Lie Algebra Commutator Series

The commutator series of a Lie algebra , sometimes called the derived series, is the sequence of subalgebras recursively defined by

(1)

with . The sequence of subspaces is always decreasing with respect to inclusion or dimension, and becomes stable when is finite dimensional. The notation means the linear span of elements of the form , where and .

When the commutator series ends in the zero subspace, the Lie algebra is called solvable. For example, consider the Lie algebra of strictly upper triangular matrices, then

			(2)
			(3)
			(4)

and . By definition, where is the term in the Lie algebra lower central series, as can be seen by the example above.

In contrast to the solvable Lie algebras, the semisimple Lie algebras have a constant commutator series. Others are in between, e.g.,

(5)

which is semisimple, because the matrix trace satisfies

(6)

Here, is a general linear Lie algebra and is the special linear Lie algebra.