Space-Charge Effects

N. Chauvin

Commissariat à  l'Énergie Atomique et aux Énergies Alternatives, Gif-sur-Yvette, France

First, this chapter introduces the expressions for the electric and magnetic space-charge internal fields and forces induced by high-intensity beams. Then, the root-mean-square equation with space charge is derived and discussed. In the third section, the one-dimensional Child–Langmuir law, which gives the maximum current density that can be extracted from an ion source, is exposed. Space-charge compensation can occur  in the low-energy beam transport lines (located after the ion source). This phenomenon, which counteracts the space-charge defocusing effect, is explained and its main parameters are presented. The fifth section presents an overview of the principal methods to perform beam dynamics numerical simulations. An example of a particles-in-cells code, SolMaxP, which takes into account space-charge compensation, is given. Finally, beam dynamics simulation results obtained with this code in the case of the IFMIF injector are presented.

§ SPACE CHARGE: BEAM SELF-GENERATED FIELDS AND FORCES

§.§ Space charge

Consider two particles of identical charge $q$. If they are at rest, the Coulomb force exerts a repulsion (see fig:SC_Forces(a).
Now, if they are travelling with a velocity $v=\beta c$, they represent two parallel currents $I = q v$, which attract each other as a result of the effect of their magnetic fields (see fig:SC_Forces(b)).

(a)     (b)

(a) Coulomb force exerts a repulsion between stationary charges. (b) There is attraction between moving charges.

Now, consider a test particle of charge $q$ in an unbunched beam of particles (charge $q_i$) with a circular cross-section (see fig:SC_Beam).
The Coulomb repulsion pushes the test particle outwards. The induced force is zero in the beam centre and increases towards the edge of the beam (fig:SC_Beam(a)). The magnetic force is radial and attractive for the test particle in a travelling beam, represented as parallel currents (fig:SC_Beam(b)).

(a)     (b)

(a) Charges: repulsion. (b) Parallel currents: attraction

§.§ Space-charge fields

Consider a continuous beam with a cylindrical symmetry distribution that moves with a constant velocity $v=\beta c$.
The beam charge density is

\begin{equation}
\rho(x,y,z) = \rho(r) ,
\end{equation}

with $r=\sqrt{(x^2+y^2)}$.
For symmetry reasons, the electric field has only a radial component $E_r$. Using the integral form of Gauss's law over a cylinder centred on the beam axis gives

\begin{equation}\label{eq:Er}
E_r(r) = \frac{1}{\epsilon_0 r}\int_0^r \rho(r') r' \, \rmd r'
\end{equation}

The beam current density is

\begin{equation}
\mathbf{J}(x,y,z) = J(r) \mathbf{u}_z ,
\end{equation}

where $\mathbf{u}_z$ is the unitary vector of the beam propagation.
If the particles of the beam have the same longitudinal speed, $\mathbf{v}_z=\beta_z c \, \mathbf{u}_z$, we have

\begin{equation}
\mathbf{J}(r) = \rho(r) \beta_z c \, \mathbf{u}_z .
\end{equation}

For symmetry reasons, the magnetic field has only an azimuthal component $B_\theta$. Using the integral form of Ampere's law over a cylinder centred on the beam axis gives

\begin{align}\label{eq:Btheta}
B_\theta(r) & = \frac{\mu_0}{r}  \int_0^r J(r') r' \, \rmd r' \nonumber \\
& = \frac{\mu_0 \beta_z c}{r}  \int_0^r \rho(r') r' \, \rmd r' .
\end{align}

From Eqs. (<ref>) and (<ref>), we obtain (as $c={1}/{\sqrt{\epsilon_0 \mu_0}}$ )

\begin{equation}\label{eq:BthetaEr}
B_\theta(r) = \frac{\beta_z}{c} E_r(r) .
\end{equation}

§.§ Space-charge forces

The space-charge fields exert a force $\mathbf{F}$ on a test particle at radius $r$ that can be expressed by

\begin{equation}
\mathbf{F} = q ( \mathbf{E} + \mathbf{v} \times \mathbf{B} ) ,
\end{equation}

which with our geometry can be written as

\begin{equation}\label{eq:Fr_Init}
F_r  =  q ( E_r + \beta_z c B_\theta ) .
\end{equation}

If the particle trajectories obey the paraxial assumption, one can write

\begin{equation}
\beta^2 = \beta_x^2 + \beta_y^2 + \beta_z^2 \approx \beta_z^2 .
\end{equation}

From Eqs. (<ref>) and (<ref>), it follows finally that

\begin{equation} \label{eq:SC_Force}
F_r  =  q E_r ( 1 - \beta^2 ) = \frac{qE_r}{\gamma^2} .
\end{equation}

The following should be noted about this derivation:

* In Eq. (<ref>), the $1$ represents the electric force and the $\beta^2$ the magnetic force.

* The electric force is defocusing for the beam; the magnetic force is focusing.

* The ratio of magnetic to electric forces, $-\beta^2$, is independent of the beam density distribution.

* For relativistic particles, the beam magnetic force almost balances the electric force.

* For non-relativistic particles (like low-energy ion beams), the space magnetic force is negligible: in the ion source extraction region and in the low-energy beam lines, the space charge has a defocusing effect.

§.§.§ Uniform beam density

A uniform beam density is expressed by

\begin{equation}
\rho(r) =
\begin{cases} \rho_0  & \text{if\ $r \leq r_0$,} \\
0 &\text{if\ $r > r_0$.}
\end{cases}
\end{equation}

The charge per unit length is

\begin{equation}
\lambda = \rho_0 \pi r_0^2.
\end{equation}

The total beam current can be expressed by

\begin{equation} \label{eq:BeamCurrent}
I = \beta c \int_0^{r_0} 2 \pi \rho(r') r' \, \rmd r'
\end{equation}

So, we obtain

\begin{equation} \label{eq:CurrentDensity}
\rho_0 = \frac{I}{\beta c \pi r_0^2}.
\end{equation}

From Eqs. (<ref>) and (<ref>):

E_r(r)     = I/ 2πϵ_0 βc r_0^2   r

E_r(r)     = I/ 2πϵ_0 βc r

The following should be noted:

* The field is linear inside the beam.

* Outside of the beam, it varies according to $1/r$.

Similarly, from Eqs. (<ref>) and (<ref>):

B_θ(r)     = μ_0I/ 2πr_0^2   r

B_θ(r)     = μ_0 I/ 2πr

§.§.§ Space-charge forces – Gaussian beam density

A Gaussian  beam density is expressed by

\begin{equation}
\rho(r) = \rho_{0 {\rm g}} \exp\left( \frac{-r^2}{2\sigma_r^2} \right) .
\end{equation}

The charge per unit length is

\begin{equation}
\lambda = 2 \rho_{0{\rm g}} \pi \sigma_r^2 .
\end{equation}

The space-charge electric field is

\begin{equation}
E_r(r) = \frac{\rho_{0{\rm g}}}{\epsilon_0} \frac{\sigma_r^2}{r}
\left[ 1- \exp\left( \frac{-r^2}{2\sigma_r^2} \right) \right].
\label{eq:Er_Gauss}
\end{equation}

The following should be noted:

* The field is nonlinear inside the beam.

* Far from the beam (several $\sigma_r$), it varies according to $1/r$.

§.§ Space-charge expansion in a drift region

Consider a particle (charge $q$, mass $m_0$) beam of current $I$, propagating at speed $v=\beta c$ in a drift region, with the following hypotheses:

* The beam has cylindrical symmetry and a radius $r_0$.

* The beam is paraxial ($\beta_r \ll \beta_z$).

* The beam has an emittance equal to 0.

* The beam density is uniform.

Newton's second law for the transverse motion of the beam particles gives

\begin{equation} \label{eq:Trans_motion}
\frac{\rmd(m_0 \gamma \beta_r c)}{\rmd t}
= m_0 \gamma \frac{\rmd^2r}{\rmd t^2} = qE_r(r) - q \beta c B_\theta(r).
\end{equation}

Using Eqs. (<ref>) for $E_r$ and (<ref>) for $B_\theta$ in Eq. (<ref>) gives

\begin{equation}
\label{eq:Trans_motion2}
m_0 \gamma \frac{\rmd^2r}{\rmd t^2}
= \frac{qIr}{2\pi \epsilon_0 r_0^2 \beta c}(1-\beta^2),
\end{equation}

\begin{equation} \label{eq:dtds}
\frac{\rmd^2r}{\rmd t^2} = \beta^2 c^2 \frac{\rmd^2r}{\rmd z^2}.
\end{equation}

Equation (<ref>) becomes

\begin{equation}\label{eq:Trans_Motion_Final}
\frac{\rmd^2r}{\rmd z^2} = \frac{qIr}{2\pi \epsilon_0 r_0^2 m_0 c^3 \beta^3 \gamma^3}.
\end{equation}

We will now introduce several parameters used in the literature on beams with space charge. The characteristic current $I_0$ is defined by

\begin{equation}
I_0 =\frac{4\pi\epsilon_0 m_0 c^3}{q}.
\end{equation}

The generalized perveance $K$, a dimensionless parameter, is defined by

\begin{equation}\label{eq:Perveance}
K =\frac{qI}{2\pi\epsilon_0 m_0 c^3\beta^3\gamma^3}.
\end{equation}

The perveance refers to the magnitude of space-charge effects in a beam. Figure <ref> compares the generalized perveance, as a function of the beam energy, for various high-intensity accelerators.

