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.

23 pages. Revised argument in section 3, results unchanged. Corrected typos

References in corpus (1)

• Deformations of W-algebras associated to simple Lie algebras