Normalization of a nonlinear representation of a Lie algebra, regular on an abelian ideal
aa r X i v : . [ m a t h . R T ] J a n Normalization of a nonlinear representation of a Lie algebra,regular on an abelian ideal
Mabrouk BEN AMMAROctober 21, 2018
Abstract
We consider a nonlinear representation of a Lie algebra which is regular on anabelian ideal, we define a normal form which generalizes that defined in [2].
The study of a vector fields in a neighborhood of a point on a complex manifold is, ofcourse, reduced to that of a vector fields T in a neighborhood of the origin of E = C n .If T is regular at the origin of E , we know that there exists a coordinate system( x , . . . , x n ) of E in which the vector fields can be expressed as T = ∂∂x , that is, there exists a local diffeomorphism φ of E such that φ (0) = 0 and φ ⋆ ( T ) = ∂∂x . If T is analytic then φ can be chosen to be analytic [9].If T is singular at the origin of E , then the situation is not so simple; generically, T islinearizable, that is, there exists a local diffeomorphism φ of E such that φ (0) = 0 and φ ⋆ ( T ) = X j,k a jk x j ∂∂x k , a jk ∈ C . But, the linearization is not always possible, so, we introduce the notion of normal form ofa 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 φ ⋆ ( T ) = φ ⋆ ( T ). 1his problem has been studied extensively since Poincar´e in formal, analytical or C i nf ty contexts. The normal form of T is then: T ′ = S + N wher S is semisimple and N is nilpotent satisfying[ S , N ] = 0 . Let us now consider the case, not of a single vector fields, but of a finite dimensionalLie 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 senseof Flato, Pinczon and Simon [5]: T : g −→ X ( E ) , X T X such that [ T X , T Y ] = T [ X,Y ] , where X ( E ) is the space of vector fields which are singular at 0. In this case, we usethe structure of the Lie algebra g to precise the convenient normal forms. The notionof normal form given in [2] generalizes those known for the nilpotent and semisimplecases ([1], [3],[6],[7]). Let r be the solvable radical of g , we know that, by the Levi-Malcevdecomposition theorem, g can be decomposed( D ) g = r ⊕ s where s is semi-simple. The nonlinear representation T of the complex Lie algebra g issaid to be normal with respect the Levi-Malcev decomposition ( D ) of g , if its restriction T | s is linear and its restriction T | r to r is in the following form T | r = D + N + X k ≥ T k | r , where the linear part D + N of T | r is such that D is diagonal with coefficients µ , . . . , µ n ,(the µ i are elements of r ∗ ) and N is strictly upper triangular. Moreover, T k has thefollowing form in the coordinates x i of ET kX = n X i =1 X | α | = k Λ αi ( X ) x α · · · x α n n ∂∂x i , X ∈ r , where any coefficient Λ αi can be nonzero only if it is resonant, that means, the correspond-ing linear form µ iα ∈ r ∗ defined by µ iα = n X r =1 α r µ r − µ i , is a root of r , (eigenvalue of the adjoint representation of r ). The representation T is saidto be normalizable with respect ( D ) if it is equivalent to a normal one with respect ( D ).2n this paper, we study the situation where the T X are not all in X ( E ). More precisely,we introduce the notion of nonlinear representation ( T, E ) of g in E such that T X = T X + X k ≥ T kX ∈ X ( E ) , X ∈ g , where X ( E ) is the space of formal vector fields on E (not necessarily vanishing at 0), inparticular, T X ∈ E . This situation seems to be more delicate in its generality, but here wetreat the case where the Lie algebra g can be written g = g + m , where m is an abelian ideal of g and g is a subalgebra of g . The Lie algebras of groupsof symmetries of simple physical systems are often of this type (Galil´ee group, Poincar´egroup, ...). Moreover, we assume that T X = 0 if X ∈ g and T X = 0 if X ∈ m \ { } . We also assume that T is analytic and we will prove in this case that we can alwaysput the representation T in the following normal form: there exists a coordinate system( x , . . . , x p , y , . . . , y q ) of E such that:(a) For all X ∈ m , T X can be expressed as T X = p X i =1 a i ∂∂x i , where a i ∈ m ∗ .