Loop group

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.