A reconstruction theorem for Connes-Landi deformations of commutative spectral triples

JOURNAL OF GEOMETRY AND PHYSICS

卷:98

Article

Cacic, Branimir

关键词: Noncommutative geometry;    Spectral triple;    Strict deformation quantisation;    Connes-Landi deformation;    Isospectral deformation;    Toric noncommutative manifold;   

DOI  :  10.1016/j.geomphys.2015.07.028

来源: Elsevier

【 摘 要 】

We formulate and prove an extension of Connes's reconstruction theorem for commutative spectral triples to so-called Connes-Landi or isospectral deformations of commutative spectral triples along the action of a compact Abelian Lie group G, also known as toric noncommutative manifolds. In particular, we propose an abstract definition for such spectral triples, where noncommutativity is entirely governed by a deformation parameter sitting in the second group cohomology of the Pontryagin dual of G, and then show that such spectral triples are well-behaved under further Connes-Landi deformation, thereby allowing for both quantisation from and dequantisation to G-equivariant abstract commutative spectral triples. We then use a refinement of the Connes-Dubois-Violette splitting homomorphism to conclude that suitable Connes-Landi deformations of commutative spectral triples by a rational deformation parameter are almost-commutative in the general, topologically non-trivial sense. (C) 2015 Elsevier B.V. All rights reserved.

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