Partial Poisson-Lie groups and groupoids. Application to Von Neumann algebras
DOI:
https://doi.org/10.17398/Keywords:
partial Poisson structure, convenient Lie algebroid, convenient partial Lie groupoid, convenient partial Poisson-Lie group, direct limit, von Neumann algebraAbstract
This paper develops a comprehensive framework for Poisson geometry in the infinite-dimensional convenient setting, as introduced by Frölicher, Kriegl and Michor, with a particular focus on Poisson-Lie groups, Lie groupoids, and applications to operator algebras. Building upon the foundational work of Weinstein on Poisson groupoids, Mackenzie and Xu on Lie bialgebroids and Poisson groupoids, Tumpach on Banach-Lie groupoids associated with operator algebras, and Odzijewicz and collaborators on groupoids arising from W*-algebras, we extend several classical finite-dimensional constructions to the convenient framework. After reviewing the fundamental properties of partial Poisson manifolds and establishing several complementary results, we introduce and study convenient partial Poisson-Lie groups. We then extend key arguments of Weinstein on Poisson-Lie groupoids to the infinite-dimensional context, identifying the results that remain valid beyond the finite-dimensional setting. Further developments include the extension of fundamental constructions from Lie groupoid theory, notably the canonical identification between the conormal bundle of the unit space and the dual of the associated Lie algebroid established by Mackenzie and Xu. Building on these foundations, we construct sub-Poisson structures on cotangent bundles of Lie groupoids and investigate their behavior under direct limits of finite-dimensional Poisson groupoids. The theory is illustrated through explicit examples. As an application, we consider groupoids associated with von Neumann algebras and endow them with natural Poisson-Lie structures. An appendix collects the background material on W*-algebras required for the development of the theory.
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