Infinitely many star products to play with

TL;DR

This paper introduces new classes of star products based on linear Lie algebras, explores their properties, and extends noncommutative geometry concepts like spectral triples to these spaces, broadening the scope of noncommutative geometry.

Contribution

It presents novel non-formal star products for linear Lie algebra deformations and develops the framework of star triples for noncompact noncommutative spaces.

Findings

Defined new star products for all three-dimensional Lie algebra cases

Introduced noncompact spectral triples and star triples

Analyzed properties of the Moyal multiplier algebra

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}^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.