The product structure of the equivariant K-theory of the based loop group of SU(2)

TL;DR

This paper computes the G-equivariant K-theory of the based loop group of SU(2), revealing a structure as an inverse limit of symmetric subalgebras, generalizing classical cohomology results.

Contribution

It provides a novel computation of the equivariant K-theory of the loop space of SU(2), extending classical cohomology descriptions to an equivariant K-theoretic setting.

Findings

K^*_G(\Omega G) is an inverse limit of symmetric subalgebras.

The structure generalizes the classical divided polynomial algebra.

The algebra is described via symmetric invariants in products of P^1.

Abstract

Let G=SU(2) and let \Omega G denote the space of continuous based loops in G, equipped with the pointwise conjugation action of G. It is a classical fact in topology that the ordinary cohomology H^*(\Omega G) is a divided polynomial algebra \Gamma[x]. The algebra \Gamma[x] can be described as an inverse limit as k goes to infinity of the symmetric subalgebra in the exterior algebra \Lambda(x_1, ...,x_k) in the variables x_1, ..., x_k. We compute the R(G)-algebra structure of the G-equivariant K-theory of \Omega G in a way which naturally generalizes the classical computation of the ordinary cohomology ring of \Omega G as a divided polynomial algebra \Gamma[x]. Specifically, we prove that K^*_G(\Omega G) is an inverse limit of the symmetric (S_{2r}-invariant) subalgebra of K^*_G((P^1)^{2r}), where the symmetric group S_{2r} acts in the natural way on the factors of the 2r-fold product (P^1)^{2r} and G acts diagonally via the standard action on each complex projective line P^1.