Loop group

Basic notions

Group homomorphisms
Subgroup Normal subgroup Group action
Quotient group (Semi-)direct product Direct sum Free product Wreath product
kernel image
simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable
Glossary of group theory List of group theory topics

Finite groups

Dihedral group D_n
Quaternion group Q

Frobenius group

Schur multiplier

cyclic
alternating
Lie type
sporadic

In mathematics, a loop group (not to be confused with a loop) is a group of loops in a topological group G with multiplication defined pointwise.

Definition

In its most general form a loop group is a group of continuous mappings from a manifold $M$ to a topological group $G$ .

More specifically,^[1] let $M = S 1$ , the circle in the complex plane, and let $LG$ denote the space of continuous maps $S 1 \to G$ , i.e.

LG=\{\gamma :S^{1}\to G|\gamma \in C(S^{1},G)\},

equipped with the compact-open topology. An element of $LG$ is called a loop in $G$ . Pointwise multiplication of such loops gives $LG$ the structure of a topological group. Parametrize $S 1$ with $θ$ ,

\gamma :\theta \in S^{1}\mapsto \gamma (\theta )\in G,

and define multiplication in $LG$ by

(\gamma _{1}\gamma _{2})(\theta )\equiv \gamma _{1}(\theta )\gamma _{2}(\theta ).

Associativity follows from associativity in $G$ . The inverse is given by

\gamma ^{-1}:\gamma ^{-1}(\theta )\equiv \gamma (\theta )^{-1},

and the identity by

e:\theta \mapsto e\in G.

The space $LG$ is called the free loop group on $G$ . A loop group is any subgroup of the free loop group $LG$ .

Examples

An important example of a loop group is the group

\Omega G\,

of based loops on $G$ . It is defined to be the kernel of the evaluation map

e_{1}:LG\to G,\gamma \mapsto \gamma (1)

and hence is a closed normal subgroup of $LG$ . (Here, $e 1$ is the map that sends a loop to its value at $1\in S^{1}$ .) Note that we may embed $G$ into $LG$ as the subgroup of constant loops. Consequently, we arrive at a split exact sequence

1\to \Omega G\to LG\to G\to 1

The space $LG$ splits as a semi-direct product,

LG=\Omega G\rtimes G

We may also think of $Ω G$ as the loop space on $G$ . From this point of view, $Ω G$ is an H-space with respect to concatenation of loops. On the face of it, this seems to provide $Ω G$ with two very different product maps. However, it can be shown that concatenation and pointwise multiplication are homotopic. Thus, in terms of the homotopy theory of $Ω G$ , these maps are interchangeable.

Loop groups were used to explain the phenomenon of Bäcklund transforms in soliton equations by Chuu-Lian Terng and Karen Uhlenbeck.^[2]

Notes

^ Bäuerle & de Kerf 1997
^ Geometry of Solitons by Chuu-Lian Terng and Karen Uhlenbeck

References

Bäuerle, G.G.A; de Kerf, E.A. (1997). A. van Groesen; E.M. de Jager; A.P.E. Ten Kroode (eds.). Finite and infinite dimensional Lie algebras and their application in physics. Studies in mathematical physics. Vol. 7. North-Holland. ISBN 978-0-444-82836-1 – via ScienceDirect.
Pressley, Andrew; Segal, Graeme (1986), Loop groups, Oxford Mathematical Monographs. Oxford Science Publications, New York: Oxford University Press, ISBN 978-0-19-853535-5, MR 0900587

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