Loop group

{{Short description|Mathematical group of loops in a Lie group}} {{for|groups of actors involved in re-recording movie dialogue during post-production|Dubbing (filmmaking)}} {{CS1 config|mode=cs2}} {{Group theory sidebar |Topological}} {{Lie groups |Other}}

In mathematics, a '''loop group''' is, in the most common Lie-theoretic sense, the group {{math|''LG'' {{=}} ''C''∞(''S''1, ''G'')}} of smooth maps from the circle {{math|''S''1}} to a Lie group {{math|''G''}}, with multiplication defined pointwise.<ref name="Segal-PS">{{cite book |last1=Pressley |first1=Andrew |last2=Segal |first2=Graeme |author-link2=Graeme Segal |title=Loop groups |series=Oxford Mathematical Monographs |publisher=Clarendon Press |location=Oxford |year=1986 |isbn=978-0-19-853535-5 |mr=0900587}}</ref> When {{math|''G''}} is a compact Lie group, {{math|''LG''}} is a basic example of an infinite-dimensional Lie group, with Lie algebra {{math|''L''𝔤 {{=}} ''C''∞(''S''1, 𝔤)}}.<ref name="Segal-survey">{{cite book |last=Segal |first=G. B. |title=Loop groups |series=Lecture Notes in Mathematics |volume=1064 |year=1984 |pages=155–168}}</ref><ref name="Freed1988">{{cite journal |last=Freed |first=Daniel S. |author-link=Dan Freed |title=The geometry of loop groups |journal=Journal of Differential Geometry |volume=28 |issue=2 |year=1988 |pages=223–276 |doi=10.4310/jdg/1214442279 |bibcode=1988JDGeo..2842279F |mr=0961515}}</ref>

The subgroup {{math|Ω''G''}} 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.<ref name="Segal-PS"/><ref name="FHT">{{cite journal |last1=Freed |first1=Daniel S. |last2=Hopkins |first2=Michael J. |last3=Teleman |first3=Constantin |title=Loop groups and twisted K-theory III |journal=Annals of Mathematics |series=2 |volume=174 |issue=2 |year=2011 |pages=947–1007 |doi=10.4007/annals.2011.174.2.5}}</ref> In algebraic geometry one also studies '''algebraic loop groups''', defined by {{math|''LG''(''R'') {{=}} ''G''(''R''((''t'')))}}, together with their associated affine Grassmannians and affine flag varieties.<ref name="PR">{{cite journal |last1=Pappas |first1=George |last2=Rapoport |first2=Michael |author2-link=Michael Rapoport |title=Twisted loop groups and their affine flag varieties |journal=Advances in Mathematics |volume=219 |issue=1 |year=2008 |pages=118–198 |doi=10.1016/j.aim.2008.04.006 |doi-access=free |mr=2435422}}</ref>

== Definition ==

Let {{math|''G''}} be a topological group. The set {{math|''C''(''S''1,''G'')}} of continuous maps from the circle to {{math|''G''}} becomes a topological group under pointwise multiplication when equipped with the compact-open topology.<ref name="Neeb-locconv">{{cite journal |last=Neeb |first=Karl-Hermann |title=Towards a Lie theory of locally convex groups |journal=Japanese Journal of Mathematics |volume=1 |issue=2 |year=2006 |pages=291–468 |doi=10.1007/s11537-006-0606-y |arxiv=1501.06269 }}</ref> Since {{math|''S''1}} is compact, this is the same as the topology of uniform convergence.<ref name="Neeb-locconv" />

In Lie theory one usually considers the group :<math>LG=C^\infty(S^1,G)</math> of smooth loops in a finite-dimensional Lie group {{math|''G''}}. It is endowed with the smooth compact-open topology, namely the initial topology induced by the iterated tangent maps :<math>C^\infty(S^1,G)\to \prod_{k\ge 0} C(T^kS^1,T^kG).</math> With this topology, {{math|''LG''}} is an infinite-dimensional Lie group.<ref name="Roberts-Whitney">{{cite journal |last=Roberts |first=David Michael |title=Extending Whitney's extension theorem: nonlinear function spaces |journal=Annales de l'Institut Fourier |volume=71 |issue=3 |year=2021 |pages=1241–1292 |doi=10.5802/aif.3424|arxiv=1801.04126 }}</ref><ref name="Segal-survey">{{cite book |last=Segal |first=G. B. |title=Loop groups |series=Lecture Notes in Mathematics |volume=1064 |year=1984 |pages=155–168}}</ref>

