Quantized Nambu–Poisson manifolds and n-Lie algebras

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}\begin{document}${\mathbbm{R}\!}^n$\end{document} 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.