Infinitely many star products to play with
By J. Gracia-Bond́ıa, F. Lizzi, G. Marmo, P. Vitale. 2001
147 citations recorded by the source corpus.
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 ] = iθ 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(θ 2n ) are elucidated.
Subfields: Algebra and Number Theory, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/697d8e7fcef665afc636fada406bc9e38ae8bc95
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In Algebra and Number Theory
In Mathematical Physics
Citations in the corpus
- Moyal Planes are Spectral Triples (2003, background): “(There have been, of course, some noteworthy partial treatments: one can mention [ 41 , 73], which identify some of the outstanding issues.) The time has come to add a new twist to the tale.”
- Smoothness and Locality for Nonunital Spectral Triples (2003, background): “We also note that similar proposals have been advanced in [10].”
- What is geometry in quantum theory (2004, unknown): “comes from Connes’ noncommutative geometry [141].”