More precisely, let F be a field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F[X]. ∂X is the derivative with respect to X. The algebra is generated by X and ∂X.
- YX − XY − 1.
The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The n-th Weyl algebra, An, is the ring of differential operators with polynomial coefficients in several variables. It is generated by Xi and ∂Xi.
Weyl algebras are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. It is a quotient of the universal enveloping algebra of the Lie algebra of the Heisenberg group.
- M. Rausch de Traubenberg, M. J. Slupinski, A. Tanasa, Finite-dimensional Lie subalgebras of the Weyl algebra, (2005) (Classifies subalgebras of the one dimensional Weyl algebra over the complex numbers; shows relationship to SL(2,C))