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Generalized Clifford theory for graded spaces. (English) Zbl 1327.15051

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

MSC:

15A66 Clifford algebras, spinors
15A63 Quadratic and bilinear forms, inner products
15A75 Exterior algebra, Grassmann algebras
17B75 Color Lie (super)algebras
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