Generalized Drinfeld-Sokolov Reductions and KdV Type Hierarchies

TL;DR

This paper generalizes Drinfeld-Sokolov reductions to construct new KdV hierarchies using Hamiltonian methods, revealing specific algebraic structures and extending known scalar cases to matrix versions.

Contribution

It introduces a Hamiltonian framework for generalized DS reductions, identifying conditions for regular elements and deriving matrix Gelfand-Dickey hierarchies from algebraic partitions.

Findings

Existence of graded regular elements only for specific partitions of n

Reduction for grade 1 regular elements yields matrix Gelfand-Dickey hierarchies

Hamiltonian approach simplifies the derivation of these hierarchies

Abstract

Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local reductions of Hamiltonian flows generated by monodromy invariants on the dual of a loop algebra. Following earlier work of De Groot et al, reductions based upon graded regular elements of arbitrary Heisenberg subalgebras are considered. We show that, in the case of the nontwisted loop algebra $\ell(gl_n)$, graded regular elements exist only in those Heisenberg subalgebras which correspond either to the partitions of $n$ into the sum of equal numbers $n=pr$ or to equal numbers plus one $n=pr+1$. We prove that the reduction belonging to the grade $1$ regular elements in the case $n=pr$ yields the $p\times p$ matrix version of the Gelfand-Dickey $r$-KdV hierarchy, generalizing the scalar case $p=1$ considered by DS. The methods of DS are utilized throughout the analysis, but formulating the reduction entirely within the Hamiltonian framework provided by the classical r-matrix approach leads to some simplifications even for $p=1$.