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

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Abstract

Thep×p matrix version of ther-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\(\widehat{gl}_{pr} \) ⊗ℂ[λ,λ−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\(\widehat{gl}_{pr + s} \) ⊗ℂ[λ,λ−1] using the natural embeddinggl pr ⊂glpr+s fors any positive integer. The hierarchies obtained admit a description in terms of ap×p matrix pseudo-differential operator comprising anr-KdV type positive part and a non-trivial negative part. This system has been investigated previously in thep=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 (W-algebra) structures related to the KdV type hierarchies.

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  1. László Fehér

    Present address: Theoretical Physics Department of Szeged University, H-6720, Szeged, Hungary

Authors and Affiliations

  1. Department of Physics, University of Wales Swansea, Singleton Park, SA2 8PP, Swansea, U.K.

    László Fehér

  2. Department of Mathematics, Leeds University, LS2 9JT, Leeds, U.K.

    Ian Marshall

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  1. László Fehér
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  2. Ian Marshall
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Communicated by M. Jimbo

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Fehér, L., Marshall, I. Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction. Commun.Math. Phys. 183, 423–461 (1997). https://doi.org/10.1007/BF02506414

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  • DOI: https://doi.org/10.1007/BF02506414

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