Quantized Nambu-Poisson manifolds and n-Lie algebras

Journal Article · Wed Dec 15 00:00:00 EST 2010 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3503773· OSTI ID:21501217

DeBellis, Joshua; Saemann, Christian; Szabo, Richard J ^[1]

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1. Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS (United Kingdom) and Maxwell Institute for Mathematical Sciences, Edinburgh (United Kingdom)

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{sup n} by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.

• OSTI ID: 21501217
• Journal Information: Journal of Mathematical Physics, Vol. 51, Issue 12; Other Information: DOI: 10.1063/1.3503773; (c) 2010 American Institute of Physics; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
97 MATHEMATICAL METHODS AND COMPUTING
BRANES
COMMUTATION RELATIONS
FIELD THEORIES
FUZZY LOGIC
GEOMETRY
LIE GROUPS
M-THEORY
PARTIAL DIFFERENTIAL EQUATIONS
QUANTIZATION
SPHERES
DIFFERENTIAL EQUATIONS
EQUATIONS
MATHEMATICAL LOGIC
MATHEMATICS
SYMMETRY GROUPS