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

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 .

See also Borel Subalgebra ,

Group Commutator Series ,

Lie Algebra ,

Lie
Algebra Representation ,

Lie Group ,

Nilpotent
Lie Group ,

Nilpotent Lie Algebra ,

Solvable Lie Group ,

Solvable
Lie Group Representation ,

Split Solvable
Lie Algebra
This entry contributed by Todd
Rowland

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Cite this as:
Rowland, Todd . "Lie Algebra Commutator Series." From MathWorld --A Wolfram Web Resource, created
by Eric W. Weisstein . https://mathworld.wolfram.com/LieAlgebraCommutatorSeries.html

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