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

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 ${\widehat{gl}}_{pr}\otimes \mathbb{C}[\lambda ,{\lambda }^{-1}]$. 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 ${\widehat{gl}}_{pr+s}\otimes \mathbb{C}[\lambda ,{\lambda }^{-1}]$ using the natural embedding $g{l}_{pr}\subset g{l}_{pr+s}$ 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 ($\mathcal{W}$-algebra) structures related to the KdV type hierarchies. Discrete reductions and modified versions of the extended $r$-KdV hierarchies are also discussed.

60 pages, plain TEX

References in corpus (8)

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Cited by in corpus (8)

• Tau-Functions generating the Conservation Laws for Generalized Integrable Hierarchies of KdV and Affine-Toda type
• Nonstandard Drinfeld-Sokolov reduction
• Virasoro symmetry of the constrained multi-component KP hierarchy and its integrable discretization
• Infinite-dimensional prolongation Lie algebras and multicomponent Landau-Lifshitz systems associated with higher genus curves
• A note on the appearance of self-dual Yang-Mills fields in integrable hierarchies
• Gauging of Geometric Actions and Integrable Hierarchies of KP Type
• Virasoro symmetries of Multi-Component Gelfand-Dickey systems
• Extended matrix Gelfand-Dickey hierarchies: reduction to classical Lie algebras