Normalization of a nonlinear representation of a Lie algebra, regular on an abelian ideal

TL;DR

This paper develops a normalization method for nonlinear Lie algebra representations that are regular on an abelian ideal, extending previous work to a broader class of representations.

Contribution

It introduces a generalized normal form for nonlinear Lie algebra representations regular on an abelian ideal, expanding the theoretical framework beyond prior models.

Findings

Defined a new normal form for nonlinear Lie algebra representations

Extended normalization techniques to representations regular on abelian ideals

Provided theoretical foundations for further applications in Lie algebra analysis

Abstract

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 [D. Arnal, M. Ben Ammar, M. Selmi, {\rm Normalisation d'une repr\'esentation non lin\'eaire d'une alg\`ebre de Lie}. {\it Annales de la Facult\'e des Sciences de Toulouse}, 5e s\'erie, tome 9, No 3, 1988, p 355--379.]

Equations

$$T=\frac{\partial }{\partial x_1},$$

$$\varphi (0)=0\text{and}{\varphi }^{\star }(T)=\frac{\partial }{\partial x_1}.$$

$$\varphi (0)=0\text{and}{\varphi }^{\star }(T)=\sum _{j,k}{a}_{jk}{x}_j\frac{\partial }{\partial x_k},a_{jk}\in \mathbb{C}.$$

$$T^{\prime }=S^1+N$$

$$[S^1,N]=0.$$

$$T:\mathfrak{g}⟶{\mathcal{X}}_0(E),X⟼T_X\text{such that}[T_X,T_Y]=T_{[X,Y]},$$

$$(\mathcal{D})\mathfrak{g}=\mathfrak{r}\oplus \mathfrak{s}$$

$$T{|}_{\mathfrak{r}}=D^1+N^1+\sum _{k\ge 2}{T}^k{|}_{\mathfrak{r}},$$

$$T_X^k=\sum _{i=1}^n\sum _{|\alpha |=k}{\Lambda }_i^{\alpha }(X)x_1^{{\alpha }_1}\cdots x_n^{{\alpha }_n}\frac{\partial }{\partial x_i},X\in \mathfrak{r},$$

$${\mu }_{\alpha }^i=\sum _{r=1}^n{\alpha }_r{\mu }_r-{\mu }_i,$$

$$T_X=T_X^0+\sum _{k\ge 1}{T}_X^k\in \mathcal{X}(E),X\in \mathfrak{g},$$

$$\mathfrak{g}={\mathfrak{g}}_0+\mathfrak{m},$$

$$T_X^0=0\text{if}X\in {\mathfrak{g}}_0\text{and}{T}_X^0\ne 0\text{if}X\in \mathfrak{m}\setminus \{0\}.$$

$$T_X=\sum _{i=1}^p{a}_i\frac{\partial }{\partial x_i},$$

$$T_X^k=\sum _{i=1}^p\sum _{|\alpha |=k}{\Lambda }_i^{\alpha }(X)y_1^{{\alpha }_1}\cdots y_q^{{\alpha }_q}\frac{\partial }{\partial x_i}+\sum _{i=1}^q\sum _{|\alpha |=k}{\Gamma }_i^{\alpha }(X)y_1^{{\alpha }_1}\cdots y_q^{{\alpha }_q}\frac{\partial }{\partial y_i}$$

$$e_i^{\alpha }\circ {\sigma }_k({\otimes }^{{\beta }_1}{e}_1\cdots {\otimes }^{{\beta }_n}{e}_n)={\delta }_{{\alpha }_1}^{{\beta }_1}\cdots {\delta }_{{\alpha }_n}^{{\beta }_n}{e}_i,$$

$${\sigma }_k(v_1,\dots ,v_k)=\frac{1}{k!}\sum _{\sigma \in {\mathcal{S}}_k}(v_{\sigma (1)},\dots ,v_{\sigma (k)}).$$

$$T=\sum _{k=1}^{\infty }{T}^k,\text{(respectively }T=\sum _{k=0}^{\infty }{T}^k)$$