Its Lie algebra is :<math>L\mathfrak g=C^\infty(S^1,\mathfrak g),</math> with pointwise bracket. Since {{math|''S''1}} is compact, the smooth compact-open topology on <math>L\mathfrak g</math> is the Fréchet topology of uniform convergence of all derivatives on {{math|''S''1}}; equivalently, after choosing an angular coordinate on {{math|''S''1}} and a norm on {{math|\mathfrak g}}, it is defined by the seminorms :<math>p_n(X)=\sup_{\theta\in S^1}\|X^{(n)}(\theta)\| \qquad (n\ge 0).</math><ref name="Neeb-diffvec">{{cite journal |last=Neeb |first=Karl-Hermann |title=On differentiable vectors for representations of infinite-dimensional Lie groups |journal=Journal of Functional Analysis |volume=259 |issue=11 |year=2010 |pages=2814–2855 |doi=10.1016/j.jfa.2010.08.002 |url=https://orbilu.uni.lu/handle/10993/13672 }}</ref>

For compact {{math|''G''}}, smooth loop groups are modeled on nuclear Fréchet spaces.<ref name="Ludewig-Waldorf">{{cite journal |last1=Ludewig |first1=Matthias |last2=Waldorf |first2=Konrad |title=Lie 2-groups from loop group extensions |journal=Journal of Geometry and Physics |volume=197 |year=2024 |pages=597–633|arxiv=2303.13176}}</ref>

== Basic constructions ==

=== Free and based loop groups ===

The '''free loop group''' of {{math|''G''}} is {{math|''LG''}} 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 {{math|Ω''G''}} is a closed normal subgroup of {{math|''LG''}}. The inclusion of constant loops gives a splitting of {{math|ev1}}, 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><ref name="Segal-PS"/>

=== Relation with loop spaces ===

As a topological space, {{math|Ω''G''}} is the based loop space of {{math|''G''}}. 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 Eckmann–Hilton argument.<ref name="Segal-PS"/>

=== Basic topology ===

The splitting of the evaluation map :<math>\operatorname{ev}_1:LG\to G</math> by constant loops identifies {{math|''LG''}} with {{math|''G'' × Ω''G''}} as a topological space: :<math>LG \cong G\times \Omega G, \qquad \gamma \mapsto \bigl(\gamma(1),\,\gamma(1)^{-1}\gamma\bigr).</math> Thus the topology of {{math|''LG''}} is determined by that of {{math|''G''}} together with the based loop space {{math|Ω''G''}}.<ref name="Segal-PS" />

In particular, the connected components of {{math|''LG''}} are classified by :<math>\pi_0(LG)\cong \pi_0(G)\times \pi_0(\Omega G).</math> If {{math|''G''}} is connected, then <math>\pi_0(\Omega G)\cong \pi_1(G)</math>, so :<math>\pi_0(LG)\cong \pi_1(G).</math> Hence {{math|''LG''}} is connected whenever {{math|''G''}} is simply connected.<ref name="Segal-PS" />

More generally, for each {{math|1=''k'' ≥ 1}}, :<math>\pi_k(LG)\cong \pi_k(G)\oplus \pi_k(\Omega G) \cong \pi_k(G)\oplus \pi_{k+1}(G).</math> Thus the homotopy groups of a loop group are determined by those of {{math|''G''}} shifted by one degree together with those of {{math|''G''}} itself.<ref name="Segal-PS" />

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 {{math|1=''LG''/''G'' ≅ Ω''G''}} makes this relation explicit.<ref name="Segal-survey" />

=== Rotation action and positive energy ===

Besides pointwise multiplication, loop groups carry a natural action of the circle by rotating the parameter: :<math>(R_\phi \gamma)(e^{i\theta})=\gamma(e^{i(\theta+\phi)}).</math> This allows one to form the semidirect product :<math>T_{\mathrm{rot}}\ltimes LG,</math> where {{math|''T''rot}} acts on {{math|''LG''}} by rotation.<ref name="Segal-survey" />

