# Weyl algebra

In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable),

${\displaystyle f_{n}(X)\partial _{X}^{n}+\cdots +f_{1}(X)\partial _{X}+f_{0}(X).}$

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.

The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension.

You can also construct the Weyl algebra as a quotient of the free algebra on two generators, X and Y, by the ideal generated by the single relation

YXXY − 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.