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

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

1994-05-02

arxiv.org/abs/hep-th/9405011v1

Int.J.Mod.Phys. A10 (1995) 2589-2614

Physics

High Energy Physics

High Energy Physics - Theory

20 pages

Scientific paper

10.1142/S0217751X95001236

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 $SL(n)$.

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