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

Abstract

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.