Correspondence Between Infinite-dimensional Poisson-Lie Groups and Lie Bialgebras

Explore a 19-minute mathematics lecture that delves into the generalization of Poisson-Lie groups and Lie bialgebras in infinite-dimensional settings. Learn how the classical Drinfeld correspondence, which establishes one-to-one relationships between Poisson structures on one-connected finite-dimensional Lie groups and Lie bialgebra structures on their Lie algebras, extends to infinite dimensions. Discover the application of this framework to regular Lie groups modeled on Fréchet or Silva locally convex topological vector spaces, including practical examples such as smooth and analytic loop groups of finite-dimensional Lie groups and diffeomorphism groups of certain manifolds. Presented as part of the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" at the Erwin Schrödinger International Institute for Mathematics and Physics.