DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Finite-dimensional Heisenberg Lie algebra

Definition

Let $k$ be a field and $n \geq 1$ . The Heisenberg Lie algebra $h_{n} (k)$ is the $(2 n + 1)$ -dimensional $k$ -vector space with basis

$X_{1}, \dots, X_{n}, P_{1}, \dots, P_{n}, Z .$

Its bracket is the bilinear extension of

$[P_{i}, X_{j}] = δ_{i j} Z, [X_{i}, P_{j}] = - δ_{i j} Z,$

with all $[X_{i}, X_{j}]$ , $[P_{i}, P_{j}]$ , $[Z, X_{i}]$ , and $[Z, P_{i}]$ equal to zero. Here $δ_{i j} = 1$ when $i = j$ and $0$ otherwise. Equivalently, for coefficient vectors $a, b, a^{'}, b^{'} \in k^{n}$ ,

$[\sum_{i} (a_{i} X_{i} + b_{i} P_{i}) + c Z, \sum_{i} (a_{i}^{'} X_{i} + b_{i}^{'} P_{i}) + c^{'} Z] = (\sum_{i} (b_{i} a_{i}^{'} - b_{i}^{'} a_{i})) Z .$

The Jacobi identity and alternation, including in characteristic $2$ , are proved in The Heisenberg bracket satisfies the Lie algebra axioms ↗.

Depends on

Lie algebras over a field

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

François Dumas, Noncommutative Invariants, §3.2.1 (standard reference, not scraped)