Evaluation modules for quantum toroidal ${\mathfrak{gl}}_n$ algebras

TL;DR

This paper studies evaluation modules for quantum toroidal ${\mathfrak{gl}}_n$ algebras, providing highest weight descriptions and decompositions related to deformations of coset theories, advancing understanding of their representation theory.

Contribution

It introduces evaluation modules for quantum toroidal ${\mathfrak{gl}}_n$ algebras, characterizes their highest weights, and analyzes Wakimoto module decompositions related to coset deformations.

Findings

Highest weights of evaluation modules are explicitly described.

Decomposition of Wakimoto modules with respect to Gelfand-Zeitlin subalgebras.

Connection to deformations of coset theories \(\widehat{\mathfrak{gl}}_n/\widehat{\mathfrak{gl}}_{n-1}\).

Abstract

The affine evaluation map is a surjective homomorphism from the quantum toroidal ${\mathfrak {gl}}_n$ algebra ${\mathcal E}'_n(q_1,q_2,q_3)$ to the quantum affine algebra $U'_q\widehat{\mathfrak {gl}}_n$ at level $\kappa$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^n=\kappa^2$. We discuss ${\mathcal E}'_n(q_1,q_2,q_3)$ 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 ${\mathcal E}'_n(q_1,q_2,q_3)$, which describes a deformation of the coset theory $\widehat{\mathfrak {gl}}_n/\widehat{\mathfrak {gl}}_{n-1}$.