Blattner's conjecture

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

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

for the derived Lie algebra of ${\mathfrak {g}}$, generated by the set of values

[x,y]

for x and y in ${\mathfrak {g}}$, 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) $\operatorname {ad} ({\mathfrak {g}})$, 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 ${\mathfrak {g}}$ is a solvable linear Lie algebra over $V$, then there exists a basis of $V$ relative to which the matrices of all elements of ${\mathfrak {g}}$ are upper triangular.

Completely solvable Lie algebras

A Lie algebra ${\mathfrak {g}}$ is called completely solvable if it has a finite chain of ideals from 0 to ${\mathfrak {g}}$ 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