Small elliptic quantum group $e_{\tau,\gamma}(\mathfrak{sl}_N)$

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