On the Generalized Enveloping Algebra of a Color Lie Algebra

Let $G$ be an abelien group, $ϵ$ an anti-bicharacter of $G$ and $L$ a $G$-graded $ϵ$ Lie algebra (color Lie algebra) over $\text{K}$ a field of characteristic zero. We prove that all $G$-graded, positive 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 $\omega \in {\mathrm{Z}}_{gr}^2(L,\text{K})$, such that $A$ is isomorphic to $U_{\omega }(L)$ the generalized enveloping algebra of $L$ associated with $\omega$. 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 color Lie algebra $L$.