This action is the starting point for the theory of positive-energy representations. A representation of {{math|''LG''}} is said to have positive energy if it is equipped with a positive action of {{math|''T''rot}} making it a representation of <math>T_{\mathrm{rot}}\ltimes LG</math>. In Segal's formulation, the action of <math>e^{i\phi}\in T_{\mathrm{rot}}</math> is positive if it is given by <math>e^{iA\phi}</math> for an operator {{math|''A''}} whose spectrum is bounded below.<ref name="Segal-survey" />

For compact {{math|''G''}}, 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.<ref name="Segal-survey" /> For this reason, the representation theory of loop groups of a compact {{math|''G''}} often resembles that of compact groups.

== Infinite-dimensional Lie group structure ==

If {{math|''G''}} is a finite-dimensional Lie group, then suitable spaces of loops in {{math|''G''}} inherit infinite-dimensional manifold structures. For smooth loops, {{math|''LG''}} is a Fréchet Lie group, and its Lie algebra is the loop algebra :<math>L\mathfrak{g} = C^\infty(S^1,\mathfrak{g}),</math> with pointwise bracket :<math>[X,Y](z)=[X(z),Y(z)].</math><ref name="Segal-survey"/><ref name="Segal-PS"/>

The exponential map is induced pointwise from that of {{math|''G''}}: :<math>\exp_{LG}(X)(z)=\exp_G(X(z)).</math> For compact {{math|''G''}}, loop groups are among the simplest and most studied examples of infinite-dimensional Lie groups.<ref name="Segal-survey"/>

To develop differential geometry on loop groups one often uses Sobolev completions {{math|''L''''s''''G''}}. 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.<ref name="Freed1988"/>

== Homogeneous spaces and factorization ==

The quotient {{math|''LG''/''G''}}, where {{math|''G''}} is embedded as the subgroup of constant loops, can be identified with {{math|Ω''G''}}. This homogeneous-space viewpoint is central in the geometry of loop groups.<ref name="Segal-survey"/><ref name="Freed1988"/>

If {{math|''G''}} is compact and {{math|''G''ℂ}} is its complexification, then the complexified loop group {{math|''LG''ℂ}} 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.<ref name="Segal-PS"/><ref name="Freed1988"/><ref name="TerngUhlenbeck">{{cite journal |last1=Terng |first1=Chuu-Lian |last2=Uhlenbeck |first2=Karen |author-link=Chuu-Lian Terng |author-link2=Karen Uhlenbeck |title=Geometry of solitons |journal=Notices of the American Mathematical Society |volume=47 |issue=1 |year=2000 |pages=17–25 |url=https://www.ams.org/notices/200001/fea-terng.pdf}}</ref>

== Central extensions and representation theory ==

Many natural representations attached to the loop group are not honest representations of {{math|''LG''}}, but of a central extension of {{math|''LG''}} by the circle group {{math|''U''(1)}}.<ref name="Segal-PS"/><ref name="FHT"/>

For a compact Lie group {{math|''G''}}, integral classes in {{math|''H''4(''BG'';'''Z''')}} give rise, by transgression, to central extensions of the loop group. Such extensions are often described by a ''level''.<ref name="FHT"/> The corresponding projective unitary representations include the integrable highest-weight or positive-energy representations, which are closely related to representations of the associated affine Kac–Moody algebra.<ref name="Segal-PS"/><ref name="FHT"/>

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 {{math|''G''}}.<ref name="FHT"/>

== Twisted loop groups ==

Let {{math|σ}} be an automorphism of {{math|''G''}} of finite order {{math|''m''}}. The corresponding '''twisted loop group''' consists of smooth maps {{math|γ:'''R'''→''G''}} satisfying :<math>\gamma(\theta+2\pi)=\sigma(\gamma(\theta)).</math> Equivalently, after passing to the circle, one may regard twisted loops as sections of a bundle over {{math|''S''1}} with monodromy {{math|σ}}.<ref name="PR"/>

