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Generalized Drinfeld-Sokolov reductions and KdV type hierarchies

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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 ℓ(gl n ), graded regular elements exist only in those Heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn=pr or to equal numbers plus onen=pr+1. We prove that the reduction belonging to the grade 1 regular elements in the casen=pr yields thep×p matrix version of the Gelfand-Dickeyr-KdV hierarchy, generalizing the scalar casep=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 forp=1.

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Author notes

  1. L. Fehér

    Present address: Bolyai Institute of Szeged University, H-6720, Szeged, Hungary

  2. I. Marshall

    Present address: Department of Mathematics, Leeds University, LS2 9JT, Leeds, UK

Authors and Affiliations

  1. Laboratoire de physique nucléaire, Université de Montréal, C.P. 6128, H3C 3J7, Montréal, Canada

    L. Fehér

  2. Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke W., H4B 1R6, Montréal, Canada

    J. Harnad & I. Marshall

  3. Centre de recherches mathématiques, Université de Montréal, C.P. 6128-A, H3C 3J7, Montréal, Canada

    L. Fehér, J. Harnad & I. Marshall

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  1. L. Fehér
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  2. J. Harnad
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  3. I. Marshall
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Communicated by A. Jaffe

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Fehér, L., Harnad, J. & Marshall, I. Generalized Drinfeld-Sokolov reductions and KdV type hierarchies. Commun.Math. Phys. 154, 181–214 (1993). https://doi.org/10.1007/BF02096838

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  • DOI: https://doi.org/10.1007/BF02096838

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