Loop group

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.

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. LG=C^\infty(S^1,G) 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 C^\infty(S^1,G)\to \prod_{k\ge 0} C(T^kS^1,T^kG). With this topology, is an infinite-dimensional Lie group. Its Lie algebra is L\mathfrak g=C^\infty(S^1,\mathfrak g), with pointwise bracket. Since is compact, the smooth compact-open topology on L\mathfrak g is the Fréchet topology of uniform convergence of all derivatives on ; equivalently, after choosing an angular coordinate on and a norm on \mathfrak g , it is defined by the seminorms p_n(X)=\sup_{\theta\in S^1}\|X^{(n)}(\theta)\| \qquad (n\ge 0). For compact , smooth loop groups are modeled on nuclear Fréchet spaces. \Omega G = \{\gamma \in LG : \gamma(1)=e\}, the kernel of the evaluation map \operatorname{ev}_1 : LG \to G, \qquad \gamma \mapsto \gamma(1). Thus is a closed normal subgroup of . The inclusion of constant loops gives a splitting of , so there is a split exact sequence 1 \to \Omega G \to LG \xrightarrow{\operatorname{ev}_1} G \to 1, and hence a semidirect product decomposition LG \cong \Omega G \rtimes G. (R_\phi \gamma)(e^{i\theta})=\gamma(e^{i(\theta+\phi)}). This allows one to form the semidirect product T_{\mathrm{rot}}\ltimes LG, 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 T_{\mathrm{rot}}\ltimes LG . In Segal's formulation, the action of e^{i\phi}\in T_{\mathrm{rot}} is positive if it is given by e^{iA\phi} 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. L\mathfrak{g} = C^\infty(S^1,\mathfrak{g}), with pointwise bracket [X,Y](z)=[X(z),Y(z)]. The exponential map is induced pointwise from that of : \exp_{LG}(X)(z)=\exp_G(X(z)). 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. \gamma(\theta+2\pi)=\sigma(\gamma(\theta)). 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 Kac–Moody types as Lie-algebraic counterparts. LG(R)=G(R((t))) on -algebras . The associated positive loop group is L^+G(R)=G(Rt). These objects underlie the theory of the affine Grassmannian \operatorname{Gr}_G = LG/L^+G and affine flag varieties, which are central in geometric representation theory, the geometric Langlands program, and the theory of local models of Shimura varieties. LG_{\mathbf C}=C^\infty(S^1,G_{\mathbf C}). 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 \gamma=\gamma_-\,\lambda\,\gamma_+, 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 Borel–Weil 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. \Pi=\exp^{-1}(1)/(2\pi)\subset \mathfrak t, then the loop group has a canonical decomposition LT \cong T \times \Pi \times U, where U=\exp(V), \qquad V=\left\{\beta:S^1\to \mathfrak t \;|\; \int_{S^1}\beta(s)\,ds=0\right\}. In particular, the connected components of are indexed by \Pi\cong \pi_1(T) . For , this gives LS^1 \cong S^1 \times \mathbf Z \times U, so the components of are indexed by the integers. C^\infty(M,G)=\{f:M\to G \mid f \text{ is smooth}\}, \qquad (f_1f_2)(x)=f_1(x)f_2(x). More generally, Segal described the mapping groups \operatorname{Map}(X,G) 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 C^\infty(M,\mathfrak g), 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.

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