In mathematics, a loop group is, in the most common Lie-theoretic sense, the group of smooth maps from the circle to a Lie group , with multiplication defined pointwise. When is a compact Lie group, is a basic example of an infinite-dimensional Lie group, with Lie algebra .
The subgroup of based loops is fundamental in homotopy theory, while central extensions of loop groups and their projective representations are closely related to affine Kac–Moody algebras, conformal field theory, and the Verlinde formula. In algebraic geometry one also studies algebraic loop groups, defined by , together with their associated affine Grassmannians and affine flag varieties.
Definition
Let be a topological group. The set of continuous maps from the circle to becomes a topological group under pointwise multiplication when equipped with the compact-open topology. Since is compact, this is the same as the topology of uniform convergence.
Its Lie algebra is
:<math>L\mathfrak g=C^\infty(S^1,\mathfrak g),</math>
with pointwise bracket. Since is compact, the smooth compact-open topology on <math>L\mathfrak g</math> is the Fréchet topology of uniform convergence of all derivatives on ; equivalently, after choosing an angular coordinate on and a norm on , it is defined by the seminorms
:<math>p_n(X)=\sup_{\theta\in S^1}\|X^{(n)}(\theta)\| \qquad (n\ge 0).</math>
For compact , smooth loop groups are modeled on nuclear Fréchet spaces.
Basic constructions
Free and based loop groups
The free loop group of is itself. The based loop group is
:<math>\Omega G = \{\gamma \in LG : \gamma(1)=e\},</math>
the kernel of the evaluation map
:<math>\operatorname{ev}_1 : LG \to G, \qquad \gamma \mapsto \gamma(1).</math>
Thus is a closed normal subgroup of . The inclusion of constant loops gives a splitting of , so there is a split exact sequence
:<math>1 \to \Omega G \to LG \xrightarrow{\operatorname{ev}_1} G \to 1,</math>
and hence a semidirect product decomposition
:<math>LG \cong \Omega G \rtimes G.</math>
Central extensions and representation theory
Many natural representations attached to the loop group are not honest representations of , but of a central extension of by the circle group .
Index theory
Loop groups enter index theory in several related ways. One of the earliest is through the determinant line bundle on the infinite-dimensional Grassmannian associated with a loop group. In the Grassmannian model used by Pressley, Segal, and others, this line bundle is tied to the basic central extension of the loop group and to geometric realizations of its representations.
A basic example is the KdV hierarchy. In their study of equations of KdV type, Segal and Wilson showed that a large class of solutions can be constructed from points of an infinite-dimensional Grassmannian associated with a loop group. In this picture the commuting flows are induced by the action of a positive loop subgroup, and the corresponding solutions are described in terms of Baker functions and tau functions.
Loop-group factorization also underlies dressing transformations and Bäcklund transformations. Terng and Uhlenbeck formulated conservation laws, scattering theory, hierarchies, and Bäcklund transformations within a common framework of loop-group actions, particularly for the ZS–AKNS hierarchy, which includes the nonlinear Schrödinger equation, modified KdV, and the -wave equation.
The same methods occur in differential geometry. Loop-group constructions are used for harmonic maps into Lie groups and symmetric spaces, the chiral model, and a range of geometric integrable systems such as constant-mean-curvature and isothermic surfaces. In compact cases, global Birkhoff- and Iwasawa-type decompositions strengthen the dressing method and lead to global Weierstrass-type representations for some geometric integrable systems.
Hodge theory
Loop groups also appear in a more specialized interaction with Hodge theory. In work of Jeremy Daniel, a loop Hodge structure is defined as an infinite-dimensional analogue of a Hodge structure incorporating features of loop-group geometry, and variations of loop Hodge structures are shown to be equivalent to harmonic bundles.
From this point of view, non-abelian Hodge theory can be expressed in terms of period maps with values in infinite-dimensional period domains related to loop-group constructions.
If is a topological group, the continuous mapping space becomes a topological group with the compact-open topology. For smooth maps one usually uses the smooth compact-open topology. Under suitable hypotheses on and , this gives a natural infinite-dimensional Lie group structure. Its Lie algebra is the corresponding current algebra
:<math>C^\infty(M,\mathfrak g),</math>
with pointwise Lie bracket.
For non-compact manifolds one often studies the compactly supported current group , or more generally section groups of bundles of Lie groups over . Gauge groups of principal bundles are of this form: if is a principal -bundle, then its gauge group is isomorphic to the section group , and the compactly supported gauge group to .
Current groups occur naturally in quantum field theory and gauge theory. Compared with loop groups, however, their general representation theory is much less fully developed; much of the recent work has focused on central extensions and on special classes of representations, such as bounded or positive-energy representations of gauge groups.