Extensions of Lie-graded algebras

Abstract

For Lie-graded algebras which are generalizations of Lie algebras with respect to a graduation, used recently in physics for the classification of elementary particles, and extension G of F by T, i.e., a short exact sequence T≳→G→≳F can be described by a Lie-graded composition on T⊕F, which is formulated in terms of a pair of mappings ∂:F→derT and Δ:F×F→T. The congruence of two extensions of F by T, i. e., the equivalence of the corresponding short exact sequences, is related to an equivalence relation on the set Z2(F,T) of such 2-cocycles (∂,Δ) such that there exists a bijection between the set of congruence classes of extensions of F by T and the set H2(F,T) of classes in Z2(F,T). This generalizes Lie algebraical results which are also known for groups. Examples for two special cases are given: the semidirect sums with Δ trivial and the almost direct sums with ∂ trivial. Both generalize the concepts of tangent and cotangent algebras of Lie algebras and their central extensions with R, the latter being used in the Bargmann theory of ray representations of these semidirect sums.