# 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*[*X*] 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}*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**,

*A*, is the ring of differential operators with polynomial coefficients in several variables. It is generated by

_{n}*X*and

_{i}*∂*.

_{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.

## References

- 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))*