Solvable Lie algebra
|Group theory → Lie groups
that consists of all Lie brackets of pairs of elements of . The derived series is the sequence of subalgebras
If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is solvable. The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory.
Any nilpotent Lie algebra is solvable, a fortiori, 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.
Let be a finite-dimensional Lie algebra over a field of characteristic 0. The following are equivalent.
- (i) is solvable.
- (ii) , the adjoint representation of , is solvable.
- (iii) There is a finite sequence of ideals of such that:
- where for all .
- (iv) is nilpotent.
Lie's Theorem states that if is a finite-dimensional vector space over an algebraically closed field of characteristic zero, and is a solvable linear Lie algebra over , then there exists a basis of relative to which the matrices of all elements of are upper triangular.
Completely solvable Lie algebras
A Lie algebra is called completely solvable if it has a finite chain of ideals from 0 to such that each has codimension 1 in the next. A finite-dimensional nilpotent Lie algebra is completely solvable, and a completely solvable Lie algebra is solvable. Over an algebraically closed field and solvable Lie algebra is completely solvable, but the 3-dimensional real Lie algebra of the group of Euclidean isometries of the plane is solvable but not completely solvable.
- Every abelian Lie algebra is solvable.
- Every nilpotent Lie algebra is solvable.
- Every Lie subalgebra, quotient and extension of a solvable Lie algebra is solvable.
- Let be a subalgebra of consisting of upper triangular matrices. Then is solvable.
Solvable Lie groups
- termination of the usual derived series, in other words taking G as an abstract group;
- termination of the closures of the derived series;
- having a solvable Lie algebra.
To have equivalence one needs to assume G connected. For connected Lie groups, these definitions are the same, and the derived series of Lie algebras are the Lie algebra of the derived series of (closed) subgroups.
- Humphreys, James E. Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1972. ISBN 0-387-90053-5