18 research outputs found
Geometric Bäcklund-Darboux transformations for the KP hierarchy
In this paper it is shown that, if you have two planes in the Sato Grassmannian that have an intersection of finite codimension, then the corresponding solutions of the KP hierarchy are linked by Bäcklund-Darboux (shortly BD-)transformations. The pseudodifferential operator that performs this transformation is shown to be built up in a geometric way from so-called elementary BD-transformations and is given here in a closed form. The corresponding action on the tau-function, associated to a plane in the Grassmannian, is also determined
CKP Hierarchy, Bosonic Tau Function and Bosonization Formulae
We develop the theory of CKP hierarchy introduced in the papers of Kyoto
school [Date E., Jimbo M., Kashiwara M., Miwa T., J. Phys. Soc. Japan 50
(1981), 3806-3812] (see also [Kac V.G., van de Leur J.W., Adv. Ser. Math.
Phys., Vol. 7, World Sci. Publ., Teaneck, NJ, 1989, 369-406]). We present
appropriate bosonization formulae. We show that in the context of the CKP
theory certain orthogonal polynomials appear. These polynomials are polynomial
both in even and odd (in Grassmannian sense) variables
Irreducible Highest Weight Representations Of The Simple n-Lie Algebra
A. Dzhumadil'daev classified all irreducible finite dimensional
representations of the simple n-Lie algebra. Using a slightly different
approach, we obtain in this paper a complete classification of all irreducible,
highest weight modules, including the infinite-dimensional ones. As a corollary
we find all primitive ideals of the universal enveloping algebra of this simple
n-Lie algebra.Comment: 24 pages, 24 figures, mistake in proposition 2.1 correcte
A geometric derivation of KdV-type hierarchies from root systems
For the root system of each complex semi-simple Lie algebra of rank two, and
for the associated 2D Toda chain , we calculate the two
first integrals of the characteristic equation on . Using the
integrals, we reconstruct and make coordinate-independent the -matrix operators in total derivatives that factor symmetries of
the chains. Writing other factorizations that involve the operators ,
we obtain pairs of compatible Hamiltonian operators that produce KdV-type
hierarchies of symmetries for \cE. Having thus reduced the problem to the
Hamiltonian case, we calculate the Lie-type brackets, transferred from the
commutators of the symmetries in the images of the operators onto
their domains. With all this, we describe the generators and derive all the
commutation relations in the symmetry algebras of the 2D Toda chains, which
serve here as an illustration for a much more general algebraic and geometric
set-up.Comment: Proc. 4th International workshop `Group analysis of differential
equations and integrable systems' (Protaras, Cyprus, October 26-29, 2008), 19
pages
Polynomial Tau-Functions for the Multicomponent KP Hierarchy
In a previous paper we constructed all polynomial tau-functions of the 1-component KP hierarchy, namely, we showed that any such tau-function is obtained from a Schur polynomial s_λ(t) by certain shifts of arguments. In the present paper we give a simpler proof of this result, using the (1-component) boson–fermion correspondence. Moreover, we show that this approach can be applied to the s-component KP hierarchy, using the s-component boson–fermion correspondence, finding thereby all its polynomial tau-functions. We also find all polynomial tau-functions for the reduction of the s-component KP hierarchy, associated to any partition consisting of s positive parts