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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[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
- 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.
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