On the Generalized Enveloping Algebra of a Color Lie Algebra

Abstract

Let G be an abelian group, ε an anti-bicharacter of G and L a G-graded ε Lie algebra (color Lie algebra) over $\mathbb{K}$ a field of characteristic zero. We prove that for all G-graded, positively filtered A such that the associated graded algebra is isomorphic to the G-graded ε-symmetric algebra S(L), there is a G- graded ε-Lie algebra L and a G-graded scalar two cocycle $\omega\in\mathrm{Z}_{gr}^2(L,\mathbb{K})$, such that A is isomorphic to U _ω (L) the generalized enveloping algebra of L associated with ω. We also prove there is an isomorphism of graded spaces between the Hochschild cohomology of the generalized universal enveloping algebra U(L) and the generalized cohomology of the color Lie algebra L.