Hilbert flag varieties and their Kähler structure (original) (raw)
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 .
Infinite-dimensional flag manifolds in integrable systems
Acta Applicandae Mathematicae, 1995
In this paper we present several instances where infinite dimensional flag varieties and their holomorphic line bundles play a role in integrable systems. As such, we give the correspondance between flag varieties and Darboux transformations for the K P-hierarchy and the n-th KdV-hierarchy. We construct solutions of the n-th M KdVhierarchy from the space of periodic flags and we treat the geometric interpretation of the Miura transform. Finally we show how the group extension connected with these line bundles shows up at integrable deformations of linear systems on ސ 1 .
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.
The structure of Hilbert flag varieties
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.
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.
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.