A graded generalization of Lie triples

Abstract

A graded generalization of Lie triples is defined such that the well-known relations between Lie triples and Lie algebras remain valid. For instance, the graded generalizations of Lie algebras, considered recently in physics and mathematics, become such triples with respect to the graded double commutator, and every such triple can be constructed by means of an involutive automorphism of degree zero on such an algebra as eigenspace of eigenvalue −1. In case of a Z 2 -graduation there is an elementary example, considered first in the second quantization in quantum field theory, which is constructed on a graded vector space V with a graded symmetric bilinear form <, >. This triple has a realization in the Clifford algebra constructed over ( V, <, >). An elementary construction of representations which in the (Lie) group case leads to inhomogenizations and tangent groups can be generalized to these triples as well.