Inmathematics, aloop group (not to be confused with aloop) is agroup ofloops in atopological groupG with multiplication definedpointwise.

In its most general form a loop group is a group ofcontinuous mappings from amanifoldM to a topological groupG.

More specifically,^[1] letM =S¹, the circle in thecomplex plane, and letLG denote thespace of continuous mapsS¹ →G, i.e.

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

equipped with thecompact-open topology. An element ofLG is called aloop inG. Pointwise multiplication of such loops givesLG the structure of a topological group. ParametrizeS¹ withθ,

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

and define multiplication inLG by

${\displaystyle ({\gamma }_1{\gamma }_2)(\theta )\equiv {\gamma }_1(\theta ){\gamma }_2(\theta ).}$

Associativity follows from associativity inG. The inverse is given by

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

and the identity by

${\displaystyle e:\theta \mapsto e\in G.}$

The spaceLG is called thefree loop group onG. A loop group is anysubgroup of the free loop groupLG.

An important example of a loop group is the group

${\displaystyle \mathrm{\Omega }G}$

of based loops onG. It is defined to be thekernel of the evaluation map

${\displaystyle e_1:LG\to G,\gamma \mapsto \gamma (1)}$,

and hence is aclosednormal subgroup ofLG. (Here,e₁ is the map that sends a loop to its value at${\displaystyle 1\in S^1}$.) Note that we may embedG intoLG as the subgroup of constant loops. Consequently, we arrive at asplit exact sequence

${\displaystyle 1\to \mathrm{\Omega }G\to LG\to G\to 1}$.

The spaceLG splits as asemi-direct product,

${\displaystyle LG=\mathrm{\Omega }G⋊G}$.

We may also think ofΩG as theloop space onG. From this point of view,ΩG is anH-space with respect to concatenation of loops. On the face of it, this seems to provideΩG with two very different product maps. However, it can be shown that concatenation and pointwise multiplication arehomotopic. Thus, in terms of the homotopy theory ofΩG, these maps are interchangeable.

Loop groups were used to explain the phenomenon ofBäcklund transforms insoliton equations byChuu-Lian Terng andKaren Uhlenbeck.^[2]

• Loop space
• Loop algebra
• Quasigroup

• De Kerf, E.A.; Bäuerle, G.G.A.; Ten Kroode, A.P.E., eds. (1997), "Representations of loop algebras",Lie Algebras - Finite and Infinite Dimensional Lie Algebras and Applications in Physics, Studies in Mathematical Physics, vol. 7, pp. 365–429,doi:10.1016/S0925-8582(97)80010-3,ISBN 978-0-444-82836-1
• 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
• Terng, Chuu-Lian; Uhlenbeck, Karen (2000),"Geometry of solitons"(PDF),Notices of the American Mathematical Society,47 (1):17–25

• Topological groups
• Solitons

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