2014/12/30 by Alexandre Bouayad, Bouayad, Alexandre
Mathematics · #16S30 #16S80 #16T20 #17B10 (Secondary) #17B37 (Primary) #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1412.8606
openalex publication_date 2014/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a parametrization of formal deformations of Verma modules of \mathfraksl2. A point in the moduli space is called a colouring. We prove that for each colouring ψ satisfying a regularity condition, there is a formal deformation Uh(ψ) of U(\mathfraksl2) acting on the deformed Verma modules. We retrieve in particular the quantum algebra Uh(\mathfraksl2) 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(\mathfraksl2). 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.