In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra
is the subalgebra of
, denoted
:
is a fortiori solvable but the converse is not true. The solvable Lie algebras and the semisimple Lie algebras form two large and generally complementary classes, as is shown by the Levi decomposition. The solvable Lie algebras are precisely those that can be obtained from Lie algebra extension, semidirect products, starting from 0 and adding one dimension at a time.
. The largest solvable
.
. The following are equivalent.
*(i)