Weyl algebra

Abstract definition

Let F be a field and V be an F-vector spaceMathworldPlanetmath with basis {Pi}iI{Qi}iI, where I is some non-empty index setMathworldPlanetmathPlanetmath. Let T be the tensor algebra of V and let J be the ideal in T generated by the set {PiQj-QjPi-δij}i,jI where δ is the Kronecker delta symbol. Then the quotientPlanetmathPlanetmath T/J is the |I|-th Weyl algebra.

A more concrete definition

If the field F has characteristic zero we have the following more concrete definition. Let R:=F[{Xi}iI] be the polynomial ringMathworldPlanetmath over F in indeterminates Xi labeled by I. For any iI, let i denote the partial differential operator with respect to Xi. Then the |I|-th Weyl algebra is the set W of all differential operators of the form


where the summation variable α is a multi-index with |I| entries, n is the degree of D, and fαR. The algebraMathworldPlanetmathPlanetmath structureMathworldPlanetmath is defined by the usual operator multiplication, where the coefficients fαR are identified with the operators of left multiplication with them for conciseness of notation. Since the derivative of a polynomialMathworldPlanetmath is again a polynomial, it is clear that W is closed underPlanetmathPlanetmath that multiplication.

The equivalence of these definitions can be seen by replacing the generatorsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath Qi with left multiplication by the indeterminates Xi, the generators Pi with the partial differential operator i, and the tensor productPlanetmathPlanetmathPlanetmath with operator multiplication, and observing that iXj-Xji=δij. If, however, the characteristic p of F is positive, the resulting homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to W is not injectivePlanetmathPlanetmath, since for example the expressions ip and Xin commute, while Pip and Qin do not.

Title Weyl algebra
Canonical name WeylAlgebra
Date of creation 2013-03-22 15:27:19
Last modified on 2013-03-22 15:27:19
Owner GrafZahl (9234)
Last modified by GrafZahl (9234)
Numerical id 5
Author GrafZahl (9234)
Entry type Definition
Classification msc 16S36
Classification msc 16S32
Related topic HeisenbergAlgebra
Related topic UniversalEnvelopingAlgebra