Beam perveance for different hadron accelerators as a function of their beam energy

Then, Eq. (<ref>) for the particle trajectories can be reduced to the form

\begin{equation}
\frac{\rmd^2r}{\rmd z^2} = \frac{K}{r_0^2}r.
\end{equation}

In the case of a laminar beam, the trajectories of all particles are similar and, in particular, the particle at $r=r_0$ will always stay at the beam boundary. Considering $r = r_0 = r_{\rm env} $, the equation of the beam radius in a drift space can be written as

\begin{equation}
\frac{\rmd^2r_{\rm env}}{\rmd z^2} = \frac{K}{r_{\rm env}}.
\end{equation}

§ ROOT-MEAN-SQUARE ENVELOPE EQUATION WITH SPACE CHARGE

§.§ Definition of root-mean-square quantities

In the following, $\mean{A}$ represents the mean of the quantity  $A$ over the beam particle distribution:

\begin{align}
&\text{r.m.s.\ size}
&&{\hspace{-25mm}} \tilde{x}  = \sqrt{\mean{x^2} - \mean{x}^2} , \label{eq:xrms} \\
&\text{r.m.s.\ divergence}
&&{\hspace{-25mm}} \widetilde{x'}  = \sqrt{\mean{x'^2} - \mean{x'}^2} , \label{eq:xprms}\\
&\text{r.m.s.\ emittance}
&&{\hspace{-25mm}} \widetilde{\epsilon_x} = \sqrt{\tilde{x}^2  \widetilde{x'}^2  - \mean {(x - \mean{x}) (x' - \mean{x'}) }^2 } . \label{eq:Emitrms}
\end{align}

The beam Twiss parameters can be expressed with the r.m.s. values as

\begin{align}
\beta_x     & = \frac{ \tilde{x}^2}{\widetilde{\epsilon_x}} , \\
\gamma_x    & = \frac{ \widetilde{x'}^2 }{\widetilde{\epsilon_x}}, \\
\alpha_x    & = -\frac{ \mean {(x - \mean{x}) (x' - \mean{x'})} }{ \widetilde{\epsilon_x}}.
\end{align}

§.§ Root-mean-square emittance and nonlinear forces

Consider an ideal particle distribution in phase space $(x,x')$ that lies on a line passing through the origin as represented on  fig:RMS_PhaseSpace. Assume that, for any particles, the divergence $x'$ is related to the position $x$ by the expression

\begin{equation}
x' = C x^n , \label{eq:Div_C_Pos_n}
\end{equation}

where $n$ is positive and $C$ is a constant. Considering the squared r.m.s. emittance (from eq:Emitrms)

\begin{equation}
\widetilde{\epsilon_x}^2  =  \mean{x^2}  \mean{x'^2}  - \mean{x x'}^2 ,
\end{equation}

and using eq:Div_C_Pos_n we obtain

\begin{equation}
\widetilde{\epsilon_x}^2  = C^2 ( \mean{x^2}  \mean{x^{2n}}  - \mean{x^{n+1}}^2 ).
\end{equation}

When $n=1$, the line is straight (fig:RMS_PhaseSpace(a)) and the r.m.s. emittance is $\widetilde{\epsilon_x}^2=0$. When $n\neq1$, the relationship is nonlinear, the line of the phase space is curved (fig:RMS_PhaseSpace(b)), and the r.m.s. emittance is in general not zero. It is interesting to note that, even if the space phase area occupied by the beam distribution is zero, the r.m.s. emittance may be not zero if the distribution lies on a curved line.

(a)     (b)

Root-mean-square emittance for distributions with zero areas: (a) $\varepsilon = 0$; and (b) $\varepsilon \neq 0$

§.§ Root-mean-square envelope equation

Consider a beam moving in the direction $s$, where individual particles satisfy the equation of motion

\begin{equation}
x'' + \kappa(s)x -F_{\rm s} = 0 ,
\end{equation}

where $\kappa(s)x$ represents a linear external focusing force (for instance quadrupole $\kappa = qB/\gamma m a \beta c$) and $F_{\rm s}$ is a space-charge force term (in general not linear).

To simplify the situation, assume that the beam is centred on the axis with no divergence, so that $\mean{x}=0$ and $\mean{x'}=0$ (i.e. $\tilde{x}^2=\mean{x^2} \equiv \overline{x^2}$ ).
The equations of motion for the second moments of the distribution can then be written as

\begin{equation}\label{eq:dx2}
\frac{\rmd\overline{x^2}}{\rmd s} = 2\overline{xx'}
\end{equation}

\begin{equation}\label{eq:dxxp}
\begin{split}
\frac{\rmd\overline{xx'}}{\rmd s} & = \overline{x'^2} + \overline{xx''} \\
& = \overline{x'^2} - \kappa(s)\overline{x^2} - \overline{xF_{\rm s}}.
\end{split}
\end{equation}

Differentiating eq:xrms twice and  using eq:dx2 gives

\begin{equation}\label{eq:xpp}
\tilde{x}'' = \frac{\overline{x''} + \overline{x^2}}{\tilde{x}}-\frac{\overline{xx'}}{\tilde{x}^3}.
\end{equation}

Using eq:dxxp and (<ref>), we have finally that

\begin{equation}\label{eq:RMS_Motion_Final}
\tilde{x}'' + \kappa(s)\tilde{x} -  \frac{\widetilde{\epsilon_x}^2}{\tilde{x}^3} - \frac{\overline{xF_{\rm s}} }{\tilde{x}} =0.
\end{equation}

[b]eq:RMS_Motion_Final is the r.m.s. envelope equation and expresses the motion of the r.m.s. beam size. In three dimensions, one can write three envelope equations that are coupled through the space charge and can depend on the beam dimension in the three directions. These equations can be useful to find an analytic solution for simple problems. Furthermore, they are very important in the space-charge community.

§.§ Example: elliptical continuous beam of uniform density

An elliptical uniform beam density is expressed by

\begin{equation}
\rho(r) =
\begin{cases} \rho_0
& \text{if\ $ {x^2}/{r_x^2} + {y^2}/{r_y^2} < 1 $,} \\
0 &\text{otherwise.}
\end{cases}
\end{equation}

As the distribution is uniform, the semi-axes of the ellipse, $r_x$ and $r_y$, are related to the r.m.s. beam sizes: $r_x=2\tilde{x}$ and $r_y=2\tilde{y}$. So, the r.m.s. envelope equations are given by

\begin{align}
\tilde{x}'' + \kappa_x(s)\tilde{x} -  \frac{ \widetilde{\epsilon_x}^2}{\tilde{x}^3} - \frac{K }{2(\tilde{x} + \tilde{y})} = 0 , \label{eq:RMS_EnvX}\\
\tilde{y}'' + \kappa_y(s)\tilde{y} -  \frac{ \widetilde{\epsilon_y}^2}{\tilde{y}^3} - \frac{K }{2(\tilde{x} + \tilde{y})} = 0 . \label{eq:RMS_EnvY}
\end{align}

The second term of eq:RMS_EnvX and (<ref>) is a focusing term. The third term is the emittance term, which is defocusing (negative sign). The fourth term is the space-charge term, which is also defocusing. It can be seen that the two planes are coupled through the space-charge term of the r.m.s. envelope equation.

§.§ Linear space-charge force and equivalent beam

The space-charge force is linear only if the beam density is uniform, which is very unlikely in the case of  practical beams. So the space-charge force is fundamentally nonlinear. Nevertheless, it was shown by Sacherer  [16] that, for ellipsoidal bunches, where the r.m.s. emittance is either constant or specified in advance, the evolution of the r.m.s. beam projection is nearly independent of the beam density. This means that for calculation of the r.m.s. dynamics, the actual distribution can be replaced by an equivalent uniform beam that has the same intensity and the same r.m.s. second moments.

§ THE CHILD–LANGMUIR LAW

The Child law states the maximum current density that can be carried by charged-particle flow across a one-dimensional extraction gap. The limit arises from the longitudinal electric fields of the beam space charge. It is a very important result in beam physics of collective effects and also in ion source fields where the extracted beam current is one of the most important parameters of an ion source.

In this section, the Child–Langmuir law for a one-dimensional extraction gap will be derived. It is enough to emphasize the fundamental physical phenomena that limit the beam current extracted from a particle source. However, for a practical extraction system, it is necessary to perform three-dimensional calculations by including form factors that depend on the geometry of the extraction electrodes.

§.§ Extraction gap

The extraction gap is the first stage of an accelerator: low-energy charged particles from a source are accelerated to moderate energy ($\sim$10keV to $\sim$1MeV) and formed into a beam. The Child law calculation applies to the one-dimensional gap of fig:SC_ChildLangmuir. A voltage $-V_0$ is applied across the vacuum gap of width $d$. Charged particles with low kinetic energy enter at the grounded boundary. The particles have rest mass $m_0$ and carry positive charge $q$.  Particles leave the right-hand boundary with kinetic energy $qV_0$. We shall assume that the exit electrode is an ideal mesh that defines an equipotential surface while transmitting all particles.

Geometry of an infinite planar extraction gap

§.§ One-dimensional Child law for non-relativistic particles

To simplify the calculation, the following assumptions are made:

* Particle motion is non-relativistic ($qV_0 \ll m_0c^2$).