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.<ref name="PR"/>

== Algebraic loop groups ==

In algebraic geometry and arithmetic geometry, one replaces smooth loops by Laurent series. If {{math|''G''}} is an algebraic group over a field {{math|''k''}}, its '''algebraic loop group''' is the functor :<math>LG(R)=G(R((t)))</math> on {{math|''k''}}-algebras {{math|''R''}}. The associated '''positive loop group''' is :<math>L^+G(R)=G(Rt).</math><ref name="PR"/>

These objects underlie the theory of the affine Grassmannian :<math>\operatorname{Gr}_G = LG/L^+G</math> and affine flag varieties, which are central in geometric representation theory, the geometric Langlands program, and the theory of local models of Shimura varieties.<ref name="PR"/>

== Complex and holomorphic loop groups ==

If {{math|''G''}} is a compact Lie group with complexification {{math|''G'''''C'''}}, then the smooth loop group {{math|''LG''}} has a complexification :<math>LG_{\mathbf C}=C^\infty(S^1,G_{\mathbf C}).</math> This is one of the special features of loop groups among infinite-dimensional Lie groups.<ref name="Segal-survey" />

A closely related role is played by subgroups of loops that extend holomorphically across one of the discs bounded by {{math|''S''1}}. Writing {{math|''D''}} for the unit disc and {{math|''D''*}} for its exterior, one defines subgroups {{math|''L''+''G'''''C'''}} and {{math|''L''-''G'''''C'''}} consisting of boundary values of holomorphic maps {{math|''D''→''G''\mathbf C}} and {{math|''D''*→''G'''''C'''}}, respectively.<ref name="Segal-PS" /><ref name="Segal-survey" />

These holomorphic subgroups enter the Birkhoff factorization theorem, according to which a loop in {{math|''G''\mathbf C}} can, on suitable strata, be written in the form :<math>\gamma=\gamma_-\,\lambda\,\gamma_+,</math> where {{math|γ− \in ''L''−''G'''''C'''}}, {{math|γ+ \in ''L''+''G'''''C'''}}, and {{math|λ:''S''1→''T''}} is a homomorphism into a maximal torus.<ref name="Segal-survey" /> This factorization is the infinite-dimensional analogue of Bruhat decomposition and underlies much of the geometry of homogeneous spaces of loop groups.<ref name="Segal-PS" />

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.<ref name="Segal-survey" />

== Examples ==

The simplest nontrivial example is {{math|1=''G'' {{=}} ''S''1}}. In this case, smooth loops are classified up to connected component by their winding number. More generally, if {{math|''T''}} is a torus with Lie algebra {{math|𝔱}} and cocharacter lattice :<math>\Pi=\exp^{-1}(1)/(2\pi)\subset \mathfrak t,</math> then the loop group has a canonical decomposition :<math>LT \cong T \times \Pi \times U,</math> where :<math>U=\exp(V), \qquad V=\left\{\beta:S^1\to \mathfrak t \;|\; \int_{S^1}\beta(s)\,ds=0\right\}.</math> In particular, the connected components of {{math|''LT''}} are indexed by <math>\Pi\cong \pi_1(T)</math>. For {{math|1=''T'' {{=}} ''S''1}}, this gives :<math>LS^1 \cong S^1 \times \mathbf Z \times U,</math> so the components of {{math|''LS''1}} are indexed by the integers.<ref name="FHT2">{{cite journal |last1=Freed |first1=Daniel S. |last2=Hopkins |first2=Michael J. |last3=Teleman |first3=Constantin |title=Loop groups and twisted K-theory II |journal=Journal of the American Mathematical Society |volume=26 |issue=3 |year=2013 |pages=595–644 |doi=10.1090/S0894-0347-2013-00761-4 |doi-access=free }}</ref>

== 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.<ref name="Segal-survey" /><ref name="Segal-Wilson-KdV" />

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.<ref name="FHT" />

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 {{math|''G''}}.<ref name="FHT" />

These index-theoretic constructions link loop-group representation theory with geometric quantization, central extensions, and the topology of the group {{math|''G''}} itself.<ref name="FHT" />

== Applications ==

Loop groups arise in several areas of mathematics and mathematical physics.

