Lie Algebra Lower Central Series

The lower central series of a Lie algebra 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 lower central series ends in the zero subspace, the Lie algebra is called nilpotent. For example, consider the Lie algebra of strictly upper triangular matrices, then

			(2)
			(3)
			(4)
			(5)

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

In contrast to the nilpotent Lie algebras, the semisimple Lie algebras have a constant lower central series. Others are in between, e.g.,

(6)

which is semisimple, because the matrix trace satisfies

(7)

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