Algebra theory MOC

Commutator

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

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

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