Commutation factors on generalized Lie algebras

Generalized Lie algebras or color algebras, as we shall call them, are described by an Abelian grading group Γ and a commutation factor ∈ defined on Γ. In this paper Γ is assumed to be finite. It is shown that color algebras with the pair (Γ,∈) can also be considered as color algebras with the different pair (Γ′,∈′) and that as a result a canonical pair (Γ_c,∈_c) is possible. It is further shown that, in fact, a unique "minimal" (Γ_c,∈_c) can be used for all algebras with the pair (Γ,∈).