Table Of ContentNormalization of a nonlinear representation of a Lie algebra,
regular on an abelian ideal
7 Mabrouk BEN AMMAR
1
0
January 17, 2017
2
n
a
J
Abstract
6
1 We consider a nonlinear representation of a Lie algebra which is regular on an
abelian ideal, we define a normal form which generalizes that defined in [2].
]
T
R
1 Introduction
.
h
t
a The study of a vector fields in a neighborhood of a point on a complex manifold is, of
m course, reduced to that of a vector fields T in a neighborhood of the origin of E =Cn.
[ If T is regular at the origin of E, we know that there exists a coordinate system
1 (x1,...,xn) of E in which the vector fields can be expressed as
v
7 ∂
T = ,
7
∂x
3 1
4
that is, there exists a local diffeomorphism φ of E such that
0
.
1 ∂
0 φ(0) = 0 and φ⋆(T) = .
∂x
7 1
1
: If T is analytic then φ can be chosen to be analytic [9].
v
If T is singular at the origin of E, then the situation is not so simple; generically, T is
i
X
linearizable, that is, there exists a local diffeomorphism φ of E such that
r
a
∂
φ(0) = 0 and φ⋆(T) = a x , a ∈ C.
jk j jk
∂x
Xj,k k
But, the linearization is not always possible, so, we introduce the notion of normal form of
a vector fields. Such a normal form, by construction, must enjoy the following properties:
(a) If T is any vector fields then there exists a local diffeomorphism φ of E such that
φ(0) = 0 and φ⋆(T) is in normal form.
(b) This normal form is unique, that is, if φ⋆(T) and φ⋆(T) are in normal form then
1 2
φ⋆(T) = φ⋆(T).
1 2
1
This problem has been studied extensively since Poincar´e in formal, analytical or Cinfty
contexts. The normal form of T is then:
T′ = S1+N
wher S1 is semisimple and N is nilpotent satisfying
[S1,N] = 0.
Let us now consider the case, not of a single vector fields, but of a finite dimensional
Lie algebra g of vector fields, or, if we prefer, not the case of a germ of local actions on E,
but rather the case of germs of local actions of a Lie group G on E.
Let us consider the formal or analytic setting. If all vectors fields are singular at 0,
then we are in the presence of a nonlinear representation T of a Lie algebra g in the sense
of Flato, Pinczon and Simon [5]:
T :g −→ X (E), X 7−→ T such that [T ,T ] = T ,
0 X X Y [X,Y]
where X (E) is the space of vector fields which are singular at 0. In this case, we use
0
the structure of the Lie algebra g to precise the convenient normal forms. The notion
of normal form given in [2] generalizes those known for the nilpotent and semisimple
cases ([1], [3],[6],[7]). Let r be the solvable radical of g, we know that, by the Levi-Malcev
decomposition theorem, g can be decomposed
(D) g = r⊕s
where s is semi-simple. The nonlinear representation T of the complex Lie algebra g is
said to be normal with respect the Levi-Malcev decomposition (D) of g, if its restriction
T| is linear and its restriction T| to r is in the following form
s r
T| = D1+N1+ Tk| ,
r r
Xk≥2
wherethelinearpartD1+N1 ofT| issuchthatD1isdiagonalwithcoefficientsµ , ..., µ ,
r 1 n
(the µ are elements of r∗) and N1 is strictly upper triangular. Moreover, Tk has the
i
following form in the coordinates x of E
i
n
∂
Tk = Λα(X)xα1···xαn , X ∈ r,
X i 1 n ∂x
Xi=1|αX|=k i
where any coefficient Λα can benonzero only if it is resonant, that means, the correspond-
i
ing linear form µi ∈ r∗ defined by
α
n
µi = α µ −µ ,
α r r i
Xr=1
is a root of r, (eigenvalue of the adjoint representation of r). The representation T is said
to be normalizable with respect (D) if it is equivalent to a normal one with respect (D).
