Document Zbl 1241.19002 - zbMATH Open

The following three facts are well-known:

1)

The irreducible representations of a compact Lie group \(G\) are classified by the lowest/highest weights.

2)

The irreducible positive energy representations of the loop group \(LG= \mathrm{Loop}(\mathbb S^1,G)\) associated with a compact Lie group \(G\), are classified, as the representations of the central extension \[ 1 \to U(1) \to \widetilde{LG} \to LG \to 1; \] the topological class of this extension is known as the level \(\tau\).

3)

The Verlinde product, as the fusion rule on \(R^\tau(LG)\) makes it to become a ring, the Verlinde ring, and its complexifictation \(R^\tau(LG)\otimes \mathbb C\) - an algebras, the Verlinde algebra.

These three ingredients are composed together to make the twisted \(K\)-theory of compact Lie groups.
The authors start with the twisted \(K\)-theory by examples in §1 and twistings in \(K\)-theory in §2. The formal theory of twisted \(K\)-groups is introduced in §3 and some computations of \(K^\tau_G(G)\) for nondegenerate \(\tau\) in §4.
The main result of the paper is Theorem 1: Let \(G\) be a compact Lie group and \(\tau\) be a level for the loop group \(LG\). The Grothendieck group \(R^\tau(LG)\) at level \(\tau\) is isomorphic to a twisted form \(K^{\zeta(\tau)}_G(G)\), of the equivariant \(K\)-theory of \(G\) acting on itself by conjugation. Under this isomorphism the fusion product, when it is defined, corresponds to the Pontryagin product. The twisting \(\zeta(\tau)\) is given in terms of the level \(\zeta(\tau) = \mathfrak{g} + \check{h} + \tau\), where \(\check{h}\) is the dual Coxeter twisting: the adjoint representation of \(G\) in the Lie algebra \(\mathfrak{g}\) makes \(\mathfrak{g}\) to be a vector bundle over a point and pulled back to \(G\) as a twisting and when \(G\) is simple and simply connected, the integer corresponding to \(\check{h}\) is the dual Coxeter number, and \(\mathfrak{g}\) is the degree shift.

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