Heisenberg algebra

In the general formulation used in conformal field theory, a Heisenberg algebra $𝔩$ over $𝕂$ is a nilpotent Lie algebra whose 1-dimensional centre is its commutator ideal #m/def/lie

𝔩 0 = 𝔷 (𝔩) = [𝔩, 𝔩] = 𝕂 𝑧

Assuming $d i m 𝔩$ is countable, one may impose a $ℤ$ -grading $𝔩 = ⨁ 𝑛 \in ℤ 𝔩 𝑛$ with $d i m 𝔩 𝑛 < \infty$ for $𝑛 \in ℤ$ and $𝔩 0$ given above, giving abelian subalgebras

𝔩 \pm = \infty ⨁ 𝑛 = 1 𝔩 \pm 𝑛

so that $𝔟 \pm = 𝔩 0 \oplus 𝔩 \pm$ are maximal abelian subalgebras of $𝔩$ . An alternating bilinear form $(\cdot, \cdot)$ on $𝔩$ is given by

[𝑥, 𝑦] = (𝑥, 𝑦) 𝑧

which is nondegenerate on $𝔩 + \oplus 𝔩 -$ and $𝔩 𝑛 \oplus 𝔩 - 𝑛$ for all $𝑛 \in ℕ$ , so one may form bases $(𝑥 𝑖) 𝑖 \in 𝐼$ of $𝔩 +$ and $(𝑦 𝑖) 𝑖 \in 𝐼$ of $𝔩 -$ satisfying the Heisenberg commutation relations

[𝑥 𝑖, 𝑧] = [𝑦 𝑖, 𝑧] = [𝑥 𝑖, 𝑥 𝑗] = [𝑦 𝑖, 𝑦 𝑗] = 0 [𝑥 𝑖, 𝑦 𝑗] = 𝛿 𝑖 𝑗 𝑧

and

d e g 𝑥 𝑖 + d e g 𝑦 𝑖 = 0

for $𝑖, 𝑗 \in 𝐼$ .

Properties

$d i m 𝔩 > 1$ , since otherwise the centre would be trivial (not 1-dimensional)
If $d i m 𝔩$ is finite, then it is odd

Heisenberg algebra

Properties

Examples

See also