2
Inthispaper,westudythesituation wheretheT arenotallinX (E).Moreprecisely,
X 0
we introduce the notion of nonlinear representation (T,E) of g in E such that
T = T0 + Tk ∈ X(E), X ∈ g,
X X X
Xk≥1
where X(E) is the space of formal vector fields on E (not necessarily vanishing at 0), in
particular, T0 ∈ E. This situation seems to be more delicate in its generality, but here we
X
treat the case where the Lie algebra g can be written
g = g +m,
0
where m is an abelian ideal of g and g is a subalgebra of g. The Lie algebras of groups
0
of symmetries of simple physical systems are often of this type (Galil´ee group, Poincar´e
group, ...). Moreover, we assume that
T0 = 0 if X ∈ g and T0 6= 0 if X ∈m\{0}.
X 0 X
We also assume that T is analytic and we will prove in this case that we can always
put the representation T in the following normal form: there exists a coordinate system
(x ,...,x ,y ,...,y ) of E such that:
1 p 1 q
(a) For all X ∈ m, T can be expressed as
X
p
∂
T = a ,
X i
∂x
Xi=1 i
where a ∈m∗.
i
(b) T| is normal au sense de [2], moreover, for all X ∈ g and k ≥ 2, Tk has the
g0 0 X
following form
p q
∂ ∂
Tk = Λα(X)yα1···yαq + Γα(X)yα1 ···yαq
X i 1 q ∂x i 1 q ∂y
Xi=1|αX|=k i Xi=1|αX|=k i
2 Notations and definitions
Let E be a complex vector space with dimension n. The space of symmetric k-linear
applications from E×···×E to E is identified with the space L(⊗kE,E) of linear maps
s
from ⊗kE to E, where ⊗kE is the space of symmetric k-tensors on E. Denote by L(E)
s s
the space L(E,E). Let (e ,...,e ) be a basis of E. For α ∈ Nn such that |α| = k and
1 n
i ∈{1,...,n} we define ei ∈ L(⊗kE,E) by:
α s
eα◦σ (⊗β1e ···⊗βn e ) = δβ1 ···δβne ,
i k 1 n α1 αn i
where δ is the Kronecker symbol, and σ is the symmetrization operator from ⊗kE to
k
⊗kE defined by
s
1
σ (v ,...,v ) = (v ,...,v ).
k 1 k k! σ(1) σ(k)
σX∈Sk
3
Obviously,(eα)isabasisofL(⊗kE,E).LetX(E)(respectively X (E))bethesetofformal
i s 0
power series (or formal vector fields)
∞ ∞
T = Tk, (respectively T = Tk)
Xk=1 Xk=0
where Tk ∈L(⊗kE,E) (with L(⊗0E,E) = E). Therefore, we can write
s s
n
Tk = eα.
i
|αX|=kXi=1
Of course, any analytical vector fields X on E which is singular in 0 admits a Taylor
expansion:
n ∞
∂
Λαxα1···xαn = Λαxα∂ .
i 1 n ∂x i i
Xi=1Xk=1|αX|=k i Xα,i
It can be identified with the formal vector fields:
n ∞
Λαeα.
i i
Xi=1Xk=1|αX|=k
If |α| = 0 we agree that eα = e .
i i
Of course, X(E) can be endowed with a Lie algebra structure defined, for eα ∈
i
L(⊗|α|E,E) and eβ ∈ L(⊗|β|E,E), by
s j s
[eα,eβ] = eα⋆eβ −eβ ⋆eα ∈L(⊗|α|+|β|−1E,E),
i j i j j i s
where
eα⋆eβ = b eα+β−1i, 1 = (0,...,0,1,0,...,0),(1 in ith place).
i j i j i
Definition 2.1 A formal vector fields T in X(E) is said to be analytic al if the power
series
Tk(⊗kv)= T (v), v ∈ E,
X
Xk≥1
converges in a neighborhood of the origin of E.
Definition 2.2 Two formal vector fields T and T′ are said to be equivalent if there exists
an element φ= φk of X (E) such that φ1 is invertible and
k≥1 0
P
ψ⋆T = T′◦ψ
where
k
T′◦φ= T′j ◦ φi1 ⊗···⊗ψij◦σk.
Xk≥1Xj=1 i1+·X··+ij=k
T and T′ are said to be analytically equivalent if φ is analytic.
4
Remark 2.3 For the composition law ◦, the map φ is invertible if and only if φ1 is an
automorphism of E.
