Non-local matrix generalizations ofW-algebras

There is a standard way to define two symplectic (hamiltonian) structures, the first and second Gelfand-Dikii brackets, on the space of ordinarym ^th-order linear differential opeatorsL=–d ^m +U ₁ d ^m–1+U ₂ d ^m–2+...+U _m . In this paper, I consider in detail the case where theU _k aren×n-matrix-valued functions, with particular emphasis on the (more interesting) second Gelfand-Dikii bracket. Of particular interest is the reduction to the symplectic submanifoldU ₁=0. This reduction gives rise to matrix generalizations of (the classical version of) thenon-linear W _m -algebras, calledV _{n, m} -algebras. The non-commutativity of the matrices leads tonon-local terms in theseV _{n, m} -algebra.s I show that these algebras contain a conformal Virasoro subalgebra and that combinationsW _k of theU _k can be formed that aren×n-matrices of conformally primary fields of spink, in analogy with the scalar casen=1. In general however, theV _{m, n} -algebras have a much richer structure than theW _m -algebras as can be seen on the examples of thenon-linear andnon-local Poisson brackets {(U _2)ab(), (U _2)cd()}, {(U _2)ab(), (W _3)cd()} and {(W _3)ab(), (W _3)cd()} which I work out explicitly for allm andn. A matrix Miura transformations is derived, mapping these complicated (second Gelfand-Dikii) brackets of theU _k to a set of much simpler Poisson brackets, providing the analogoue of the free-field representation of theW _m -algebras.