* The source on the left-hand boundary can supply an unlimited flux of particles. Flow restriction would result only from space-charge effects.

* The transverse dimension of the gap is large compared with $d$.

* The transverse magnetic force generated by particle current across the gap is negligible compared with the axial electric force. Consequently, the particle trajectories are straight across the gap. This assumption is valid for an ion beam, but can be violated in the case of a high-current relativistic electron gun.

* Particles flow continuously; the electric field and particle density at all $z$ are constant.

The steady-state condition means that the space-charge density, $\rho(z)$, is constant in time

\begin{equation}
\frac{\partial\rho(z)}{\partial t} = 0.
\end{equation}

This equation implies that the current density, $J_0$, is the same at all positions in the gap. The charge density can be expressed by

\begin{equation}\label{eq:PartDensity}
\rho(z) = \frac{J_0}{v_z(z)}.
\end{equation}

According to the above assumptions, the particle velocity in the gap is

\begin{equation}\label{eq:PartSpeed}
v_z^2(z) = \frac{2q\phi(z)}{m_0}.
\end{equation}

Using eq:PartDensity and (<ref>) we obtain

\begin{equation}\label{eq:rho_z}
\rho(z) = \frac{J_0}{\sqrt{2q\phi(z)/m_0}}.
\end{equation}

Now, consider the one-dimensional Poisson equation:

\begin{equation}\label{eq:PoissonEq}
\nabla^2 \phi = \frac{\rmd^2\phi(z)}{\rmd z^2} = -\frac{\rho(z)}{\epsilon_0}.
\end{equation}

Substituting eq:rho_z into eq:PoissonEq gives

\begin{equation}\label{eq:d2Phiddz2}
\frac{\rmd^2\phi(z)}{\rmd z^2} = -\frac{J_0}{\epsilon_0\sqrt{2q/m_0}}\frac{1}{\phi^{1/2}}.
\end{equation}

If we introduce the dimensionless variables $\zeta = z/d$ and $\Phi=-\phi/V_0$, eq:d2Phiddz2 becomes

\begin{equation}\label{eq:Phid2}
\frac{\rmd^2\Phi(z)}{\rmd \zeta^2} = -\frac{\alpha}{\phi^{1/2}},
\end{equation}

\begin{equation}\label{eq:alphChild}
\alpha =\frac{J_0 d^2}{\epsilon_0 V_0\sqrt{2qV_0/m_0}}.
\end{equation}

Three boundary conditions are needed to integrate eq:Phid2: $\Phi(0) = 0$, $\Phi(1) = 1$ and $\rmd\Phi(0)/\rmd\zeta =0$.
Multiplying both sides of eq:Phid2 by $\Phi' = \rmd\Phi/\rmd\zeta$, we can integrate and obtain

\begin{equation}\label{eq:Phip2}
(\Phi')^2 = 4\alpha \sqrt{\Phi(\zeta)}.
\end{equation}

A second integration gives

\begin{equation}\label{eq:Phidezeta}
\Phi^{3/4} = \tfrac34 \sqrt{4\alpha}\zeta,
\end{equation}

or, coming back to the initial variables,

\begin{equation}\label{eq:Potential_with_SC}
\phi(z) = -V_0 \left(\frac{z}{d}\right)^{4/3}.
\end{equation}

In eq:Phidezeta, the condition $\Phi(1) = 1$ implies $\alpha = \frac{4}{9}$. Substituting in eq:alphChild gives

\begin{equation}\label{eq:Child_Law}
J_0 = \frac{4}{9}\epsilon_0\left(\frac{2q}{m_0}\right)^{1/2}\frac{V_0^{3/2}}{d^2}.
\end{equation}

This is the Child–Langmuir law, which represents the maximum current density that can be achieved in the diode by increasing the ion supply by the anode. This limitation is only due to space charge. The only way to increase the extracted current is to increase the electric field in the gap (i.e. increase the gap voltage or decrease the cathode–anode spacing).  In a real ion source, the extracting electric field cannot be increased infinitely, as electrodes break down at fields exceeding a value around 10MVm$^{-1}$.

For a given gap voltage and geometry, the current density is proportional to the square root of the charge-to-mass ratio of the particles, $\sqrt{q/m_0}$.

Let us compare the electrostatic potential for a space-charge-limited flow given by eq:Potential_with_SC, with the potential in the same gap without space charge. In that case, the potential between the anode and the cathode of the extraction gap (see fig:SC_ChildLangmuir) can be calculated from Laplace's equation (as there is no space charge, $\rho = 0$),

\begin{equation}
\nabla^2 \phi = \frac{\rmd^2\phi(z)}{\rmd z^2} = 0,
\end{equation}

with the solution

\begin{equation}\label{eq:Potential_wo_SC}
\phi(z) = -\frac{V_0}{d}\,z.
\end{equation}

By comparing eq:Potential_with_SC with eq:Potential_wo_SC, it can be seen that the space charge of the extracted particle lowers the potential (in absolute value) at any given point between the two electrodes of the planar diode.

§.§ Child–Langmuir law, orders of magnitudes

The Child–Langmuir current for non-relativistic electrons is

\begin{equation}
J_0 = 2.33\times10^{-6} \, \frac{V_0^{3/2}}{d^2}\qquad \mathrm{(A{\u}m^{-2})},
\end{equation}

where $V_0$ is in volts and $d$ in metres. For ions, eq:Child_Law becomes

\begin{equation}
J_0 = 5.44\times10^{-8} \, \sqrt{\frac{Z}{A}} \frac{V_0^{3/2}}{d^2}  \qquad \mathrm{(A{\u}m^{-2})},
\end{equation}

where $Z$ is the charge state of the ion and $A$ is its atomic number.
For a given extraction voltage and geometry, the possible  current density of electrons is around 43 times higher than that of protons.

Normalized space-charge-limited current densities for different charged species

[b]fig:SC_ChildLangmuirplot shows a plot the variation of $J_0d^2$ (in amperes) as a function of $V_0$ for electrons and some singly charged ions. The values for electrons are plotted only up to 100kV, as relativistic corrections should be introduced for higher voltages.

§ SPACE-CHARGE COMPENSATION AND BEAM TRANSPORT IN LOW-ENERGY BEAM TRANSPORT LINES

§.§ Space-charge compensation

§.§.§ Space-charge compensation principle

Space-charge compensation (SCC), or space-charge neutralization, occurs when a beam
is propagating through the residual gas  of the low-energy beam transport (LEBT) line (or some additional gas)
and subsequently induces ionization of the molecules of this gas (Fig. <ref>).
The secondary particles produced by ionization (i.e. electrons or ions),
which have an opposite polarity to the particles of the beam, are trapped by the beam potential until a steady
state is reached. Thus, the low-energy beam can be considered as a
plasma that creates a focusing effect that counteracts the space-charge effect.

Space-charge compensation process

As an example, consider a proton beam propagating through a residual gas of H$_2$. It induces the production of e$^-$/H$_2^+$ pairs by ionization according to the following reaction:

\begin{equation} \label{eq:Ionisation}
\mathrm{  p + H_2 \rightarrow p + e^- + H_2^+ }
\end{equation}

The created electron will be trapped by the beam potential and will contribute to reducing it. The created H$_2^+$ ion will be expelled from the proton beam.

In this section, basic expressions for the degree of space-charge compensation
and characteristic time are given to obtain some orders of magnitude for design
and experimental considerations. More elaborate descriptions of the SCC evolution can be found in detailed analytical (see [17, 4]) and numerical (see [4, 22]) works.

§.§.§ Degree of space-charge compensation

Now, consider a uniform cylindrical beam of radius $r_{\rm B}$ with an intensity $I_{\rm B}$ and a longitudinal speed $v=\beta_{\rm B} c$ propagating into a cylindrical beam pipe of radius $r_{\rm P}$, which is assumed to be grounded (see fig:SCC_BeamPotential).

Uniform cylindrical beam propagating through a beam pipe

If we assume that there is no space-charge compensation, the space-charge electric fields inside and outside the beam are given by eq:Er_Uniform_In and (<ref>)

E_r(r)     = I_B r/ 2πϵ_0 βc  r_0^2

E_r(r)     = I_B/ 2πϵ_0 βc r

By integrating these equations with the boundary condition, $\phi(r_{\rm P}) = 0$, we obtain

ϕ(r)     = I_B/ 4πϵ_0 β_B c [ 1 + 2ln(r_P/r_B) - r/r_B^2 ]

ϕ(r)     = I_B/ 2πϵ_0 β_B c  lnr_P/r

The potential on the beam axis (i.e. potential well) created by a uniform beam, without SCC, is given by  eq:Pot_Uniform_In for $r=0$:

\begin{equation}
\phi_{0}=\frac{I_{B}}{4\pi\varepsilon_{0}\beta_{B}c}
\left[1+2\ln\left(\frac{r_{P}}{r_{B}}\right)\right]
\label{eq:Phi0}
\end{equation}

During the SCC process, the neutralizing particles created by gas ionization
are trapped by this potential well. [b]eq:Phi0 shows that the potential well (i.e. the space-charge force) increases if the radius of the beam decreases.
So, achieving a beam waist in a LEBT could be critical for the quality of the beam.

