Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction

TL;DR

This paper extends the matrix Gelfand-Dickey hierarchy using generalized Drinfeld-Sokolov reduction, revealing new integrable systems with local Poisson structures and conformal algebra connections, and explores their reductions and modifications.

Contribution

It introduces a new class of extended matrix Gelfand-Dickey hierarchies via generalized Drinfeld-Sokolov reduction for larger Lie algebras, with detailed structural analysis.

Findings

Derived new extended hierarchies from $ ext{gl}_{pr+s}$ Lie algebra.

Established local Poisson brackets and $ ext{W}$-algebra structures.

Discussed discrete reductions and modified hierarchies.

Abstract

The $p\times p$ matrix version of the $r$-KdV hierarchy has been recently treated as the reduced system arising in a Drinfeld-Sokolov type Hamiltonian symmetry reduction applied to a Poisson submanifold in the dual of the Lie algebra $\widehat{gl}_{pr}\otimes {\Complex}[\lambda, \lambda^{-1}]$. Here a series of extensions of this matrix Gelfand-Dickey system is derived by means of a generalized Drinfeld-Sokolov reduction defined for the Lie algebra $\widehat{gl}_{pr+s}\otimes {\Complex}[\lambda,\lambda^{-1}]$ using the natural embedding $gl_{pr}\subset gl_{pr+s}$ for $s$ any positive integer. The hierarchies obtained admit a description in terms of a $p\times p$ matrix pseudo-differential operator comprising an $r$-KdV type positive part and a non-trivial negative part. This system has been investigated previously in the $p=1$ case as a constrained KP system. In this paper the previous results are considerably extended and a systematic study is presented on the basis of the Drinfeld-Sokolov approach that has the advantage that it leads to local Poisson brackets and makes clear the conformal ($\cal W$-algebra) structures related to the KdV type hierarchies. Discrete reductions and modified versions of the extended $r$-KdV hierarchies are also discussed.