy variety defined by A.
Reviewer: V.Yanchevskij

MSC:

16Kxx Division rings and semisimple Artin rings
20H25 Other matrix groups over rings
15A72 Vector and tensor algebra, theory of invariants
20G05 Representation theory for linear algebraic groups
14F22 Brauer groups of schemes
12G05 Galois cohomology
14M20 Rational and unirational varieties

Citations:

Zbl 0676.00006