In mathematics, a loop group is, in the most common Lie-theoretic sense, the group LG = C∞(S1, G) of smooth maps from the circle S1 to a Lie group G, with multiplication defined pointwise. When G is a compact Lie group, LG is a basic example of an infinite-dimensional Lie group, with Lie algebra L𝔤 = C∞(S1, 𝔤). The subgroup ΩG 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 LG(R) = G(R((t))), together with their associated affine Grassmannians and affine flag varieties.
Definition Let G be a topological group. The set C(S1,G) of continuous maps from the circle to G becomes a topological group under pointwise multiplication when equipped with the compact-open topology. Since S1 is compact, this is the same as the topology of uniform convergence. In Lie theory one usually considers the group
L G = C ∞ ( S 1 , G ) {\displaystyle LG=C^{\infty }(S^{1},G)}
of smooth loops in a finite-dimensional Lie group G. It is endowed with the smooth compact-open topology, namely the initial topology induced by the iterated tangent maps
C ∞ ( S 1 , G ) → ∏ k ≥ 0 C ( T k S 1 , T k G ) . {\displaystyle C^{\infty }(S^{1},G)\to \prod _{k\geq 0}C(T^{k}S^{1},T^{k}G).}
With this topology, LG is an infinite-dimensional Lie group. Its Lie algebra is
L g = C ∞ ( S 1 , g ) , {\displaystyle L{\mathfrak {g}}=C^{\infty }(S^{1},{\mathfrak {g}}),}
with pointwise bracket. Since S1 is compact, the smooth compact-open topology on L g {\displaystyle L{\mathfrak {g}}} is the Fréchet topology of uniform convergence of all derivatives on S1; equivalently, after choosing an angular coordinate on S1 and a norm on g {\displaystyle {\mathfrak {g}}} , it is defined by the seminorms
p n ( X ) = sup θ ∈ S 1 ‖ X ( n ) ( θ ) ‖ ( n ≥ 0 ) . {\displaystyle p_{n}(X)=\sup _{\theta \in S^{1}}\|X^{(n)}(\theta )\|\qquad (n\geq 0).}
For compact G, smooth loop groups are modeled on nuclear Fréchet spaces.
Basic constructions
Free and based loop groups The free loop group of G is LG itself. The based loop group is
Ω G = { γ ∈ L G : γ ( 1 ) = e } , {\displaystyle \Omega G=\{\gamma \in LG:\gamma (1)=e\},}
the kernel of the evaluation map
ev 1 : L G → G , γ ↦ γ ( 1 ) . {\displaystyle \operatorname {ev} _{1}:LG\to G,\qquad \gamma \mapsto \gamma (1).}
Thus ΩG is a closed normal subgroup of LG. The inclusion of constant loops gives a splitting of ev1, so there is a split exact sequence
1 → Ω G → L G → ev 1 G → 1 , {\displaystyle 1\to \Omega G\to LG\xrightarrow {\operatorname {ev} _{1}} G\to 1,}
and hence a semidirect product decomposition
L G ≅ Ω G ⋊ G . {\displaystyle LG\cong \Omega G\rtimes G.}
Relation with loop spaces As a topological space, ΩG is the based loop space of 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.
Basic topology The splitting of the evaluation map
ev 1 : L G → G {\displaystyle \operatorname {ev} _{1}:LG\to G}
by constant loops identifies LG with G × ΩG as a topological space:
L G ≅ G × Ω G , γ ↦ ( γ ( 1 ) , γ ( 1 ) − 1 γ ) . {\displaystyle LG\cong G\times \Omega G,\qquad \gamma \mapsto {\bigl (}\gamma (1),\,\gamma (1)^{-1}\gamma {\bigr )}.}
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