We start with an overview of the "generalized Hamiltonian dynamics"
introduced in 1973 by Y. Nambu, its motivations, mathematical background and
subsequent developments -- all of it on the classical level. This includes the
notion (not present in Nambu's work) of a generalization of the Jacobi identity
called Fundamental Identity. We then briefly describe the difficulties
encountered in the quantization of such n-ary structures, explain their
reason and present the recently obtained solution combining deformation
quantization with a "second quantization" type of approach on Rn. The
solution is called "Zariski quantization" because it is based on the
factorization of (real) polynomials into irreducibles. Since we want to
quantize composition laws of the determinant (Jacobian) type and need a Leibniz
rule, we need to take care also of derivatives and this requires going one step
further (Taylor developments of polynomials over polynomials). We also discuss
a (closer to the root, "first quantized") approach in various circumstances,
especially in the case of covariant star products (exemplified by the case of
su(2)). Finally we address the question of equivalence and triviality of such
deformation quantizations of a new type (the deformations of algebras are more
general than those considered by Gerstenhaber).Comment: 23 pages, LaTeX2e with the LaTeX209 option. To be published in the
proceedings of the Ascona meeting. Mathematical Physics Studies, volume 20,
Kluwe