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

  • symplectomorphism group, quantomorphism group

• Butcher group

References

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

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

  (with an eye towards application in mathematical physics)

• Josef Teichmann, Innite dimensional Lie Theory from the point of view of Functional Analysis (pdf)

  (with an eye towards functional analysis)

• 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)

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