Heisenberg algebra

Let $R$ be a commutative ring. Let $M$ be a module over $R$ freely 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

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

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 $U/\<c-1\>$ is isomorphic to the $|I|$ -th Weyl algebra over $R$ .