Colored Kac-Moody Algebras, Part I

TL;DR

This paper introduces a new parametrization of deformations of Verma modules for rak{sl}_2 using colourings, establishing a broad family of formal deformations and laying groundwork for future research on Kac-Moody algebras and Langlands duality.

Contribution

It develops a novel framework using colourings to parametrize deformations of rak{sl}_2} modules, connecting to quantum groups and setting the stage for advanced studies in Kac-Moody algebra deformations.

Findings

Regular colourings parametrize formal deformations of rak{sl}_2}

Quantum algebra U_h(rak{sl}_2) is recovered from a specific colouring

Framework paves way for future work on Kac-Moody algebra deformations

Abstract

We introduce a parametrization of formal deformations of Verma modules of $\mathfrak{sl}_2$. A point in the moduli space is called a colouring. We prove that for each colouring $\psi$ satisfying a regularity condition, there is a formal deformation $U_h(\psi)$ of $U(\mathfrak{sl}_2)$ acting on the deformed Verma modules. We retrieve in particular the quantum algebra $U_h(\mathfrak{sl}_2)$ from a colouring by $q$-numbers. More generally, we establish that regular colourings parametrize a broad family of formal deformations of the Chevalley-Serre presentation of $U(\mathfrak{sl}_2)$. The present paper is the first of a series aimed to lay the foundations of a new approach to deformations of Kac-Moody algebras and of their representations. We will employ in a forthcoming paper coloured Kac-Moody algebras to give a positive answer to E. Frenkel and D. Hernandez's conjectures on Langlands duality in quantum group theory.