Classical \(W\)-algebras
The authors reconsider the relation between classical \(W\)-algebras and deformations of differential operators, emphasizing the consistency with diffeomorphisms. Generators of the \(W\)-algebra that are \(k\)-differentials are constructed by a systematic procedure. The method extends, following Drinfeld and Sokolov, to \(W\)-algebras based on arbitrary simple Lie algebras. The discussion is carried in three steps. In the case of \(sl(n)\), we first decompose a normalized \(n\)th order differential operator \(D\) mapping \(-{n-1\over2 }\) differentials into \({n+1\over 2}\) differentials into a canonical sum involving currents \(w_ j\) of weight \(j\) ranging between 2 and \(n\) in correspondence with the generators of the \(A_{n- 1}\) \(W\)-algebra. We then define associated deformations of \(D\), each one depending on an arbitrary function, which generalize the KdV flows and in the simplest \(W_ 3\)-instance, amount to localizing the latter. In the last step, using the second Adler-Gelfand-Dikij Hamiltonian structure one is able to compute the Poisson brackets between the \(w\) generators. This endows the polynomials in the \(w\)'s and their derivatives with a Lie algebra structure containing the Virasoro algebra generated by \(w_ 2\). In a last section we succinctly present the generalization to an arbitrary simple Lie algebra and discuss in more detail the slightly more involved case of the \(D_ n\) Lie algebra.
scientific article; zbMATH DE number 25828
scientific article