Quantized Nambu-Poisson Manifolds and n-Lie Algebras

Abstract

We investigate the geometric interpretation of quantized Nambu-Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu-Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin-Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras, as well as the approach based on harmonic analysis. We find an interpretation of Nambu-Heisenberg n-Lie algebras in terms of foliations of R^n by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.

43 pages, minor corrections, presentation improved, references added

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• Higher Dimensional Quantum Hall Effect as A-Class Topological Insulator
• Groupoids, Loop Spaces and Quantization of 2-Plectic Manifolds
• Super tensor models, super fuzzy spaces and super n-ary transformations
• Tensor models and hierarchy of n-ary algebras
• Chiral Topological Insulator on Nambu 3-Algebraic Geometry
• Non-Compact Hopf Maps and Fuzzy Ultra-Hyperboloids
• $G_2$-structures and quantization of non-geometric M-theory backgrounds
• Higher (Odd) Dimensional Quantum Hall Effect and Extended Dimensional Hierarchy
• Quantized Nambu-Poisson Manifolds in a 3-Lie Algebra Reduced Model
• On current algebras, generalised fluxes and non-geometry
• Reduction of multisymplectic manifolds
• Lie 2-algebra models
• Graded Hopf Maps and Fuzzy Superspheres
• $L_{\infty }$-Algebra Models and Higher Chern-Simons Theories
• Nonassociative geometry and twist deformations in non-geometric string theory
• $SO(5)$ Landau Models and Nested Nambu Matrix Geometry
• D1-Strings in Large RR 3-Form Flux, Quantum Nambu Geometry and M5-Branes in C-Field
• Hydrodynamics on non-commutative space --A step toward hydrodynamics of granular materials--
• Quantization of Polysymplectic Manifolds
• Cartan-Weyl 3-algebras and the BLG Theory II: Strong-Semisimplicity and Generalized Cartan-Weyl 3-algebras
• Matrix Quantization of Classical Nambu Brackets and Super $p$-Branes
• Higher bracket structure of density operators in Weyl fermion systems and topological insulators
• Branes, Quantization and Fuzzy Spheres
• Reduced Chern-Simons Quiver Theories and Cohomological 3-Algebra Models
• Quantum matrix geometry in the lowest Landau level and higher Landau levels
• Non-Commutative Geometry in Higher Dimensional Quantum Hall Effect as A-Class Topological Insulator