Infinitely many star products to play with

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_θ^{2n) are elucidated.

LaTeX, 36 pages. Minor corrections, references added

References in corpus (25)

• String Theory and Noncommutative Geometry
• Noncommutative Field Theory
• The quantum structure of spacetime at the Planck scale and quantum fields
• Noncommutative Geometry and Matrix Theory: Compactification on Tori
• Gravity coupled with matter and foundation of non-commutative geometry
• Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations
• Non-commutative World-volume Geometries: Branes on SU(2) and Fuzzy Spheres
• A Universal Action Formula
• Algebras of distributions suitable for phase-space quantum mechanics. I
• Nonabelian noncommutative gauge theory via noncommutative extra dimensions
• Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra
• Quantum symmetry groups of noncommutative spheres
• *-TREK: The One-Loop N=4 Noncommutative SYM Action
• Fuzzy Spheres in AdS/CFT Correspondence and Holography from Noncommutativity
• Map of Witten's * to Moyal's *
• Geometry of the Gauge Algebra in Noncommutative Yang-Mills Theory
• Non-commutative world-volume interactions on D-brane and Dirac-Born-Infeld action
• On summability of distributions and spectral geometry
• On the ultraviolet behaviour of quantum fields over noncommutative manifolds
• Connes duality in Lorentzian geometry
• Twists and spectral triples for isospectral deformations
• Noncommutative perturbation in superspace
• A generalization of the Jordan-Schwinger map: classical version and its q--deformation
• Contracting the Wigner-Kernel of a Spin to the Wigner-Kernel of a Particle
• Spectral Quadruples

Cited by in corpus (48)

• Cosmological perturbations and short distance physics from Noncommutative Geometry
• Star products, duality and double Lie algebras
• Quantization maps, algebra representation and non-commutative Fourier transform for Lie groups
• Twisted Noncommutative Field Theory with the Wick-Voros and Moyal Products
• Generalized Weyl systems and kappa-Minkowski space
• Noncommutative field theory with the Wick-Voros product
• Noncommutative field theory on ${\mathbb{R}}_{\lambda }^3$
• Noncommutative field theories on $R_{\lambda }^3$: Towards UV/IR mixing freedom
• Noncommutative field theory from angular twist
• Duality symmetry for star products
• Translation Invariance, Commutation Relations and Ultraviolet/Infrared Mixing
• Quantum gauge theories on noncommutative 3-d space
• Local covariant quantum field theory over spectral geometries
• Noncommutative $R^d$ via closed star product
• Gauge theories on quantum spaces
• kappa-Minkowski representations on Hilbert spaces
• Non-commutative deformation of Chern-Simons theory
• Matrix Bases for Star Products: a Review
• Noncommutative differential calculus for Moyal subalgebras
• Moyal multiplier algebras of the test function spaces of type S
• Position-dependent noncommutative products: classical construction and field theory
• Time Discretization From Noncommutativity
• Non-singular Brans-Dicke collapse in deformed phase space
• Noncommutative gauge theories on ${\mathbb{R}}_{\lambda }^3$: Perturbatively finite models
• Late time acceleration in a non-commutative model of modified cosmology
• Slavnov-Taylor identities, non-commutative gauge theories and infrared divergences
• A novel approach to non-commutative gauge theory
• Closed star product on noncommutative ${\mathbb{R}}^3$ and scalar field dynamics
• $\kappa$-Minkowski star product in any dimension from symplectic realization
• Localization and observers in $\varrho$-Minkowski spacetime
• Feynman problem in the noncommutative case
• Star Product and Invariant Integration for Lie type Noncommutative Spacetimes
• Involutive representations of coordinate algebras and quantum spaces
• Double Quantization
• Lie-Poisson gauge theories and $\kappa$-Minkowski electrodynamics
• Recurrence relations for symplectic realization of (quasi)-Poisson structures
• $\mathcal{P}\mathcal{T}$-symmetric quantum field theory on the noncommutative spacetime
• Deformed Quantum Phase Spaces, Realizations, Star Products and Twists
• Derivation based differential calculi for noncommutative algebras deforming a class of three dimensional spaces
• The action functional for Moyal planes
• Isometric Embeddings and Noncommutative Branes in Homogeneous Gravitational Waves
• On the L${}_{\infty }$ structure of Poisson gauge theory
• Field theories on $\rho$-deformed Minkowski space-time
• Self-dual road to noncommutative gravity with twist: a new analysis
• Orbital angular momentum from marginals of quadrature distributions
• Invariant star-products on S^2 and the canonical trace
• Isospectral Deformations of Eguchi-Hanson Spaces as Nonunital Spectral Triples
• Quantum spaces, central extensions of Lie groups and related quantum field theories