2010/01/31 by Joshua DeBellis, Christian Sämann, Christian Saemann +1 · 38 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Axiom #Canonical quantization #Extension (predicate logic) #Fuzzy logic #Homotopy and Cohomology in Algebraic Topology #Noncommutative and Quantum Gravity Theories #Noncommutative geometry #Noncommutative quantum field theory #Quantization (signal processing) #Quantum #hep-th #math-ph #math.MP #math.QA
paper · pdf · doi:10.1063/1.3503773
published in Journal of Mathematical Physics 51(12) (American Institute of Physics) · 43 pages, minor corrections, presentation improved, references added
arxiv created 2010/02/18 · openalex publication_date 2010/12/01 · arxiv updated 2011/01/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 \documentclass[12pt]minimal\begindocument\mathbbmR n\enddocumentRn by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.