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Math examination for admission at the ENS-LYON (Ecole Normale Supérieure), in 1990.
Warning, statement error at the end

FIRST MATHEMATICS EXAMINATION
Duration: 4 hours

Throughout the questions, V is supposed to be a n-dimensional vector space on the Field K (which will always be R or C). A subset F of L(V), set of endomorphisms of V is said to be triangulable if there exist a basis in V in which the matrix of ANY element of F is upper triangular. It is reminded that a subspace W of V is said stable by F if, for all u ö F, W is stable by u, i.e. u(x) ö W for all x ö W.

The main topic of this problem consists in looking for eigenvectors that are common to all elements of a subset F of L(V), endowed with suitable properties, the main application being to obtain some sufficient conditions for triangulation.

PART I
In the first five questions of this part, K=C.
1¯) Show that, for a subset F of L(V) to be triangulable, it is necessary that the elements of F have at least one common eigenvector.

We assume in the remaining of this part, that F is a subset of L(V) such that, for any u ö F and v ö F, one have u¯ v = v ¯ u. The purpose of the following questions is to show that F is triangulable.
2¯) Let u ö F, l is a eigenvalue of u and let Vu(l) be the corresponding eigenspace. Show that Vu(l) is stable by F.
3¯) Show that the elements of F have a common eigenvector.
4¯) Show that F is triangulable.
5¯) We additionaly assume that any element of F is diagonalizable. Can we find a basis in V in which the matrix of any element of F is diagonal?
6¯) Re-do the problem raised in question 2¯, replacing C by R.

PART II
In this part, K = C.

Given u ö L(V) and v ö L(V), let [u,v]=u¯ v - v¯ u. A subset F of L(V) is called a Lie algebra (made of endomorphisms of V) if the following conditions are fulfilled :

(i)  F is a vector subspace of L(V)
(ii)  For any u ö F and v ö F, [u,v] ö F.

We will call dimension of a Lie algebra F, which will be denoted by dim(F), its dimension as a subspace over the field K.

Let F be a Lie albegra, we call ideal of F any subspace I of F such that [u,v] ö I for any u ö F and v ö I.
1¯) Let F be a Lie algebra of dimension 2, such that there exists u0 ö F and v0 ö F obeying [u0,v0] ¿ 0 . Let also FÂ ba another Lie algebra of dimension 2, obeying the same property. Show that there exists an isomorphism (vector space isomorphism) f from F to FÂ such that f([u,v])=[f(u),f(v)] for any u ö F and v ö F.

Let F be a Lie algebra and I an ideal of F. Given a linear form l on I, we denote by W the subspace of V of vectors x such that v(x)=l(v)x for any v ö I. The purpose of questions 2¯ to 5¯ is to show that W is stable by F.

Let u ö F, and x be a non-vanishing element of W ; we define by recursion a sequence (xk) in the following way: x0=x and xk = u(xk-1) for any integer k ° 1.
2¯) Show that, for any k ö N and any v ö I, v(xk)-l(v)xk lies in the subspace generated by {x0,x1,¥,xk-1}.
3¯) Let U be the subspace of V generated by the vectors xk, where k is any positive integer. Show that U is stable by I à u.
4¯) Give a relation between l([u,v]) and the trace (i.e. the sum of eigenvalues) of the restriction to U of the endomorphism [u,v].
5¯) Show that W is stable by F.

A Lie algebra F is said to be solvable if their exists an increasing sequence
 {0} = F0 ä F1 ä ¥ ä Fp = F
of subspaces F such that, for any integer k obeying 1 È k È p, one has :
 [u,v] ö Fk-1 for any u ö Fk and v ö Fk
6¯) Show that any Lie algebra of dimension È 2 is solvable.

The purpose of the following questions is to proove the "Lie Theorem" which states that any solvable Lie algebra is traingulable. We consequently let F be a solvable Lie algebra.
7¯) Let d=dim(F). Show that there exists an ideal I of F, of dimension d-1. Show that I is also a solvable Lie algebra.
8¯) Show that the elements of F have a common eigenvector.
9¯) Show that F is triangulable.
10¯) Show that, conversely, any triangulable Lie algebra is solvable.
11¯) Show that the result of question I.4¯ is a corrolary of the "Lie Theorem".

PART III

In this part, the field K can be either R or C. For any u ö L(V), we will denote by adu the element of L(L(V)) defined by adu(v)=[u,v] for any v ö L(V).
2¯) Show that, if u is a nilpotent operator in L(V), then adu is a nilpotent operator in L(L(V)).
3¯) Let F and G be two Lie algebras (made of endomorphisms of V) such that G ä F. Let H be a complementary subspace of G in F, and let q be the projector on H parallel to G, i.e. the application from F to H which maps any u ö F to the unique v ö H such that u-v ö G. Show that there exists one and only one linear application
 p: G Û L(H)
such that, for any g ö G and any u ö F,
 p(g)(q(u)) = q([g,u]).

We now denote by F a Lie algebra (made of endomorphisms of V), such that any element of F is a nilpotent operator in V. The purpose of the following questions is to demonstrate the "Engel's theorem", which states that their exists a non vanishing vector x ö V such that u(x)=0 for any u ö F.
4¯) Let G be a second Lie algebra (made of endomorphisms of V), such that G ä F and G ¿ F. We keep the same notations as in the previous question and we let FÂ = p(G), VÂ=H. Show that FÂ is a Lie algebra ( made of endomorphisms of VÂ), that dim(FÂ) < dim(F) and that any element of FÂ is nilpotent.
5¯) Let d=dim(F). We assume that, for any finite dimensional vector space W on the field K, and any Lie algebra B ,made of nilpotent endomorphisms of W, and such that dim(B) È d-1, there exists a non-vanishing vector x ö W such that u(x)=0 for any u ö B. We also keep the notations and hypothesis of question 4¯. Show that there exists a Lie algebra G1,made of endomorphisms of V, obeying the following properties:
 G ä G1 ä F
 dim(G1) = dim(G) + 1
 G1 is an ideal of G.
Conclude that there exists an ideal F1 of F, such that dim(F1)=d-1.
6¯) Proove Engel's theorem.
7¯) Show that any Lie algebra made of nilpotent operators of de the vector space V is triangulable.