$$T^k=\sum _{|\alpha |=k}\sum _{i=1}^n{e}_i^{\alpha }.$$

$$\sum _{i=1}^n\sum _{k=1}^{\infty }\sum _{|\alpha |=k}{\Lambda }_i^{\alpha }{x}_1^{{\alpha }_1}\cdots x_n^{{\alpha }_n}\frac{\partial }{\partial x_i}=\sum _{\alpha ,i}{\Lambda }_i^{\alpha }{x}^{\alpha }{\partial }_i.$$

$$\sum _{i=1}^n\sum _{k=1}^{\infty }\sum _{|\alpha |=k}{\Lambda }_i^{\alpha }{e}_i^{\alpha }.$$

$$[e_i^{\alpha },e_j^{\beta }]=e_i^{\alpha }\star e_j^{\beta }-e_j^{\beta }\star e_i^{\alpha }\in \mathrm{L}({\otimes }_s^{|\alpha |+|\beta |-1}E,E),$$

$$\begin{gathered} e_i^{\alpha }\star e_j^{\beta }= \\ {b}_i{e}_j^{\alpha +\beta -1_i},1_i=(0,\dots ,0,1,0,\dots ,0),(1\text{ in }i\text{th place}). \end{gathered}$$

$$\sum _{k\ge 1}{T}^k({\otimes }^{k}v)=T_X(v),v\in E,$$

$$\psi \star T=T^{\prime }\circ \psi$$

$$T^{\prime }\circ \varphi =\sum _{k\ge 1}\sum _{j=1}^k{T}^{\prime j}\circ \left(\sum _{i_1+\cdots +i_j=k}{\varphi }^{i_1}\otimes \cdots \otimes {\psi }^{i_j}\right)\circ {\sigma }_k.$$

$$T:\mathfrak{g}\to \mathcal{X}(E),X\mapsto T_X,$$

$$[T_X,T_Y]=T_{[X,Y]},X,Y\in \mathfrak{g}.$$

$$\varphi \star T_X=T_X^{\prime }\circ \varphi ,X\in \mathfrak{g}.$$

$$\sum _{k\ge 1}{T}_X^k({\otimes }^{k}e)=T_X(v),v\in E$$

Full text

1 Introduction

The study of a vector fields in a neighborhood of a point on a complex manifold is, of course, reduced to that of a vector fields $T$ in a neighborhood of the origin of $E=\mathbb{C}^{n}$.

If $T$ is regular at the origin of $E$, we know that there exists a coordinate system $(x_{1},\dots,x_{n})$ of $E$ in which the vector fields can be expressed as

[TABLE]

that is, there exists a local diffeomorphism $\phi$ of $E$ such that

[TABLE]

If $T$ is analytic then $\phi$ can be chosen to be analytic [9].

If $T$ is singular at the origin of $E$, then the situation is not so simple; generically, $T$ is linearizable, that is, there exists a local diffeomorphism $\phi$ of $E$ such that

[TABLE]

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 $\phi$ of $E$ such that $\phi(0)=0$ and $\phi^{\star}(T)$ is in normal form.

• (b)

This normal form is unique, that is, if $\phi^{\star}_{1}(T)$ and $\phi^{\star}_{2}(T)$ are in normal form then $\phi^{\star}_{1}(T)=\phi^{\star}_{2}(T)$.

This problem has been studied extensively since Poincaré in formal, analytical or $C^{i}nfty$ contexts. The normal form of $T$ is then:

[TABLE]

wher $S^{1}$ is semisimple and $N$ is nilpotent satisfying

[TABLE]

Let us now consider the case, not of a single vector fields, but of a finite dimensional Lie algebra $\mathfrak{g}$ 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 $\mathfrak{g}$ in the sense of Flato, Pinczon and Simon [5]:

[TABLE]

where $\mathcal{X}_{0}(E)$ is the space of vector fields which are singular at 0. In this case, we use the structure of the Lie algebra $\mathfrak{g}$ 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 $\mathfrak{r}$ be the solvable radical of $\mathfrak{g}$, we know that, by the Levi-Malcev decomposition theorem, $\mathfrak{g}$ can be decomposed

