Solvable Lie algebra

Lie groups
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Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Table of Lie groups

In mathematics, a Lie algebra g is solvable if its derived series terminates in the zero subalgebra. That is, writing

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

for the derived Lie algebra of g, generated by the set of values

[x,y]

for x and y in g, the derived series

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

becomes constant eventually at 0.

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 is called the radical.

Properties

Let $\mathfrak{g}$ be a finite dimensional Lie algebra over a field of characteristic 0. The following are equivalent.

• (i) $\mathfrak{g}$ is solvable.
• (ii) $\operatorname{ad}(g)$, the adjoint representation of $\mathfrak{g}$, is solvable.
• (iii) There is a finite sequence of ideals $\mathfrak{a}_i$ of $\mathfrak{g}$ such that:

  $\mathfrak{g} = \mathfrak{a}_0 \supset \mathfrak{a}_1 \supset ... \mathfrak{a}_r = 0$ where $[\mathfrak{a}_i, \mathfrak{a}_i] \subset \mathfrak{a}_{i+1}$ for all $i$.

• (iv) $[\mathfrak{g}, \mathfrak{g}]$ is nilpotent.

Lie's Theorem states that if $V$ is a finite-dimensional vector space over an algebraically closed field of characteristic zero, and $g$ is a solvable linear Lie algebra over $V$, then there exists a basis of $V$ relative to which the matrices of all elements of $g$ are upper triangular.

Completely solvable Lie algebras

A Lie algebra g is called completely solvable if it has a finite chain of ideals from 0 to g 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.

Example

• 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 $\mathfrak{b}_k$ be a subalgebra of $\mathfrak{gl}_k$ consisting of upper triangular matrices. Then $\mathfrak{b}_k$ is solvable.

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.

See also

• Cartan's criterion
• Killing form
• Lie-Kolchin theorem
• Solvmanifold
• Dixmier mapping

External links

• EoM article Lie algebra, solvable
• EoM article Lie group, solvable

References

• 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