Now, consider a compensated beam at steady state.  If we define by $\phi_{\rm c}$ its potential on axis, the SCC degree is given by

\begin{equation}
\eta=1-\frac{\phi_{\rm c}}{\phi_{0}} . \label{eq:SCCD}
\end{equation}

The beam potential well of the compensated beam can be experimentally
measured. The values found for the 75keV, 130mA proton beam
of the LEDA range from 95% to 99% [9]. Along the
LEBT, the SCC degree is not constant, as the neutralizing particle
trajectories can be modified by external fields of the focusing element,
for example. This phenomenon induces strongly non-uniform space-charge forces that can lead to beam emittance growth

§.§.§ Space-charge compensation time

The characteristic SCC transient time, $\tau_{\rm scc}$, can be determined
by considering the time it takes for a particle of the beam to produce
a neutralizing particle on the residual gas. It is given by

\begin{equation}
\tau_{\rm scc}=\frac{1}{\sigma_{\rm ioniz}n_{\rm g}\beta_{\rm B}c} , \label{eq:SCCTime}
\end{equation}

where $\sigma_{\rm ioniz}$ is the ionization cross-section
of the incoming particles on the residual gas and $n_{\rm g}$ is the gas
density in the beam line. The space-charge compensation is expected to reach a steady state after two or three $\tau_{\rm scc}$ times.
As an example, the SCC transient time for a 95keV proton beam propagating in H$_{2}$ gas of pressure $5\times10^{-5}$hPa is 15$\rmmu$s.

§.§ Beam transport

Only the transport in the LEBT (Low Energy Beam Transport) will be considered here. Nevertheless, the source extraction system is a critical part, especially for a high-intensity injector, as the beam has to be properly formed under strong space-charge forces to be correctly transported in the LEBT.
Thus, it seems mandatory to perform simultaneously both the design and simulations
of the ion source extraction and the LEBT.

Once the beam is created and extracted from the ion source,
it has to be transported and matched by
the LEBT to the first accelerating structure like a radio frequency quadrupole (RFQ). The focus
can be achieved with electrostatic or magnetic elements. After the ion
source, because of the geometry of the extraction system, the beam
usually presents a cylindrical symmetry. In order to preserve this
symmetry and to simplify the beam tuning, magnetic solenoid lenses
or electrostatic Einzel lenses are more commonly used than quadrupoles.

§.§.§ Electrostatic LEBT

In an electrostatic LEBT, the beam is propagating without any space-charge compensation because the neutralizing particles are attracted
(or repelled) by the electric field induced by the focusing elements.
This kind of beam line is compatible with beam chopping, as there is
no transient time for the SCC. Furthermore, the design of electrostatic
LEBTs is simplified by the fact that no repelling electrodes for trapping
neutralizing particles are needed. So, the beam lines are
very compact, which tends to minimize the beam losses by charge exchange.
As an example, fig:SNS_LEBT shows the SNS ion source with
the 12cm long LEBT equipped with two Einzel lenses [19].

The SNS ion source and electrostatic LEBT

On the down side, electrostatic LEBTs are vulnerable to beam losses
that can lead to high-voltage breakdowns and beam trips. Besides,
the Einzel lenses intrinsically induce optical aberrations that create
a beam halo and emittance growth. To limit this effect, the beam radius
should not exceed two-thirds of the lens aperture radius. Finally, the design
of the electrostatic LEBTs is intensity-limited. As the beam is not
compensated, its divergence and size will increase rapidly with its
intensity (especially for currents of several tens of milliamps). So, it
seems difficult to operate the LEBT with a higher current than the
design current without expecting beam losses or dramatic emittance

§.§.§ Magnetostatic LEBT

In this case, the beam is fully compensated by the ionization on the
residual gas, as explained in the previous section. The gas in the
LEBT comes mainly from the ion source, but it has been shown experimentally
that the beam emittance can be improved with a higher pressure in
the beam line [11]. Besides, the nature of the
injected gas has an influence on this emittance improvement. An emittance
reduction of a factor of 2 has been reported by replacing
H$_{2}$ gas by the same partial pressure of Kr [3].
Nevertheless, the gas injection into the beam line has to be done carefully:
the higher the pressure, the higher the beam losses by charge exchange.
For example, a Kr partial pressure of $4\times10^{-5}$hPa
in a 2m LEBT leads to a H$^{+}$ (100keV) loss rate due to electron
capture of around 2.4%.

For a positive ion beam, another source of
compensation particles should be mentioned, even if it is less significant:
secondary electrons are produced when a beam hits the beam pipes.

At the end of the LEBT, the electric field of the RFQ tends to
penetrate through the injection hole and have a significant effect
on the SCC by attracting the neutralizing particles. Moreover, this
region is critical from the space-charge point of view, because a
beam waist is performed to match the beam for its injection into the
RFQ. So, a polarized electrode is placed as close as possible to the
RFQ entrance to repel the neutralizing
particles in the LEBT and to minimize the uncompensated zone (see fig:SCC_RFQCone).

(a)     (b)

(a) No electron repeller: neutralizing particles (electrons in this case) are attracted by the RFQ electric field. (b) Electrode repeller located before the RFQ: some neutralizing particles  are repelled into the LEBT; the uncompensated zone is minimized.

In a magnetic LEBT, the rise time of the pulsed beams is dominated
by the SCC transient time (i.e. several tens of microseconds). A fast chopping
system has to be inserted to reach a rise time of the order of
hundreds of nanoseconds. In the case of H$^{-}$ ion beams, a phenomenon
of overcompensation occurs during the SCC transient time  [2].
When the beam is fully compensated, neutralizing particles (in that case H$^{+}$)
are still created, but, as they are significantly slower than the electrons,
the SCC degree can be greater than 1 during the time it takes for the
excess H$^{+}$ to be expelled from the beam. During that time,
the beam is over-focused and instabilities can be observed.

§ BEAM DYNAMICS SIMULATION CODES WITH SPACE CHARGE

§.§ Numerical codes for ion source extraction systems

Some 2D- or 3D-like codes AXCEL-INP [18], PBGUNS [20] and
IBSimu [13] have been successfully used to design
sophisticated ion source extraction systems as well as electrostatic LEBTs.

With these codes, one can shape the electrodes, compute the generated electric field and track the particle in the defined domain. Over the past few years, elaborate
optimization of the geometry of the extraction system has been performed
to increase the extracted beam intensity while minimizing the optical
aberration and the beam divergence. As an example, the extraction system
of the SILHI source, which has an intermediate and a repelling electrode, forming,
together with the plasma and grounded electrodes, a pentode extraction
system (see fig:SILHI_Extraction) [7], has been developed using AXCEL-INP.

Five-electrode beam extraction system of the SILHI source

§.§ Numerical codes for beam dynamics transport

In a classical numerical code, the beam is represented by $N$ macro-particles ($N$ is normally less than the actual particle number in the beam)  that can be considered as a statistical sample of the beam with the same dynamics as the real particles. The macro-particles are transported through the accelerator step by step and at each time step $\rmd t$:

* external forces acting on each macro-particle are calculated,

* space-charge fields and the resulting forces are calculated, and

* tyhe equation of motion is solved for each macro-particle.

The space-charge electric field can be computed by a particle–particle interaction (PPI) method or a particles-in-cells (PIC) method, which are briefly described in the next subsections.

§.§.§ Particle–particle interaction method

For each macro-particle $i$ of charge $q$, it is assumed that the applied space-charge electrostatic field, $\vec{E}_i$, is the sum of all the fields induced by all the other macro-particles:

\begin{equation}
\vec{E}_i = \frac{q}{4\pi\varepsilon_0}\sum\limits_{i\neq j} \frac{\vec{r}_j - \vec{r}_i }{ \| \vec{r}_j - \vec{r}_i \| ^3} .
\end{equation}

The advantages of this method are that it is easy to code and the electric field is directly computed on the macro-particles. The main drawbacks are that the method is time-consuming for calculation (proportional to the square of the number of macro-particles) and the obtained space-charge field map is not smooth (the lower the macro-particle number, the more granularity), which can lead to non-physical emittance growth.

§.§.§ Particles-in-cells method

In this case, the physical simulated space is meshed. The mesh geometry depends on the situation. The meshing can be in one, two or three  dimensions, depending on the symmetry of the  simulated geometry and the beam.

The average beam density at each node of the mesh is obtained by counting the number of particles that are located close to it (an interpolation can induce smoothing). Once the density function is obtained, the field at each node is computed by solving the Poisson equation. Several techniques can be used to solve this equation at each node of the mesh.

* A direct method. The field is directly calculated at each node of the mesh. The calculation time is proportional to the square of the mesh number.

* A fast Fourier transform (FFT) method. The field at one node is given by the convolution product of the density and a Green's function. This can be solved by using the fact that a Fourier transform of a convolution product is equal to the product of the Fourier transforms. If $n$ is the mesh number, the calculation time is proportional to $n \log(n)$. One drawback is that  the FFT method does not take into account the boundary conditions of the conductors.

* A relaxation method [14], which is an iterative method. If $n$ is the mesh number, the calculation time is proportional to $n \log(n)$. It can take into account the particular boundary conditions as conductive items, for instance.

