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

• Final Report Year: 2023

Final Report Abstract

In the project, the regularity problem was completely solved in the asymptotic estimate (AE) context – only in the k = 0 case, additionally sequentially completeness of the Lie group had to be assumed. • It was worked out that (in the general context) the essential part of the regularity problem is semiregularity. Indeed, C k -semiregularity for k ∈ N ∪ {∞} (i.e., definedness of the evolution map on all Lie algebra-valued C k -curves) already implies Mackey k-continuity of the evolution map, which in turn implies its differentiability (the given continuity properties then carry over to the differential of the evolution map as this explicitly given in in terms of differentials of the Lie group operations as well as Riemann integrals over Lie algebra-valued curves). In the Fréchet case, Mackey k-continuity is even equivalent to C k -continuity, so that C k -regularity and C k -semiregularity are equivalent in this case. Altogether, this also indicates that the continuity property that is core relevant to the regularity problem is Mackey k-continuity (which is in the generic case significantly weaker than C k -continuity). A conceptionally different approach was used to obtain semiregularity results. Based on the (comprehensive) investigations of the Lax-equation as well as the introduction of a certain integral transformation between spaces of curves (continuous to smooth Lie algebra-valued curves), it was shown that in the sequentially complete AE context C ∞ -semiregularity already implies C 0 - semiregularity. In a certain sense, the additional condition for k = 0 on the Lie group, namely that the Lie group is sequentially complete, had been replaced by the (analogous) condition that the Lie algebra is sequentially complete (in the Banach case, it is indeed easy to see that sequentially completeness of the modeling space implies sequentially completeness of the Lie group. But, already in the generic AE-case this is not ad hoc at all). Also the first (and most fundamental) step was done to generalize Duistermaat and Kolk’s Lie group construction (Lie’s third theorem) to the sequentially complete AE context. Specifically, the results obtained for the Lax equation are applied in the same paper to equip the space of Lie algebra-valued C k -curves with a Lie group structure analogous to that in a paper from 2023. They following results were not part of the original project plan. They naturally emerged from the investigations of the Lax equation: • The extensive analysis of the Lax equation in the sequentially complete AE context (e.g., Banach-case), including existence, uniqueness and algebraic compatibility properties of the solutions, as well as an explicit representation of the solutions in terms of iterated Riemann integrals over nested commutators. • The generalization of the Baker-Campbell-Dynkin-Hausdorff formula in the Banach/weakly regular nilpotent case (even in finite dimensions unknown so far). Additionally, the resulting explicit representation of the product integral in terms of iterated Riemann integrals over nested commutators (e.g., holonomies in principal fibre bundles). • The generalization of Seeley’s classic extension theorem to infinite dimensions (including additional enhancements).

Publications

• Differentiability of the evolution map and Mackey continuity. Forum Mathematicum, 31(5), 1139-1177.
  Hanusch, Maximilian
  
• The regularity problem for Lie groups with asymptotic estimate Lie algebras. Indagationes Mathematicae, 31(1), 152-176.
  Hanusch, Maximilian
  
• The Strong Trotter Property for Locally µ-convex Lie Groups, J. Lie Theory 30 (2020), 1, 025-032
  M. Hanusch
  
• A -seeley-extension-theorem for Bastiani’s differential calculus. Canadian Journal of Mathematics, 75(1), 170-201.
  Hanusch, Maximilian
  
• Regularity of Lie groups. Communications in Analysis and Geometry, 30(1), 53-152.
  Hanusch, Maximilian
  
• Decompositions of analytic 1-manifolds. Indagationes Mathematicae, 34(4), 752-811.
  Hanusch, Maximilian
  
• The Lax equation and weak regularity of asymptotic estimate Lie groups. Annals of Global Analysis and Geometry, 63(3).
  Hanusch, Maximilian