Evaluation Modules for Quantum Toroidal $$\mathfrak {gl}_n$$ Algebras

Abstract

The affine evaluation map is a surjective homomorphism from the quantum toroidal \(\mathfrak {gl}_n\) algebra \({\mathcal E}^{\prime }_n(q_1,q_2,q_3)\) to the quantum affine algebra \(U^{\prime }_q\widehat {\mathfrak {gl}}_n\) at level κ completed with respect to the homogeneous grading, where q2 = q2 and \(q_3^n=\kappa ^2\). We discuss \({\mathcal E}^{\prime }_n(q_1,q_2,q_3)\) evaluation modules. We give highest weights of evaluation highest weight modules. We also obtain the decomposition of the evaluation Wakimoto module with respect to a Gelfand–Zeitlin-type subalgebra of a completion of \({\mathcal E}^{\prime }_n(q_1,q_2,q_3)\), which describes a deformation of the coset theory \(\widehat {\mathfrak {gl}}_n/\widehat {\mathfrak {gl}}_{n-1}\).