Once the field at each node is known, the field at the macro-particle location is calculated by interpolation from the closest nodes. After the evaluation of the beam density and the field for each particle, the computing time is proportional to the macro-particle number. The algorithm of a PIC code is presented in fig:SC_PIC_Code.

The PIC codes are the most commonly used for space-charge calculations, as they are the fastest and most efficient. A compromise has to be found between the mesh size and the particle number in order reach a sufficient resolution while avoiding some numerical noise that can lead to non-physical effects.

PIC codes that are commonly used for beam dynamics simulations with space charge are: TRACK [1], IMPACT [15] and TraceWin[21].

Algorithm of a PIC code dedicated to particle transport with space charge

§.§ Beam dynamics numerical codes with space-charge compensation

In order to achieve realistic beam transport simulations of high-intensity
ion beams at low energy ($\leq$100keV), it is necessary to
take into account the space-charge compensation of the beam by ionization of the
residual gas. For that, it is necessary to use self-consistent codes,
like WARP [12] or SolMaxP [5].
For example, SolMaxP, has recently been developed at CEA-Saclay and
is now used to design and simulate high-intensity injectors.

SolMaxP is a PIC code with an additional module (Monte Carlo algorithm) to simulate the beam interactions with  gas (ionization, neutralization, scattering) and beam line elements (secondary emission). The dynamics of the main beam is calculated as well as the dynamics of the secondary particles.
[b]fig:SC_PIC_Code_Compensation shows the SolMaxP algorithm.

Algorithm of SolMaxP, a self-consistent code for particle transport with
space-charge compensation

The SolMaxP code inputs are the particle distribution of the beam,
the applied external fields (e.g. focusing elements, electron repeller)
and the beam line geometry and gas pressure.
The outputs are the particle distributions (ions, electrons, neutral) all
along the beam line and the electric field map derived from the potential map
created by the space charge along the beam line.

§.§ Example of  numerical simulations of a high-intensity injector: IFMIF

§.§.§ Simulation conditions

First, the modelling of the IFMIF/EVEDA ion  source extraction system
[8] has been done with AXCEL-INP.
The D$^{+}$, D$_{2}^{+}$ and D$_{3}^{+}$ particle distributions coming
from this model are the input of these simulations.

Then, the simulations presented  have been done under the following
conditions or hypotheses:

* D$^{+}$, D$_{2}^{+}$ and D$_{3}^{+}$ beams are transported.

* The electric field map of the source extraction system is included
to get relevant boundary conditions.

* The gas pressure is considered to be homogeneous in the beam line.

* The gas ionization is produced by ion beam and electron impact.

* No beam scattering on gas is considered.

* No secondary electrons are created when the beam hits the beam pipe.

§.§.§ IFMIF injector parameters and layout

The IFMIF injector has to deliver a 140mA, 100keV CW D$^{+}$ beam
of 0.25$\rmpi$mmmrad emittance. It is composed by a 2.45GHz electron cyclotron resonance (ECR) source based on the SILHI design
and an LEBT with dual solenoid lens focusing system with integrated dipole correcting coils (see fig:SC_Simu_IFMIF_Injector)  [10].

IFMIF source and LEBT layout

§.§.§ Simulation results

SCC transient time

SolMaxP simulations make it possible to determine the SCC transient
time. The simulation starts at the time $t=0$ with no beam in the line
but only a fixed gas pressure. Then, the beam starts
to propagate in the beam line and its evolution can be followed.
The evolution of the D$^+$ beam emittance measured
between the two solenoids is represented on fig:EmitTime_Gas_Plot.
In this plot, the emittance evolution is given for different pressure conditions
in the beam line. Assuming that the D$_{2}$ gas contribution to the total pressure
in the beam line is $10^{-5}$hPa (coming from the ion source),
two simulations were done by adding a partial pressure ($2$ or $4\times10^{-5}$hPa)
of either D$_{2}$ gas or krypton, all the other parameters remaining constant.

Emittance evolution versus time for different gas pressure in the IFMIF LEBT between the two solenoids

As expected (see eq:SCCTime), the SCC transient time decreases when the beam is transported through a gas with a higher ionization cross-section ($\sigma_{\rm ioniz\,Kr} > \sigma_{\rm ioniz\,D_2}$). Thus, the simulated  space-charge compensation transient time is around 20$\rmmu$s for a krypton pressure of $2\times10^{-5}$hPa.

SCC potential map

A cut in the $(z0y)$ plane of the space-charge potential in the LEBT,
when the space-charge compensation has reached its steady state,
is represented on fig:PotentialMap_Plot.
In this plot, the abscissa $z=0$ represents the position of the repelling
electrode of the source extraction system, while $z=2.05$m is the RFQ
entrance. The solenoids are respectively located at $z=0.75$m and $z=1.65$m.

Two-dimensional cut  in the $(z0y)$ plane of a space-charge potential map in the IFMIF/EVEDA LEBT

From the space-charge potential map, the SCC degree can be calculated
along the IFMIF/EVEDA LEBT.
The potential on the uncompensated beam axis $\phi_{0}$ is calculated
with eq:SCCD and the SolMaxP simulations give
the potential on the compensated beam axis (see fig:PotentialMap_Plot
for $y=0$). The result is shown in fig:SCCDegree_Plot.
It can be observed that, in the ion source extraction region
and after the repelling electrode at the RFQ injection, the SCC is
poor because the electrons are attracted out of the beam. In the central
part of the LEBT, where the solenoids and a drift are located, the
SCC degree reaches around 95%, which is compatible with the experimentally
measured values [9, 11].

Space-charge compensation degree along
the IFMIF/EVEDA LEBT

IFMIF LEBT beam dynamics

The extensive calculations that have been performed for the IFMIF injector led to
a very compact design of the LEBT and to an optimization of some parameters, like the position
of electron repeller electrode in the injection cone  [6].

The beam dynamics simulations showed that the IFMIF/EVEDA deuteron
beam can be transported and injected into the RFQ with optimized emittance
and Twiss parameters. Under these conditions, the RFQ transmission
is above 95%. The transport of the D$^+$ beam in the injector and in the
first section of the IFMIF RFQ is shown on  fig:IFMIF_LEBT_Beam.

Beam transport simulation in the IFMIF injector and in the first RFQ cells

The SolMaxP code is in qualitatively good agreement with experimental results, but some
quantitative  confrontations will be done with experimental results obtained with the
beams of the IFMIF injector. Besides, the code will be improved in order to take into account
more physical phenomena, like a precise calculation of the gas pressure along the beam line
(the pressure appears to be a critical parameter for the SCC) or the beam scattering on the gas.

[1]
V.N. Aseev, P.N. Ostroumov, E.S. Lessner and B. Mustapha, TRACK: the
new beam dynamics code, Proc. Particle Accelerator Conf.,
Knoxville, USA, 2005.

[2]
R. Baartman and D.H. Yuan, Space charge neutralization studies of an
H$^-$ beam, Proc. European Particle Accelerator Conf.,
Rome, Italy, 1988.

[3]
P.-Y. Beauvais, R. Ferdinand, R. Gobin, J.M. Lagniel, P.-A. Leroy, L. Celona,
G. Ciavola, S. Gammino, B. Pottin and J. Sherman, Emittance
improvement of the electron cyclotron resonance high intensity light ion
source proton beam by gas injection in the low energy beam transport, Rev.
Sci. Instrum. 71(3) (2000) 1413–1416.

[4]
A. BenIsmail, Etude des effets transitoires de la compensation de charge
d'espace sur la dynamique d'un faisceau intense, Ph.D. thesis, Université
Paris XI, Orsay, 2005.

[5]
A. Chancé, N. Chauvin and R. Duperrier, The SolMaxP code,
Proc. Int. Particle Accelerator Conf., New Orleans, USA, 2012.

[6]
N. Chauvin, M. Comunian, O. Delferrière, R. Duperrier, R. Gobin, P.A.P.
Nghiem and D. Uriot, Final design of the IFMIF-EVEDA low energy beam
transport line, Proc. Particle Accelerator Conf.,
Vancouver, Canada, 2009.

[7]
O. Delferriere and D. De Menezes, Extraction design and low energy beam
transport optimization of space charge dominated multispecies ion beam
sources, Rev. Sci. Instrum. 75(5) (2004) 1659–1661.

[8]
O. Delferriere, D. De Menezes, R. Gobin, F. Harrault and O. Tuske,
Electron cyclotron resonance 140mA D$^+$ beam extraction
optimization for IFMIF-EVEDA accelerator, Rev. Sci. Instrum.
79(2) (2008) 02B723.

[9]
R. Ferdinand, J. Sherman, R.R. Stevens Jr. and T. Zaugg,
Space-charge neutralization measurement of a 75-keV, 130-mA
hydrogen-ion beam, Proc. Particle Accelerator Conf.,
Vancouver, Canada, 1997.

[10]
R. Gobin, P.-Y. Beauvais, O. Delferriere, D. De Menezes, O. Tuske, G. Adroit,
Y. Gauthier and F. Harrault, A 140mA CW deuteron electron cyclotron
resonance source for the IFMIF-EVEDA project, Rev. Sci.
Instrum. 79(2) (2008) 02B303.

