On a symmetry of the category of integrable modules

TL;DR

This paper explores a symmetry in the category of integrable modules for affine Lie algebra-related vertex operator algebras, identifying how certain operators transform modules and fully characterizing their action.

Contribution

It explicitly determines the action of elta-operators associated with miniscule coweights on all irreducible modules of L(k,0).

Findings

Identified highest weight vectors for transformed modules.

Fully characterized the elta-operator action on module classes.

Extended Li's symmetry results to all irreducible modules.

Abstract

Haisheng Li showed that given a module (W,Y_W(\cdot,x)) for a vertex algebra (V,Y(\cdot,x)), one can obtain a new V-module W^{\Delta} = (W,Y_W(\Delta(x)\cdot,x)) if \Delta(x) satisfies certain natural conditions. Li presented a collection of such \Delta-operators for V=L(k,0) (a vertex operator algebra associated with an affine Lie algebras, k a positive integer). In this paper, for each irreducible L(k,0)-module W, we find a highest weight vector of W^{\Delta} when \Delta is associated with a miniscule coweight. From this we completely determine the action of these \Delta-operators on the set of isomorphism equivalence classes of L(k,0)-modules.