GENERALIZED HIROTA EQUATIONS AND REPRESENTATION THEORY I: THE CASE OF SL(2) AND SLq(2)

Abstract

This paper begins investigation of the concept of a “generalized τ function,” defined as a generating function of all the matrix elements of a group element g∈G in a given highest weight representation of a universal enveloping algebra [Formula: see text]. In the generic situation, the time variables correspond to the elements of maximal nilpotent subalgebras rather than Cartanian elements. Moreover, in the case of quantum groups such τ “functions” are not c numbers but take their values in noncommutative algebras (of functions on the quantum group G). Despite all these differences from the particular case of conventional τ functions of integrable (KP and Toda lattice) hierarchies [which arise when G is a Kac-Moody (one-loop) algebra of level k=1], these generic τ functions also satisfy bilinear Hirota-like equations, which can be deduced from manipulations with intertwining operators. The most important applications of the formalism should be to k>1 Kac-Moody and multiloop algebras, but this paper contains only illustrative calculations for the simplest case of ordinary (zero-loop) algebra SL (2) and its quantum counterpart SL q(2), as well as for the system of fundamental representations of SL (n).