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 ${ \frak g}_R$.)
Categories: math.QA math.RT