SOLVABLE CROSSED PRODUCT ALGEBRAS REVISITED

Christian Brown,Susanne Pumplün

doi:10.1017/s0017089519000089

Abstract

AbstractFor any central simple algebra over a field F which contains a maximal subfield M with non-trivial automorphism group G = AutF(M), G is solvable if and only if the algebra contains a finite chain of subalgebras which are generalized cyclic algebras over their centers (field extensions of F) satisfying certain conditions. These subalgebras are related to a normal subseries of G. A crossed product algebra F is hence solvable if and only if it can be constructed out of such a finite chain of subalgebras. This result was stated for division crossed product algebras by Petit and overlaps with a similar result by Albert which, however, was not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F.

Highlights

Crossed product algebras play an important role in the theory of central simple algebras: every element in the Brauer group of F is similar to a crossed product algebra, their multiplicative structure can be described by a group action
In order to do so, we extend the classical definition of a generalized cyclic algebra (D, σ, d) as we do not assume that D needs to be a division algebra
We note that it is known that given a finite-dimensional central division algebra D over F, Matm(D) is a crossed product algebra over a maximal subfield if and only if there is an irreducible subgroup G of Matm(D) and a normal abelian subgroup G0 of G, such that the centralizer CG (G0) of G0 in G is G0, and such that the F subalgebra F [G0] of Matm(D) generated by elements of G0 over F does not contain zero divisors [9, Theorem 1]

Summary

Introduction

The first results on the structure of central simple algebras which contain a maximal subfield with non-trivial solvable group G = AutF (M ) are stated in Section 3 (Theorems 7 and 13). These algebras have certain chains of generalized cyclic algebras (with centers larger than F ) as subalgebras. Most of the results presented here are part of the first author’s PhD thesis [4] written under the supervision of the second author

Preliminaries

Cyclic and crossed product subalgebras of central simple algebras

Solvable crossed product algebras

Some simple consequences for admissible groups