Heisenberg algebra

Let $R$ be a commutative ring. Let $M$ be a http://planetmath.org/node/5420moduleover $R$ 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$[\cdot ,\cdot ]:M\times M\to M$ by bilinear extension by setting

$$[c,c]=[c,P_i]=[P_i,c]=[c,Q_i]=[Q_i,c]=[P_i,P_j]=[Q_i,Q_j]=0\text{ forall }i,j\in I,$$

$$=[Q_i,P_j]=0\text{ for all distinct }i,j\in I,$$

$$=-[Q_i,P_i]=c\text{ for all }i\in I.$$

The module $M$ together with this product is called a Heisenbergalgebra. The element $c$ is called the central element.

It is easy to see that the product $[\cdot ,\cdot ]$ also fulfills theJacobi identity, so a Heisenberg algebra is actually a Lie algebra ofrank $|I|+1$ (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 $R=ℂ$ andtypically $I=\{1,2,3\}$, but also in the theory of vertex with $I=\mathbb{Z}$.

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