The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting $\W$-algebra is a one-parameter deformation of $\W_{ \rm KP}$ admitting a central extension for generic values of the parameter, reducing naturally to $\W_n$ for special values of the parameter, and contracting to the centrally extended $\W_{ 1+\infty}$, $\W_\infty$ and further truncations. In the classical limit, all algebras in the one-parameter family are equivalent and isomorphic to $\w_{ \rm KP}$. The reduction induced by setting the spin-one field to zero yields a one-parameter deformation of $\widehat{ \W}_\infty$ which contracts to a new nonlinear algebra of the $\W_\infty$-type.
Categories: hep-th