(b) T | g is normal au sense de [2], moreover, for all X ∈ g and k ≥ T kX has thefollowing form T kX = p X i =1 X | α | = k Λ αi ( X ) y α · · · y α q q ∂∂x i + q X i =1 X | α | = k Γ αi ( X ) y α · · · y α q q ∂∂y i Let E be a complex vector space with dimension n . The space of symmetric k -linearapplications from E × · · · × E to E is identified with the space L( ⊗ ks E, E ) of linear mapsfrom ⊗ ks E to E , where ⊗ ks E is the space of symmetric k -tensors on E . Denote by L( E )the space L( E, E ). Let ( e , . . . , e n ) be a basis of E . For α ∈ N n such that | α | = k and i ∈ { , . . . , n } we define e iα ∈ L( ⊗ ks E, E ) by: e αi ◦ σ k ( ⊗ β e · · · ⊗ β n e n ) = δ β α · · · δ β n α n e i , where δ is the Kronecker symbol, and σ k is the symmetrization operator from ⊗ k E to ⊗ ks E defined by σ k ( v , . . . , v k ) = 1 k ! X σ ∈S k ( v σ (1) , . . . , v σ ( k ) ) . e αi ) is a basis of L( ⊗ ks E, E ). Let X ( E ) (respectively X ( E )) be the set of formalpower series (or formal vector fields) T = ∞ X k =1 T k , (respectively T = ∞ X k =0 T k )where T k ∈ L( ⊗ ks E, E ) (with L( ⊗ s E, E ) = E ). Therefore, we can write T k = X | α | = k n X i =1 e αi . Of course, any analytical vector fields X on E which is singular in 0 admits a Taylorexpansion: n X i =1 ∞ X k =1 X | α | = k Λ αi x α · · · x α n n ∂∂x i = X α,i Λ αi x α ∂ i . It can be identified with the formal vector fields: n X i =1 ∞ X k =1 X | α | = k Λ αi e αi . If | α | = 0 we agree that e αi = e i .Of course, X ( E ) can be endowed with a Lie algebra structure defined, for e αi ∈ L( ⊗ | α | s E, E ) and e βj ∈ L( ⊗ | β | s E, E ), by[ e αi , e βj ] = e αi ⋆ e βj − e βj ⋆ e αi ∈ L( ⊗ | α | + | β |− s E, E ) , where e αi ⋆ e βj = b i e α + β − i j , i = (0 , . . . , , , , . . . , , (1 in i th place) . Definition 2.1
A formal vector fields T in X ( E ) is said to be analytic al if the powerseries X k ≥ T k ( ⊗ k v ) = T X ( v ) , v ∈ E, 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 existsan element φ = P k ≥ φ k of X ( E ) such that φ is invertible and ψ ⋆ T = T ′ ◦ ψ where T ′ ◦ φ = X k ≥ k X j =1 T ′ j ◦ X i + ··· + i j = k φ i ⊗ · · · ⊗ ψ i j ◦ σ k .T and T ′ are said to be analytically equivalent if φ is analytic. emark 2.3 For the composition law ◦ , the map φ is invertible if and only if φ is anautomorphism 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 T X , such that [ T X , T Y ] = T [ X,Y ] , X, Y ∈ g . Definition 2.5
We say that two representations ( T, E ) and ( T ′ , E ) are equivalent if thereexists φ ∈ X ( E ) such that φ is invertible and φ ⋆ T X = T ′ X ◦ φ, X ∈ g . Definition 2.6
A nonlinear formal representation T of g is said to be analytic if thepower series X k ≥ T kX ( ⊗ k e ) = T X ( v ) , v ∈ E 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 complexfinite dimensional Lie algebra g in E = C n , such that g = g + m , where m is an abelian ideal of g and g is a subalgebra of g . Moreover, we assume that T X = 0 if X ∈ g and T X = 0 if X ∈ m \ { } , and then we normalize T step by step. T | m Proposition 3.1
There exists a coordinate system ( x , . . . , x p , y , . . . , y q ) , p + q = n , of E such that, for all X ∈ m , T X can be expressed as T X = p X i =1 a i ∂∂x i , where a i ∈ m ∗ .Proof. Since the representation (
T, E ) is analytic, then, there exists a neighborhood U ofthe origin of E such that T x ( v ) ∈ E, for any v in U and X in g . So, for any v in U , we consider the subspace E v spanned bythe vectors T X ( v ), where X browses m . If ( X , . . . , X p ) is a basis of m , then the T X i ( v ) are5enerators of E v , thus, dim E v ≤ p . But, the system ( T X (0) , . . . , T X p (0)) is independent,indeed, the condition α T X (0) + · · · + α p T X p (0) = T α X (0)+ ··· + α p X p = 0implies that α X (0)+ · · · + α p X p = 0 since T does not vanish on m \{ } . Thus, dim E = p .The determinant map is a continuous map, then there exists a neighborhood V ⊂ U suchthat dim E v ≥ p , for all v ∈ V . Therefore, dim E v = p , for all v ∈ V .Thus, the map v E v is an involutive integrable distribution, therefore, the Frobinius theorem ensures the ex-istence of an analytic coordinate system ( x , . . . , x p , y , . . . , y q ) of E such that the ∂∂x i , i = 1 , . . . , p , generate this distribution. (cid:3) T | g Now, we begin the second step to normalize the representation (
T, E ). We know that, forany X in g , we have T X = X k ≥ T kX . We consider the coordinate system ( x , . . . , x p , y , . . . , y q ) of E defined in the previoussection and we prove the following results Proposition 4.1
For any X in g , T X has the following form T X = X i,j a ij ( X ) y j ∂∂x i + X i,j b ij ( X ) y j ∂∂y i + X i,j c ij ( X ) x j ∂∂x i and for k ≥ , T kX has the following form T kX = X i, | α | = k A αi ( X ) y α ∂∂x i + X i | α | = k B αi ( X ) y α ∂∂y i . Proof.
