A finite number of defining relations and a UCE theorem of the elliptic Lie algebras and superalgebras with rank $\ge 2$

Abstract

In this paper, we give a finite number of defining relations satisfied by a finite number of generators for the elliptic Lie algebras and superalgebras ${\mathfrak{g}}_R$ with rank $\ge 2$. Here the $R$'s denote the reduced and non-reduced elliptic root systems with rank $\ge 2$. We also show that if $\mathcal{L}$ is an extended affine Lie algebra (EALA) whose non-isotropic roots form the $R$, then there exists a natural homomorphism $\mathcal{F}:{\mathfrak{g}}_R\to \mathcal{L}$, which also give a universal central extension (UCE) surjective map from $[{\mathfrak{g}}_R,{\mathfrak{g}}_R]$ to the core of $\mathcal{L}$. (More precisely, we take a ${\bar{\mathfrak{g}}}_R$ instead of the ${\mathfrak{g}}_R$.)

30 pages