Hierarchy of Hamilton equations on Banach Lie–Poisson spaces related to restricted Grassmannian

Goliński, Tomasz; Odzijewicz, Anatol

doi:10.1016/j.jfa.2010.01.019

Cited by 11 publications

(15 citation statements)

References 23 publications

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“…where the trace is defined on L 1,2 (H) by Tr A = Tr p + A| H + + Tr p − A| H − (called the restricted trace in [GO10]). According to Proposition 2.1 in [GO10], one has Tr AB = Tr BA for any A ∈ L 1,2 (H) and any B ∈ L res (H).…”

Section: Schatten Ideal L 1 (H) Of Trace Class Operators For a Boundmentioning

confidence: 99%

Banach Poisson–Lie Groups and Bruhat–Poisson Structure of the Restricted Grassmannian

Tumpach

2020

Commun. Math. Phys.

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In this paper, we construct a (generalized) Banach Poisson-Lie group structure on the unitary restricted Banach Lie group acting transitively on the restricted Grassmannian. A "dual" Banach Lie group consisting of (a class of) upper triangular bounded operators admits also a Poisson-Lie group structure. We show that the restricted Grassmannian inherits a Bruhat-Poisson structure from the unitary Banach Lie group, and that the action of the dual Banach Lie group on it (by "dressing transformations") is a Poisson map. This action generates the KdV hierarchy as explained in [SW85], and its orbits are the Schubert cells of the restricted Grassmannian as described in [PS88].

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Section: Schatten Ideal L 1 (H) Of Trace Class Operators For a Boundmentioning

confidence: 99%

Banach Poisson–Lie Groups and Bruhat–Poisson Structure of the Restricted Grassmannian

Tumpach

2020

Commun. Math. Phys.

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“…One of the key features of this new theory is that Banach Poisson manifolds provide an appropriate setting for an unified approach to the Hamiltonian and the quantum mechanical description of physical systems. In addition to the seminal article [20], we also refer the reader to the monograph [19] for a detailed exposition on Banach Poisson manifolds; meanwhile several interesting examples and applications can be found in [5,6,15]. An important class of infinite dimensional linear Poisson manifolds is given by Banach Lie-Poisson spaces: a Banach space b is called a Banach Lie-Poisson space if b * is a Banach Lie algebra endowed with a Lie bracket [ · , · ] such that ad *…”

Section: Finite Rank Orbits As Symplectic Leavesmentioning

confidence: 99%

Section: Finite Rank Orbits As Symplectic Leavesmentioning

confidence: 99%

The compatible Grassmannian

Andruchow

Chiumiento

Lucero

2014

Differential Geometry and its Applications

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Let A be a positive injective operator in a Hilbert space (H, < , >), and denote by [ , ] the inner product defined by A: [f, g] =< Af, g >. A closed subspace S ⊂ H is called Acompatible if there exists a closed complement for S, which is orthogonal to S with respect to the inner product [ , ]. Equivalently, if there exists a necessarily unique idempotent operator Q S such that R(Q S ) = S, which is symmetric for this inner product. The compatible Grassmannian Gr A is the set of all A-compatible subspaces of H. By parametrizing it via the one to one correspondence S ↔ Q S , this set is shown to be a differentiable submanifold of the Banach space of all operators in H which are symmetric with respect to the form [ , ]. A Banach-Lie group acts naturally on the compatible Grassmannian, the group of all invertible operators in H which preserve the form [ , ]. Each connected component in Gr A of a compatible subspace S of finite dimension, turns out to be a symplectic leaf in a Banach Lie-Poisson space. For 1 ≤ p ≤ ∞, in the presence of a fixed [ , ]-orthogonal decomposition of H, H = S 0+ N 0 , we study the restricted compatible Grassmannian (an analogue of the restricted, or Sato Grassmannian). This restricted compatible Grassmannian is shown to be a submanifold of the Banach space of p-Schatten operators which are symmetric for the form [ , ]. It carries the locally transitive action of the Banach-Lie group of invertible operators which preserve [ , ], and are of the form G = 1 + K, with K in p-Schatten class. The connected components of this restricted Grassmannian are characterized by means of of the Fredholm index of pairs of projections. Finsler metrics which are isometric for the group actions are introduced for both compatible Grassmannians, and minimality results for curves are proved.2010 MSC: 58B10, 58B20, 47B10.

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“…In the paper [GO10] we have constructed a hierarchy of Hamiltonian integrable systems on L 2 (H − , H + ), where H − and H + are complex separable Hilbert spaces (finite or infinite dimensional) and L 2 denotes the class of Hilbert-Schmidt operators. In this paper we will restrict our considerations to the subcase when both H − and H + are finite dimensional.…”

Section: Hierarchy Of Hamiltonian Integrable Systems On Mat M ×N (C)mentioning

confidence: 99%

“…In this paper we study a hierarchy of integrable Hamiltonian systems on the space of linear maps L(H − , H + ) between two complex finite dimensional Hilbert spaces H − and H + which is the particular case of a more general hierarchy defined on the Banach Lie-Poisson space related to the restricted Grassmannian, see [GO10]. This hierarchy, as we will show, can be used to describtion of the variation of plenary wave envelopes through the dielectric nonlinear medium.…”

Section: Introductionmentioning

confidence: 99%