Fernand Pelletier

On partial Banach-Lie algebroid structure: some motivations

In finite dimension it is well known that the cotangent bundle of a Poisson manifold is provided with a natural structure of Lie algebroid. On the other hand the prolongation of a Lie algebroid can be provided with a structure of Banach Lie algebroid. After looking for problems we meet for an adaptation for such results in the Banach setting, first, we will propose a generalization of finite Poisson manifold by the notion partial Poisson Banach manifold

$M$ , which is defined by a Poisson anchor on a weak subbundle

$T^\flat M$ of

$T^*M$ . In this context, it will appear that we obtain only a "partial structure of Banach-Lie algebroid" on

$T^\flat M$ and not a Banach-Lie algebroid structure even if

$T^\flat M=T^*M$ .

For analog reasons, although the prolongation of a Banach Lie algebroid is naturally provided with an anchor, the Lie bracket can be lift onto the prolongation but gives rise only a "partial of Banach-Lie algebroid" structure on this prolongation but is not a Banach-Lie algebroid structure.

XXXIX Workshop on Geometric Methods in Physics	19.06–25.06.2022
XI School on Geometry and Physics	27.06–1.07.2022

Event sponsored by:
University of Bialystok