Heisenberg algebra

Let R be a commutative ring. Let M be a http://planetmath.org/node/5420module over R http://planetmath.org/node/5420freely generated by two sets {Pi}iI and {Qi}iI, where I is an index setMathworldPlanetmathPlanetmath, and a further element c. Define a productPlanetmathPlanetmath [,]:M×MM by bilinear extension by setting

[c,c]=[c,Pi]=[Pi,c]=[c,Qi]=[Qi,c]=[Pi,Pj]=[Qi,Qj]=0 for all i,jI,
=[Qi,Pj]=0 for all distinct i,jI,
=-[Qi,Pi]=c for all iI.

The module M together with this product is called a Heisenberg algebra. The element c is called the central element.

It is easy to see that the product [,] also fulfills the Jacobi identityMathworldPlanetmathPlanetmath, so a Heisenberg algebra is actually a Lie algebra of rank |I|+1 (opposed to the rank of M as free modulePlanetmathPlanetmath, which is 2|I|+1) with one-dimensional center generated by c.

Heisenberg algebras arise in quantum mechanics with R= and typically I={1,2,3}, but also in the theory of vertex with I=.

In the case where R is a field, the Heisenberg algebra is related to a Weyl algebra: let U be the universal enveloping algebra of M, then the quotient U/c-1 is isomorphicPlanetmathPlanetmathPlanetmath to the |I|-th Weyl algebra over R.

Title Heisenberg algebra
Canonical name HeisenbergAlgebra
Date of creation 2013-03-22 15:24:57
Last modified on 2013-03-22 15:24:57
Owner GrafZahl (9234)
Last modified by GrafZahl (9234)
Numerical id 6
Author GrafZahl (9234)
Entry type Definition
Classification msc 17B99
Related topic WeylAlgebra
Defines central element