PlanetMath: Heisenberg algebra

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Heisenberg algebra

(Definition)

Let be a commutative ring. Let be a module over freely generated by two sets $\{P_i\}_{i\in I}$ and $\{Q_i\}_{i\in I}$ , where is an index set, and a further element . Define a product $[\cdot,\cdot]\colon M\times M\to M$ by bilinear extension by setting

$\displaystyle [c,c]=[c,P_i]=[P_i,c]=[c,Q_i]=[Q_i,c]=[P_i,P_j]=[Q_i,Q_j]=0$ for all $\displaystyle i,j\in I,$
$\displaystyle [P_i,Q_j]=[Q_i,P_j]=0$ for all distinct $\displaystyle i,j\in I,$
$\displaystyle [P_i,Q_i]=-[Q_i,P_i]=c$ for all $\displaystyle i\in I.$

The module

together with this product is called a Heisenberg algebra. The element

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 $\vert I\vert+1$ (opposed to the rank of as free module, which is $2\vert I\vert+1$ ) with one-dimensional center generated by .

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

In the case where is a field, the Heisenberg algebra is related to a Weyl algebra: let be the universal enveloping algebra of , then the quotient $U/\< c-1\>$ is isomorphic to the $\vert I\vert$ -th Weyl algebra over .

"Heisenberg algebra" is owned by GrafZahl.

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