In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form
More precisely, let F be the underlying 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, and the algebra is generated by X and ∂X. For example:where .
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.
More generally, let be a partial differential ring with commuting derivatives . The Weyl algebra associated to is the noncommutative ring satisfying the relations for all . The previous case is the special case where and where is a field. This article discusses only the special case unless otherwise stated.
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