Small elliptic quantum group \(e_{\tau,\gamma}(\mathfrak{sl}_N)\) (Q2762318)

In this paper the authors define a dynamical quantum group \(e_{\tau,\gamma}({\mathfrak sl}_N)\), which is an elliptic dynamical analogue of the universal enveloping algebra \(U({\mathfrak sl}_N)\). It is called a small elliptic quantum group in comparison to the elliptic quantum group \(E_{\tau,\gamma}({\mathfrak sl}_N)\) introduced by \textit{G. Felder} [Proceedings of the XIth International Congress on Mathematical Physics, Paris, 1994, Internat. Press, Cambridge, MA, 211-218 (1995; Zbl 0998.17015)].NEWLINENEWLINENEWLINEThe authors give a procedure which assigns to every \(e_{\tau,\gamma}({\mathfrak sl}_N)\)-module an \(E_{\tau,\gamma}({\mathfrak sl}_N)\)-module structure on the same underlying vector space analogous to pulling back \(U({\mathfrak sl}_N)\)-modules to modules over the corresponding Yangian \(Y({\mathfrak sl}_N)\). These \(E_{\tau,\gamma}({\mathfrak sl}_N)\)-modules are called evaluation modules and play a role in the theory of qKZ and qKZB difference equations. The authors also define highest weight modules and Verma modules over \(e_{\tau,\gamma}({\mathfrak sl}_N)\). Then they show that for every finite-dimensional \({\mathfrak sl}_N\)-module and for every Verma module over \({\mathfrak sl}_N\) a corresponding \(e_{\tau,\gamma}({\mathfrak sl}_N)\)-module structure can be defined on the same underlying vector space. NEWLINENEWLINENEWLINEThe small elliptic quantum group \(e_{\tau,\gamma}({\mathfrak sl}_N)\) admits rational and trigonometric degenerations which are closely related to the exchange quantum group \(F(\text{SL}(N))\). The authors construct a functor from the category of admissible \({\mathfrak sl}_N\)-modules to the category of semistandard modules over the rational dynamical quantum group \(e_{\text{rat}}({\mathfrak sl}_N)\) as well as an equivalence between the category of semistandard \(e_{\text{rat}}({\mathfrak sl}_N)\)-modules and the category of rational dynamical \(F(\text{SL}(N))\)-modules. In particular, this gives a new construction of highest weight modules for \(F(\text{SL}(N))\).