SCHUR–WEYL DUALITY FOR HEISENBERG COSETS

Abstract

Let V be a simple vertex operator algebra containing a rank n Heisenberg vertex algebra H and let C = Com(H;V) be the coset of H in V. Assuming that the module categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of C are found and every simple C-module is shown to be contained in at least one V-module. A corollary of this is that if V is rational, C 2 -cofinite and CFT-type, and Com(C;V) is a rational lattice vertex operator algebra, then C is likewise rational. These results are illustrated with many examples and the C..