Infinite-dimensional flag manifolds in integrable systems (original) (raw)
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Theoretical and Mathematical Physics, 2010
We develop a group theory approach for constructing solutions of integrable hierarchies corresponding to the deformation of a collection of commuting directions inside the Lie algebra of upper-triangular Z×Z matrices. Depending on the choice of the set of commuting directions, the homogeneous space from which these solutions are constructed is the relative frame bundle of an infinite-dimensional flag variety or the infinite-dimensional flag variety itself. We give the evolution equations for the perturbations of the basic directions in the Lax form, and they reduce to a tower of differential and difference equations for the coefficients of these perturbed matrices. The Lax equations follow from the linearization of the hierarchy and require introducing a proper analogue of the Baker-Akhiezer function.
Hilbert flag varieties and their K hler structure
Journal of physics, 2002
In this paper we introduce the infinite-dimensional flag varieties associated with integrable systems of the KdV-and Toda-type and we discuss the structure of these manifolds. As an example we treat the Fubini-Study metric on the projective space associated with a separable complex Hilbert space and we conclude by showing that all flag varieties introduced before possess a Kähler structure .
Hilbert flag varieties and their Kähler structure
Journal of Physics A-mathematical and General, 2002
In this paper, we introduce the infinite-dimensional flag varieties associated with integrable systems of the KdV- and Toda-type and discuss the structure of these manifolds. As an example we treat the Fubini-Study metric on the projective space associated with a separable complex Hilbert space and conclude by showing that all flag varieties introduced before possess a Kähler structure.
The Gelfand–Zeitlin integrable system and K-orbits on the flag variety
Progress in Mathematics, 2014
In this expository paper, we provide an overview of the Gelfand-Zeiltin integrable system on the Lie algebra of n × n complex matrices gl(n, C) introduced by Kostant and Wallach in 2006. We discuss results concerning the geometry of the set of strongly regular elements, which consists of the points where Gelfand-Zeitlin flow is Lagrangian. We use the theory of K n = GL(n−1, C)×GL(1, C)-orbits on the flag variety B n of GL(n, C) to describe the strongly regular elements in the nilfiber of the moment map of the system. We give an overview of the general theory of orbits of a symmetric subgroup of a reductive algebraic group acting on its flag variety, and illustrate how the general theory can be applied to understand the specific example of K n and GL(n, C).
The Structure of Hilbert Flag Varieties Dedicated to the memory of our father
Publications of The Research Institute for Mathematical Sciences, 1994
In this paper we present a geometric realization of infinite dimensional analogues of the finite dimensional representations of the general linear group. This requires a detailed analysis of the structure of the flag varieties involved and the line bundles over them. In general the action of the restricted linear group can not be lifted to the line bundles and thus leads to central extensions of this group. It is determined exactly when these extensions are non-trivial. These representations are of importance in quantum field theory and in the framework of integrable systems. As an application, it is shown how the flag varieties occur in the latter context.
New approaches to integrable hierarchies of topological type
Russian Mathematical Surveys
This survey is devoted to a large class of systems of partial differential equations which on the one hand appear in classical problems of mathematical physics and on the other hand provide an efficient tool for the description of enumerative invariants in algebraic geometry. Particular attention is paid to new approaches to these systems, in particular, to the approach proposed in a recent paper of the author. Bibliography: 57 titles.
Holomorphic line bundles over Hilbert flag varieties
Proceedings of symposia in pure mathematics, 1994
In this contribution we present a geometric realization of an infinite dimensional analogue of the irreducible representations of the unitary group. This requires a detailed analysis of the structure of the flag varieties involved and the line bundles over it. These constructions are of importance in quantum field theory and in the framework of integrable systems. As an application, it is shown how they occur in the latter context.
Poisson surfaces and algebraically completely integrable systems
Journal of Geometry and Physics, 2015
One can associate to many of the well known algebraically integrable systems of Jacobians (generalized Hitchin systems, Sklyanin) a ruled surface which encodes much of its geometry. If one looks at the classification of such surfaces, there is one case of a ruled surface that does not seem to be covered. This is the case of projective bundle associated to the first jet bundle of a topologically nontrivial line bundle. We give the integrable system corresponding to this surface; it turns out to be a deformation of the Hitchin system.
On a Theorem of Halphen and its Application to Integrable Systems
Journal of Mathematical Analysis and Applications, 2000
We extend Halphen's theorem which characterizes the solutions of certain nth-order differential equations with rational coefficients and meromorphic fundamental systems to a first-order n × n system of differential equations. As an application of this circle of ideas we consider stationary rational algebro-geometric solutions of the KdV hierarchy and illustrate some of the connections with completely integrable models of the Calogero-Moser-type. In particular, our treatment recovers the complete characterization of the isospectral class of such rational KdV solutions in terms of a precise description of the Airault-McKean-Moser locus of their poles.
Spectral curves, algebraically completely integrable Hamiltonian systems, and moduli of bundles
1995
This is the expanded text of a series of CIME lectures. We present an algebro-geometric approach to integrable systems, starting with those which can be described in terms of spectral curves. The prototype is Hitchin's system on the cotangent bundle of the moduli space of stable bundles on a curve. A variant involving meromorphic Higgs bundles specializes to many familiar