The loop space of a topological group $G$ inherits the structure of a group under pointwise group multiplication of loops. This is called a loop group of $G$.
(Notice that this is a group structure in addition to the infinity-group-structure of any loop space under composition of loops.)
If $G$ is a Lie group, then there is a smooth version of the loop group consisting of smooth functions $S^1 \to G$. By the discussion at manifold structure of mapping spaces the collection of such smooth maps is itself an infinite-dimensional smooth manifold and so the smooth loop group of a Lie group is an infinite-dimensional Lie group.
Among all infinite-dimensional Lie groups, loop groups are a most well behaved class. In particular their representation theory is similar to that of compact Lie groups.
Some of these nice properties are solely due to the circle $S^1$ being a compact manifold. For $X$ any other compact manifold there is similarly an infinite-dimensional Lie group $[X,G]$ of smooth functions $X \to G$ under pointwise multiplication in $G$.
Such mapping groups appear in physics notably as groups of gauge transformations over a spacetime/worldvolume $X$. Accordingly, loop groups play a prominint role in 1- and 2-dimensional quantum field theory, notably the WZW model describing the propagation of a string on $G$. The current algebras (affine algebras) which arise as Lie algebras of (centrally extended) loop groups derive their name from this relation to physics. Accordingly, as for compact Lie groups, the representation theory of loop groups is naturally understood in terms of their geometric quantization (by a loop variant of the orbit method).
On the other hand, for $X$ of dimension greater that 1 there are very few known results about the properties of the mapping group $[X,G]$.
Let $G$ be a compact Lie group. Write $\mathfrak{g}$ for its Lie algebra.
The Lie algebra of $L G$ is the loop Lie algebra?
Let $G$ be a compact Lie group.
The complexification? of $L G$ is the loop group of the complexification of $G$
Loop groups of compact Lie groups have canonical central extensions, often called Kac-Moody central extensions . A detailed discussion is in (PressleySegal). A review is in (BCSS)
Write
for the automorphism which rotates loops by an angle? $\theta$.
The corresponding semidirect product group we write $S^1 \rtimes L G$
Let $V$ be a topological vector space. A linear representation
of the circle group is called positive if $\exp(i \theta)$ acts by $\exp(i A \theta)$ where $A \in End(V)$ is a linear operator with positive spectrum.
A linear representation
is said to have positive energy or to be a positive energy representation if it extends to a representation of the semidirect product group $S^1 \rtimes L G$ such that the restriction to $S^1$ is positive.
We discuss the quantization of loop groups in the sense of geometric quantization of their canonical prequantum bundle.
Let $G$ be a compact Lie group. Let $T \hookrightarrow G$ be the inclusion of a maximal torus. There is a fiber sequence
By the discussion at orbit method, if $G$ is a semisimple Lie group, then $G/T$ is isomorphic to the coadjoint orbit of an element $\langle \lambda , -\rangle \in \mathfrak{g}^*$ for which $T \simeq G_\lambda$ is the stabilizer subgroup.
If moreover $G$ is simply connected, then the weight lattice $\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t}$ of the Lie group $G$ is isomorphic to the group of group characters
The irreducible projective positive energy representations of $L G$ correspond precisley to the possible geometric quantizations of $L G / T$ (as in the orbit method).
More in detail:
The degree-2 integral cohomology of $L G / T$ is
Writing $L_{n,\lambda}$ for the corresponding complex line bundle with level $n \in \mathbb{Z}$ and weight $\lambda \in \hat T$ we have that
the space of holomorphic sections of $L_{n,\lambda}$ is either zero or is an irreducible positive energy representation;
every such arises this way;
and is non-zero precisely if $(n,\lambda)$ is positive in the sense that for each positive coroot? $h_\alpha$ of $G$
This appears for instance as (Segal, prop. 4.2).
Under mild conditions (but over the complex numbers) the representation ring of a loop group $L G$ is equivalent to the $G$-equivariant elliptic cohomology (see there for more) of the point (Ando 00, theorem 10.10).
This is a higher analog of how $G$-equivariant K-theory of the point gives the representation ring of $G$.
The standard textbook on loop groups is
A review talk is
A review of some aspects with an eye towards loop groups as part of the crossed module of groups representing a string 2-group is in
The relation between representations of loop groups and twisted K-theory over the group is the topic of
The relation between representations of loop groups an equivariant elliptic cohomology of the point is discussed in
Discussion with respect to flag varieties is in