Semi-infinite cohomology and the linkage principle for W-algebras

2019/05/16 by Gurbir Dhillon, Dhillon, Gurbir
Mathematics · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1905.06477

Abstract

Let \mathfrakg be a simple Lie algebra, and let W_κ be the affine W-algebra associated to a principal nilpotent element of \mathfrakg and level κ. We explain a duality between the categories of smooth W modules at levels κ+ κ_c and -κ+ κ_c, where κ_c is the critical level. Their pairing amounts to a construction of semi-infinite cohomology for the W-algebra. As an application, we determine all homomorphisms between the Verma modules for W, verifying a conjecture from the conformal field theory literature of de Vos--van Driel. Along the way, we determine the linkage principle for Category \mathscrO of the W-algebra.

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