Small Elliptic Quantum Group $e_{tau,{\hat{I}}^3}(s{l}_N)$

Abstract

The small elliptic quantum group $e_{\ddot{I}„,{\hat{I}}^3}(s{l}_N)$, introduced in the paper, is an elliptic dynamical analogue of the universal enveloping algebra $U(s{l}_n)$. We define highest weight modules, Verma modules and contragradient modules over $e_{\ddot{I}„,{\hat{I}}^3}(s{l}_N)$, the dynamical Shapovalov form for $e_{\ddot{I}„,{\hat{I}}^3}(s{l}_N)$ and the contravariant form for highest weight $e_{\ddot{I}„,{\hat{I}}^3}(s{l}_N)$-modules. We show that any finite-dimensional $s{l}_N$-module and any Verma module over $s{l}_N$ can be lifted to the corresponding $e_{\ddot{I}„,{\hat{I}}^3}(s{l}_N)$-module on the same vector space. For the elliptic quantum group $E_{\ddot{I}„,{\hat{I}}^3}(s{l}_N)$ we construct the evaluation morphism $E_{\ddot{I}„,{\hat{I}}^3}(s{l}_N)\to e_{\ddot{I}„,{\hat{I}}^3}(s{l}_N)$, thus making any $e_{\ddot{I}„,{\hat{I}}^3}(s{l}_N)$-module into an evaluation $E_{\ddot{I}„,{\hat{I}}^3}(s{l}_N)$-module.

34 pages, amstex.tex (ver. 2.1), misprints are corrected