commutator series

The commutator series of a Lie algebra g, sometimes called the derived series, is the sequence of subalgebras recursively defined by g^(k + 1) = [g^k, g^k], 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 element a and B element b. When the commutator series ends in the zero subspace, the Lie algebra is called solvable.

Lie algebra | Lie algebra representation | Lie group | nilpotent Lie algebra | nilpotent Lie group | solvable Lie group