Dual Pairs and Tensor Categories of Modules over Lie Algebras ∞and 1 + ∞

We introduce a tensor category + (resp. − ) of certain modules of ∞with nonnegative (resp. nonpositive) integral central charges with the usual tensor product. We also introduce a tensor category ( f, ) consisting of certain modules overGL(N) for allN. We show that the tensor categories ± and fare semisimple abelian and all equivalent to each other. We give a formula to decompose a tensor product of two modules and obtain a reciprocity law in each of these categories. We also introduce a tensor category wof certain modules over 1 + ∞with nonnegative integral central charges. We show that wis semisimple abelian and give an explicit formula to decompose a tensor product of two modules in w.