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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Equivariant spectral triples and Poincaré duality for $ SU_q(2)$

Authors: Partha Sarathi Chakraborty and Arupkumar Pal
Journal: Trans. Amer. Math. Soc. 362 (2010), 4099-4115
MSC (2010): Primary 58B34, 46L87, 19K35
Published electronically: March 23, 2010
MathSciNet review: 2608397
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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.

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Additional Information

Partha Sarathi Chakraborty
Affiliation: Institute of Mathematical Sciences, CIT Campus, Chennai–600 113, India

Arupkumar Pal
Affiliation: Indian Statistical Institute, 7, SJSS Marg, New Delhi–110 016, India

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
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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