Quantum Deformations of $\tau$-functions, Bilinear Identities and Representation Theory

Abstract

This paper is a brief review of recent results on the concept of ``generalized $\tau$-function'', defined as a generating function of all the matrix elements in a given highest-weight representation of a universal enveloping algebra ${\cal G}$. Despite the differences from the particular case of conventional $\tau$-functions of integrable (KP and Toda lattice) hierarchies, these generic $\tau$-functions also satisfy bilinear Hirota-like equations, which can be deduced from manipulations with intertwining operators. The main example considered in details is the case of quantum groups, when such $\tau$-``functions'' are not $c$-numbers but take their values in non-commutative algebras (of functions on the quantum group $G$). The paper contains only illustrative calculations for the simplest case of the algebra SL(2) and its quantum counterpart $SL_q(2)$, as well as for the system of fundamental representations of SL(n).