  We classify positive energy representations with finite degeneracies of the
Lie algebra $W_{1+\infty}\/$ and construct them in terms of representation
theory of the Lie algebra $\hatgl ( \infty R_m )\/$ of infinite matrices with
finite number of non-zero diagonals over the algebra $R_m = \C [ t ] / ( t^{m +
1} )\/$. The unitary ones are classified as well. Similar results are obtained
for the sin-algebras.
