Torus equivariant spectral triples for odd dimensional quantum spheres coming from $C^*$-extensions

Details Torus equivariant spectral triples for odd dimensional quantum spheres coming from $C^*$-extensions Torus equivariant spectral triples for odd dimensional quantum spheres coming from $C^*$-extensions

2007-01-25

arxiv.org/abs/math/0701738v1

Mathematics

K-Theory and Homology

LaTeX2e, 12 pages

Scientific paper

The torus group $(S^1)^{\ell+1}$ has a canonical action on the odd dimensional sphere $S_q^{2\ell+1}$. We take the natural Hilbert space representation where this action is implemented and characterize all odd spectral triples acting on that space and equivariant with respect to that action. This characterization gives a construction of an optimum family of equivariant spectral triples having nontrivial $K$-homology class thus generalizing our earlier results for $SU_q(2)$. We also relate the triple we construct with the $C^*$-extension \[ 0\longrightarrow \clk\otimes C(S^1)\longrightarrow C(S_q^{2\ell+3}) \longrightarrow C(S_q^{2\ell+1}) \longrightarrow 0. \]

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