Small Elliptic Quantum Group $e_{tau,\gamma}(sl_N)$

Abstract

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