Tau-functions as highest weight vectors for W_{1+infty} algebra

TL;DR

This paper constructs highest weight modules of the W_{1+infty} algebra using tau-functions from the Gelfand--Dickey hierarchy, providing a detailed classification and singular vector formulas.

Contribution

It introduces a new class of highest weight modules for W_{1+infty} linked to tau-functions, with a complete reducibility and singular vector description.

Findings

Modules are quasifinite

Complete reducibility classification provided

Formulas for singular vectors included

Abstract

For each r = (r_1, r_2,...,r_N) we construct a highest weight module M_r of the Lie algebra W_{1+infty}. The highest weight vectors are specific tau-functions of the N-th Gelfand--Dickey hierarchy. We show that these modules are quasifinite and we give a complete description of the reducible ones together with a formula for the singular vectors. This paper is the first of a series of papers (q-alg/9602010, q-alg/9602011, q-alg/9602012) on the bispectral problem.