Quantized Nambu-Poisson Manifolds and n-Lie Algebras

Understand

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.