Consider ( X , . . . , X p ) ∈ m p such that T X i = ∂∂x i . Since m is an ideal of m , then [ m , g ] ⊂ m , and therefore, for any X in g and j = 1 , . . . , p ,we have T [ X j ,X ] = (cid:20) ∂∂x j , T X (cid:21) = (cid:20) ∂∂x j , T X (cid:21) = p X i =1 α i ∂∂x i , for some α i ∈ C . In particular, for k ≥
2, we have (cid:20) ∂∂x j , T kX (cid:21) = 0 , (cid:3) Now, for any X in g , we define A X = P i,α A αi ( X ) y α ∂∂x i = P k ≥ A kX , ( A j i = a ij ) ,B X = P i,α B αi ( X ) y α ∂∂x i + P i,j c ij ( X ) x j ∂∂x i = P k ≥ B kX ,H X = P i,j b ij ( X ) y j ∂∂y i ,K X = P i,j c ij ( X ) x j ∂∂x i , and we consider the subspaces E and E corresponding, respectively, to coordinate sys-tems ( x , . . . , x p ) and ( y , . . . , y q ). Proposition 4.2 i) ( H, E ) and ( K, E ) are two linear representations of g .ii) For any X in g we have A [ X,Y ] = [ A X , B Y ] + [ B X , A Y ] ,B [ X,Y ] = [ B X , B Y ] . Thus, ( B, E ) is a nonlinear representation of g in E .Proof. i) Obvious.ii) For any X in g , we have T X = A X + B X , then we have T [ X,Y ] = A [ X,Y ] + B [ X,Y ] = [ A X + B X , A Y + B Y ] = [ A X , B Y ] + [ B X , A Y ] + [ B X , B Y ] . We easily check that A [ X,Y ] = [ A X , B Y ] + [ B X , A Y ] ,B [ X,Y ] = [ B X , B Y ] . (cid:3) Now, we consider a Levi-Malcev decomposition of g :( D ) g = r ⊕ s where s is semi-simple and r is the solvable radical of g . We triangularize simultaneouslythe H X and K X , where X browses r (Lie theorem [8]). Thus, we define the linear forms: µ , . . . , µ p ∈ r ∗ and ν , . . . , ν q ∈ r ∗ . The µ ( X ) , . . . , µ p ( X ) , ν ( X ) , . . . , ν q ( X ) are the eigen-values of B ( X ), indeed, B X = H X + K X . They are also the eigenvalues of T ( X ), since A X ∈ L( E , E ). Thus, the simultaneous triangulation of H and K leads to that of T byretaining the first p components in E and the q other in E .Let us denote by λ , . . . , λ n the roots of T | r , and consider the elements λ αj of r ∗ definedby λ αj ( X ) = n X i =1 α i λ i ( X ) − λ j ( X ) , where j = 1 , . . . , n and α ∈ N . 7 efinition 4.3 We say that ( α, j ) is resonant if λ αj is a root of r . Definition 4.4
An element X in r is said to be vector resonance if, for any α , for any i and for any j , λ αj ( X ) = v i ( X ) ⇒ λ αj = v i , where the v i are the roots of r . 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 . The resonant pairs ( α, j ) appearing here are of two types:(1) P pi =1 α i ν i − µ j is a root of r .(2) P pi =1 α i ν i − ν j is a root of r .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 | r is normalizable in the sense of [2], that means, there exists an analyticoperator φ = P φ k in X ( E ) , such that φ is invertible and, for any X in r , φ ⋆ T X ◦ φ − is in the form: X ( α,i ) ∈ R Λ αi ( X ) y α ∂∂x i + X ( α,i ) ∈ R ′ Γ αi ( X ) y α ∂∂y i . Proof.
