Weyl algebra

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

<math>

 f_n(X) \partial_X^n + \cdots + f_1(X) \partial_X + f_0(X).

</math>

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]. ∂_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, A_n, is the ring of differential operators with polynomial coefficients in several variables. It is generated by X_i and <math>\part_{X_i}</math>.

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))