Weyl Algebra

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

More precisely, let F be a field, and let F be the ring of polynomials in one variable, X, with coefficients in F. Then each f_i lies in F. ∂_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.

The Weyl algebra is a quotient of the free algebra on two generators, X and Y, by the ideal generated by elements of the form

The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The n-th Weyl algebra, A_n, is the ring of differential operators with polynomial coefficients in n variables. It is generated by X_i and .

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 Heisenberg algebra, the Lie algebra of the Heisenberg group, by setting the element 1 of the Lie algebra equal to the unit 1 of the universal enveloping algebra.

The Weyl algebra is also referred to as the symplectic Clifford algebra. Weyl algebras represent the same structure for bilinear forms that (orthogonal) Clifford algebras represent for quadratic forms.