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

Abstract

Abstract. The p × 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 gl pr ⊗ C I [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 gl pr+s ⊗ C I [λ, λ −1 ] using the natural embedding gl pr ⊂ gl pr+s for s any positive integer. The hierarchies obtained admit a description in terms of a p × 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 (W-algebra) structures related to the KdV type hierarchies.Discrete reductions and modified versions of the extended r-KdV hierarchies are also discussed.

Extended matrix Gelfand - Dickey hierarchies: reduction to classical Lie algebras

“…Many of the results are described fully in [9], whilst at the same time much of this theory is standard and goes back to the work of Adler, Gelfand-Dickey and DrinfeldSokolov, see [18], [19] and [10].…”

“…We next apply a two-step factorisation procedure to the Lax operator L that leads to modifications of the flows in (9). By modification, we mean that there is a non-invertible Poisson map given in terms of a differential polynomial formula, from the Poisson space of the new (modified) variables to M DS .…”

“…Lemma 3 was proved in [9]. The modified flows are defined on the phase space Θ by pulling back the Hamiltonians H Q j in (8) by means of the map µ.…”

“…The modified flows are defined on the phase space Θ by pulling back the Hamiltonians H Q j in (8) by means of the map µ. For reasons explained in [9] (see also [1,2]), if s = 0 the factor…”

“…It was shown in [9] how hierarchies with a Lax operator of the form in (1) can be obtained by the Drinfeld-Sokolov (DS) reduction method. Specifically, with the partition n = pr + s = p times r + · · · + r + s times 1 + · · · + 1 (2) is associated [11] a graded Heisenberg subalgebra of the loop algebra gl n ⊗ C I [λ, λ −1 ], and a generalised KdV hierarchy having the Lax operator in (1) results from application of the DS reduction procedure [10] (see also [12,13,14]) with respect to a grade-one element from this Heisenberg subalgebra, if r > 1.…”