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

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 ${\frak g}_R$ with rank $\geq 2$. Here the $R$'s denote the reduced and non-reduced elliptic root systems with rank $\geq 2$. We also show that if ${\cal L}$ is an extended affine Lie algebra (EALA) whose non-isotropic roots form the $R$, then there exists a natural homomorphism ${\cal F}:{\frak g}_R \to{\cal L}$, which also give a universal central extension (UCE) surjective map from $[{\frak g}_R,{\frak g}_R]$ to the core of ${\cal L}$. (More precisely, we take a ${\bar {\frak g}}_R$ instead of the ${\frak g}_R$.)