Generalized Clifford theory for graded spaces

Abstract Assume that Γ is a finite abelian group and e is an antisymmetric bicharacter on Γ. Let V be a Γ-graded space with a non-degenerate e -symmetric bilinear form of degree zero. The goal of this paper is to develop a generalized Clifford theory on V . We first introduce the e -Clifford algebra C ( V ) and the e -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 e -Lie algebra.