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 Î F,
W is stable by u, i.e. u(x) Î W for all x Î 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.
We assume in the remaining of this part, that F is a subset
of L(V) such that, for any u Î F and
v Î F, one have
u° v = v ° u. The purpose of the following questions is to show
that F is triangulable.
2^{°}) Let u Î F, l is a eigenvalue of u
and let V_{u}(l) be the corresponding eigenspace. Show that
V_{u}(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.
Given u Î L(V) and v Î
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] Î I for any u Î F and
v Î I.
1^{°}) Let F be a Lie algebra of dimension 2,
such that there exists u_{0} Î F and v_{0}
Î F obeying [u_{0},v_{0}] ¹ 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 Î F and v Î 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 Î I. The purpose of questions
2^{°} to 5^{°}
is to show that W is stable by F.
Let u Î F, and x be a non-vanishing element of W ; we define by recursion
a sequence (x_{k}) in the following way: x_{0}=x and x_{k} = u(x_{k-1}) for any integer k
³ 1.
2^{°}) Show that, for any k Î N and any
v Î I,
v(x_{k})-l(v)x_{k} lies in the subspace generated by
{x_{0},x_{1},¼,x_{k-1}}.
3^{°}) Let U be the subspace of V generated by the vectors x_{k}, where
k is any positive integer. Show that U is stable by I È 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} = F_{0} Ì F_{1} Ì ¼ Ì F_{p} = F |
[u,v] Î F_{k-1} for any u Î F_{k} and v Î F_{k} |
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".
In this part, the field K can be either R or C.
For any u Î L(V), we will denote by ad_{u}
the element of L(L(V)) defined by ad_{u}(v)=[u,v] for any
v Î L(V).
1^{°}) Show that, for u Î L(V) and
v Î L(V), we have
ad_{[u,v]}=[ad_{u},ad_{v}].
2^{°}) Show that, if u is a nilpotent operator in L(V), then
ad_{u} 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 Ì 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 Î F
to the unique v Î H such that u-v Î G.
Show that there exists one and only one linear application
p: G ® L(H) |
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
Î V such that u(x)=0 for any u Î F.
4^{°}) Let G be a second Lie algebra (made of endomorphisms of V), such
that G Ì F and G ¹ 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 Î W such that u(x)=0 for any u
Î B. We also keep the notations and hypothesis of
question 4^{°}. Show that there exists
a Lie algebra G_{1},made of endomorphisms of V,
obeying the following properties:
G Ì G_{1} Ì F |
dim(G_{1}) = dim(G) + 1 |
G_{1} is an ideal of G. |