In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...

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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In mathematics, a derivative is defined as the instantaneous rate of change of a function and the process of finding the derivative is called differentiation. ...

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. In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. ... In abstract algebra the matrix ring M(n,R) is set of all n-by-n matrices over an arbitrary ring R. This forms a ring under matrix addition and multiplication. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... In abstract algebra, a domain is the noncommutative analogue of an integral domain. ... In mathematics, the Ore condition in ring theory is a condition introduced by Oystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more general localization of a ring. ...

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 In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In abstract algebra, a free algebra is the noncommutative analogue of a polynomial ring. ... In mathematics, the term ideal has multiple meanings. ...

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 n variables. It is generated by X_i and .

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, by setting the element 1 of the Lie algebra equal to the unit 1 of the universal enveloping algebra. Hermann Weyl Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics, and acknowledged to be one of the most important physicists of the twentieth century. ... In quantum physics, the Heisenberg uncertainty principle or the Heisenberg indeterminacy principle — the latter name given to it by Niels Bohr — states that when measuring conjugate quantities, which are pairs of observables of a single elementary particle, increasing the accuracy of the measurement of one quantity increases the uncertainty of... Fig. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, the Heisenberg group is a group of 3×3 upper triangular matrices of the form Elements a,b,c can be taken from some (arbitrary) commutative ring. ...

One may give an abstract construction of the algebras A_n in terms of generators and relations. We do so in a more sophisticated way: Start with an abstract vector space V (of dimension 2n) equipped with a symplectic form ω. Define the Weyl algebra W(V) to be In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...

where the notation means "the ideal generated by". In other words, W(V) is the algebra generated by V subject only to the relation vw − wv = ω(v,w). Then, W(V) is isomorphic to (it does not depend on the choice of ω). In this form, one sees that 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 symmetric algebra Sym(V) equipped with the deformed Moyal product (considering the symmetric algebra to be polynomial functions on V ^* , where the variables span the vector space V, and replacing 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 and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by X_i and (as is frequently done in quantum mechanics). In mathematics, the term ideal has multiple meanings. ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... In mathematics, the Moyal product is an example for an associative, non-commutative product on the functions of a Poisson manifold. ... Fig. ...

Thus, the Weyl algebra is a [[quantization (physics}|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. In mathematics, the Moyal product is an example for an associative, non-commutative product on the functions of a Poisson manifold. ... In mathematics, the exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ... Clifford algebras are a type of associative algebra in mathematics. ...

For more details about this quantization in the case n = 1 (and an extension using the Fourier transform to integrable ("most") functions, not just polynomial functions), see Weyl quantization. In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for associating a quantum mechanical Hermitian operator with a classical distribution in phase space. ...

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))