Solvable Lie algebra

In mathematics, a Lie algebra g is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra is the subalgebra of g, denoted

$[{\mathfrak {g}},{\mathfrak {g}}]$ that consists of all Lie brackets of pairs of elements of g. The derived series is the sequence of subalgebras

${\mathfrak {g}}\geq [{\mathfrak {g}},{\mathfrak {g}}]\geq [[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]\geq [[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]],[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]]\geq ...$ 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.

A maximal solvable subalgebra is called a Borel subalgebra. The largest solvable ideal of a Lie algebra is called the radical.

Characterizations

Let g be a finite-dimensional Lie algebra over a field of characteristic 0. The following are equivalent.

with each ai + 1 an ideal in ai. A sequence of this type is called an elementary sequence.
such that gi + 1 is an ideal in gi and gi/gi + 1 is abelian.

Properties

Lie's Theorem states that if V is a finite-dimensional vector space over an algebraically closed field K of characteristic zero, and g is a solvable linear Lie algebra over a subfield k of K, and if π is a representation of g over V, then there exists a simultaneous eigenvector vV of the matrices π(X) for all elements Xg. More generally, the result holds if all eigenvalues of π(X) lie in K for all Xg.

• Every Lie subalgebra, quotient and extension of a solvable Lie algebra is solvable.
• A solvable nonzero Lie algebra has a nonzero abelian ideal, the last nonzero term in the derived series.
• A homomorphic image of a solvable Lie algebra is solvable.
• If a is a solvable ideal in g and g/a is solvable, then g is solvable.
• If g is finite-dimensional, then there is a unique solvable ideal rg containing all solvable ideals in g. This ideal is the radical of g, denoted rad g.
• If a, bg are solvable ideals, then so is a + b.
• A solvable Lie algebra g has a unique largest nilpotent ideal n, the set of all Xg such that adX is nilpotent. If Template:Mvar is any derivation of g, then D(g) ⊂ n.

Completely solvable Lie algebras

A Lie algebra g is called completely solvable or split solvable if it has a elementary sequence of ideals in g from 0 to g. 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.

• (a) A solvable Lie algebra g is split solvable if and only if the eigenvalues of adX are in k for all X in g.

Examples

$X=\left({\begin{matrix}0&\theta &x\\-\theta &0&y\\0&0&0\end{matrix}}\right),\quad \theta ,x,y\in \mathbb {R} .$ Then g is solvable, but not split solvable. It is isomorphic with the Lie algebra of the group of translations and rotations in the plane.

Solvable Lie groups

The terminology arises from the solvable groups of abstract group theory. There are several possible definitions of solvable Lie group. For a Lie group G, there is

• 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.