30 research outputs found
An analytic description of the vector constrained KP hierarchy
In this paper we give a geometric description in terms of the Grassmann
manifold of Segal and Wilson, of the reduction of the KP hierarchy known as the
vector -constrained KP hierarchy. We also show in a geometric way that these
hierarchies are equivalent to Krichever's general rational reductions of the KP
hierarchy.Comment: 15 pages, Latex2
The vector k-constrained KP hierarchy and Sato's Grassmannian
We use the representation theory of the infinite matrix group to show that
(in the polynomial case) the --vector --constrained KP hierarchy has a
natural geometrical interpretation on Sato's infinite Grassmannian. This
description generalizes the the --reduced KP or Gelfand--Dickey hierarchies.Comment: 15 pages, AMSTe
Darboux transformations for the KP hierarchy in the Segal-Wilson setting
In this paper it is shown that inclusions inside the Segal-Wilson Grassmannian give rise to Darboux transformations between the solutions of the -hierarchy corresponding to these planes. We present a closed form of the operators that procure the transformation and express them in the related geometric data. Further the associated transformation on the level of -functions is given. \u
The construction of Frobenius manifolds from KP tau-functions
Frobenius manifolds (solutions of WDVV equations) in canonical coordinates are determined by the system of Darboux–Egoroff equations. This system of partial differential equations appears as a specific subset of the n-component KP hierarchy. KP representation theory and the related Sato infinite Grassmannian are used to construct solutions of this Darboux–Egoroff system and the related Frobenius manifolds. Finally we show that for these solutions Dubrovin's isomonodromy tau-function can be expressed in the KP tau-function
Denominator identities for finite-dimensional Lie superalgebras and Howe duality for compact dual pairs
We provide formulas for the denominator and superdenominator of a basic
classical type Lie superalgebra for any set of positive roots. We establish a
connection between certain sets of positive roots and the theory of reductive
dual pairs of real Lie groups. As an application of our formulas, we recover
the Theta correspondence for compact dual pairs. Along the way we give an
explicit description of the real forms of basic classical type Lie
superalgebras.Comment: Latex, 75 pages. Minor corrections. Final version, to appear in the
Japanese Journal of Mathematic
Extremal projectors for contragredient Lie (super)symmetries (short review)
A brief review of the extremal projectors for contragredient Lie
(super)symmetries (finite-dimensional simple Lie algebras, basic classical Lie
superalgebras, infinite-dimensional affine Kac-Moody algebras and
superalgebras, as well as their quantum -analogs) is given. Some
bibliographic comments on the applications of extremal projectors are
presented.Comment: 21 pages, LaTeX; typos corrected, references adde