Equivariant spectral triples for SU_q(l+1) and the odd dimensional quantum spheres

Abstract

We formulate the notion of equivariance of an operator with respect to a covariant representation of a C^∗-dynamical system. We then use a combinatorial technique used by the authors earlier in characterizing spectral triples for SU_q(2) to investigate equivariant spectral triples for two classes of spaces: the quantum groups SU_q(l+1) for l>1, and the odd dimensional quantum spheres S_q^2l+1 of Vaksman & Soibelman. In the former case, a precise characterization of the sign and the singular values of an equivariant Dirac operator acting on the L₂ space is obtained. Using this, we then exhibit equivariant Dirac operators with nontrivial sign on direct sums of multiple copies of the L₂ space. In the latter case, viewing S_q^2l+1 as a homogeneous space for SU_q(l+1), we give a complete characterization of equivariant Dirac operators, and also produce an optimal family of spectral triples with nontrivial K-homology class.