Definition 2.4 Let g be a Lie algebra with finite dimension over C. A nonlinear (formal)
representation (T,E) of g in E is a linear map:
T : g → X(E), X 7→ T ,
X
such that
[T ,T ]= T , X, Y ∈ g.
X Y [X,Y]
Definition 2.5 We say that two representations (T,E) and (T′,E) are equivalent if there
exists φ∈ X (E) such that φ is invertible and
0 1
φ⋆T = T′ ◦φ, X ∈ g.
X X
Definition 2.6 A nonlinear formal representation T of g is said to be analytic if the
power series
Tk(⊗ke) = T (v), v ∈E
X X
Xk≥1
converges in a neighborhood of 0 in E, for all X ∈ g.
In the following, we consider a nonlinear analytic representation (T,E) of a complex
finite dimensional Lie algebra g in E = Cn, such that
g = g +m,
0
where m is an abelian ideal of g and g is a subalgebra of g. Moreover, we assume that
0
T0 = 0 if X ∈ g and T0 6= 0 if X ∈m\{0},
X 0 X
and then we normalize T step by step.
3 Normalization of T|
m
Proposition 3.1 There exists a coordinate system (x ,...,x ,y ,...,y ), p+q = n, of
1 p 1 q
E such that, for all X ∈ m, T can be expressed as
X
p
∂
T = a ,
X i
∂x
Xi=1 i
where a ∈ m∗.
i
Proof. Since the representation (T,E) is analytic, then, there exists a neighborhood U of
the origin of E such that
T (v) ∈ E,
x
for any v in U and X in g. So, for any v in U, we consider the subspace E spanned by
v
the vectors T (v), whereX browses m. If (X ,...,X ) is a basis of m, then the T (v) are
X 1 p Xi
5
generators of E , thus, dimE ≤ p. But, the system (T (0),...,T (0)) is independent,
v v X1 Xp
indeed, the condition
α T (0)+···+α T (0) = T0 = 0
1 X1 p Xp α1X1(0)+···+αpXp
impliesthatα X (0)+···+α X = 0sinceT0doesnotvanishonm\{0}.Thus,dimE = p.
1 1 p p 0
The determinant map is a continuous map, then there exists a neighborhood V ⊂ U such
that dimE ≥ p, for all v ∈ V. Therefore, dimE = p, for all v ∈ V.
v v
Thus, the map
v 7→ E
v
is an involutive integrable distribution, therefore, the Frobinius theorem ensures the ex-
istence of an analytic coordinate system (x ,...,x ,y ,...,y ) of E such that the ∂ ,
1 p 1 q ∂xi
i = 1,...,p, generate this distribution.
(cid:3)
4 Normalization of T|
g0
Now, we begin the second step to normalize the representation (T,E). We know that, for
any X in g , we have
0
T = Tk.
X X
Xk≥1
We consider the coordinate system (x ,...,x ,y ,...,y ) of E defined in the previous
1 p 1 q
section and we prove the following results
Proposition 4.1 For any X in g , T1 has the following form
0 X
∂ ∂ ∂
T1 = a (X)y + b (X)y + c (X)x
X ij j∂x ij j∂y ij j∂x
Xi,j i Xi,j i Xi,j i
and for k ≥ 2, Tk has the following form
X
∂ ∂
Tk = Aα(X)yα + Bα(X)yα .
X i ∂x i ∂y
X i X i
i,|α|=k i|α|=k
Proof. Consider (X ,...,X ) ∈ mp such that
1 p
∂
T = .
Xi ∂x
i
Since m is an ideal of m, then [m,g ] ⊂ m, and therefore, for any X in g and j = 1,...,p,
0 0
we have
p
∂ ∂ ∂
T = ,T = ,T1 = α ,
[Xj,X] (cid:20)∂x X(cid:21) (cid:20)∂x X(cid:21) i∂x
j j Xi=1 i
for some α ∈C. In particular, for k ≥ 2, we have
i
∂
,Tk = 0,
(cid:20)∂x X(cid:21)
j
6
and then the result follows.
