arXiv subjects

Fernand Pelletier

Publications and source records attributed to Fernand Pelletier.

At least 19 recordsLinked to original sources

arXiv2026

Flows and Homotopies of Banach Lie algebroids

We establish flow and homotopy tools for Banach Lie algebroids that do not rely on compactly supported extensions. Under suitable hypotheses, we construct parameter-dependent local extensions of sections along algebroid paths, complete lifts and their fibrewise-linear evolutions, and a Bochner-integral variation-of-constants formula. We then prove that an admissible-path homotopy is equivalently a Lie algebroid morphism from the parameter square and a solution of the associated transport equation. This provides the extension, transport and infinitesimal variation theory required for a local integration program of Banach Lie algebroids to be developed in a sequel.

math.DG↗

arXiv2025

Partial Poisson Lie groups and groupoids. Application to Von Neumann algebras

The purpose of this paper is to propose a version of the notion of convenient Lie groupoid as a generalization of this concept in finite dimension. The authors point out which obstructions appear in the infinite dimensional context and how an adapted notion of "bi-algebroid " in finite dimension (cf. \cite{MaXu94}) can be, nevertheless, recovered. This paper is self-contained and recalls some important properties of "partial Poisson manifolds" (cf. \cite{CaPe23}, Chapter~7) and "Banach Poisson Lie groups" (cf. \cite{Tum20}) needed for their purpose. The paper also gives an illustration of these concepts from all the results of A.~Odzijewicz and his collaborators on Lie groupoids and Von Neumann algebras.

math.DG↗

arXiv2025

Partial Dirac Structures and Dynamical Systems

In a previous paper (PeCa24), the notion of Dirac structure in finite dimension was extended to the convenient setting. In particular, we introduce the notion of \emph{partial Dirac structure on a convenient manifold} and look for which all geometrical results in finite dimension which are still true in this infinite dimensional framework. Note that this context is justified by many mechanical infinite dimensional examples which recover all the classical ones, such as Hilbert, Banach, Fr\'{e}chet context or direct limits of Banach spaces. Another reason is that if we want to extend the variational technics, as in YoMa06II, to the infinite dimensional setting, the category of convenient vector spaces is cartesian closed, which is not the case for the category of locally convex vector spaces and so this variational approach does not work in this last setting. In this second part, first, we look for an adaptation of the results obtained in YoMa06II for a variational approach of constraint Lagrangian on a subbundle of a Banach manifold. Then we study the same type of problem but for constraint Lagrangians on a \textit{singular} distribution. Theses results are finally applied to the characterization of normal geodesics for a conical Finsler metric on a Banach manifold.

math.DG↗

arXiv2024

Partial Dirac structures in infinite dimension

In this paper the notion of Dirac structure in finite dimension is extended to the convenient setting. In particular, we introduce the notion of partial Dirac structure on convenient Lie algebroids and manifolds. We then look for those structures whose classical geometrical results in finite dimension can be extended to this infinite dimensional context. Finally, we are interested in the projective and direct limits of such structures.

math.DG↗

arXiv2024

Transitive Lie algebroids and \textbf{Q}-manifolds

We introduce the notions of locally trivial \textbf{Q}-groupoid and principal \textbf{Q}-bundle, which are the appropriate versions of locally trivial Lie groupoids and principal fibre bundles in the framework of R. Barre's \textbf{Q}-manifolds. As an application, we prove that every transitive Lie algebroid over a second countable, smooth manifold arises from the Atiyah sequence of a certain principal \Q-bundle. Consequently, transitive Lie algebroids over second countable, smooth manifolds are integrated to locally trivial \textbf{Q}-groupoids.

math.DG↗

arXiv2024

Partial Nambu-Poisson structures on a convenient manifold

This paper offers an adaptation to the convenient setting of finite dimensional Nambu-Poisson structures. In particular, for partial Nambu structures, we look for those whose classical geometrical results in finite dimension can be extended to this infinite dimensional context. Finally, we are interested in the projective and direct limits of such structures.

math.DG↗

arXiv2023

Structure of finite dimensional linear bundles morphisms

Let $p_E : E \to M$ be a fibre bundle over the $m$-dimensional manifold $M$ whose typical fibre is the vector space $\R^e$ and let $p_F : F \to N$ be a fibre bundle over the $n$-dimensional manifold $N$ whose typical fibre is the vector space $\R^f$. We are interested in the structure of the set $\Morph(E,F)$ of smooth linear bundle morphisms $\Phi : E \to F$ above smooth maps $\varphi : M \to N$.

math.DG↗

arXiv2023

Smooth Banach structure on orbit spaces and leaf spaces

We investigate the quotients of Banach manifolds with respect to free actions of pseudogroups of local diffeomorphisms. These quotient spaces are called H-manifolds since the corresponding simply transitive action of the pseudogroup on its orbits is regarded as a homogeneity condition. The importance of these structures stems from the fact that for every regular foliation without holonomy of a Banach manifold, the corresponding leaf space has the natural structure of an H-manifold. This is our main technical result, and one of its remarkable consequences is an infinite-dimensional version of Sophus Lie's third fundamental theorem, to the effect that every real Banach-Lie algebra can be integrated to an H-group, that is, a group object in the category of H-manifolds. In addition to these general results we discuss a wealth of examples of H-groups which are not Banach-Lie groups.

