Equivariant spectral triples and Poincaré duality for $SU_q(2)$
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- by Partha Sarathi Chakraborty and Arupkumar Pal PDF
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Abstract:
Let $\mathcal {A}$ be the $C^*$-algebra associated with $SU_q(2)$, let $\pi$ be the representation by left multiplication on the $L_2$ space of the Haar state and let $D$ be the equivariant Dirac operator for this representation constructed by the authors earlier. We prove in this article that there is no operator other than the scalars in the commutant $\pi (\mathcal {A})’$ that has bounded commutator with $D$. This implies that the equivariant spectral triple under consideration does not admit a rational Poincaré dual in the sense of Moscovici, which in particular means that this spectral triple does not extend to a $K$-homology fundamental class for $SU_q(2)$. We also show that a minor modification of this equivariant spectral triple gives a fundamental class and thus implements Poincaré duality.References
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Additional Information
- Partha Sarathi Chakraborty
- Affiliation: Institute of Mathematical Sciences, CIT Campus, Chennai–600 113, India
- MR Author ID: 670986
- Email: parthac@imsc.res.in
- Arupkumar Pal
- Affiliation: Indian Statistical Institute, 7, SJSS Marg, New Delhi–110 016, India
- Email: arup@isid.ac.in
- Received by editor(s): October 29, 2007
- Received by editor(s) in revised form: December 20, 2007
- Published electronically: March 23, 2010
- Additional Notes: The first author acknowledges support from Endeavour India Executive Award 2007, DEST, Government of Australia
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 4099-4115
- MSC (2010): Primary 58B34, 46L87, 19K35
- DOI: https://doi.org/10.1090/S0002-9947-10-05139-1
- MathSciNet review: 2608397