[TABLE]

where $\mathfrak{s}$ is semi-simple. The nonlinear representation $T$ of the complex Lie algebra $\mathfrak{g}$ is said to be normal with respect the Levi-Malcev decomposition $(\mathcal{D})$ of $\mathfrak{g}$, if its restriction $T|_{\mathfrak{s}}$ is linear and its restriction $T|_{\mathfrak{r}}$ to $\mathfrak{r}$ is in the following form

[TABLE]

where the linear part $D^{1}+N^{1}$ of $T|_{\mathfrak{r}}$ is such that $D^{1}$ is diagonal with coefficients $\mu_{1},\,\dots,\,\mu_{n}$, (the $\mu_{i}$ are elements of $\mathfrak{r}^{*}$) and $N^{1}$ is strictly upper triangular. Moreover, $T^{k}$ has the following form in the coordinates $x_{i}$ of $E$

[TABLE]

where any coefficient $\Lambda^{\alpha}_{i}$ can be nonzero only if it is resonant, that means, the corresponding linear form $\mu_{\alpha}^{i}\in\mathfrak{r}^{*}$ defined by

[TABLE]

is a root of $\mathfrak{r}$, (eigenvalue of the adjoint representation of $\mathfrak{r}$). The representation $T$ is said to be normalizable with respect $(\mathcal{D})$ if it is equivalent to a normal one with respect $(\mathcal{D})$.

In this paper, we study the situation where the $T_{X}$ are not all in $\mathcal{X}_{0}(E)$. More precisely, we introduce the notion of nonlinear representation $(T,E)$ of $\mathfrak{g}$ in $E$ such that

[TABLE]

where $\mathcal{X}(E)$ is the space of formal vector fields on $E$ (not necessarily vanishing at 0), in particular, $T^{0}_{X}\in E$. This situation seems to be more delicate in its generality, but here we treat the case where the Lie algebra $\mathfrak{g}$ can be written

[TABLE]

where $\mathfrak{m}$ is an abelian ideal of $\mathfrak{g}$ and $\mathfrak{g}_{0}$ is a subalgebra of $\mathfrak{g}$. The Lie algebras of groups of symmetries of simple physical systems are often of this type (Galilée group, Poincaré group, …). Moreover, we assume that

[TABLE]

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 $(x_{1},\dots,x_{p},y_{1},\dots,y_{q})$ of $E$ such that:

• (a)

For all $X\in\mathfrak{m}$, $T_{X}$ can be expressed as

[TABLE]

where $a_{i}\in\mathfrak{m}^{*}$.

• (b)

$T|_{\mathfrak{g}_{0}}$ is normal au sense de [2], moreover, for all $X\in\mathfrak{g}_{0}$ and $k\geq 2$, $T^{k}_{X}$ has the following form

[TABLE]

2 Notations and definitions

Let $E$ be a complex vector space with dimension $n$. The space of symmetric $k$-linear applications from $E\times\cdots\times E$ to $E$ is identified with the space $\mathrm{L}(\otimes_{s}^{k}E,E)$ of linear maps from $\otimes_{s}^{k}E$ to $E$, where $\otimes_{s}^{k}E$ is the space of symmetric $k$-tensors on $E$. Denote by $\mathrm{L}(E)$ the space $\mathrm{L}(E,E)$. Let $(e_{1},\dots,e_{n})$ be a basis of $E$. For $\alpha\in\mathbb{N}^{n}$ such that $|\alpha|=k$ and $i\in\{1,\dots,n\}$ we define $e_{\alpha}^{i}\in\mathrm{L}(\otimes_{s}^{k}E,E)$ by:

[TABLE]

where $\delta$ is the Kronecker symbol, and $\sigma_{k}$ is the symmetrization operator from $\otimes^{k}E$ to $\otimes^{k}_{s}E$ defined by

[TABLE]

Obviously, $(e^{\alpha}_{i})$ is a basis of $\mathrm{L}(\otimes_{s}^{k}E,E)$. Let $\mathcal{X}(E)$ (respectively $\mathcal{X}_{0}(E)$) be the set of formal power series (or formal vector fields)

[TABLE]

where $T^{k}\in\mathrm{L}(\otimes_{s}^{k}E,E)$ (with $\mathrm{L}(\otimes_{s}^{0}E,E)=E$). Therefore, we can write

[TABLE]

Of course, any analytical vector fields $X$ on $E$ which is singular in 0 admits a Taylor expansion:

[TABLE]

It can be identified with the formal vector fields:

[TABLE]

If $|\alpha|=0$ we agree that $e^{\alpha}_{i}=e_{i}$.

