The Heisenberg algebra and the canonical commutator relations

The Heisenberg Lie algebra \mathcal{H} is an algebra over \mathbb{R} of dimension 3. It is generated by p, q, z satisfying the relations [pq]=pq-qp = \frac{h}{2\pi i}. Here h is the Planck constant.

The centre of \mathcal{H} is generated by z. It is then easy to show that \mathcal{H} is nilpotent. (Recall that a simple Lie algebra has only trivial centre.)

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