Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2001-12-11
JHEP0204:026,2002
Physics
High Energy Physics
High Energy Physics - Theory
LaTeX, 36 pages. Minor corrections, references added
Scientific paper
10.1088/1126-6708/2002/04/026
While there has been growing interest for noncommutative spaces in recent times, most examples have been based on the simplest noncommutative algebra: [x_i,x_j]=i theta_{ij}. Here we present new classes of (non-formal) deformed products associated to linear Lie algebras of the kind [x_i,x_j]=ic_{ij}^k x_k. For all possible three-dimensional cases, we define a new star product and discuss its properties. To complete the analysis of these novel noncommutative spaces, we introduce noncompact spectral triples, and the concept of star triple, a specialization of the spectral triple to deformations of the algebra of functions on a noncompact manifold. We examine the generalization to the noncompact case of Connes' conditions for noncommutative spin geometries, and, in the framework of the new star products, we exhibit some candidates for a Dirac operator. On the technical level, properties of the Moyal multiplier algebra M(R_\theta^{2n) are elucidated.
Gracia-Bondia Jose M.
Lizzi Fedele
Marmo Giuseppe
Vitale Patrizia
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