  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.