Of course, $\mathcal{X}(E)$ can be endowed with a Lie algebra structure defined, for $e^{\alpha}_{i}\in\mathrm{L}(\otimes^{|\alpha|}_{s}E,E)$ and $e^{\beta}_{j}\in\mathrm{L}(\otimes^{|\beta|}_{s}E,E)$, by

[TABLE]

where

[TABLE]

Definition 2.1

A formal vector fields $T$ in $\mathcal{X}(E)$ is said to be analytic al if the power series

[TABLE]

converges in a neighborhood of the origin of $E$.

Definition 2.2

Two formal vector fields $T$ and $T^{\prime}$ are said to be equivalent if there exists an element $\phi=\sum_{k\geq 1}\phi^{k}$ of $\mathcal{X}_{0}(E)$ such that $\phi^{1}$ is invertible and

[TABLE]

where

[TABLE]

$T$* and $T^{\prime}$ are said to be analytically equivalent if $\phi$ is analytic.*

Remark 2.3

For the composition law $\circ$, the map $\phi$ is invertible if and only if $\phi^{1}$ is an automorphism of $E$.

Definition 2.4

Let $\mathfrak{g}$ be a Lie algebra with finite dimension over $\mathbb{C}$. A nonlinear (formal) representation $(T,E)$ of $\mathfrak{g}$ in $E$ is a linear map:

[TABLE]

such that

[TABLE]

Definition 2.5

We say that two representations $(T,E)$ and $(T^{\prime},E)$ are equivalent if there exists $\phi\in\mathcal{X}_{0}(E)$ such that $\phi_{1}$ is invertible and

[TABLE]

Definition 2.6

A nonlinear formal representation $T$ of $\mathfrak{g}$ is said to be analytic if the power series

[TABLE]

converges in a neighborhood of 0 in $E$, for all $X\in\mathfrak{g}$.

In the following, we consider a nonlinear analytic representation $(T,E)$ of a complex finite dimensional Lie algebra $\mathfrak{g}$ in $E=\mathbb{C}^{n}$, such that

[TABLE]

where $\mathfrak{m}$ is an abelian ideal of $\mathfrak{g}$ and $\mathfrak{g}_{0}$ is a subalgebra of $\mathfrak{g}$. Moreover, we assume that

[TABLE]

and then we normalize $T$ step by step.

3 Normalization of $T|_{\mathfrak{m}}$

Proposition 3.1

There exists a coordinate system $(x_{1},\dots,x_{p},y_{1},\dots,y_{q})$, $p+q=n$, of $E$ such that, for all $X\in\mathfrak{m}$, $T_{X}$ can be expressed as

[TABLE]

where $a_{i}\in\mathfrak{m}^{*}$.

Proof. Since the representation $(T,E)$ is analytic, then, there exists a neighborhood $\mathcal{U}$ of the origin of $E$ such that

[TABLE]

for any $v$ in $\mathcal{U}$ and $X$ in $\mathfrak{g}$. So, for any $v$ in $\mathcal{U}$, we consider the subspace $E_{v}$ spanned by the vectors $T_{X}(v)$, where $X$ browses $\mathfrak{m}$. If $(X_{1},\dots,X_{p})$ is a basis of $\mathfrak{m}$, then the $T_{X_{i}}(v)$ are generators of $E_{v}$, thus, $\mathrm{dim}E_{v}\leq p$. But, the system $(T_{X_{1}}(0),\dots,T_{X_{p}}(0))$ is independent, indeed, the condition

[TABLE]

implies that $\alpha_{1}{X_{1}}(0)+\cdots+\alpha_{p}{X_{p}}=0$ since $T^{0}$ does not vanish on $\mathfrak{m}\setminus\{0\}$. Thus, $\mathrm{dim}E_{0}=p$. The determinant map is a continuous map, then there exists a neighborhood $\mathcal{V}\subset\mathcal{U}$ such that $\mathrm{dim}E_{v}\geq p$, for all $v\in\mathcal{V}$. Therefore, $\mathrm{dim}E_{v}=p$, for all $v\in\mathcal{V}$.

