J.M. Gracia-Bondía, F. Lizzi, G. Marmo, P. Vitale
Infinitely Many Star Products to Play with
Preprint series: ESI preprints

MSC:

58B30 Noncommutative differential geometry and topology, See also {46L30, 46L87, 46L89}

81R50 Quantum groups and related algebraic methods, See Also {16W30, 17B37}

81T30 String and superstring theories; other extended objects , See also {83E30}

PACS: 02.40.Gh,03.65.Fd,11.25.-w
Abstract: 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
$[x_i,x_j]=ic_{ij}^kx_k$. 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 {\em noncompact\/}
spectral triples, and the concept of {\em 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.

Keywords: non-commutative geometry, spacetime symmetry, differential and algebraic geometry