(cid:3)
Now, for any X in g , we define
0
A = Aα(X)yα ∂ = Ak , (A1j = a ),
X i,α i ∂xi k≥1 X i ij
P P
B = Bα(X)yα ∂ + c (X)x ∂ = Bk ,
X i,α i ∂xi i,j ij j∂xi k≥1 X
P P P
H = b (X)y ∂ ,
X i,j ij j∂yi
P
K = c (X)x ∂ ,
X i,j ij j∂xi
P
and we consider the subspaces E and E corresponding, respectively, to coordinate sys-
1 2
tems (x ,...,x ) and (y ,...,y ).
1 p 1 q
Proposition 4.2 i) (H,E ) and (K,E ) are two linear representations of g .
1 2 0
ii) For any X in g we have
0
A = [A ,B ]+[B ,A ],
[X,Y] X Y X Y
B = [B ,B ].
[X,Y] X Y
Thus, (B,E) is a nonlinear representation of g in E.
0
Proof. i) Obvious.
ii) For any X in g , we have
0
T = A +B ,
X X X
then we have
T = A +B = [A +B ,A +B ] = [A ,B ]+[B ,A ]+[B ,B ].
[X,Y] [X,Y] [X,Y] X X Y Y X Y X Y X Y
We easily check that
A = [A ,B ]+[B ,A ],
[X,Y] X Y X Y
B = [B ,B ].
[X,Y] X Y
(cid:3)
Now, we consider a Levi-Malcev decomposition of g :
0
(D) g = r⊕s
where s is semi-simple and r is the solvable radical of g. We triangularize simultaneously
the H and K , where X browses r (Lie theorem [8]). Thus, we define the linear forms:
X X
µ ,...,µ ∈ r∗ and ν ,...,ν ∈ r∗. The µ (X),...,µ (X),ν (X),...,ν (X) are the eigen-
1 p 1 q 1 p 1 q
values of B1(X), indeed, B1 = H +K . They are also the eigenvalues of T1(X), since
X X X
A1 ∈ L(E ,E ). Thus, the simultaneous triangulation of H and K leads to that of T1 by
X 2 1
retaining the first p components in E and the q other in E .
1 2
Letusdenotebyλ ,...,λ therootsofT1| ,andconsidertheelements λα ofr∗ defined
1 n r j
by
n
λα(X) = α λ (X)−λ (X),
j i i j
Xi=1
where j = 1,...,n and α ∈N.
7
Definition 4.3 We say that (α,j) is resonant if λα is a root of r.
j
Definition 4.4 An element X in r is said to be vector resonance if, for any α, for any
0
i and for any j,
λα(X )= v (X ) ⇒ λα = v ,
j 0 i 0 j i
where the v are the roots of r.
i
For the two following lemmas see, for instance, [1] and [2].
Lemma 4.5 The set of vector resonance is dense in r.
Lemma 4.6 There exists a resonance vector in r such that [X ,s] = 0.
0
The resonant pairs (α,j) appearing here are of two types:
p
(1) α ν −µ is a root of r.
i=1 i i j
P
(2) p α ν −ν is a root of r.
i=1 i i j
P
Let us denote by
R = {(α,j) | (α,j) resonant of type (1)}
and by
R′ = {(α,j) | (α,j) resonant of type (2)}
Theorem 4.7 T| is normalizable in the sense of [2], that means, there exists an analytic
r
operator φ= φk in X (E), such that φ1 is invertible and, for any X in r, φ⋆T ◦φ−1
0 X
is in the formP:
∂ ∂
Λα(X)yα + Γα(X)yα .
i ∂x i ∂y
(αX,i)∈R i (αX,i)∈R′ i
Proof. Let X be a resonance vector of r such that [X ,s] = 0. It is well known that T is
0 0 X0
analytically normalizable: there exists an analytic operator φ= φk in X (E), such that
0
φ1 is invertible and, for any X in r, P
φ⋆T ◦φ−1 = T′ ,
X0 X0
where T′ = S1 +N with
X0 X0 X0
p q
∂ ∂
S1 = µ (X )x + ν (X )y ,
X0 i 0 i∂x i 0 i∂y
Xi=1 i Xi=1 i
and
[S1 ,N ] = 0.
X0 X0
Therefore, N has the following form
X0
∂ ∂
N = Λα(X )yα + Γα(X )yα ,
X0 i 0 ∂x i 0 ∂y
(α,Xi)∈R0 i (α,Xi)∈R′0 i
8
where
q q
R = {(α,i) | α ν −µ = 0} and R′ = {(α,i) | α ν −ν = 0}.
