Some invariants for equivalent G-algebras

Abstract

In an earlier paper (Clifford theory with Schur indices, J. Algebra 170 (1994) 661–677), the author introduced a generalization of the Brauer-Wall group. It is defined for any given finite group G and any field F of characteristic 0. Each element of this generalized Brauer-Wall group is an equivalence class of central simple G-algebras. He showed that given a finite group H with a normal subgroup N such that H N ≅ G , and an irreducible character χ of H, there corresponds naturally an element of this generalized Brauer-Wall group, and that this element alone controls the Clifford theory (including Schur indices) of χ with respect to N. The present paper studies some invariants for G-algebras which are preserved under equivalence of G-algebras in the above sense. These invariants are the basis of a characterization of each equivalence class of central simple G-algebras in some cases, as is described in a forthcoming paper of the author. The present paper also includes a brief comparison of this generalization of the Brauer-Wall group with others that have appeared in the literature.