Equivariant spectral triples and Poincaré duality for $SU_q(2)$

Details Equivariant spectral triples and Poincaré duality for $SU_q(2)$ Equivariant spectral triples and Poincaré duality for $SU_q(2)$

2002-11-23

arxiv.org/abs/math/0211367v2

Mathematics

Operator Algebras

v2: main result strengthened, a new section added, title changed; 21 pages, LaTeX v1: 9 pages, Latex2e

Scientific paper

Let $\mathcal{A}$ be the $C^*$-algebra associated with $SU_q(2)$, $\pi$ be the representation by left multiplication on the $L_2$ space of the Haar state and let $D$ 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 $\pi(\cla)'$ that has bounded commutator with $D$. This implies that the equivariant spectral triple under consideration does not admit a rational Poincar\'e dual in the sense of Moscovici, which in particular means that this spectral triple does not extend to a $K$-homology fundamental class for $SU_q(2)$. We also show that a minor modification of this equivariant spectral triple gives a fundamental class and thus implements Poincar\'e duality.

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