Thus, the map

[TABLE]

is an involutive integrable distribution, therefore, the Frobinius theorem ensures the existence of an analytic coordinate system $(x_{1},\dots,x_{p},y_{1},\dots,y_{q})$ of $E$ such that the $\frac{\partial}{\partial x_{i}}$, $i=1,\dots,p$, generate this distribution.

$\Box$

4 Normalization of $T|_{\mathfrak{g}_{0}}$

Now, we begin the second step to normalize the representation $(T,E)$. We know that, for any $X$ in $\mathfrak{g}_{0}$, we have

[TABLE]

We consider the coordinate system $(x_{1},\dots,x_{p},y_{1},\dots,y_{q})$ of $E$ defined in the previous section and we prove the following results

Proposition 4.1

For any $X$ in $\mathfrak{g}_{0}$, $T^{1}_{X}$ has the following form

[TABLE]

and for $k\geq 2$, $T^{k}_{X}$ has the following form

[TABLE]

Proof. Consider $(X_{1},\dots,X_{p})\in\mathfrak{m}^{p}$ such that

[TABLE]

Since $\mathfrak{m}$ is an ideal of $\mathfrak{m}$, then $[\mathfrak{m},\mathfrak{g}_{0}]\subset\mathfrak{m}$, and therefore, for any $X$ in $\mathfrak{g}_{0}$ and $j=1,\dots,p$, we have

[TABLE]

for some $\alpha_{i}\in\mathbb{C}$. In particular, for $k\geq 2$, we have

[TABLE]

and then the result follows.

$\Box$

Now, for any $X$ in $\mathfrak{g}_{0}$, we define

[TABLE]

and we consider the subspaces $E_{1}$ and $E_{2}$ corresponding, respectively, to coordinate systems $(x_{1},\dots,x_{p})$ and $(y_{1},\dots,y_{q})$.

Proposition 4.2

i) $(H,E_{1})$ and $(K,E_{2})$ are two linear representations of $\mathfrak{g}_{0}$.

ii) For any $X$ in $\mathfrak{g}_{0}$ we have

[TABLE]

Thus, $(B,E)$ is a nonlinear representation of $\mathfrak{g}_{0}$ in $E$.

Proof. i) Obvious.

ii) For any $X$ in $\mathfrak{g}_{0}$, we have

[TABLE]

then we have

[TABLE]

We easily check that

[TABLE]

$\Box$

Now, we consider a Levi-Malcev decomposition of $\mathfrak{g}_{0}$:

[TABLE]

where $\mathfrak{s}$ is semi-simple and $\mathfrak{r}$ is the solvable radical of $\mathfrak{g}$. We triangularize simultaneously the $H_{X}$ and $K_{X}$, where $X$ browses $\mathfrak{r}$ (Lie theorem [8]). Thus, we define the linear forms: $\mu_{1},\dots,\mu_{p}\in\mathfrak{r}^{*}$ and $\nu_{1},\dots,\nu_{q}\in\mathfrak{r}^{*}$. The $\mu_{1}(X),\dots,\mu_{p}(X),\nu_{1}(X),\dots,\nu_{q}(X)$ are the eigenvalues of $B^{1}(X)$, indeed, $B^{1}_{X}=H_{X}+K_{X}$. They are also the eigenvalues of $T^{1}(X)$, since $A^{1}_{X}\in\mathrm{L}(E_{2},E_{1})$. Thus, the simultaneous triangulation of $H$ and $K$ leads to that of $T^{1}$ by retaining the first $p$ components in $E_{1}$ and the $q$ other in $E_{2}$.

