Algebra theory MOC

Commutator

Let be an K-monoid. The commutator is a Lie bracket defined by falg

which together with the associative product of forms a Poisson algebra. The commutator algebra or associated Lie algebra is sometimes denoted , and a version with a renormalized product is denoted .

Proof of Poisson algebra

Clearly for all and

hence the commutator is an alternating multilinear map. Now

hence the commutator is a Lie bracket. Finally

as required.

See also Anticommutator and Supercommutator.

Properties

1. (see above)
3. Every Unital subalgebra is a Lie subalgebra under the commutator.

Graded structure

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

Examples

• Linear commutator

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