[11]
R. Gobin, P.-Y. Beauvais, R. Ferdinand, P.-A. Leroy, L. Celona, G. Ciavola and
S. Gammino, Improvement of beam emittance of the CEA high intensity
proton source SILHI, Rev. Sci. Instrum. 70(6) (1999)

[12]
D.P. Grote, A. Friedman, J.-L. Vay and I. Haber, The WARP
code: modeling high intensity ion beams, AIP Conf. Proc.
749(1) (2005) 55–58.

[13]
T. Kalvas, O. Tarvainen, T. Ropponen, O. Steczkiewicz, J. Ärje and H. Clark,
Ibsimu: a three-dimensional simulation software for charged particle
optics, Rev. Sci. Instrum. 81(2) (2010) 02B703.

[14]
W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical
Recipes: The Art of Scientific Computing, 3rd edn. (Cambridge University Press, Cambridge, 2007).

[15]
J. Qiang, R.D. Ryne, S. Habib and V. Decyk, An
object-oriented parallel particle-in-cell code for beam dynamics simulation
in linear accelerators, J. Comput. Phys. 163(2)
(2000) 434–451.

[16]
F.J. Sacherer, RMS envelope equations with space charge, IEEE
Trans. Nucl. Sci. 18(3) (1971) 1105–1107.

[17]
I.A. Soloshenko, Space charge compensation of technological ion beams,
IEEE Trans. Plasma Sci. 27(4) (1999) 1097–1100.

[18]
Axcel INP code, http://www.aetassociates.com/software.php?inp.

[19]
M.P. Stockli, B. Han, S.N. Murray, T.R. Pennisi, M. Santana and R.F.
Welton, Ramping up the spallation neutron source beam power with the
H$^-$ source using 0mgCs/day, Rev. Sci. Instrum.
81(2) (2010) 02A729.

[20]
PBGINS code, http://far-tech.com/pbguns/index.html.

[21]
TraceWin code, http://irfu.cea.fr/Sacm/en/logiciels/index3.php.

[22]
J.L. Vay and C. Deutsch, Intense ion beam propagation in a reactor sized
chamber, Nucl. Instrum. Methods Phys. Res. A464(1–3)
(2001) 293–298.

§ FURTHER READINGS

=2em =-2em

* A. BenIsmail, R. Duperrier, D. Uriot and N. Pichoff, Space charge compensation studies of hydrogen ion beams in a drift section, Phys. Rev. ST Accel. Beams 10(7) (2007) 070101.

* M. Reiser, Theory and Design of Charged Particle Beams (John Wiley & Sons, New York, 2008).

* F. Sacherer, RMS envelope equations with space charge, CERN Internal Report, SI/DL/70-12 (1970).

* K. Schindl, Space charge, CERN Accelerator School 2003: Intermediate Course on Accelerators, CERN-2006-002 (2006).

* T.P. Wangler, RF Linear Accelerators  (Wiley-VCH, Weinheim, 2008).


# Centralizers of linear and locally nilpotent derivations

L. Bedratyuk  Department of Information Technology, Khmelnytsky National
University, Khmelnytsky, Instytutska ,11, 29016, Ukraine
<EMAIL_ADDRESS>,  Y.Chapovskyi  Department of Algebra and
Computer Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko
National University of Kyiv, 64, Volodymyrska street, 01033 Kyiv, Ukraine
<EMAIL_ADDRESS>and  A. Petravchuk  Department of Algebra and Computer
Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National
University of Kyiv, 64, Volodymyrska street, 01033 Kyiv, Ukraine
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $K$ be an algebraically closed field of characteristic zero,
$A=K[x_{1},\dots,x_{n}]$ the polynomial ring, $R=K(x_{1},\dots,x_{n})$ the
field of rational functions, and let $W_{n}(K)=\operatorname{Der}_{K}A$ be the
Lie algebra of all $K$-derivations on $A$. If $D\in W_{n}(K),$ $D\not=0$ is
linear (i.e. of the form
$D=\sum_{i,j=1}^{n}a_{ij}x_{j}\frac{\partial}{\partial x_{i}}$) we give a
description of the centralizer of $D$ in $W_{n}(K)$ and point out an algorithm
for finding generators of $C_{W_{n}(K)}(D)$ as a module over the ring of
constants in case when $D$ is the basic Weitzenboeck derivation. In more
general case when the ring $A$ is a finitely generated domain over $K$ and $D$
is a locally nilpotent derivation on $A,$ we prove that the centralizer
$C_{{\rm Der}A}(D)$ is a "large" subalgebra in ${\rm Der}_{K}A$, namely
$\operatorname{rk}_{A}C_{\operatorname{Der}A}(D):=\dim_{R}RC_{\operatorname{Der}A}(D)$
equals ${\rm tr}.\deg_{K}R,$ where $R$ is the field of fraction of the ring
$A.$

###### Key words and phrases:

locally nilpotent derivations, basic Weitzenboeck derivation, Lie algebra,
centralizer, kernel of derivation

###### 2000 Mathematics Subject Classification:

Primary 17B66; Secondary 17B05, 13N15

##  1\. Introduction

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero,
$A=\mathbb{K}[x_{1},\dots,x_{n}]$ the polynomial algebra and
$R=\mathbb{K}(x_{1},\dots,x_{n})$ the field of rational functions in $n$
variables. Recall that a $\mathbb{K}$-linear map $D:A\to A$ is called a
$\mathbb{K}$-derivation of the algebra $A$ if $D(fg)=D(f)g+fD(g)$ for any
$f,g\in A$. All $\mathbb{K}$-derivations of the $\mathbb{K}$-algebra $A$ form
the Lie algebra $W_{n}(\mathbb{K})$ over the field $\mathbb{K}$ with respect
to the commutation and every element $D\in W_{n}(\mathbb{K})$ can be uniquely
written in the form $D=f_{1}\partial_{1}+\dots+f_{n}\partial_{n}$, where
$f_{i}\in A$ і $\partial_{i}:=\frac{\partial}{\partial x_{i}}$ are partial
derivatives. Note that every derivation of the algebra $A$ can be uniquely
extended on the field of fractions $R=\mathbb{K}(x_{1},\ldots,x_{n}),$ we will
denote this extension by the same letter when no confusion can arise. The Lie
algebra of all $\mathbb{K}$-derivations of the field $R$ will be denoted by
${\widetilde{W}}_{n}(\mathbb{K}),$ elements of this algebra can be uniquely
written in the form $D=\varphi_{1}\partial_{1}+\dots+\varphi_{n}\partial_{n},$
where $\varphi_{i}\in R,i=1,\ldots n.$

The structure of the Lie algebra $W_{n}(\mathbb{K})$ is of great interest
because, from the geometrical point of view, $W_{n}(\mathbb{K})$ is the Lie
algebra of all vector fields on $\mathbb{K}^{n}$ with polynomial coefficients;
from the viewpoint of differential equations, any derivation
$D=f_{1}\partial_{1}+\dots+f_{n}\partial_{n}$ can be considered as an
autonomous system of ordinary differential equations

$\dot{x}=F(x),\ {\rm where}\ x=x(t)=(x_{1}(t),\ldots,x_{n}(t)),\
F(x)=(f_{1}(x),\dots,f_{n}(x)).$

For a given derivation $D\in W_{n}(\mathbb{K})$ one can consider its
centralizer $C_{W_{n}(\mathbb{K})}(D)$ in $W_{n}(\mathbb{K})$, i.e the set of
all elements of the Lie algebra $W_{n}(\mathbb{K})$ that commute with $D$; we
will denote it briefly by $C_{W_{n}}(D)$). The centralizer $C_{W_{n}}(D)$ is
obviously a subalgebra of the Lie algebra $W_{n}(\mathbb{K})$ and $D\in
C_{W_{n}}(D).$ The problem of describing centralizers of elements $D\in
W_{n}(\mathbb{K})$ is closely related to some problems from theory of
differential equations and geometry, it is solved only in some special cases
(see for example, [12, 9, 3, 4, 13]).

The paper is organized as follows. In Sect. 2, we consider centralizers of
linear derivations, i.e. derivations of the form

$D=\sum_{i=1}^{n}f_{i}\partial_{i},\;f_{i}=\sum_{j=1}^{n}a_{ij}x_{j},\;a_{ij}\in\mathbb{K}.$

Such a derivation is completely defined by the matrix
$A=(a_{ij})_{i,j=1}^{n}$, the commutator of derivations $D$ і
$D^{\prime}=\sum_{i,j=1}^{n}b_{ij}x_{j}\partial_{i}$ with the matrix
$B=(b_{ij})$ is determined by the matrix $[A,B]=AB-BA$. Therefore the Lie
algebra of all linear derivations is isomorphic to the general linear Lie
algebra $\mathfrak{gl}_{n}(\mathbb{K}).$ It will cause no confusion if we use
the same notation to these isomorphic algebras and assume that
$\mathfrak{gl}_{n}(\mathbb{K})\subset W_{n}(\mathbb{K}).$

For a linear derivation $D=\sum_{i,j=1}^{n}a_{ij}x_{j}\partial_{i}$ one can
consider two centralizers: $C_{0}=C_{\mathfrak{gl}_{n}(\mathbb{K})}(D)$ and
$C=C_{W_{n}(\mathbb{K})}(D)$ (obviously, $C_{0}\subseteq C$). The structure of
the centralizer $C_{0}$ is well known, since $C_{0}$ consists of linear
derivations determined by matrices that commute with $A=(a_{ij})$. Description
of such matrices was obtained many years ago (see, for example, [6]). It
turned out that in some sense the subalgebra $C$ is near to $C_{0}$. This was
proved in Theorem 1 which states that
$C_{W_{n}(\mathbb{K})}(D)=(FC_{\mathfrak{gl}_{n}(\mathbb{K})}(D))\cap
W_{n}(\mathbb{K}),$ where $F$ is the kernel of $D$ in
$R=\mathbb{K}(x_{1},\dots,x_{n})$ Unfortunately, this theorem does not allow
in general to find $C_{W_{n}}(D)$ since the intersection $(FC_{0})\cap
W_{n}(\mathbb{K})$ is very hard to calculate.

