Graded Casimir elements and central extensions of color Lie algebras

A color Lie algebra is a generalization of a Lie (super)algebra by an Abelian group $\mathrm{\Gamma }$. The underlying vector space and defining relations of the algebra are graded by $\mathrm{\Gamma }$, and the color Lie algebra admits graded Casimir elements. Furthermore, its loop algebra admits graded central extensions. We present a general method for constructing 2nd order graded Casimir elements and graded central extensions for a given color Lie algebra and its loop algebra, respectively. We also show that there exists a large class of color Lie algebras admitting such graded Casimir elements or central extensions by providing three examples, namely, $\mathfrak{s}\mathfrak{l}(2)$ for $\mathrm{\Gamma }={\mathbb{Z}}_3^2$, and $\mathfrak{q}(n)$ and $\mathfrak{o}\mathfrak{s}\mathfrak{p}(m\mathrm{\mid }2n)$ for $\mathrm{\Gamma }={\mathbb{Z}}_2^2$.