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

TL;DR

This paper demonstrates that the K-homology class of the Dirac operator on q-deformed compact Lie groups aligns with the classical case under KK-equivalence, revealing deep structural similarities.

Contribution

It establishes the correspondence of Dirac operator classes on quantum and classical groups and constructs continuous Drinfeld twists relating their algebraic structures.

Findings

K-homology class of Dirac operator matches classical case under KK-equivalence

Constructs continuous family of Drinfeld twists for quantum groups

Shows structural similarity between deformed and classical algebraic objects

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