2-Representations of sl2 from Quasi-Maps

2023/01/06 by Raphaël Rouquier, Rouquier, Raphael
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2301.02644

Abstract

We describe a new type of 2-representations, using coherent sheaves. Feigin, Finkelberg, Kuznetsov, Mirković and Braverman have provided a construction of Verma modules for complex semi-simple Lie algebras using based quasi-map spaces from ℙ¹ to flag varieties (zastavas). We consider here the case of \mathfraksl₂, where the zastavas are smooth, and are mere affine spaces. We show that coherent sheaves on zastavas provide a 2-Verma module for \mathfraksl₂ in the sense of Naisse-Vaz. Adding a superpotential and considering matrix factorizations, we obtain a realization of simple 2-representations of \mathfraksl₂.

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