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

Abstract

Abstract. Let A be the C * -algebra associated with SU q (2), let π 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 π(A) that has bounded commutator with 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-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 Poincaré duality.

K-groups of the quantum homogeneous space SUq(n)∕SUq(n− 2)

“…It is natural to expect that, for compact quantum groups and their homogeneous spaces, there should be associated canonical spectral triples. Chakraborty and Pal [2003] showed that indeed that is the case for quantum SU(2). In fact for odd-dimensional quantum spheres, one can construct finitely summable spectral triples that display Poincaré duality [Chakraborty and Pal 2010].…”

K-groups of the quantum homogeneous space $SU_{q}(n)/SU_{q}(n-2)$

“…Chakraborty and Pal showed that ( [3]) indeed that is the case for quantum SU (2). In fact for odd dimensional quantum spheres one can construct finitely summable spectral triples that witnesses Poincare duality ( [4]). A natural question in this connection is, are these examples somewhat singular or in general one can construct finitely summable spectral triples with further properties like Poincare duality on quantum groups associated with Lie groups or their homogeneous spaces.…”