# 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 HeisenbergAlgebra 2013-03-22 15:24:57 2013-03-22 15:24:57 GrafZahl (9234) GrafZahl (9234) 6 GrafZahl (9234) Definition msc 17B99 WeylAlgebra central element