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Let be a partial differential ring with commuting derivatives . The Weyl algebra associated to is the noncommutative ring satisfying the relations for all .
![{\displaystyle R[\partial _{1},\ldots ,\partial _{m}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3af487da3b36cd62ef3d951a3d970f7b4ac59327)
This article focuses on the special case where and where is a field which is, in some references, called "the" Weyl algebra.
![{\displaystyle R=F[x_{1},\ldots ,x_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4897a04e475f16e6c598202242efcae280a7e63)
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.
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, An, is the ring of differential operators with polynomial coefficients in n variables. It is generated by Xi and ∂Xi, i = 1, ..., n.
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 for symplectic bilinear forms the same structure that Clifford algebras represent for non-degenerate symmetric bilinear forms.[1]
- ^ a b Helmstetter, Jacques; Micali, Artibano (2008). "Introduction: Weyl algebras". Quadratic Mappings and Clifford Algebras. Birkhäuser. p. xii. ISBN 978-3-7643-8605-4.
- ^ 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.