Solvable Lie algebra

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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.

External links

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

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