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. 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. Weyl algebras represent the same structure for symplectic bilinear forms that Clifford algebras represent for non-degenerate symmetric bilinear forms. One may give an abstract construction of the algebras An in terms of generators and relations. Start with an abstract vector space V (of dimension 2n) equipped with a symplectic form ω. Define the Weyl algebra W(V) to be 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 An via the choice of a Darboux basis for ω.

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Heisenberg group
In mathematics, the Heisenberg group , named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form under the operation of matrix multiplication. Elements a, b and c can be taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group"). The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem.
Symmetric algebra
In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is a commutative algebra over K that contains V, and is, in some sense, minimal for this property. Here, "minimal" means that S(V) satisfies the following universal property: for every linear map f from V to a commutative algebra A, there is a unique algebra homomorphism g : S(V) → A such that f = g ∘ i, where i is the inclusion map of V in S(V).
Tensor algebra
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T^•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property (see below). The tensor algebra is important because many other algebras arise as quotient algebras of T(V).
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