0 j j i 0 j j i
Xj=1 Xj=1
The operator φ normalizes T| (see [2]).
r
(cid:3)
Proposition 4.8 The normalization operator φ of T| leaves invariant T| .
r m
Proof. To prove that φ leaves invariant T| we will prove that
m
φ= ···(I +W )◦···◦(I +W ),
k 1
where I is the identity of E and W ∈ L(⊗kE,E).
k s
(a) B1 = H +K being decomposed into a semisimpe part S1 and a nilpotent
X0 X0 X0 X0
part, therefore, to reduce T1 it suffices to reduce A1 . We decompose A1 :
X0 X0 X0
A1 = A1 +A1 ,
X0 0X0 1X0
where
∂
A1 = a (X )y ∈ ker adS1
0X0 ij 0 i∂x X0
X i
(i,j),νj=µi
and
∂
A1 = a (X )y ∈ Im adS1 .
1X0 ij 0 i∂x X0
X i
(i,j),νj6=µi
We reduce A1 by removing A1 . Indeed, for any X and Y in r,
X0 1X0
A1 = [A1 ,B1]+[B1 ,A1 ],
[X,Y] X Y X Y
therefore, there exists W1 in L(E) and a 1-cocycle V1 such that
A1 = [B1 ,W ]+V1.
X X 1
In particular, A1 = [B1 ,W ]+V1, then we can choose, for instance,
X0 X0 1
W = (ad B1 (A1 ) and V1 = A1 .
1 X0 1X0 X0 0X0
The stability of the eigenvector subspaces of ad S1 by B1 proves that W1 ∈ L(E ,E ),
X0 X0 2 1
therefore, (I +W ) is invertible and
1
(I +W )⋆T1 ◦(I +W1)−1 = B1 +A1 , (1)
1 X0 X0 0X0
∂ ∂
(I +W )⋆ ◦(I +W )−1 = , i= 1,...,p. (2)
1 1
∂x ∂x
i i
T| is then stable by (i+W ).
m 1
9
b) Now, we assume that T is normalized up to order k−1, we write
X0
T = S1 + Nj +Tk + Tj
X0 X0 X0 X0 X0
1≤Xj<k Xj>k
and we decompose Tk :
X0
Tk = Tk +Tk ,
X0 0X0 1X0
where
Tk ∈ ker adS1 and Tk ∈ Im adS1 .
0X0 X0 1X0 X0
Therefore, Tk has the following form
0X0
∂ ∂
Tk =Nk = Λα(X )yα + Γα(X )yα .
0X0 X0 i 0 ∂x i 0 ∂y
(α,Xi)∈R0 i (α,Xi)∈R′0 i
We seek an operator W in L(⊗kE,E) such that
k s
(1+W )⋆T = (S1 + Nj + Tj )◦(1+W ).
k X0 X0 X0 X0 k
1≤Xj≤k Xj>k
We can choose
W = (ad T1 )−1(T1 ).
k X0 1X0
The eigenvector subspaces of ad S1 are stable by ad T1 , therefore, W has the following
X0 X0 k
form:
∂ ∂
W = Wαyα + W′αyα .
k i ∂x i ∂y
(α,Xi)∈R0 i (α,Xi)∈R′0 i
Thus,
∂ ∂
(I +W )⋆ ◦(I +W )−1 = , i = 1,...,p.
k k
∂x ∂x
i i
We proceed by induction, so, we construct
φ= ···(I +W )◦···◦(I +W )
k 1
which normalizes T| and leaves invariant T| .
r m
5 Linearization of T|
s
Let us now consider the representation (T′,E) of g in E, defined on g by
T′ = φ⋆T ◦φ−1.
X X
The representation being T′| , as in [2], we construct an analytic invertible operator ψ ∈
r
X (E), which linearizes T′| and leaves T′| in normal form. This is possible through the
0 s r
choice of X switching with s: [X ,s] = 0.
0 0
On the other hand, by construction, the components of ψ are independent of the
coordinates x ,...,x , then ψ leaves invariant T′| = T| .
1 p m m
10
