Vertex operators and 2-representations of quantum affine algebras

Abstract

We construct 2-representations of quantum affine algebras from 2-representations of quantum Heisenberg algebras. The main tool in this construction are categorical vertex operators, which are certain complexes in a Heisenberg 2-representation that recover vertex operators after passing to the Grothendieck group. As an application we categorify the Frenkel-Kac-Segal homogeneous realization of the basic representation of (simply laced) quantum affine algebras. This gives rise to categorical actions of quantum affine (and toroidal) algebras on derived categories of coherent sheaves on Hilbert schemes of points of ALE spaces.

50 pages. Strengthened main result and simplified exposition

References in corpus (5)

• 2-Kac-Moody algebras
• Geometric construction of representations of affine algebras
• Heisenberg algebra and a graphical calculus
• Vertex Operators and Moduli Spaces of Sheaves
• Loop realizations of quantum affine algebras

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• A diagrammatic categorification of a Clifford algebra
• A categorical action of the shifted $q=0$ affine algebra
• Symmetric self-adjoint Hopf categories and a categorical Heisenberg double
• A Littlewood-Richardson rule for the MacDonald inner product and bimodules over wreath products