Kashiwara Algebras and Imaginary Verma Modules for $U_q(\hat{\mathfrak{g}})$

TL;DR

This paper constructs Kashiwara operators and algebras for imaginary Verma modules of quantum affine algebras, demonstrating that a specific quotient is a simple module, advancing understanding of their structure.

Contribution

It introduces Kashiwara type operators and the algebra al_q for imaginary Verma modules of quantum affine algebras, showing a quotient is a simple module.

Findings

Construction of Kashiwara operators and algebra al_q

Identification of a simple module al_q-module

Advancement in understanding of imaginary Verma modules

Abstract

We consider imaginary Verma modules for quantum affine algebra $U_q(\hat{\mathfrak{g}})$, where $\hat{\mathfrak{g}}$ is of type 1 i.e. of non-twisted type, and construct Kashiwara type operators and the Kashiwara algebra $\mathcal K_q$. We show that a certain quotient $\mathcal N_q^-$ of $U_q(\hat{\mathfrak{g}})$ is a simple $\mathcal K_q$-module.