Solvable Lie algebra

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.

Contents

Properties

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

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.

Example

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

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

External links

References