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

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 SUq(2) to investigate equivariant spectral triples for two classes of spaces: the quantum groups SUq(l+1) for l > 1, and the odd dimensional quantum spheres S q 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 as a homogeneous space for SUq(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. AMS Subject Classification No.: 58B34, 46L87, 19K33