Dual pairs and tensor categories of modules over Lie algebras gl_{\infty} and W_{1 +\infty}

We introduce a tensor category O_+ (resp. O_{-}) of certain modules of gl_{\infty} with non-negative (resp. non-positive) integral central charges with the usual tensor product. We also introduce a tensor category O_f consisting of certain modules over GL(N) for all N. We show that the tensor categories O_+, O_{-} and O_f are semisimple abelian and all equivalent to each other. We give a formula to decompose a tensor product of two modules in each of these categories. We also introduce a tensor category O^w of certain modules over W_{1 +\infty} with non-negative integral central charges. We show that O^w is semisimple abelian and give an explicit formula to decompose a tensor product of two modules in O^w.