Weyl algebra

rdf:langString Weyl algebra

rdf:langString 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.