Let us denote by $\lambda_{1},\dots,\lambda_{n}$ the roots of $T^{1}|_{\mathfrak{r}}$, and consider the elements $\lambda^{\alpha}_{j}$ of $\mathfrak{r}^{*}$ defined by

[TABLE]

where $j=1,\dots,n$ and $\alpha\in\mathbb{N}$.

Definition 4.3

We say that $(\alpha,j)$ is resonant if $\lambda^{\alpha}_{j}$ is a root of $\mathfrak{r}$.

Definition 4.4

An element $X_{0}$ in $\mathfrak{r}$ is said to be vector resonance if, for any $\alpha$, for any $i$ and for any $j$,

[TABLE]

where the $v_{i}$ are the roots of $\mathfrak{r}$.

For the two following lemmas see, for instance, [1] and [2].

Lemma 4.5

The set of vector resonance is dense in $\mathfrak{r}$.

Lemma 4.6

There exists a resonance vector in $\mathfrak{r}$ such that $[X_{0},\mathfrak{s}]=0$.

The resonant pairs $(\alpha,j)$ appearing here are of two types:

• (1)

$\sum_{i=1}^{p}\alpha_{i}\nu_{i}-\mu_{j}$ is a root of $\mathfrak{r}$.

• (2)

$\sum_{i=1}^{p}\alpha_{i}\nu_{i}-\nu_{j}$ is a root of $\mathfrak{r}$.

Let us denote by

[TABLE]

and by

[TABLE]

Theorem 4.7

$T|_{\mathfrak{r}}$* is normalizable in the sense of [2], that means, there exists an analytic operator $\phi=\sum\phi^{k}$ in $\mathcal{X}_{0}(E)$, such that $\phi^{1}$ is invertible and, for any $X$ in $\mathfrak{r}$, $\phi\star T_{X}\circ\phi^{-1}$ is in the form:*

[TABLE]

Proof. Let $X_{0}$ be a resonance vector of $\mathfrak{r}$ such that $[X_{0},\mathfrak{s}]=0$. It is well known that $T_{X_{0}}$ is analytically normalizable: there exists an analytic operator $\phi=\sum\phi^{k}$ in $\mathcal{X}_{0}(E)$, such that $\phi^{1}$ is invertible and, for any $X$ in $\mathfrak{r}$,

[TABLE]

where $T^{\prime}_{X_{0}}=S^{1}_{X_{0}}+N_{X_{0}}$ with

[TABLE]

and

[TABLE]

Therefore, $N_{X_{0}}$ has the following form

[TABLE]

where

[TABLE]

The operator $\phi$ normalizes $T|_{\mathfrak{r}}$ (see [2]).

$\Box$

Proposition 4.8

The normalization operator $\phi$ of $T|_{\mathfrak{r}}$ leaves invariant $T|_{\mathfrak{m}}$.

Proof. To prove that $\phi$ leaves invariant $T|_{\mathfrak{m}}$ we will prove that

[TABLE]

where $I$ is the identity of $E$ and $W_{k}\in\mathrm{L}(\otimes^{k}_{s}E,E)$.

(a) $B^{1}_{X_{0}}=H_{X_{0}}+K_{X_{0}}$ being decomposed into a semisimpe part $S^{1}_{X_{0}}$ and a nilpotent part, therefore, to reduce $T^{1}_{X_{0}}$ it suffices to reduce $A^{1}_{X_{0}}$. We decompose $A^{1}_{X_{0}}$:

[TABLE]

where

[TABLE]

and

[TABLE]

We reduce $A^{1}_{X_{0}}$ by removing $A^{1}_{1X_{0}}$. Indeed, for any $X$ and $Y$ in $\mathfrak{r}$,

[TABLE]

therefore, there exists $W^{1}$ in $\mathrm{L}(E)$ and a 1-cocycle $V^{1}$ such that

[TABLE]

In particular, $A^{1}_{X_{0}}=[B^{1}_{X_{0}},W_{1}]+V^{1}$, then we can choose, for instance,

