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Integrating central extensions of Lie algebras via Lie 2-groups. (English) Zbl 1344.22009

The vanishing of \(\pi_2\) for finite dimensional Lie groups implies that central extensions of finite dimensional Lie algebras integrate to central extensions of Lie groups. The paper under review studies the corresponding phenomenon in the infinite dimensional setup. It is shown that, in the infinite dimensional setup, the obstruction given by non-trivial \(\pi_2\) may be overcome by integrating central extensions of Lie algebras not to Lie groups but to central extensions of étale Lie \(2\)-groups. As an application, a generalization of Lie’s Third Theorem to infinite dimensional Lie algebras is obtained. This generalization asserts that each locally exponential Lie algebra with topologically split center integrates to an étale Lie \(2\)-group.

MSC:

22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
58H05 Pseudogroups and differentiable groupoids
58B25 Group structures and generalizations on infinite-dimensional manifolds
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
17B55 Homological methods in Lie (super)algebras
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