COMPLEX PARAMETRIZED W-ALGEBRAS - THE GL CASE

We define deformations of the Poisson bracket algebras of functions on manifolds L(alpha) = {partial derivative(alpha) + SIGMA(i=1)infinity w(i)partial derivative(alpha-i)\w(i) is-an-element-of C(infinity)(S1)} of pseudodifferential symbols on the circle (alpha is-an-element-of C ), arising in the works of Rosly and Khesin-Zakharevich. These deforma tions have vertex operator algebraic (VOA) counterparts, which have (f or alpha = n integer) a quotient isomorphic to the W-algebra W(n) asso ciated by Fateev and Lukyanov to gl(n). The product operation of symbo ls defines a Lie-Poisson structure on PI(alpha is-an-element-of C) L(a lpha)BAR (Rosly, Khesin-Zakharevich); we show that this structure has also a VOA counterpart.