Lie Algebra Lower Central Series

The Lie algebra 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 induction, , where is the term in the Lie algebra commutator series.

In contrast to the nilpotent Lie algebras, the semisimple Lie algebras over a field of field characteristic zero have a constant lower central series. An intermediate example for is

(6)

where is semisimple. The matrix trace identity

(7)

shows that every commutator has matrix trace zero. Here, is a general linear Lie algebra and is the special linear Lie algebra.