Lie Algebra Commutator Series

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


with g^0=g. The sequence of subspaces is always decreasing with respect to inclusion or dimension, and becomes stable when g is finite dimensional. The notation [a,b] means the linear span of elements of the form [A,B], where A in a and B in b.

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

g^0=[0 a_(12) a_(13) a_(14) a_(15); 0 0 a_(23) a_(24) a_(25); 0 0 0 a_(34) a_(35); 0 0 0 0 a_(45); 0 0 0 0 0]
g^1=[0 0 a_(13) a_(14) a_(15); 0 0 0 a_(24) a_(25); 0 0 0 0 a_(35); 0 0 0 0 0; 0 0 0 0 0]
g^2=[0 0 0 0 a_(15); 0 0 0 0 0; 0 0 0 0 0; 0 0 0 0 0; 0 0 0 0 0],

and g^3=0. By definition, g^k subset g_k where g_k 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.,


which is semisimple, because the matrix trace satisfies


Here, gl_n is a general linear Lie algebra and sl_n 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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Rowland, Todd. "Lie Algebra Commutator Series." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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