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 KacâÂÂMoody 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.
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.
In Lie theory one usually considers the group
of smooth loops in a finite-dimensional Lie group . It is endowed with the smooth compact-open topology, namely the initial topology induced by the iterated tangent maps
With this topology, is an infinite-dimensional Lie group.
Its Lie algebra is
with pointwise bracket. Since is compact, the smooth compact-open topology on 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
For compact , smooth loop groups are modeled on nuclear Fréchet spaces.
The free loop group of is itself. The based loop group is
the kernel of the evaluation map
Thus is a closed normal subgroup of . The inclusion of constant loops gives a splitting of , so there is a split exact sequence
and hence a semidirect product decomposition
As a topological space, is the based loop space of . Its pointwise product and the usual concatenation of based loops are different operations, but they induce the same multiplication up to homotopy; this is a manifestation of the EckmannâÂÂHilton argument.
The splitting of the evaluation map
by constant loops identifies with as a topological space:
Thus the topology of is determined by that of together with the based loop space .
In particular, the connected components of are classified by
If is connected, then , so
Hence is connected whenever is simply connected.
More generally, for each ,
Thus the homotopy groups of a loop group are determined by those of shifted by one degree together with those of itself.
These elementary identifications are one reason loop groups are important in algebraic topology. In the unitary case they are closely related to Bott periodicity, and Segal's Grassmannian model of the homogeneous space makes this relation explicit.
Besides pointwise multiplication, loop groups carry a natural action of the circle by rotating the parameter:
This allows one to form the semidirect product
where acts on by rotation.
This action is the starting point for the theory of positive-energy representations. A representation of is said to have positive energy if it is equipped with a positive action of making it a representation of . In Segal's formulation, the action of is positive if it is given by for an operator whose spectrum is bounded below.
For compact , irreducible positive-energy representations are a distinguished class of projective representations of loop groups. They extend holomorphically to the complexified loop group and decompose into finite-dimensional energy eigenspaces. For this reason, the representation theory of loop groups of a compact often resembles that of compact groups.
If is a finite-dimensional Lie group, then suitable spaces of loops in inherit infinite-dimensional manifold structures. For smooth loops, is a Fréchet Lie group, and its Lie algebra is the loop algebra
with pointwise bracket
The exponential map is induced pointwise from that of :
For compact , loop groups are among the simplest and most studied examples of infinite-dimensional Lie groups.
To develop differential geometry on loop groups one often uses Sobolev completions . In particular, based loop groups of compact, connected, simply connected, simple Lie groups carry natural geometric structures, including Kähler metrics in the Hilbert-manifold setting.
The quotient , where is embedded as the subgroup of constant loops, can be identified with . This homogeneous-space viewpoint is central in the geometry of loop groups.
If is compact and is its complexification, then the complexified loop group admits factorization phenomena analogous to Birkhoff factorization and Bruhat decomposition. These decompositions play a major role in the geometry of loop groups, the theory of Toeplitz operators, and the construction of solutions to integrable systems.
A central feature of loop-group theory is that many natural representations are not honest representations of , but of a central extension of by the circle group .
For a compact Lie group , integral classes in give rise, by transgression, to central extensions of the loop group. Such extensions are often described by a level. The corresponding projective unitary representations include the integrable highest-weight or positive-energy representations, which are closely related to representations of the associated affine KacâÂÂMoody algebra.
The representation theory of loop groups is also linked to the Verlinde ring and to twisted equivariant K-theory. In work of Freed, Hopkins, and Teleman, the Verlinde ring of positive-energy representations is identified with an appropriate twisted equivariant K-group of .
Let be an automorphism of of finite order . The corresponding twisted loop group consists of smooth maps satisfying
Equivalently, after passing to the circle, one may regard twisted loops as sections of a bundle over with monodromy .
Twisted loop groups occur naturally in the theory of affine Dynkin diagrams, in representation theory, and in the theory of affine flag varieties. They include the twisted affine KacâÂÂMoody types as Lie-algebraic counterparts.
In algebraic geometry and arithmetic geometry, one replaces smooth loops by Laurent series. If is an algebraic group over a field , its algebraic loop group is the functor
on -algebras . The associated positive loop group is
These objects underlie the theory of the affine Grassmannian
and affine flag varieties, which are central in geometric representation theory, the geometric Langlands program, and the theory of local models of Shimura varieties.
If is a compact Lie group with complexification , then the smooth loop group has a complexification
This is one of the special features of loop groups among infinite-dimensional Lie groups.
A closely related role is played by subgroups of loops that extend holomorphically across one of the discs bounded by . Writing for the unit disc and for its exterior, one defines subgroups and consisting of boundary values of holomorphic maps and , respectively.
These holomorphic subgroups enter the Birkhoff factorization theorem, according to which a loop in can, on suitable strata, be written in the form
where , , and is a homomorphism into a maximal torus. This factorization is the infinite-dimensional analogue of Bruhat decomposition and underlies much of the geometry of homogeneous spaces of loop groups.
Holomorphic methods also enter representation theory. In Segal's BorelâÂÂWeil picture, positive-energy representations are realized as spaces of holomorphic sections of line bundles over homogeneous spaces attached to loop groups, and every irreducible positive-energy representation extends to a holomorphic representation of the complexified loop group.
The simplest nontrivial example is . In this case, smooth loops are classified up to connected component by their winding number. More generally, if is a torus with Lie algebra and cocharacter lattice
then the loop group has a canonical decomposition
where
In particular, the connected components of are indexed by . For , this gives
so the components of are indexed by the integers.
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.
Analytic index theory also appears through families of Toeplitz operators and through spectral flow. In the torus case, Freed, Hopkins, and Teleman describe a family of Dirac operators whose spectral flow gives the basic topological class needed for their construction of twisted K-theory classes associated with loop-group representations.
A deeper connection comes from the Dirac family attached to a positive-energy representation of a loop group. In the work of Freed, Hopkins, and Teleman, such families of Fredholm operators produce classes in twisted equivariant K-theory, and this construction is one of the ingredients in their identification of the Verlinde ring with twisted equivariant K-theory of .
These index-theoretic constructions link loop-group representation theory with geometric quantization, central extensions, and the topology of the group itself.
Loop groups arise in several areas of mathematics and mathematical physics.
Loop groups play a central role in the modern theory of integrable systems. A large class of nonlinear evolution equations can be written in Lax pair or zero-curvature form with a complex spectral parameter . When the coefficients depend rationally or Laurent-polynomially on and take values in a Lie algebra , they may be viewed as elements of a loop algebra . Splittings of loop algebras into positive and negative parts, together with factorization in the corresponding loop groups, then produce commuting hierarchies of flows and solution-generating procedures.
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 ZSâÂÂAKNS 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.
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.
A current group generalizes a loop group, replacing the circle with a smooth manifold . Thus a current group is the group of smooth mappings from into , with multiplication defined pointwise:
More generally, Segal described the mapping groups for compact manifolds as higher-dimensional analogues of loop groups, noting that in mathematical physics they occur as current groups and gauge groups.
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
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.