7 Solvable Lie groups Recall that a solvable group is one all of whose composition factors are abelian The term comes from Galois theory where a polynomial is solvable by radicals and Artin Schrier extensions in positive characteristic if and only if its Galois group is solvable For Lie groups the term solvable has the same meaning and for Lie algebras it means the obvious variation the Lie algebra is solvable if all composition factors are abelian Lie algebras The main goal of this section is to prove Lie s theorem that a complex solvable Lie algebra of matrices is conjugate to an algebra of upper triangular matrices Lie s theorem fails in positive charactersitic so in proving it we need to make use of some property of matrices that holds in characteristic 0 but not in positive characteristic One such property is that if the trace of I vanishes then so does this is used in the following lemma Lemma 79 Suppose that the Lie algebra G over a field of characteristic 0 has an ideal H and acts on the finite dimensional vector space V Then G acts on each eigenspace of H Proof Recall that an eigenvalue of H is given by some linear form on H and the corresponding eigenspace consists of vectors v such that h v h v for all h H Pick some eigenvector of H with eigenvalue and pick some g G We need to show that g v also has eigenvalue Look at the space W spanned by g gv g 2 v which has an increasing filtration 0 W0 W1 Wn W where Wi is spanned by Wi 1 and g i v Then each Wi Wi 1 is at most 1dimensional and is acted on by H with eigenvalue because g h is in H So on W any element h of H has trace n h In particular g h has trace n g h so g h 0 because g h has trace 0 and n is invertible this is where we use the characteristic 0 assumption But g h 0 implies that hgv ghv g h v so gv is an eigenvalue of H with eigenvalue which is what we were trying to prove This lemma really does fail in infinite dimensions or in characteristic p 0 For example we can take the nilpotent Lie algebra spanned by the operators 1 x d dx which acts on k x Then 1 is an eigenvalue of d dx but x1 is not In characteristic p we can take the finite dimensional quotient k x xp It is clear from the proof that it holds in characteristic p 0 provided the vector space V has dimension less than p this is quite a common phenomenon results true in characteristic 0 are often true in characteristic p 0 provided we stick to vector spaces of dimension less than p Theorem 80 Lie s theorem If a solvable Lie algebra G over an algebraically closed field of characteristic 0 acts on a non zero finite dimensional vector space it has an eigenvector Proof If G is nonzero then as it is solvable we can find an ideal H of codimension 1 By induction on the dimension of G there is an eigenspace W of H for some eigenvalue of H If g is any element of G not in H then by the previous lemma g acts on W and as we are working over an algebraically closed field we 37 can find some eigenvector of g on W This is an eigenvector of G because G is spanned by g and H By repeatedly applying this theorem we see that the Lie algebra fixes a flag So solvable Lie subalgebras of Mn C are conjugate to subalgebras of the Lie algebra of upper triangular matrices Another way of stating Lie s theorem is that any irreducible representation of a finite dimensional complex solvable Lie algebra is 1 dimensional This does not mean that their representation theory is trivial Non abelian solvable complex Lie algebras have plenty of infinite dimensional irreducible representations And even finite dimensional representations are hard to study because there are plently of indecomposbale representations that are not irreducible In fact even for abelian Lie algebras of dimension 2 the finite dimensional indecomposable representations are very hard to classify We will see later that the representation theory of simple Lie algebras is much easier because we do not ahve this problem all indecomposable representations are irreducible If we examine the proof we see that Lie s theorem still holds in positive characteristic provided the dimension of the vector space is less than the characteristic Example 81 The solvable in fact nilpotent Lie algebra spanned by the operators 1 x d dx acts on k x and has no eigenvectors In characteristic p is acts on the finite dimensional quotient k x xp but has no eigenvectors in fact the action is irreducible So Lie s theorem fails in characteristic p 0 and in infinite dimensions Corollary 82 The derived subalgebra of a finite dimensional solvable Lie algebra over a field of characteristic 0 is nilpotent Proof We can extend the field to be algebraically closed In this case the corollary follows from Lie s theorem because the Lie algebra can be assume to be upper triangular in which case its derived algebra consists of strictly upper triangular matrices and is therefore nilpotent Although Lie s theorem fails in positive characteristic for Lie algebras it still holds for solvable algebraic groups in any characteristic this is Kolchin s theorem However it fails for solvable connected Lie groups these are not necessarily isomorphic to groups of upper triangular matrices More generally still Borel proved that any solvable algebraic group acting on a projective variety over an algebraically closed field has a fixed point The special case when the projective variety is projective space is Kolchin s theorem Example 83 There are obvious analogues of Lie s theorems for connected solvable Lie groups of matrices However for disconnected solvable groups the conclusions do not hold For example the symmetric group S3 acting on its irreducible 2 dimensional representation has no eigenvectors And the derived subgroup of a solvable finite group is usually not nilpotent an example is the solvable symmetric group S4 whose derived subgroup is the alternating group A4 Example 84 Lie s theorem shows that in some sense solvable connected Lie groups are not too far from nilpotent ones they are given by sticking an abelian 38 group on top of a nilpotent one For disconnected finite groups the solvable ones can be much more complicated than nilpotent ones For example a typical example of a smallish solvable group is GL2 F3 of order 48 with the chain of normal subgroups 1 Z 2Z Q8 SL2 F3 GL2 F3 with quotients of orders 2 4 3 2 Larger finite solvable groups tend to be a similar but more complicated mess and are rather hard to work with The relatively easy structure of solvable connected