THE PRODUCT STRUCTURE OF THE EQUIVARIANT K-THEORY OF THE BASED LOOP GROUP OF SU(2) Academic Article uri icon

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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.

publication date

  • June 1, 2014