* In algebraic topology, the based loop group {{math|Ω''G''}} is a basic example of an H-space and is closely related to the homotopy type of {{math|''G''}}.<ref name="Segal-PS"/> * In representation theory, loop groups and their central extensions lead to affine Kac–Moody algebras and positive-energy representations.<ref name="Segal-PS"/><ref name="FHT"/> * In algebraic geometry, algebraic loop groups give rise to affine Grassmannians and affine flag varieties.<ref name="PR"/> * In mathematical physics, loop groups occur in conformal field theory, Chern–Simons theory, and twisted equivariant K-theory.<ref name="FHT"/>

=== Integrable systems ===

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'' {{math|λ}}. When the coefficients depend rationally or Laurent-polynomially on {{math|λ}} and take values in a Lie algebra {{math|𝔤}}, they may be viewed as elements of a loop algebra {{math|''L''𝔤}}. 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.<ref name="Terng-integrable">{{cite journal |last=Terng |first=Chuu-Lian |author-link=Chuu-Lian Terng |title=Geometries and symmetries of soliton equations and integrable elliptic equations |journal=Advanced Studies in Pure Mathematics |series=Surveys on Geometry and Integrable Systems |volume=51 |year=2008 |pages=401–488 |doi=10.2969/aspm/05110401 |isbn=978-4-86497-001-3 }}</ref>

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.<ref name="Segal-Wilson-KdV">{{cite journal |last1=Segal |first1=Graeme |last2=Wilson |first2=George |title=Loop groups and equations of KdV type |journal=Publications Mathématiques de l'IHÉS |volume=61 |year=1985 |pages=5–65 |doi=10.1007/BF02698802 |mr=0783348 |url=http://www.numdam.org/item/PMIHES_1985__61__5_0/ }}</ref>

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 {{math|''n''}}-wave equation.<ref name="Terng-Uhlenbeck-Poisson">{{cite journal |last1=Terng |first1=Chuu-Lian |last2=Uhlenbeck |first2=Karen |author-link=Chuu-Lian Terng |author-link2=Karen Uhlenbeck |title=Poisson actions and scattering theory for integrable systems |journal=Surveys in Differential Geometry |volume=4 |year=1998 |pages=315–402 |doi=10.4310/SDG.1998.v4.n1.a7}}</ref>

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.<ref name="Terng-integrable" /><ref name="Dorfmeister-Inoguchi-Kobayashi">{{cite journal |last1=Dorfmeister |first1=Josef F. |last2=Inoguchi |first2=Jun-ichi |last3=Kobayashi |first3=Shimpei |title=A loop group method for affine harmonic maps into Lie groups |journal=Advances in Geometry |volume=16 |issue=3 |year=2016 |pages=379–399 |doi=10.1515/advgeom-2015-0008}}</ref> 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.<ref name="Brander-loop-decomp">{{cite journal |last=Brander |first=David |title=Loop group decompositions in almost split real forms and applications to soliton theory and geometry |journal=Journal of Geometry and Physics |volume=58 |issue=12 |year=2008 |pages=1792–1800 |doi=10.1016/j.geomphys.2008.09.003 |arxiv=0805.1979 |bibcode=2008JGP....58.1792B }}</ref>

=== 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.<ref name="Daniel-loopHodge">{{cite journal |last=Daniel |first=Jeremy |title=Loop Hodge structures and harmonic bundles |journal=Algebraic Geometry |volume=4 |issue=5 |year=2017 |pages=603–643 |doi=10.14231/AG-2017-030 |doi-access=free }}</ref>

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.<ref name="Daniel-loopHodge" />

== Current groups ==

A '''current group''' generalizes a loop group, replacing the circle with a smooth manifold {{math|''M''}}. Thus a current group is the group of smooth mappings from {{math|''M''}} into {{math|''G''}}, with multiplication defined pointwise: :<math>C^\infty(M,G)=\{f:M\to G \mid f \text{ is smooth}\}, \qquad (f_1f_2)(x)=f_1(x)f_2(x).</math> More generally, Segal described the mapping groups <math>\operatorname{Map}(X,G)</math> for compact manifolds {{math|''X''}} as higher-dimensional analogues of loop groups, noting that in mathematical physics they occur as current groups and gauge groups.<ref name="Segal1984-current">{{cite book |last=Segal |first=G. B. |title=Loop groups |series=Lecture Notes in Mathematics |volume=1064 |year=1984 |pages=155–168}}</ref>

