The present paper is a lucid survey article devoted to the quantization procedures involving suitable deformations of the commutative algebra \(C^\infty(M)\) representing classical observables on a phase space \((M,\omega)\) of a classical dynamical system, commutativity being referred to the product. The basic idea consists in replacing the latter by a noncommutative one (star product) depending on the Planck constant in such a way that, among other things, it reproduces the original one when the latter goes to zero (classical limit). This scheme differs from other traditional approaches (e.g. the canonical or the geometric quantization ones) in that it does not emphasize the construction of the quantum Hilbert space. The Moyal construction of the star product is discussed in detail on \(\mathbb{R}^{2n}\), starting from the Fourier-Weyl transform, making the ensuing link with Wigner functions clear by means of the Grossmann-Royer operators. A generalization to coadjoint orbits of Lie groups via the Stratonovich Weyl kernel is presented and illustrated by means of the Galilei and Newton-Hooke groups.
The same procedure is used, with suitable modifications, to get quantum groups from ordinary (Poisson-)Lie groups. A Poisson structure (Sklyanin) is introduced on \(C^\infty(G)\) (\(G\) being a Lie group) via a classical \(r\) matrix verifying the classical Yang-Baxter equation. Then, Drinfel’d’s theorem about the existence of a universal object fulfilling the quantum Yang-Baxter equation is used to quantize (in the above sense) the ensuing Poisson-Lie group. The standard but fundamental case given by \(\text{SL}(2,\mathbb{C})\) is developed in detail.
In summary, although no detailed proofs of theorems are given throughout, the basic ideas are appropriately emphasized through the examples, making this survey useful for a general audience.