Generalized Hirota Equations and Representation Theory. I. The case of $SL(2)$ and $SL_q(2)$"

TL;DR

This paper explores a generalized concept of tau-functions linked to group representations, extending to quantum groups and non-commutative settings, with initial examples for SL(2) and SL_q(2).

Contribution

It introduces a new framework for tau-functions as generating functions in representation theory, applicable to quantum groups and multi-loop algebras, with initial illustrative cases.

Findings

Generalized tau-functions satisfy Hirota-like bilinear equations.

Tau-functions for quantum groups take values in non-commutative algebras.

Illustrative calculations provided for SL(2), SL_q(2), and fundamental representations of SL(n).

Abstract

This paper begins investigation of the concept of ``generalized $\tau$-function'', defined as a generating function of all the matrix elements of a group element $g \in G$ in a given highest-weight representation of a universal enveloping algebra ${\cal G}$. 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 $\tau$-``functions'' are not $c$-numbers but take their values in non-commutative algebras (of functions on the quantum group $G$). Despite all these differences from the particular case of conventional $\tau$-functions of integrable (KP and Toda lattice) hierarchies (which arise when $G$ is a Kac-Moody (1-loop) algebra of level $k=1$), these generic $\tau$-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 multi-loop algebras, but this paper contains only illustrative calculations for the simplest case of ordinary (0-loop) algebra $SL(2)$ and its quantum counterpart $SL_q(2)$, as well as for the system of fundamental representations of $SL(n)$.