The Geometry of some Quantum Homogeneous Spaces and the Weak Heat Kernel Expansion[HBNI Th29]

Abstract

We study the non-commutative geometry of some quantum homogeneous spaces associated with the quantum group SUq(n). First we consider the quantum space SUq(n)/SUq(n − 1) called the odd dimensional quantum spheres and denoted S2n−1 q . We consider two spectral triples associated to the odd dimensional quantum spheres S2n−1 q . We show that the spectral triples satisfy the hypothesis of the local index formula. A conceptual explanation is given by considering a property which we call the weak heat kernel asymptotic expansion property of spectral triples. We show that a spectral triple having the weak heat kernel asymptotic expansion property satisfies the hypothesis of the local index formula. We also show that this property is stable under quantum double suspension. Finally we compute the K-groups of the quantum homogeneous space SUq(n)/SUq(n − 2).