The third section of the paper provides a method for finding generators of
$C=C_{W_{n}}(D)$ (as a module over ${\rm Ker}D$) in the case when the matrix
of $D$ is a single Jordan block (Theorem 3). To solve this problem we describe
the kernels of powers $D^{n},n\geq 2$ of the Weitzenb$\ddot{\rm o}$ck
derivation under condition that we know the generators of ${\rm Ker}D$ as a
subalgebra of $A$ (Theorem 2). Note that many papers were devoted to finding
generators of kernels of locally nilpotent derivations (see, for example, [2],
[8], [11]).

The fourth section of the paper deals with centralizers of locally nilpotent
derivations $D$ on finitely generated domains $A$ over an arbitrary field
$\mathbb{K}$ of characteristic zero. In the case, when $D$ has a slice in $A$,
(i.e. there exists $s\in A$ such that $D(s)=1$) we prove that
$C_{\operatorname{Der}_{\mathbb{K}}A}(D)\simeq
C^{(-)}\rightthreetimes\operatorname{Der}C,$ where $C=\operatorname{Ker}D$ в
$A$, $C^{(-)}$ is a vector space over $\mathbb{K}$ with zero multiplication,
i.e an abelian Lie algebra and elements of $\operatorname{Der}_{\mathbb{K}}C$
act on $C^{(-)}$ as derivations of the algebra $C$ (Proposition 2). In general
case, it was proved for a locally nilpotent derivation $D$ of a domain $A$
over $\mathbb{K}$ with ${\rm tr}.\deg_{\mathbb{K}}A=n$ that the centralizer
$C_{\operatorname{Der}_{\mathbb{K}}A}(D)$ has rank $n$ over $A$ (see Theorem
4). Note that centralizers of locally nilpotent derivations were studied in
[4] from another point of view.

We use standard notation. If $L$ is a subalgebra of the Lie algebra
$\operatorname{Der}_{\mathbb{K}}(A)$ and $R$ the field of fractions of $A$
then the rank of $L$ over $A$ is the dimension $\dim_{R}RL$; we denote it by
$\operatorname{rk}_{R}L$. The index of a derivation $D\in W_{n}(\mathbb{K})$
is, by definition, the least number of nonzero coefficients in
$D=f_{1}\partial_{1}+\dots+f_{n}\partial_{n}.$ A derivation
$D\in\operatorname{Der}_{\mathbb{K}}(A)$ is called locally nilpotent
derivation if for any $a\in A$ there exists $k=k(a)$ such that $D^{k}(a)=0$.
The smallest natural $k$ with $D^{k}(a)\neq 0$, but $D^{k+1}(a)=0$ will be
called $D$-order of the element $a$ and denoted by
$\operatorname{ord}_{D}(a)$. For the sake of brevity, we will write
$C_{W{n}}(D)$ i $C_{\mathfrak{gl}_{n}}(D),$ without indicating the fixed
ground field $\mathbb{K}.$

##  2\. Centralizers of linear derivations

Throughout this section, $A=\mathbb{K}[x_{1},\ldots,x_{n}]$,
$W_{n}(\mathbb{K})=\operatorname{Der}_{\mathbb{K}}(A).$

######  Lemma 1.

Let $D\in W_{n}(\mathbb{K}),\;f\in\operatorname{Ker}D$ and let $\varphi$ be an
arbitrary automorphism of the ring $A$. Then $\varphi^{-1}D\varphi\in
W_{n}(\mathbb{K})$ and

  1. (1)

$\operatorname{Ker}_{A}(\varphi^{-1}D\varphi)=\varphi^{-1}(\operatorname{Ker}D)$
;

  2. (2)

$\varphi^{-1}(fD)\varphi=\varphi^{-1}(f)\varphi^{-1}D\varphi$ ;

  3. (3)

$C_{W_{n}}(D)=\varphi(C_{W_{n}}(\varphi^{-1}D\varphi))\varphi^{-1}$.

###### Proof.

The proof is straightforward. ∎

######  Lemma 2.

Let
$D=\sum_{i=1}^{n}f_{i}\partial_{i},\;T=\sum_{i=1}^{n}g_{i}\partial_{i}\;f_{i},\,g_{i}\in
A$ be derivations from $W_{n}(\mathbb{K})$. The derivations $T$ and $D$
commute if and only if $D(g_{i})=T(f_{i}),\;i=1,\dots,n$.

###### Proof.

It suffices to note that $g_{i}=T(x_{i}),\;f_{i}=D(x_{i}),\;i=1,\dots,n$. ∎

######  Lemma 3.

Let $D=\sum_{i=1}^{n}f_{i}\partial_{i}\in W_{n}(\mathbb{K})$ be such a
derivation that $f_{1},\dots,f_{k}\in\mathbb{K}[x_{1},\dots,x_{k}]$ and
$f_{k+1},\dots,f_{n}\in\mathbb{K}[x_{k+1},\dots,x_{n}]$. If $T\in
C_{W_{n}}(D),\;T=\sum_{i=1}^{n}g_{i}\partial_{i},$ then the derivations
$T_{1}=\sum_{i=1}^{k}g_{i}\partial_{i}$ and
$T_{2}=\sum_{i=k+1}^{n}g_{i}\partial_{i}$ belong to $C_{W_{n}}(D)$ and
$T=T_{1}+T_{2}$.

###### Proof.

Put $T_{1}=\sum_{i=1}^{n}t_{i}\partial_{i}$, where
$t_{i}=g_{i},\;i=1,\dots,k,$ and $t_{i}=0,\;i=k+1,\dots,n.$ Let us show that
$[T_{1},D]=0$. According to Lemma 2, it suffices to show that
$T_{1}(f_{i})=D(t_{i}),\;i=1,\dots,n$. Let first $i=1,\dots,k$. Then
$t_{i}=g_{i}$ and $T_{1}(f_{i})=T(f_{i})=D(g_{i})=D(t_{i})$. Let now
$i=k+1,\dots,n$. Then obviously $T_{1}(f)=0$ and, by conditions of the lemma,
we have $D(t_{i})=D(0)=0$. Thus, $D(t_{i})=T_{1}(f_{i})$ for $i=1,\dots,n$ and
by Lemma 2 we have $[D,T_{1}]=0$. Similarly, one can show that $[D,T_{2}]=0$.
∎

######  Remark 1.

One can easily check that the statements of Lemmas 2-3 are true for
derivations of the field $R=\mathbb{K}(x_{1},\ldots,x_{n})$, i.e. elements of
the Lie algebra ${\widetilde{W}}_{n}(\mathbb{K}).$

######  Theorem 1.

Let $D=\sum_{i,j=1}^{n}a_{ij}x_{j}\partial_{i}\in W_{n}(\mathbb{K})$ be a
linear derivation, $F=\operatorname{Ker}_{R}D$ be a field of constants of $D$
in $R=\mathbb{K}(x_{1},\ldots,x_{n}).$ If $C_{\mathfrak{gl}_{n}}(D)$ and
$C_{W_{n}}(D)$ are centralizers of $D$ in
$\mathfrak{gl}_{n}(\mathbb{K})\subset W_{n}(\mathbb{K})$ and in
$W_{n}(\mathbb{K}),$ respectively, then
$C_{W_{n}}(D)=(FC_{\mathfrak{gl}_{n}}(D))\cap W_{n}(\mathbb{K})$.

