Generalized Clifford theory for graded spaces

Abstract

Assume that Γ is a finite abelian group and ε is an antisymmetric bicharacter on Γ. Let V be a Γ-graded space with a non-degenerate ε-symmetric bilinear form of degree zero. The goal of this paper is to develop a generalized Clifford theory on V. We first introduce the ε-Clifford algebra C(V) and the ε-exterior algebra Λ(V), and then establish an analogue of Chevalley identification between C(V) and Λ(V). Secondly, we extend the non-degenerate bilinear form of degree zero on V to a non-degenerate bilinear form on Λ(V). Finally, as an application, we give a realization of the orthosymplectic ε-Lie algebra.