$SL(n,R)$ KDV Hierarchy and its Nonpolynomial Realization Through Kac-Moody Currents

TL;DR

This paper demonstrates that the $SL(n,R)$ KdV hierarchy can be represented through nonpolynomial functions of Kac-Moody currents, using Borel subgroup actions, and constructs a Lax pair confirming a Hamiltonian reduction.

Contribution

It introduces a nonpolynomial realization of the $SL(n,R)$ KdV hierarchy via Kac-Moody currents and constructs a Lax pair, extending the understanding of integrable systems and symplectic structures.

Findings

Representation of $SL(n,R)$ KdV hierarchy as nonpolynomials in currents

Construction of Lax pair confirming Hamiltonian reduction

Identification of a moduli space with symplectic structure

Abstract

It is shown that $SL(n,R)$ KdV hierarchy can be expressed as definite nonpolynomials in Kac Moody currents and their derivatives by the action of Borel subgroup of $SL(n,R)$ on the phase space of centrally extended $sl(n,R)$ Kac Moody currents. Construction of Lax pair is shown, confirming Drinfeld Sokolov type Hamiltonian reduction. This suggests an example of a moduli space with symplectic structure corresponding to extended conformal symmetries.