Banach-Lie quotients, enlargibility, and universal complexifications. (English) Zbl 1029.22029

Glöckner, Helge; Neeb, Karl-Hermann

J. Reine Angew. Math. 560, 1-28 (2003).

The authors study various questions about the existence of Banach-Lie groups with prescribed Lie algebras. Their first main result asserts that the quotient of a Banach-Lie group by a Lie subgroup always carries the structure of a Banach-Lie group. So far one had to make extra hypotheses on the subgroup to obtain this conclusion. This result they use to reprove a characterization (due to van Est) of enlargible Lie algebras, i.e. those Banach-Lie algebras which occur as Lie algebras of Banach-Lie groups.
There are two kinds of universal objects considered for Banach-Lie algebras: a universal “enlargible envelope”, which is a Banach-Lie algebra through which all homomorphisms into enlargible Lie algebras factor, and a universal complexification which is defined in analogy to the finite-dimensional case. In general, neither of these objects needs to exist. The authors give necessary and sufficient conditions for the existence of both.

Reviewer: Joachim Hilgert (Clausthal)

MSC:

22E65	Infinite-dimensional Lie groups and their Lie algebras: general properties
22E15	General properties and structure of real Lie groups
22E10	General properties and structure of complex Lie groups

Keywords:

Banach-Lie group; period groups

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