# 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^{}
$[\cdot ,\cdot ]:M\times M\to M$ by bilinear extension by setting

$$[c,c]=[c,{P}_{i}]=[{P}_{i},c]=[c,{Q}_{i}]=[{Q}_{i},c]=[{P}_{i},{P}_{j}]=[{Q}_{i},{Q}_{j}]=0\text{for all}i,j\in I,$$ | ||

$$=[{Q}_{i},{P}_{j}]=0\text{for all distinct}i,j\in I,$$ | ||

$$=-[{Q}_{i},{P}_{i}]=c\text{for all}i\in I.$$ |

The module $M$ together with this product is called a *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 $R=\u2102$ 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 $U/\u27e8c-1\u27e9$ 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 |