Examples of poisson modules, II

Caressa, Paolo

doi:10.1007/bf02921426

Cited by 2 publications

(10 citation statements)

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“…Let (A, •, { , } 0 ) be a Poisson algebra and P 1 ⊂ A a Poisson ideal. If P 0 = A/P 1 is the quotient Poisson algebra, then the trivial extension algebra of P 0 by P 1 is an admissible Poisson algebra [3] with Poisson structure defined by…”

Section: Admissible Poisson Algebrasmentioning

confidence: 99%

“…Here, we define Remark 3.6 An admissible Poisson structure on P 0 ⋉ P 1 does not always exist. A necessary condition is that the P 0 -module P 1 admits a contravariant derivative (Theorem 4.4), which is generally not the case [3].…”

Section: Admissible Poisson Algebrasmentioning

confidence: 99%

“…In this paper, we study Poisson structures on trivial extension algebras of a given Poisson algebra which are compatible with the given Poisson structure in a natural way. Such a class generalizes some known Poisson algebra structures induced by Poisson modules [2,3,4,15]. One of our main motivations is to provide a general algebraic framework for the study of (infinitesimal) Poisson algebras that arise in the problem of constructing first-order approximations of Poisson structures around Poisson submanifolds [12] (see also [11]).…”

Section: Introductionmentioning

confidence: 99%

“…Recently, trivial extension algebras were characterized as some algebras on which every derivation is inner [1]. As is known, Poisson modules are closely related with the notion of flat contravariant derivative [2,3]. This fact was exploited for the study of the semiclassical geometry associated with the quantization of the deformation of vector bundles over Poisson manifolds [2] (see also [4]).…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the data, which we call Poisson triples (Definition 4.1), parameterize admissible Poisson structures and the associated objects. In the case when the contravariant derivative is flat and is the only non-trivial data Poisson structures on P 0 ⋉ P 1 induced by Poisson modules [2,3,4,15] are recovered. We also show that there exists a one-to-one correspondence between admissible Poisson algebras and Lie algebroids [9,10] on Ω 1 P0 ⊕ P 1 satisfying the condition that the natural projection Ω 1 P0 ⊕ P 1 Ω 1 P0 is a Lie algebroid morphism (Theorem 4.4).…”

Section: Introductionmentioning

confidence: 99%

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Poisson Structures on Trivial Extension Algebras

García-Beltrán¹,

Ruíz-Pantaleón

Vorobiev

2021

Preprint

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We present a class of Poisson structures on trivial extension algebras which generalize some known structures induced by Poisson modules and allow us to give an algebraic setting for the so-called infinitesimal Poisson algebras. We show that there exists a one-to-one correspondence between such class of Poisson structures and some data involving (not necessarily flat) contravariant derivatives, and then we give a formulation of this result in terms of Lie algebroids. Some examples coming from Poisson modules and Poisson submanifolds are given.

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Section: Admissible Poisson Algebrasmentioning

confidence: 99%

Section: Admissible Poisson Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Poisson Structures on Trivial Extension Algebras

García-Beltrán¹,

Ruíz-Pantaleón

Vorobiev

2021

Preprint

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Lagrange formal calculus as applied to Lagrange mechanics: An exercise in anachronism

Bersani

Bersani²,

Caressa

2022

Mathematics and Mechanics of Solids

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When Lagrange wrote his masterpiece Mécanique Analytique, the foundations of analysis were not completely understood: to erect the great building of Analytical Mechanics upon solid foundations, the Piedmontese mathematician tried to lay the foundations of differential calculus in a purely algebraic way, using power series instead of functions, regardless about convergence and uniqueness issues. While this foundation was unsatisfactory as shown by Cauchy some decades later, it can shed light on how Lagrange considered the analytical objects (curves, energies, etc.) he dealt with in Mechanics. In this paper, we review these Lagrangian foundations of analysis, and we try to adopt its obvious modern counterpart, i.e., formal power series, to express some results in Analytical Mechanics related to Helmholtz conditions and Rayleigh description of dissipation. By means of purely algebraic manipulations, we will easily recover results otherwise proved by means of modern analysis.

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Examples of poisson modules, II

Cited by 2 publications

References 3 publications

Poisson Structures on Trivial Extension Algebras

Poisson Structures on Trivial Extension Algebras

Lagrange formal calculus as applied to Lagrange mechanics: An exercise in anachronism

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