K-Homology Class of the Dirac Operator on a Compact Quantum Group

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 $U_q\g$ and $U\g$ a family of Drinfeld twists relating the deformed and classical coproducts can be chosen to be continuous in $q$.

2010 Mathematics Subject Classification: 58B34, 58B32, 46L80

Keywords and Phrases: Quantum groups, Dirac operator, K-homology

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