K-homology class of the Dirac operator on a compact quantum group

Abstract

By a result of Nagy, the C^*-algebra of continuous functions on the $q$-deformation $G_q$ of a simply connected semisimple compact Lie group $G$ is KK-equivalent to $C(G)$. We show that under this equivalence the K-homology class of the Dirac operator on $G_q$, which we constructed in an earlier paper, corresponds to that of the classical Dirac operator. Along the way we prove that for an appropriate choice of isomorphisms between completions of $U_q\mathfrak{g}$ and $U\mathfrak{g}$ a family of Drinfeld twists relating the deformed and classical coproducts can be chosen to be continuous in $q$.