Z_2-gradings of Clifford algebras and multivector structures

TL;DR

This paper characterizes Z_2-gradings of real Clifford algebras compatible with multivector structures, classifies their even subalgebras, and explores implications for spinor spaces and signature changes.

Contribution

It provides a complete classification of compatible Z_2-gradings of Clifford algebras and relates them to multivector structures and signature modifications.

Findings

Classified all compatible Z_2-gradings of Cl(V,g)

Derived relations between subalgebras and the standard even part

Parametrized signature changes via Z_2-gradings

Abstract

Let Cl(V,g) be the real Clifford algebra associated to the real vector space V, endowed with a nondegenerate metric g. In this paper, we study the class of Z_2-gradings of Cl(V,g) which are somehow compatible with the multivector structure of the Grassmann algebra over V. A complete characterization for such Z_2-gradings is obtained by classifying all the even subalgebras coming from them. An expression relating such subalgebras to the usual even part of Cl(V,g) is also obtained. Finally, we employ this framework to define spinor spaces, and to parametrize all the possible signature changes on Cl(V,g) by Z_2-gradings of this algebra.