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 $\{P_{i}\}_{i\in I}$ and $\{Q_{i}\}_{i\in I}$ , where $I$ is an index set, and a further element $c$ . Define a product $[\cdot,\cdot]\colon M\times M\to M$ by bilinear extension by setting

	$\displaystyle[c,c]=[c,P_{i}]=[P_{i},c]=[c,Q_{i}]=[Q_{i},c]=[P_{i},P_{j}]=[Q_{i% },Q_{j}]=0\text{ for all }i,j\in I,$
	$\displaystyle=[Q_{i},P_{j}]=0\text{ for all distinct }i,j\in I,$
	$\displaystyle=-[Q_{i},P_{i}]=c\text{ for all }i\in I.$

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 $[\cdot,\cdot]$ also fulfills the Jacobi identity, so a Heisenberg algebra is actually a Lie algebra of rank $|I|+1$ (opposed to the rank of $M$ as free module, which is $2|I|+1$ ) with one-dimensional center generated by $c$ .

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

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 $U/\langle c-1\rangle$ is isomorphic 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