If {{math|''G''}} is a topological group, the continuous mapping space {{math|''C''(''M'',''G'')}} becomes a topological group with the compact-open topology. For smooth maps one usually uses the smooth compact-open topology.<ref name="NeebWagemann2008-current">{{cite journal |last1=Neeb |first1=Karl-Hermann |last2=Wagemann |first2=Friedrich |title=Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds |journal=Geometriae Dedicata |volume=134 |year=2008 |pages=17–60 |doi=10.1007/s10711-008-9244-2 |arxiv=math/0703460 }}</ref> Under suitable hypotheses on {{math|''M''}} and {{math|''G''}}, this gives {{math|1=''C''∞(''M'',''G'')}} 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.<ref name="Neeb2006-current">{{cite journal |last=Neeb |first=Karl-Hermann |title=Towards a Lie theory of locally convex groups |journal=Japanese Journal of Mathematics |volume=1 |issue=2 |year=2006 |pages=291–468 |doi=10.1007/s11537-006-0606-y |arxiv=1501.06269 }}</ref>

For non-compact manifolds one often studies the compactly supported current group {{math|1=''C''∞''c''(''M'',''G'')}}, or more generally section groups {{math|Γ''c''(''M'',𝒢)}} of bundles of Lie groups over {{math|''M''}}. Gauge groups of principal bundles are of this form: if {{math|Ξ→''M''}} is a principal {{math|''G''}}-bundle, then its gauge group is isomorphic to the section group {{math|Γ(''M'',Ad(Ξ))}}, and the compactly supported gauge group to {{math|Γ''c''(''M'',Ad(Ξ))}}.<ref name="JanssensNeeb2024-current">{{cite book |last1=Janssens |first1=Bas |last2=Neeb |first2=Karl-Hermann |title=Positive Energy Representations of Gauge Groups I: Localization |series=Memoirs of the European Mathematical Society |volume=9 |publisher=EMS Press |location=Berlin |year=2024 |doi=10.4171/mems/9 |isbn=978-3-98547-067-9 }}</ref>

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.<ref name="Segal1984-current" /><ref name="JanssensNeeb2024-current" />

== Notes == {{reflist}}

== References ==

* {{cite journal |last=Freed |first=Daniel S. |author-link=Dan Freed |title=The geometry of loop groups |journal=Journal of Differential Geometry |volume=28 |issue=2 |year=1988 |pages=223–276 |doi=10.4310/jdg/1214442279 |bibcode=1988JDGeo..2842279F |mr=0961515}} * {{cite journal |last1=Freed |first1=Daniel S. |last2=Hopkins |first2=Michael J. |last3=Teleman |first3=Constantin |title=Loop groups and twisted K-theory III |journal=Annals of Mathematics |series=2 |volume=174 |issue=2 |year=2011 |pages=947–1007 |doi=10.4007/annals.2011.174.2.5}} * {{cite journal |last1=Pappas |first1=George |last2=Rapoport |first2=Michael |title=Twisted loop groups and their affine flag varieties |journal=Advances in Mathematics |volume=219 |issue=1 |year=2008 |pages=118–198 |doi=10.1016/j.aim.2008.04.006 |mr=2435422}} * {{cite book |last1=Pressley |first1=Andrew |last2=Segal |first2=Graeme |author-link2=Graeme Segal |title=Loop groups |series=Oxford Mathematical Monographs |publisher=Clarendon Press |location=Oxford |year=1986|mr=0900587|isbn=0-19-853535-X}} * {{cite book |last=Segal |first=G. B. |title=Loop groups |series=Lecture Notes in Mathematics |volume=1064 |year=1984 |pages=155–168}} * {{cite journal |last1=Terng |first1=Chuu-Lian |last2=Uhlenbeck |first2=Karen |author-link=Chuu-Lian Terng |author-link2=Karen Uhlenbeck |title=Geometry of solitons |journal=Notices of the American Mathematical Society |volume=47 |issue=1 |year=2000 |pages=17–25 |url=https://www.ams.org/notices/200001/fea-terng.pdf}}

Category:Lie groups Category:Topological groups Category:Representation theory Category:Algebraic groups