We develop a formalism of multicomponent BKP hierarchies using elementary geometry of spinors. The multicomponent KP and the modified KP hierarchy (hence all their reductions like KdV, NLS, AKNS or DS) are reductions of the multicomponent BKP.
The remarkable link between the soliton theory and the group GL ∞ was discovered in the early 1980s by Sato [S] and developed, making use of the spinor formalism, by Date, Jimbo, Kashiwara and Miwa [DJKM1,2,3], [JM]. The basic object that they considered is the KP hierarchy of partial differential equations, which they study through a sequence of equivalent formulations that we describe below. The first formulation is a deformation (or Lax) equation of a formal pseudo-differential operator L = ∂ + u 1 ∂ −1 + u 2 ∂ −2 + . . .
Abstract. 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.
We define Lie algebroids over infinite jet spaces and establish their equivalent representation through homological evolutionary vector fields. Keywords: Lie algebroid, BRST-differential, Poisson structure, integrable systems, string theory.Introduction. The construction of Lie algebroids over smooth manifolds is important in differential geometry (particularly, in Poisson geometry) and appears in various models of mathematical physics (e.g., in Poisson sigmamodels). We extend the classical definition of Lie algebroids over smooth manifolds [1] to the construction of variational Lie algebroids over infinite jet spaces. We define these structures in a standard way via vector bundles and also through homological vector fields Q 2 = 0 on infinite jet super-bundles, then proving the equivalence. Our generalization of the classical construction manifestly respects the geometry which appears under mappings between smooth manifolds. For this reason, the variational picture, which we develop here, more fully grasps the geometry of strings in space-time [2].First, we very briefly recall the definition of classical Lie algebroids; we refer to [1] for more details (see also [3] or the surveys [4] and references therein). The standard examples of Lie algebroids over usual manifolds are the tangent bundle or the Poisson algebroid structure of the cotangent bundle
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