[TABLE]

The stability of the eigenvector subspaces of $\mathrm{ad}~{}S^{1}_{X_{0}}$ by $B^{1}_{X_{0}}$ proves that $W^{1}\in\mathrm{L}(E_{2},E_{1})$, therefore, $(I+W_{1})$ is invertible and

[TABLE]

[TABLE]

$T|_{\mathfrak{m}}$ is then stable by $(i+W_{1})$.

b) Now, we assume that $T_{X_{0}}$ is normalized up to order $k-1$, we write

[TABLE]

and we decompose $T^{k}_{X_{0}}$:

[TABLE]

where

[TABLE]

Therefore, $T^{k}_{0X_{0}}$ has the following form

[TABLE]

We seek an operator $W_{k}$ in $\mathrm{L}(\otimes^{k}_{s}E,E)$ such that

[TABLE]

We can choose

[TABLE]

The eigenvector subspaces of $\mathrm{ad}~{}S^{1}_{X_{0}}$ are stable by $\mathrm{ad}~{}T^{1}_{X_{0}}$, therefore, $W_{k}$ has the following form:

[TABLE]

Thus,

[TABLE]

We proceed by induction, so, we construct

[TABLE]

which normalizes $T|_{\mathfrak{r}}$ and leaves invariant $T|_{\mathfrak{m}}$.

5 Linearization of $T|_{\mathfrak{s}}$

Let us now consider the representation $(T^{\prime},E)$ of $\mathfrak{g}$ in $E$, defined on $\mathfrak{g}$ by

[TABLE]

The representation being $T^{\prime}|_{\mathfrak{r}}$, as in [2], we construct an analytic invertible operator $\psi\in\mathcal{X}_{0}(E)$, which linearizes $T^{\prime}|_{\mathfrak{s}}$ and leaves $T^{\prime}|_{\mathfrak{r}}$ in normal form. This is possible through the choice of $X_{0}$ switching with $\mathfrak{s}$: $[X_{0},\mathfrak{s}]=0$.

On the other hand, by construction, the components of $\psi$ are independent of the coordinates $x_{1},\dots,x_{p}$, then $\psi$ leaves invariant $T^{\prime}|_{\mathfrak{m}}=T|_{\mathfrak{m}}$.

6 Recapitulation

Consider a nonlinear analytic representation $(T,E)$ of a complex finite dimensional Lie algebra $\mathfrak{g}$ in $E=\mathbb{C}^{n}$, such that

[TABLE]

where $\mathfrak{m}$ is a $p$-dimensional abelian ideal of $\mathfrak{g}$ and $\mathfrak{g}_{0}$ is a subalgebra of $\mathfrak{g}$. Moreover, we assume that

[TABLE]

We consider a Levi-Malcev decomposition of $\mathfrak{g}_{0}$:

[TABLE]

where $\mathfrak{s}$ is semi-simple and $\mathfrak{r}$ is the solvable radical of $\mathfrak{g}$. Then, we have

Theorem 6.1

The representation $(T,E)$ of $\mathfrak{g}$ can be normalized with respect the decompositions $(\mathcal{D}_{1})$ and $(\mathcal{D}_{2})$. That means, there exists a coordinate system $(x_{1},\dots,x_{p},y_{1},\dots,y_{q})$ of $E$ such that

• a)

For all $X\in\mathfrak{m}$, $T_{X}$ can be expressed as

[TABLE]

where $a_{i}\in\mathfrak{m}^{*}$.

• b)

$T|_{\mathfrak{s}}$* is a linear representation of $\mathfrak{s}$ in $E$.*

• c)

For any $X$ in $\mathfrak{r}$, $T_{X}$ is in the form:

[TABLE]

where $R_{0}=\{(\alpha,i)~{}|~{}\sum_{j=1}^{q}\alpha_{j}\nu_{j}-\mu_{i}=0\}$ and $R^{\prime}_{0}=\{(\alpha,i)~{}|~{}\sum_{j=1}^{q}\alpha_{j}\nu_{j}-\nu_{i}=0\}$.