W-ALGEBRAS

W-algebras are defined as polynomial extensions of the Virasoro algebr a by primary fields, and they occur in a natural manner in the context of two-dimensional integrable systems, notably in the KdV and Toda sy stems. Their occurrence in those theories can be traced to their being the residual symmetry algebras when certain first-class constraints a re placed on Kac-Moody algebras. In particular, their occurrence in 2- dimensional Toda theories is explained by the fact that the Toda theor ies can be regarded as constrained Wess-Zumino-Novikov-Witten (WZNW) t heories. The general form of such first-class constraint for WZNW theo ries is investigated, and is shown to lead to a wider class of two-dim ensional integrable systems, all of which have W-algebras as symmetry algebras.