Colored Kac-Moody Algebras, Part I

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.