nLab infinite-dimensional Lie group

Idea

A group object in infinite-dimensional smooth manifolds is an infinite-dimensional Lie group.

If the underlying type of manifolds is Banach manifold, then one speaks of Banach-Lie groups, etc.

Examples

• U(ℋ) in its norm topology is a Banach Lie group (but not in its operator topology)

• stable unitary group

• loop group

• diffeomorphism group

  • volume-preserving diffeomorphism group

  • symplectomorphism group

  • quantomorphism group

• Butcher group

References

• Vladimir Arnold: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Annales de l’Institut Fourier 16 1 (1966) 319–361 [numdam:AIF_1966__16_1_319_0/]

  (application to hydrodynamics)

• John Milnor: Remarks on infinite-dimensional Lie Groups, in: Relativity, Groups, and Topology II, Proceedings of Les Houches Session XL (1983) 1007-1058 [pdf, inSpire:217235]

• R. S. Ismagilov: Representations of Infinite-Dimensional Groups, Translations of Mathematical Monographs 152 (1996) [ISBN:978-0-8218-0418-6]

  (on the representation theory)

• Andreas Kriegl, Peter Michor, Regular infinite dimensional Lie groups, Journal of Lie Theory Volume 7 (1997) 61-99 [pdf]

• Karl-Hermann Neeb: Monastir summer school: Infinite-dimensional Lie groups (2005) [pdf, pdf]

• Karl-Hermann Neeb: Towards a Lie theory of locally convex groups, Japanese Journal of Math. 1 (2006) 291-468 [arXiv:1501.06269]

• Rudolf Schmid: Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics, Advances in Mathematical Physics 2010 [doi:10.1155/2010/280362, pdf]

  (with an eye towards application in mathematical physics)

• Alexander Schmeding, Chaper 3 of: An introduction to infinite-dimensional differential geometry, Cambridge University Press [arXiv:2112.08114]