Editing Lie group (section)

== Infinite-dimensional Lie groups ==
Lie groups are often defined to be finite-dimensional, but there are many groups that resemble Lie groups, except for being infinite-dimensional. The simplest way to define infinite-dimensional Lie groups is to  model them locally on [[Banach space]]s (as opposed to [[Euclidean space]] in the finite-dimensional case), and in this case much of the basic theory is similar to that of finite-dimensional Lie groups. However this is inadequate for many applications, because many natural examples of infinite-dimensional Lie groups are not [[Banach manifold]]s. Instead one needs to define Lie groups modeled on more general [[Locally convex space|locally convex]] topological vector spaces. In this case the relation between the Lie algebra and the Lie group becomes rather subtle, and several results about finite-dimensional Lie groups no longer hold.

The literature is not entirely uniform in its terminology as to exactly which properties of infinite-dimensional groups qualify the group for the prefix ''Lie'' in ''Lie group''. On the Lie algebra side of affairs, things are simpler since the qualifying criteria for the prefix ''Lie'' in ''Lie algebra'' are purely algebraic. For example, an infinite-dimensional Lie algebra may or may not have a corresponding Lie group. That is, there may be a group corresponding to the Lie algebra, but it might not be nice enough to be called a Lie group, or the connection between the group and the Lie algebra might not be nice enough (for example, failure of the exponential map to be onto a neighborhood of the identity). It is the "nice enough" that is not universally defined.

Some of the examples that have been studied include:
* The group of [[diffeomorphism]]s of a manifold. Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the [[Witt algebra]], whose [[Lie algebra extension|central extension]] the [[Virasoro algebra]] (see [[Lie algebra extension#Virasoro algebra|Virasoro algebra from Witt algebra]] for a derivation of this fact) is the symmetry algebra of [[two-dimensional conformal field theory]]. Diffeomorphism groups of compact manifolds of larger dimension are [[Convenient vector space#Regular Lie groups|regular Fréchet Lie groups]]; very little about their structure is known.
* The diffeomorphism group of spacetime sometimes appears in attempts to [[Quantization (physics)|quantize]] gravity.
* The group of smooth maps from a manifold to a finite-dimensional Lie group is an example of a [[gauge group]] (with operation of [[pointwise multiplication]]), and is used  in [[quantum field theory]] and [[Donaldson theory]]. If the manifold is a circle these are called [[loop group]]s, and have central extensions whose Lie algebras are (more or less) [[Kac&ndash;Moody algebra]]s.
* There are infinite-dimensional analogues of general linear groups, orthogonal groups, and so on.<ref>{{cite book |doi=10.1016/S0925-8582(97)80009-7 |chapter=Lie algebras of infinite matrices |title=Lie Algebras - Finite and Infinite Dimensional Lie Algebras and Applications in Physics |series=Studies in Mathematical Physics |date=1997 |volume=7 |pages=305–364 |isbn=978-0-444-82836-1 |editor1-first=E.A. |editor1-last=De Kerf |editor2-first=G.G.A. |editor2-last=Bäuerle |editor3-first=A.P.E. |editor3-last=Ten Kroode }}</ref> One important aspect is that these may have ''simpler'' topological properties: see for example [[Kuiper's theorem]]. In [[M-theory]], for example, a 10-dimensional SU(''N'') gauge theory becomes an 11-dimensional theory when ''N'' becomes infinite.