Homological properties of color Lie superalgebras

Let $\mathcal{L}={\mathcal{L}}_{+}\oplus {\mathcal{L}}_{-}$ be a finite dimensional color Lie superalgebra over a field of characteristic 0 with universal enveloping algebra $U(\mathcal{L})$. We show that $\limfunc{gldim}(U($\mathcal{L}${+}))= \limfunc{lFPD}(U($\mathcal{L}$))= \limfunc{rFPD}(U($\mathcal{L}$))= \limfunc{injdim}{U($\mathcal{L}$)}(U($\mathcal{L}$))= \dim ($\mathcal{L}$_{+})$. We also prove that $U(\mathcal{L})$ is Auslander-Gorenstein and Cohen-Macaulay and thus that it has a QF classical quotient ring.