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loop group

Contents

Idea

The loop space of a topological group GG inherits the structure of a group under pointwise group multiplication of loops. This is called a loop group of GG.

(Notice that this is a group structure in addition to the infinity-group-structure of any loop space under composition of loops.)

If GG is a Lie group, then there is a smooth version of the loop group consisting of smooth functions S 1GS^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 1S^1 being a compact manifold. For XX any other compact manifold there is similarly an infinite-dimensional Lie group [X,G][X,G] of smooth functions XGX \to G under pointwise multiplication in GG.

Such mapping groups appear in physics notably as groups of gauge transformations over a spacetime/worldvolume XX. 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 GG. 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 XX of dimension greater that 1 there are very few known results about the properties of the mapping group [X,G][X,G].

Properties

Lie algebra

Let GG be a compact Lie group. Write 𝔤\mathfrak{g} for its Lie algebra.

Proposition

The Lie algebra of LGL G is the loop Lie algebra?

Lie(LG)LLie(G)=L𝔤. Lie(L G) \simeq L Lie(G) = L \mathfrak{g} \,.

Complexification

Let GG be a compact Lie group.

Proposition

The complexification? of LGL G is the loop group of the complexification of GG

(LG) L(G ). (L G)_{\mathbb{C}} \simeq L (G_\mathbb{C}) \,.

Central extensions

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)

Representations

Positive energy

Write

t θ:LGLG t_\theta \colon L G \to L G

for the automorphism which rotates loops by an angle? θ\theta.

The corresponding semidirect product group we write S 1LGS^1 \rtimes L G

Definition

Let VV be a topological vector space. A linear representation

S 1Aut(V) S^1 \to Aut(V)

of the circle group is called positive if exp(iθ)\exp(i \theta) acts by exp(iAθ)\exp(i A \theta) where AEnd(V)A \in End(V) is a linear operator with positive spectrum.

A linear representation

ρ:LGAut(V) \rho : L G \to Aut(V)

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 1LGS^1 \rtimes L G such that the restriction to S 1S^1 is positive.

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 GG be a compact Lie group. Let TGT \hookrightarrow G be the inclusion of a maximal torus. There is a fiber sequence

G/T LG/T LG/G ΩG. \array{ G/T &\to& L G / T \\ && \downarrow \\ && L G / G & \simeq \Omega G } \,.
Remark

By the discussion at orbit method, if GG is a semisimple Lie group, then G/TG/T is isomorphic to the coadjoint orbit of an element λ,𝔤 *\langle \lambda , -\rangle \in \mathfrak{g}^* for which TG λT \simeq G_\lambda is the stabilizer subgroup.

If moreover GG is simply connected, then the weight lattice Γ wt𝔱 *𝔱\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t} of the Lie group GG is isomorphic to the group of group characters

Γ wtHom LieGrp(G,U(1)). \Gamma_{wt} \stackrel{\simeq}{\to} Hom_{LieGrp}(G,U(1)) \,.
Proposition

The irreducible projective positive energy representations of LGL G correspond precisley to the possible geometric quantizations of LG/TL G / T (as in the orbit method).

More in detail:

The degree-2 integral cohomology of LG/TL G / T is

H 2(LG/T)H 2(G/T,)T^. H^2(L G / T) \simeq \mathbb{Z} \oplus H^2(G / T, \mathbb{Z}) \simeq \mathbb{Z} \oplus \hat T \,.

Writing L n,λL_{n,\lambda} for the corresponding complex line bundle with level nn \in \mathbb{Z} and weight λT^\lambda \in \hat T we have that

  1. the space of holomorphic sections of L n,λL_{n,\lambda} is either zero or is an irreducible positive energy representation;

  2. every such arises this way;

  3. and is non-zero precisely if (n,λ)(n,\lambda) is positive in the sense that for each positive coroot? h αh_\alpha of GG

    0λ(h α)nh α,h α. 0 \leq \lambda(h_\alpha) \leq n \langle h_\alpha, h_\alpha\rangle \,.

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 LGL G is equivalent to the GG-equivariant elliptic cohomology (see there for more) of the point (Ando 00, theorem 10.10).

This is a higher analog of how GG-equivariant K-theory of the point gives the representation ring of GG.

References

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)

Revised on March 28, 2014 05:23:41 by Urs Schreiber (89.204.154.44)