Equivariant spectral triples and Poincaré duality for 𝑆𝑈_{𝑞}(2)

Abstract

Let A \mathcal {A} be the C ∗ C^* -algebra associated with S U q ( 2 ) SU_q(2) , let π \pi be the representation by left multiplication on the L 2 L_2 space of the Haar state and let D D be the equivariant Dirac operator for this representation constructed by the authors earlier. We prove in this article that there is no operator other than the scalars in the commutant π ( A ) ′ \pi (\mathcal {A})’ that has bounded commutator with D D . This implies that the equivariant spectral triple under consideration does not admit a rational Poincaré dual in the sense of Moscovici, which in particular means that this spectral triple does not extend to a K K -homology fundamental class for S U q ( 2 ) SU_q(2) . We also show that a minor modification of this equivariant spectral triple gives a fundamental class and thus implements Poincaré duality.