Topological groups, Lie groups Lie groups: basic notions Example

Loop group

Builds on Lie group Fréchet manifold Circle group

Example

Let $G$ be a Lie group and let $S^1$ be the circle. The loop group of $G$ is the set $LG = \set{\gamma \colon S^1 \to G : \gamma \text{ smooth}},$ with pointwise multiplication $(\gamma\delta)(z) = \gamma(z)\delta(z)$ . It is a group. The pointwise product and the pointwise inverse $z \mapsto \gamma(z)^{-1}$ of smooth loops are smooth, because multiplication and inversion of $G$ are smooth. The identity element is the constant loop at $e$ , and associativity holds pointwise.

For compact $G$ , $LG$ is an infinite-dimensional Lie group and a Fréchet manifold. Its Lie algebra is the loop algebra $L\mathfrak{g}$ of smooth maps $S^1 \to \mathfrak{g}$ with the pointwise bracket.

Remarks

The constant loops form a copy of $G$ inside $LG$ . For $G$ the circle group $U(1)$ , a loop $\gamma \colon S^1 \to U(1)$ has a winding number, and $LU(1)$ has one connected component for each integer. Central extensions of loop groups of compact Lie groups are closely tied to affine Lie algebras and their representations.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Binary operation definition
Fréchet manifold concept
Group concept
Lie group concept
Commutativity definition
Abelian group definition
Circle group object
Loop group example

Tag	`8MEDVAEERQ`
URI	`m2://8MEDVAEERQ`
Kind	example
Writing	W0 stub
MSC	22E67, 22E65
Instance of	Group
Uses	Lie group, Fréchet manifold, Circle group
Same as	Wikidata Q6675827 Wikipedia Loop group
Changed	2 November 2026

Loop group

Example

Remarks

Reading path

Further reading

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