Generalized Drinfeld-Sokolov reductions and KdV type hierarchies

L. Fehér, J. Harnad, I. Marshall

Research output: Contribution to journalArticle

42 Citations (Scopus)


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 ℓ(gln), 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×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.

Original languageEnglish
Pages (from-to)181-214
Number of pages34
JournalCommunications in Mathematical Physics
Issue number1
Publication statusPublished - May 1 1993

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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