Regularity of Lie Groups and Lie's Third Theorem

• Applicant: Dr. Maximilian Hanusch
• Subject Area: Mathematics
• Term: from 2018 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 405865003

Besides their relevance in mathematics, infinite-dimensional Lie groups play an important role in physics, where they occur as phase spaces and symmetry groups. Although particular classes like diffeomorphism-, gauge-, and operator groups are well-understood, the general theory is still in the initial stage. This is essentially due to the absence of a general theory of ODE's in the (Hausdorff) locally convex case. Besides the question under which circumstances a given Lie algebra is enlargible (Lie's third Theorem), regularity properties of Lie groups play a central role in this context. Regularity is concerned with the domain-, continuity-, and smoothness properties of the product integral -- a notion that naturally generalizes the concept of the Riemann integral to Lie groups (Lie algebra-valued curves are thus integrated to Lie group elements). Due to the preparatory work done in prior to, and within the scope of my project, essentially the domain problem as well as parts of the continuity problem are left open in this context. A further issue to be investigated in this project is the question of higher differentiability (not smoothness) of the product integral in the semiregular context. Semiregularity means that all Lie algebra-valued curves of a fixed differentiability class are contained in the domain of the product integral (are integrable). One of my recent results states that the product integral is automatically differentiable in the semiregular case, and the guess is that this statement extends to higher orders of differentiability. Related to this is another aim of this project, namely to generalize Milnor's integrability result for Lie algebra homomorphisms (that originally motivated him to introduce the notion of regularity in the context of infinite-dimensional Lie groups) from the regular to the semiregular case. A further focus of my project is on the enlargibility problem for Lie algebras (Lie's third theorem) in the infinite-dimensional asymptotic estimate and sequentially complete context. The enlargibility problem is concerned with the question under which circumstances a given infinite-dimensional Lie algebra arises as the Lie algebra of a (simply connected) Lie group.

• DFG Programme: Research Grants