Solvable Lie algebra
|Group theory → Lie groups
that consists of all Lie brackets of pairs of elements of g. The derived series is the sequence of subalgebras
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.
Let g be a finite-dimensional Lie algebra over a field of characteristic 0. The following are equivalent.
- (i) g is solvable.
- (ii) ad(g), the adjoint representation of g, is solvable.
- (iii) There is a finite sequence of ideals ai of g:
- (iv) [g, g] is nilpotent.
- (v) For g n-dimensional, there is a finite sequence of subalgebras ai of g:
- with each ai + 1 an ideal in ai. A sequence of this type is called an elementary sequence.
- (vi) There is a finite sequence of subalgebras gi of g,
- such that gi + 1 is an ideal in gi and gi/gi + 1 is abelian.
- (vii) g is solvable if and only if its Killing form B satisfies B(X, Y) = 0 for all X in g and Y in [g, g]. This is Cartan's criterion for solvability.
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 v ∈ V of the matrices π(X) for all elements X ∈ g. More generally, the result holds if all eigenvalues of π(X) lie in K for all X ∈ g.
- Every Lie subalgebra, quotient and extension[clarification needed] 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 r ⊂ g containing all solvable ideals in g. This ideal is the radical of g, denoted rad g.
- If a, b ⊂ g are solvable ideals, then so is a + b.
- A solvable Lie algebra g has a unique largest nilpotent ideal n, the set of all X ∈ g such that adX is nilpotent. If D 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 an elementary sequence[definition needed] 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 a 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 solvable Lie algebra g is split solvable if and only if the eigenvalues of adX are in k for all X in g.
- A semisimple Lie algebra is never solvable.
- Every abelian Lie algebra is solvable.
- Every nilpotent Lie algebra is solvable.
- Let bk be the subalgebra of glk consisting of upper triangular matrices. Then bk is solvable.
- Let g be the set of matrices on the form
- 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
- termination of the usual derived series of the group 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 the Lie algebra is the Lie algebra of the derived series of (closed) subgroups.
- Fulton, W.; Harris, J. (1991). Representation theory. A first course. Graduate Texts in Mathematics 129. New York: Springer-Verlag. ISBN 978-0-387-97527-6. MR 1153249.
- Humphreys, James E. (1972). Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics 9. New York: Springer-Verlag. ISBN 0-387-90053-5.
- Knapp, A. W. (2002). Lie groups beyond an introduction. Progress in Mathematics 120 (2nd ed.). Boston·Basel·Berlin: Birkhäuser. ISBN 0-8176-4259-5..