Equivariant spectral triples for $SU_q(\ell+1)$ and the odd dimensional quantum spheres

Chakraborty, Partha Sarathi; Pal, Arupkumar

Equivariant spectral triples for $SU_q(\ell+1)$ and the odd dimensional quantum spheres

Authors

Description

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(\ell+1) for \ell>1, and the odd dimensional quantum spheres S_q^{2\ell+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 L_2 space is obtained. Using this, we then exhibit equivariant Dirac operators with nontrivial sign on direct sums of multiple copies of the L_2 space. In the latter case, viewing S_q^{2\ell+1} as a homogeneous space for SU_q(\ell+1), we give a complete characterization of equivariant Dirac operators, and also produce an optimal family of spectral triples with nontrivial K-homology class.
v2: some small mistakes resulting from an error in an expression for CG coefficient corrected; three references added. v1: LaTeX2e, uses xy-pic and eepic