Extensions of Banach Lie–Poisson spaces

Abstract

The extension of Banach Lie-Poisson spaces is studied and linked to the extension of a special class of Banach Lie algebras. The case of W Ã -algebras is given particular attention. Semidirect products and the extension of the restricted Banach Lie-Poisson space by the Banach Lie-Poisson space of compact operators are given as examples. r

Induced and coinduced Banach Lie–Poisson spaces and integrability

“…This paper continues the investigation of Banach Lie-Poisson spaces introduced in [12] and also studied in [4,13]. The theory of Banach Lie-Poisson spaces gives a natural generalization in the functional analytical context of the Poisson geometry of finite-dimensional integrable Hamiltonian systems.…”

Applications of Hypocontinuous Bilinear Maps in Infinite-Dimensional Differential Calculus

“…Definition 4.9 can be adapted to spaces which are not S-reflexive, along the lines of [29], [30]: Definition 4.5 A locally convex Poisson vector space with respect to S is a locally convex space E that is a k ∞ -space and whose evaluation homomorphism…”

“…If S = c, then η E is a topological embedding automatically in the situation of Definition 4.5 as we assume that E is a k ∞ -space (and hence a k-space). In fact, η E : E → (E Remark 4.8 Since reflexive Banach spaces are rather rare, the more complicated nonreflexive theory cannot be avoided in the study of Banach-Lie-Poisson vector spaces (as in [29], [30]). By contrast, typical non-Banach locally convex spaces are reflexive and hence fall within the simple, basic framework of Definition 4.2.…”

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

“…If we forget about the underlying Banach Lie groups and consider their Banach Lie algebras only, then the maps ϕ and ω satisfying conditions (A.18)-(A.19) with additional smoothness conditions, define the structure of Banach Lie algebra on n ⊕ h, see [1,12].…”

“…The first aim of this paper is to investigate Banach Lie-Poisson spaces related to the restricted Grassmannian Gr res , see [2,12]. The restricted Grassmannian Gr res has its own long story as one of the most important infinite dimensional Kähler manifolds in mathematical physics.…”