math.DG↗

arXiv2022

Structures bihamiltoniennes partielles

We introduce three notions of partial bihamiltonian structures ($\operatorname{PQ}$, $\operatorname{PN}$ et $\operatorname{P\Omega}$) in the convenient setting defined by Fr\"{o}licher, Kriegl and Michor. We study geometrical objects linked with these structures and give examples in finite and infinite dimensions.

math.DG↗

arXiv2021

A survey on partial Nambu-Poisson structures in the convenient setting

This paper is a survey (may be incomplete) on partial Nambu-Poisson structures in infinite dimension, mainly in the convenient setting. These ones can be seen as a generalization of both partial Poisson and Nambu-Poisson structures. We also study the properties of the associated characteristic distribution. Finally, we are interested in the projective and direct limits of such structures.

math.DG↗

arXiv2021

Regulated curves on a Banach manifold and singularities of endpoint map. I. Banach manifold structure

We consider regulated curves in a Banach bundle whose projection on the basis is continuous with regulated derivative. We build a Banach manifold structure on the set of such curves. This result was previously obtained for the case of strong Riemannian Banach manifold and absolutely continuous curves in arXiv:1612.02604. The essential argument used was the existence of a "local addition" on such a manifold. Our proof is true for any Banach manifold. In the second part of the paper the problems of controllability will be discussed.

math.FA↗

arXiv2021

An integrability criterion for a projective limit of Banach distributions

We give an integrability criterion for a projective limit of Banach distributions on a Fr\'echet manifold which is a projective limit of Banach manifolds. This leads to a result of integrability of projective limit of involutive bundles on a projective sequence of Banach manifolds. This can be seen as a version of Frobenius Theorem in Fr\'echet setting. As consequence, we obtain a version of the third Lie theorem for a Fr\'echet-Lie group which is a submersive projective limit of Banach Lie groups. We also give an application to a sequence of prolongations of a Banach Lie algebroid.

math.DG↗

arXiv2020

Prolongations of convenient Lie algebroids

We first define the concept of Lie algebroid in the convenient setting. In reference to the finite dimensional context, we adapt the notion of prolongation of a Lie algebroid over a fibred manifold to a convenient Lie algebroid over a fibred manifold. Then we show that this construction is stable under projective and direct limits under adequate assumptions.

math.DG↗

arXiv2020

Projective limit of a sequence of compatible weak symplectic forms on a sequence of Banach bundles and Darboux Theorem

Given a projective sequence of Banach bundles, each one provided with a of weak symplectic form, we look for conditions under which, the corresponding sequence of weak symplectic forms gives rise to weak symplectic form on the projective limit bundle. Then we apply this results to the tangent bundle of a projective limit of Banach manifolds. This naturally leads to ask about conditions under which the Darboux Theorem is also true on the projective limit of Banach manifolds. We will give some necessary and some sufficient conditions so that such a result is true. Then we discuss why, in general, the Moser's method can not work on projective limit of Banach weak symplectic Banach manifolds without very strong conditions like Kumar 's results ([17]). In particular we give an example of a projective sequence of weak symplectic Banach manifolds on which the Darboux Theorem is true on each manifold, but is not true on the projective limit of these manifolds.

math.DG↗

arXiv2018

Symmetries of special 2-flags

This work is a continuation of authors' research interrupted in the year 2010. Derived are recursive relations describing for the first time all infinitesimal symmetries of special 2-flags (sometimes also misleadingly called `Goursat 2-flags'). When algorithmized to the software level, they will give an answer filling in the gap in knowledge as of 2010: on one side the local finite classification of special 2-flags known in lengths not exceeding four, on the other side the existence of a continuous numerical modulus of that classification in length seven.

math.DG↗

arXiv2018

Banach-Lie groupoids and generalized inversion

We study a few basic properties of Banach-Lie groupoids and algebroids, adapting some classical results on finite dimensional Lie groupoids. As an illustration of the general theory, we show that the notion of locally transitive Banach-Lie groupoid sheds fresh light on earlier research on some infinite-dimensional manifolds associated with Banach algebras.

math.FA↗

arXiv2017

Integrability on Direct Limit Banach manifolds

This paper is devoted to the framework of direct limit of anchored Banach bundles over a convenient manifold which is a direct limit of Banach manifold. In particular we give a criterion of integrability for distributions on such convenient manifolds which are locally direct limits of particular sequences of Banach anchor ranges.

math.DG↗

arXiv2014

Configurations of an Articulated Arm and Singularities of Special Multi-Flags

P. Mormul has classified the singularities of special multi-flags in terms of "EKR class" encoded by sequences $j_1,\dots, j_k$ of integers (see [Singularity Theory Seminar, Warsaw University of Technology, Vol. 8, 2003, 87-100] and [Banach Center Publ., Vol. 65, Polish Acad. Sci., Warsaw, 2004, 157-178]). However, A.L. Castro and R. Montgomery have proposed in [Israel J. Math. 192 (2012), 381-427] a codification of singularities of multi-flags by RC and RVT codes. The main results of this paper describe a decomposition of each "EKR" set of depth $1$ in terms of RVT codes as well as characterize such a set in terms of configurations of an articulated arm. Indeed, an analogue description for some "EKR" sets of depth $2$ is provided. All these results give rise to a complete characterization of all "EKR" sets for $1\leq k\leq 4$.

math.DS↗