Dual pairs and tensor categories of modules over Lie algebras \(\widehat{gl}_{\infty}\) and \({\mathcal W}_{1+\infty}\) (Q1293110)

The author introduces two tensor categories of certain irreducible highest weight representations of \(\widehat{gl}\) with nonnegative and nonpositive integral central charges, respectively. He shows that a tensor of two irreducible modules in each category decomposes into an (infinite) direct sum of irreducible modules with finite multiplicities. A reciprocity law has been obtained. Moreover, the author proves that these two categories are abelian tensor categories and are equivalent to each other. Similar results on a related tensor category of certain modules of \({\mathcal W}_{1+\infty}\) are obtained.