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 ${\mathcal{E}}_n^{\prime }(q_1,q_2,q_3)$ to the quantum affine algebra $U_q^{\prime }{\widehat{\mathfrak{gl}}}_n$ at level $\kappa$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^n={\kappa }^2$. We discuss ${\mathcal{E}}_n^{\prime }(q_1,q_2,q_3)$ 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 ${\mathcal{E}}_n^{\prime }(q_1,q_2,q_3)$, which describes a deformation of the coset theory ${\widehat{\mathfrak{gl}}}_n/{\widehat{\mathfrak{gl}}}_{n-1}$.

Latex, 24 pages. Section 5.3 and Appendix are added

Cited by in corpus (1)

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