Cohomology and Extensions
The loop space of a topological group inherits the structure of a group under pointwise group multiplication of loops. This is called a loop group of .
(Notice that this is a group structure in addition to the infinity-group-structure of any loop space under composition of loops.)
If is a Lie group, then there is a smooth version of the loop group consisting of smooth functions . 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 being a compact manifold. For any other compact manifold there is similarly an infinite-dimensional Lie group of smooth functions under pointwise multiplication in .
Such mapping groups appear in physics notably as groups of gauge transformations over a spacetime/worldvolume . 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 . 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 of dimension greater that 1 there are very few known results about the properties of the mapping group .
Let be a compact Lie group. Write for its Lie algebra.
The Lie algebra of is the loop Lie algebra?
Let be a compact Lie group.
The complexification? of is the loop group of the complexification of
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)
for the automorphism which rotates loops by an angle? .
The corresponding semidirect product group we write
By geometric quantization (looped orbit method)
We discuss the quantization of loop groups in the sense of geometric quantization of their canonical prequantum bundle.
Let be a compact Lie group. Let be the inclusion of a maximal torus. There is a fiber sequence
The irreducible projective positive energy representations of correspond precisley to the possible geometric quantizations of (as in the orbit method).
More in detail:
The degree-2 integral cohomology of is
Writing for the corresponding complex line bundle with level and weight we have that
the space of holomorphic sections of is either zero or is an irreducible positive energy representation;
every such arises this way;
and is non-zero precisely if is positive in the sense that for each positive coroot? of
This appears for instance as (Segal, prop. 4.2).
Relation to equivariant elliptic cohomology
Under mild conditions (but over the complex numbers) the representation ring of a loop group is equivalent to the -equivariant elliptic cohomology (see there for more) of the point (Ando 00, theorem 10.10).
This is a higher analog of how -equivariant K-theory of the point gives the representation ring of .
The standard textbook on loop groups is
- Andrew Pressley, Graeme Segal, Loop groups Oxford University Press (1988)
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
- BCSS, From loop groups to 2-groups (web)
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
- Matthew Ando, Power operations in elliptic cohomology and representations of loop groups Transactions of the American Mathematical Society 352, 2000, pp. 5619-5666. (JSTOR, pdf)
Discussion with respect to flag varieties is in
- Shrawan Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Birkhäuser 2002