The passage from quantum to classical mechanics is quite well defined by taking the limit as Planck's constant h tends to 0. The passage in the other direction is not so clear cut, and may not be uniquely determined. On the algebraic side, “quantization” involves deforming commutative algebras to non-commutative ones:

Usually we deal with q = eh rather than h, so classical results correspond to the case q = 1. Quantum spaces correspond to more general k-algebras, not necessarily commutative.

Let k be a fixed field and fix q ∈ k with q ≠ 0. Write k〈x1, …, xn〉 for the k-algebra of polynomials in non-commuting indeterminates x1, …, xn. As a vector space over k, a basis is given by those elements

for which r ∈ ℕ and m1, …, mr ∈ ℤ+ and ξ : {1, …, r} → {1, …, n} is any function. Notice that

The coordinate algebra of the space of quantum 2 × 2 matrices is defined by

where R is the system of equations

(mnemonic)

The monomials a m1bm2cm3dm4 form a basis for the algebra, as a vector space over k.

is an A-point of Mq−1(2).

The above result can be proved by direct calculation, but this gives little insight into the special nature of the relations R. Examples such as this arose in work of L. D. Fadde′ev [FRT88] and his school on the quantum inverse scattering transform (QIST) method.