Theoretical Statistics and Mathematics Unit, ISI Delhi

Regularity and dimension spectrum of the equivariant spectral triple for the odd dimensional quantum spheres

by Arupkumar Pal and S. Sundar

The odd dimensional quantum sphere $S_q^{2\ell+1}$ is a homogeneous
space for the quantum $SU(\ell+1)$ group. A generic equivariant spectral
triple for $S_q^{2\ell+1}$ on its $L_2$ space was constructed by
Chakraborty \& Pal. We prove
regularity for that spectral triple here. We also compute its
dimension spectrum and show that it is simple.
We give detailed construction of its smooth function
algebra and some related algebras that help proving
regularity and in the computation of the dimension spectrum.
Following the idea of Connes for $SU_q(2)$,
we first study another spectral triple for $S_q^{2\ell+1}$
equivariant under torus group action
constructed by Chakraborty \& Pal.
We then derive the results for the $SU_q(\ell+1)$-equivariant
triple in the $q=0$ case from those for the torus equivariant
triple. For the $q\neq 0$ case, we deduce regularity and dimension
spectrum from the the $q=0$ case.

isid/ms/2008/09 [fulltext]

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