Commutator

Let $𝐴$ be an $𝕂$ -monoid. The commutator is a Lie bracket defined by #m/def/ralg

[𝑥, 𝑦] = 𝑥 𝑦 - 𝑦 𝑥

which together with the associative product of $𝐴$ forms a Poisson algebra. The commutator algebra or associated Lie algebra is sometimes denoted $𝐴 -$ , and a version with a renormalized product $(-) \times (-) = 12 [-, -]$ is denoted $𝐴 - 1 / 2$ .

Proof of Poisson algebra

Clearly for all $𝑥, 𝑦, 𝑧 \in 𝐴$ and $𝜆, 𝜇 \in 𝕂$

$[𝑥, 𝜆 𝑦 + 𝜇 𝑧] = 𝜆 [𝑥, 𝑦] + 𝜇 [𝑥, 𝑧]$
$[𝜆 𝑥 + 𝜇 𝑦, 𝑧] = 𝜆 [𝑥, 𝑧] + 𝜇 [𝑦, 𝑧]$
$[𝑥, 𝑥] = 𝑥 𝑥 - 𝑥 𝑥 = 0$

hence the commutator is an alternating multilinear map. Now

0 = [𝑥, [𝑦, 𝑧]] + [𝑦, [𝑧, 𝑥] + [𝑧, [𝑥, 𝑦]] = [𝑥, 𝑦 𝑧 - 𝑧 𝑦] + [𝑦, 𝑧 𝑥 - 𝑥 𝑧] + [𝑧, 𝑥 𝑦 - 𝑦 𝑥] = 𝑥 𝑦 𝑧 - 𝑥 𝑧 𝑦 - 𝑦 𝑧 𝑥 + 𝑧 𝑦 𝑥 + 𝑦 𝑧 𝑥 - 𝑦 𝑥 𝑧 - 𝑧 𝑥 𝑦 + 𝑥 𝑧 𝑦 + 𝑧 𝑥 𝑦 - 𝑧 𝑦 𝑥 - 𝑥 𝑦 𝑧 + 𝑦 𝑥 𝑧

hence the commutator is a Lie bracket. Finally

[𝑥, 𝑦 𝑧] = 𝑥 𝑦 𝑧 - 𝑦 𝑧 𝑥 = 𝑥 𝑦 𝑧 - 𝑦 𝑥 𝑧 + 𝑦 𝑥 𝑧 - 𝑦 𝑧 𝑥 = [𝑥, 𝑦] 𝑧 + 𝑦 [𝑥, 𝑧]

as required.

See also Anticommutator and Supercommutator.

Properties

$[𝑥, 𝑦 𝑧] = [𝑥, 𝑦] 𝑧 + 𝑦 [𝑥, 𝑧]$ (see above)
$𝑥 𝑦 = 𝑦 𝑥 + [𝑥, 𝑦]$
Every Unital subalgebra is a Lie subalgebra under the commutator.

Graded structure

If the associative algebra $𝐴$ is $𝔄$ -graded where $𝔄$ is an abelian monoid, then the commutator forms a $𝔄$ -graded Lie algebra.

Examples

Linear commutator

#state/tidy | #lang/en | #SemBr