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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 *f _{i}* lies in

*∂ _{X}* is the derivative with respect to

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 isomorphic to the quotient of the free algebra on two generators, *X* and *Y*, by the ideal generated by the element

The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The ** n-th Weyl algebra**,

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 central element of the Heisenberg algebra (namely [*X*,*Y*]) equal to the unit of the universal enveloping algebra (called 1 above).

The Weyl algebra is also referred to as the **symplectic Clifford algebra**.^{[1]}^{[2]}^{[3]} Weyl algebras represent the same structure for symplectic bilinear forms that Clifford algebras represent for non-degenerate symmetric bilinear forms.^{[1]}

One may give an abstract construction of the algebras *A _{n}* in terms of generators and relations. Start with an abstract vector space

where *T*(*V*) is the tensor algebra on *V*, and the notation means "the ideal generated by".

In other words, *W*(*V*) is the algebra generated by *V* subject only to the relation *vu* − *uv* = *ω*(*v*, *u*). Then, *W*(*V*) is isomorphic to *A _{n}* via the choice of a Darboux basis for ω.

The algebra *W*(*V*) is a quantization of the symmetric algebra Sym(*V*). If *V* is over a field of characteristic zero, then *W*(*V*) is naturally isomorphic to the underlying vector space of the symmetric algebra Sym(*V*) equipped with a deformed product – called the Groenewold–Moyal product (considering the symmetric algebra to be polynomial functions on *V*^{∗}, where the variables span the vector space *V*, and replacing *iħ* in the Moyal product formula with 1).

The isomorphism is given by the symmetrization map from Sym(*V*) to *W*(*V*)

If one prefers to have the *iħ* and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by *X*_{i} and *iħ∂ _{Xi}* (as per quantum mechanics usage).

Thus, the Weyl algebra is a quantization of the symmetric algebra, which is essentially the same as the Moyal quantization (if for the latter one restricts to polynomial functions), but the former is in terms of generators and relations (considered to be differential operators) and the latter is in terms of a deformed multiplication.

In the case of exterior algebras, the analogous quantization to the Weyl one is the Clifford algebra, which is also referred to as the *orthogonal Clifford algebra*.^{[2]}^{[4]}

In the case that the ground field F has characteristic zero, the *n*th Weyl algebra is a simple Noetherian domain. It has global dimension *n*, in contrast to the ring it deforms, Sym(*V*), which has global dimension 2*n*.

It has no finite-dimensional representations. Although this follows from simplicity, it can be more directly shown by taking the trace of *σ*(*X*) and *σ*(*Y*) for some finite-dimensional representation *σ* (where [*X*,*Y*] = 1).

Since the trace of a commutator is zero, and the trace of the identity is the dimension of the representation, the representation must be zero dimensional.

In fact, there are stronger statements than the absence of finite-dimensional representations. To any finitely generated *A _{n}*-module

An even stronger statement is Gabber's theorem, which states that Char(*M*) is a co-isotropic subvariety of *V* × *V*^{∗} for the natural symplectic form.

The situation is considerably different in the case of a Weyl algebra over a field of characteristic *p* > 0.

In this case, for any element *D* of the Weyl algebra, the element *D ^{p}* is central, and so the Weyl algebra has a very large center. In fact, it is a finitely generated module over its center; even more so, it is an Azumaya algebra over its center. As a consequence, there are many finite-dimensional representations which are all built out of simple representations of dimension

The center of Weyl algebra is the field of constants. For any element in the center, implies for all and implies for . Thus is a constant.

For more details about this quantization in the case *n* = 1 (and an extension using the Fourier transform to a class of integrable functions larger than the polynomial functions), see Wigner–Weyl transform.

Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.

Weyl algebras also generalize in the case of algebraic varieties. Consider a polynomial ring

then a differential operator is defined as a composition -linear derivations of . This can be described explicitly as the quotient ring

- de Traubenberg, M. Rausch; Slupinski, M. J.; Tanasa, A. (2006). "Finite-dimensional Lie subalgebras of the Weyl algebra".
*J. Lie Theory*.**16**: 427–454. arXiv:math/0504224.*(Classifies subalgebras of the one-dimensional Weyl algebra over the complex numbers; shows relationship to SL(2,C))* - Tsit Yuen Lam (2001).
*A first course in noncommutative rings*. Graduate texts in mathematics.**131**(2nd ed.). Springer. p. 6. ISBN 978-0-387-95325-0. - Coutinho, S.C. (1997). "The many avatars of a simple algebra".
*American Mathematical Monthly*.**104**(7): 593–604. doi:10.1080/00029890.1997.11990687. - Traves, Will (2010). "Differential Operations on Grassmann Varieties". In Campbell, H.; Helminck, A.; Kraft, H.; Wehlau, D. (eds.).
*Symmetry and Spaces*. Progress in Mathematics.**278**. Birkhäuse. pp. 197–207. doi:10.1007/978-0-8176-4875-6_10. ISBN 978-0-8176-4875-6.

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^{a}^{b}Helmstetter, Jacques; Micali, Artibano (2008). "Introduction: Weyl algebras".*Quadratic Mappings and Clifford Algebras*. Birkhäuser. p. xii. ISBN 978-3-7643-8605-4. - ^
^{a}^{b}Abłamowicz, Rafał (2004). "Foreword".*Clifford algebras: applications to mathematics, physics, and engineering*. Progress in Mathematical Physics. Birkhäuser. pp. xvi. ISBN 0-8176-3525-4. **^**Oziewicz, Z.; Sitarczyk, Cz. (1989). "Parallel treatment of Riemannian and symplectic Clifford algebras". In Micali, A.; Boudet, R.; Helmstetter, J. (eds.).*Clifford algebras and their applications in mathematical physics*. Kluwer. pp. 83–96 see p.92. ISBN 0-7923-1623-1.**^**Oziewicz & Sitarczyk 1989, p. 83