Let X be a resonance vector of r such that [ X , s ] = 0. It is well known that T X isanalytically normalizable: there exists an analytic operator φ = P φ k in X ( E ), such that φ is invertible and, for any X in r , φ ⋆ T X ◦ φ − = T ′ X , where T ′ X = S X + N X with S X = p X i =1 µ i ( X ) x i ∂∂x i + q X i =1 ν i ( X ) y i ∂∂y i , and [ S X , N X ] = 0 . Therefore, N X has the following form N X = X ( α,i ) ∈ R Λ αi ( X ) y α ∂∂x i + X ( α,i ) ∈ R ′ Γ αi ( X ) y α ∂∂y i , R = { ( α, i ) | q X j =1 α j ν j − µ i = 0 } and R ′ = { ( α, i ) | q X j =1 α j ν j − ν i = 0 } . The operator φ normalizes T | r (see [2]). (cid:3) Proposition 4.8
The normalization operator φ of T | r leaves invariant T | m .Proof. To prove that φ leaves invariant T | m we will prove that φ = · · · ( I + W k ) ◦ · · · ◦ ( I + W ) , where I is the identity of E and W k ∈ L( ⊗ ks E, E ).(a) B X = H X + K X being decomposed into a semisimpe part S X and a nilpotentpart, therefore, to reduce T X it suffices to reduce A X . We decompose A X : A X = A X + A X , where A X = X ( i,j ) ,ν j = µ i a ij ( X ) y i ∂∂x i ∈ ker ad S X and A X = X ( i,j ) ,ν j = µ i a ij ( X ) y i ∂∂x i ∈ Im ad S X . We reduce A X by removing A X . Indeed, for any X and Y in r , A X,Y ] = [ A X , B Y ] + [ B X , A Y ] , therefore, there exists W in L( E ) and a 1-cocycle V such that A X = [ B X , W ] + V . In particular, A X = [ B X , W ] + V , then we can choose, for instance, W = (ad B X ( A X ) and V X = A X . The stability of the eigenvector subspaces of ad S X by B X proves that W ∈ L( E , E ),therefore, ( I + W ) is invertible and( I + W ) ⋆ T X ◦ ( I + W ) − = B X + A X , (1)( I + W ) ⋆ ∂∂x i ◦ ( I + W ) − = ∂∂x i , i = 1 , . . . , p. (2) T | m is then stable by ( i + W ). 9) Now, we assume that T X is normalized up to order k −
1, we write T X = S X + X ≤ j
Consider a nonlinear analytic representation (
T, E ) of a complex finite dimensional Liealgebra g in E = C n , such that ( D ) g = g + m , where m is a p -dimensional abelian ideal of g and g is a subalgebra of g . Moreover, weassume that T X = 0 if X ∈ g and T X = 0 if X ∈ m \ { } . We consider a Levi-Malcev decomposition of g :( D ) g = r ⊕ s where s is semi-simple and r is the solvable radical of g . Then, we have Theorem 6.1
The representation ( T, E ) of g can be normalized with respect the decompo-sitions ( D ) and ( D ) . That means, there exists a coordinate system ( x , . . . , x p , y , . . . , y q ) of E such thata) For all X ∈ m , T X can be expressed as T X = p X i =1 a i ∂∂x i , where a i ∈ m ∗ .b) T | s is a linear representation of s in E .c) For any X in r , T X is in the form: T X = X ( α,i ) ∈ R Λ αi ( X ) y α ∂∂x i + X ( α,i ) ∈ R ′ Γ αi ( X ) y α ∂∂y i , where R = { ( α, i ) | P qj =1 α j ν j − µ i = 0 } and R ′ = { ( α, i ) | P qj =1 α j ν j − ν i = 0 } . References [1] D. Arnal, M. Ben Ammar, G. Pinczon, The Poincar´e–Dulac theorem for nonlinearrepresentation of nilpotent Lie algebras, LMP, t 8, 1984, pp467–476.[2] D. Arnal, M. Ben Ammar, M. Selmi, Normalisation d’une repr´esentation non lin´eaired’une alg`ebre de Lie.
Annales de la Facult´e des Sciences de Toulouse , 5e s´erie, tome9, No 3, 1988, p 355–379.[3] M. Ben Ammar, Nonlinear representations of connected nilpotent real groups. LMP,t 8, 1984, p 119–126.[4] A. Brjuno, Formes analytiques des ´equations diff´erentielles,
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