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


(version en francais)

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 F, W is stable by u, i.e. u(x) W for all x 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 F and v F, one have u v = v u. The purpose of the following questions is to show that F is triangulable.
2) Let u 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 L(V) and v 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 :

     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] I for any u F and v I.
1) Let F be a Lie algebra of dimension 2, such that there exists u0 F and v0 F obeying [u0,v0] 0 . Let also F ba another Lie algebra of dimension 2, obeying the same property. Show that there exists an isomorphism (vector space isomorphism) f from F to F such that f([u,v])=[f(u),f(v)] for any u F and v 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 I. The purpose of questions 2 to 5 is to show that W is stable by F.

     Let u 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 N and any v 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 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 F1 Fp = F
of subspaces F such that, for any integer k obeying 1 k p, one has :
[u,v] Fk-1 for any u Fk and v Fk
6) Show that any Lie algebra of dimension 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 L(V), we will denote by adu the element of L(L(V)) defined by adu(v)=[u,v] for any v L(V).
1) Show that, for u L(V) and v L(V), we have ad[u,v]=[adu,adv].
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 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 F to the unique v H such that u-v G. Show that there exists one and only one linear application
p: G L(H)
such that, for any g G and any u 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 V such that u(x)=0 for any u F.
4) Let G be a second Lie algebra (made of endomorphisms of V), such that G F and G F. We keep the same notations as in the previous question and we let F = p(G), V=H. Show that F is a Lie algebra ( made of endomorphisms of V), that dim(F) < dim(F) and that any element of F 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) d-1, there exists a non-vanishing vector x W such that u(x)=0 for any u 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 G1 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.