Fundamental problems in the theory of infinite-dimensional Lie groups. (English) Zbl 1109.22014

Glöckner, Helge

J. Geom. Symmetry Phys. 5, 24-35 (2006).

In the preprint [On infinite-dimensional Lie groups, Preprint, Institute for Advanced Studies, Princeton, 1982] J. Milnor formulated various fundamental questions concerning infinite-dimensional Lie groups. In the paper under review, some of the answers (and partial answers) obtained in the preceding years are described. The list of the questions which are discussed in the paper consists of the following:
1. Does every Lie group have a smooth exponential map?
2. If a Lie group \(G\) is locally exponential (or analytic), does it follow that \(g\) is BCH?
3. If two simply connected Lie groups \(G\) and \(H\) have isomorphic Lie algebras, does it follow that \(G\cong H\)?
4. Is a continuous homomorphism between Lie groups necessarily smooth?
5. Is the kernel of a homomorphism necessarily a Lie subgroup?
6. Does to every closed subalgebra \(L(G)\) correspond some immersed Lie subgroup of \(G\)?
7. If \(G\) is a BCH-Lie group, does it follow that \(L(G)_\mathbb{C}\) is integrable to a Lie group?

Reviewer: Alois Klíč (Praha)

MSC:

22E65

Infinite-dimensional Lie groups and their Lie algebras: general properties

Keywords:

locally convex spaces; exponential map; BCH-Lie group; regular Lie groups