Theoretical Statistics and Mathematics Unit, ISI Delhi

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

by Partha Sarathi Chakraborty and Arupkumar Pal

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.

isid/ms/2005/03 [fulltext]

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