mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a loop group (not to be confused with a loop) is a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

of loops in a

topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...

''G'' with multiplication defined

pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some Function (mathematics), function f. An important class of pointwise concepts are the ''pointwise operations'', that ...

Definition

In its most general form a loop group is a group of continuous mappings from a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

to a topological group . More specifically, let , the circle in the

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

, and let denote the

space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...

of continuous maps , i.e. :

LG = \,

equipped with the

compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory ...

. An element of is called a ''loop'' in . Pointwise multiplication of such loops gives the structure of a topological group. Parametrize with , :

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

and define multiplication in by :

(\gamma_1 \gamma_2)(\theta) \equiv \gamma_1(\theta)\gamma_2(\theta).

Associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...

follows from associativity in . The inverse is given by :

\gamma^:\gamma^(\theta) \equiv \gamma(\theta)^,

and the identity by :

e:\theta \mapsto e \in G.

The space is called the free loop group on . A loop group is any

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

of the free loop group .

Examples

An important example of a loop group is the group :

\Omega G \,

of based loops on . 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 In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...

of . (Here, is the map that sends a loop to its value at

1 \in S^1

.) Note that we may embed into as the subgroup of constant loops. Consequently, we arrive at a

split exact sequence The term split exact sequence is used in two different ways by different people. Some people mean a short exact sequence that right-splits (thus corresponding to a semidirect product) and some people mean a short exact sequence that left-splits ( ...

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

. The space splits as a semi-direct product, :

LG = \Omega G \rtimes G

. We may also think of as the

loop space In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topolog ...

on . From this point of view, is an

H-space In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed. Definition An H-space consists of a topological space , together wit ...

with respect to concatenation of loops. On the face of it, this seems to provide with two very different product maps. However, it can be shown that concatenation and pointwise multiplication are

homotopic In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...

. Thus, in terms of the homotopy theory of , these maps are interchangeable. Loop groups were used to explain the phenomenon of Bäcklund transforms in

soliton In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...

equations by Chuu-Lian Terng and

Karen Uhlenbeck Karen Keskulla Uhlenbeck ForMemRS (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W ...

Notes

References

* * *{{citation , last1=Terng , first1=Chuu-Lian , first2=Karen , last2=Uhlenbeck , title=Geometry of solitons , journal=Notices of the American Mathematical Society , volume=47 , issue=1 , date=2000 , pages=17–25 , url=https://www.ams.org/notices/200001/fea-terng.pdf Topological groups Solitons

Definition

Examples

See also

Notes

References