###### Proof.

Without loss of generality one can assume that the matrix $A=(a_{ij})$ of the
derivation $D$ is written in Jordan normal form

(1) $A=\bigoplus_{i=1}^{t}J_{n_{i}-n_{i-1}}(\lambda_{i})\quad\text{for
some}\quad
0=n_{0}<n_{1}<\dots<n_{t}=n,\;\lambda_{1},\dots,\lambda_{t}\in\mathbb{K}.$

(we consider lower triangular Jordan blocks). To prove this, it is enough to
apply Lemma 1. Further, since the sum (1) is direct, we have
$D=D_{1}+\dots+D_{t}$, where $D_{i}\in W_{n}(\mathbb{K}),i=1,\ldots t$ and is
defined by the matrix

$O_{n_{1}}\oplus\dots\oplus J_{n_{i}-n_{i-1}}(\lambda_{i})\oplus\dots\oplus
O_{n_{t}-n_{t-1}}$

with the zero matrices $O_{n_{j}-n_{j-1}},j\not=i.$ Let $T\in C_{W_{n}}(D)$ be
an arbitrary element of the centralizer. Then, by Lemma 3, which can be easily
generalized to finitely many of terms, the equality $T=T_{1}+\dots+T_{t}$
holds, where the summands $T_{i}=\sum_{s=n_{i-1}+1}^{n_{i}}g_{s}\partial_{s}$
belong to $C_{W_{n}}(D)$ (note that in general case the polynomials $g_{j}$
depend on all variables $x_{1},\dots,x_{n}$). First, let us show that

$T_{1}=\sum_{s=1}^{n_{1}}g_{s}\partial_{s}\in(FC_{\mathfrak{gl}_{n}}(D))\cap
W_{n}(\mathbb{K}).$

Here the derivation $D_{1}$ is defined by the matrix of the form

$J_{n_{1}}(\lambda_{1})\oplus O_{n_{2}-n_{1}}\oplus\cdots\oplus
O_{n_{t}-n_{t-1}},$

where $O_{n_{i}-n_{i-1}},i=2,\ldots,t$ is a square zero matrix of suitable
order, and in our notation

$J_{n_{1}}(\lambda_{1})=\left(\begin{array}[]{ccccc}\lambda_{1}&0&\cdots&0&0\\\
1&\lambda_{1}&0&\cdots&0\\\ \cdots&\cdots&\cdots&\cdots&\cdots\\\
0&\cdots&1&\lambda_{1}&0\\\ 0&0&\cdots&1&\lambda_{1}\\\ \end{array}\right).$

By Lemma 2 the equality $[D,T_{1}]=0$ implies

$D(g_{1})=\lambda_{1}g_{1},D(g_{2})=\lambda_{1}g_{2}+g_{1},\dots,\;D(g_{n_{1}})=\lambda_{1}g_{n_{1}}+g_{n_{1}-1}.$

Let $r$ be the index of the derivation $T_{1}$, i.e. the smallest natural
number such that $g_{r}\neq 0$. Then $g_{r-1}=0$ and we have
$D(g_{r})=\lambda_{1}g_{r}.$ On the other hand, it is obvious that

$D(x_{1})=\lambda_{1}x_{1},D(x_{2})=\lambda_{1}x_{2}+x_{1},\cdots,D(x_{n_{1}})=\lambda_{1}x_{n_{1}}+x_{{n_{1}}-1}.$

Taking into account the mentioned above inequality $g_{r}\neq 0$, we obtain
$g_{r}/x_{1}\not=0$ and $D(g_{r}/x_{1})=0$, i.e. $g_{r}/x_{1}\in F$, where
$F=\operatorname{Ker}_{R}D$ is the field of constant of $D$ in $R$.

Next, let us consider the linear derivation
$D_{1}\in\mathfrak{gl}_{n}(\mathbb{K})\subset W_{n}(\mathbb{K})$ that is
defined by the matrix

$J_{n_{1}}(\lambda_{1})\oplus O_{n_{2}-n_{1}}\oplus\dots\oplus
O_{n_{t}-n_{t-1}},$

where $O_{n_{i}-n_{i-1}}$ are square zero matrices of suitable orders.

Obviously, $[D_{1},D]=0$ and $D_{1}=W_{1}+V_{1}$, where the linear derivation
$W_{1}$ is defined by the diagonal matrix

$d_{n_{1}}(\lambda_{1})\oplus O_{n_{2}-n_{1}}\oplus\dots\oplus
O_{n_{t}-n_{t-1}},$

and $V_{1}$ is given by the lower triangular matrix $M_{1}=J_{n_{1}}(0)\oplus
O_{n_{2}-n_{1}}\oplus\dots\oplus O_{n_{t}-n_{t-1}}$. It is easy to see that
$[W_{1},D]=0,$ and therefore also $[V_{1},D]=0.$ One can directly verify that
the linear derivation $V_{1}^{(r-1)},$ defined by the matrix $M_{1}^{r-1}$, is
of the form $x_{1}\partial_{r}+\dots+x_{n_{1}-r+1}\partial_{n_{1}}$ and
commutes wіth $D$. Then

$T_{1}^{(2)}:=(T_{1}-\frac{g_{r}}{x_{1}}V_{1}^{(r-1)})\in
FC_{W_{n}}(D)\subset{\widetilde{W}}_{n}(\mathbb{K}),$

where $F$ is the field of constants of $D$ in the field $R$ (recall that
$g_{r}/x_{1}\in F$). The derivation $T_{1}^{(2)}$ obviously has a greater
index than $T_{1}$ (recall that the index is the smallest number of the non-
zero coefficients of derivation). Repeating these considerations a sufficient
number of times and taking into account the Remark 1 we obtain a derivation
$T_{1}^{(s)},s\geq 2$ which commutes with $D$ and is of the form
$h_{n_{1}}\partial_{n_{1}}$ for some rational function $h_{n_{1}}\in F\cdot
A\subset R$.

But then $D(h_{n_{1}})=0,$ that is, the derivation $h_{n_{1}}\partial_{n_{1}}$
lies in $FC_{\mathfrak{gl}_{n}}(D).$ It follows that the derivation $T_{1}$ is
a sum of some derivations of $FC_{\mathfrak{gl}_{n}}(D),$ because at each step
we subtract from $T_{1}$ a derivation from $FC_{\mathfrak{gl}_{n}}(D).$
Therefore $T_{1}\in(FC_{\mathfrak{gl}_{n}}(D))\cap W_{n}(\mathbb{K})$. It can
be similarly shown that $T_{i}\in(FC_{\mathfrak{gl}_{n}}(D))\cap
W_{n}(\mathbb{K})$ for $i=2,\dots,t$. This completes the proof. ∎

##  3\. Centralizer of the basic Weitzenboeck derivation

Recall that a linear derivation $D$ on $A=\mathbb{K}[x_{1},\dots,x_{n}]$ is
called the basic Weitzenb$\ddot{\rm o}$ck derivation if $D$ is defined by the
matrix $J_{n}(0)$ consisting of one Jordan block with the eigenvalue
$\lambda=0$. (we consider lower triangular Jordan blocks). This derivation can
be written in coordinates as $D=x_{1}\partial_{2}+\dots+x_{n-1}\partial_{n}$.
If $T=\sum_{i=1}^{n}g_{i}\partial_{i}$ is an arbitrary derivation on $A$ that
commutes with $D,$ then we have by Lemma 2

$D(g_{1})=0,\;D(g_{2})=g_{1},\;\dots,\;D(g_{n})=g_{n-1}.$

For any polynomial $f\in A,$ let us denote by $D_{f}$ the derivation on $A$ of
the form

$D_{f}=D^{n-1}(f)\partial_{1}+D^{n-2}(f)\partial_{2}+\dots+D(f)\partial_{n-1}+f\partial_{n}.$

There is a convenient criterion for verifying whether a derivation $T\in
W_{n}(\mathbb{K})$ commutes with the basic Weitzenb$\ddot{\rm o}$ck
derivation.

######  Proposition 1.

A derivation $T\in W_{n}(\mathbb{K})$ commutes with the basic
Weitzenb$\ddot{\rm o}$ck derivation $D$ if and only if there is a polynomial
$f\in A$ with $D$-order $\leq n-1$ such that $T=D_{f}$.

###### Proof.

Take any derivation $T\in C_{W_{n}}(D)$ and write down it in the form
$T=\sum_{i=1}^{n}g_{i}\partial_{i}$ for some polynomials $g_{i}\in A$. Then
from the condition $[T,D]=0$ we obtain (by Lemma 2) the equalities

$D(g_{1})=0,\;D(g_{2})=g_{1},\;\dots,\;D(g_{n})=g_{n-1}.$

Hence $g_{n}$ is a polynomial of $D$-order $\leq n-1$. Letting $f=g_{n}$ we
have $T=D_{f}$ and $\operatorname{ord}_{D}(f)\leq n-1$. The other way around,
if $T=\sum_{i=1}^{n-1}D^{n-i}(f)\partial_{i}+f\partial_{n}$ and
$\operatorname{ord}_{D}(f)\leq n-1$, then again by Lemma 2 we get that
$[T,D]=0$. ∎

The Weitzenb$\ddot{\rm o}$ck derivation $D$ on
$A=\mathbb{K}[x_{1},\dots,x_{n}]$ determines the filtration

(2) $0=A_{0}\subset A_{1}\subset\dots\subset A_{n}\subset\dots,$

where $A_{1}=\operatorname{Ker}D,\;A_{i}=\\{f\in A\ |\ D(f)\in
A_{i-1}\\},\;i=2,\dots.$ It follows from Proposition 1 that for finding
$C_{W_{n}}(D)$ it is sufficient to study the terms $A_{i}$ of the filtration
for $i\leq n-1$, since $T\in C_{W_{n}}(D)$ are of the form
$T=\sum_{i=1}^{n}D^{n-i}(f)\partial_{i}$ for $f\in A_{n-1}$. Note that
$A_{n-1}$ is a module over the algebra of constants
$A_{1}=\operatorname{Ker}D$ of the derivation $D$.

For studying $A_{n-1}$ we apply some ideas from the paper [2], where the
Weitzenb$\ddot{\rm o}$ck derivation $D$ was embedded in a $3$-dimensional Lie
subalgebra $L=\mathbb{K}\left<D,\widehat{D},H\right>\subset
W_{n}(\mathbb{K});$ the Lie algebra $L$ is constructed in such a way to be
isomorphic to the simple Lie algebra $\mathfrak{sl}_{2}(\mathbb{K})$.

Linear derivations $\widehat{D}$ and $H$ on $A$ we define by the rule:

$\widehat{D}(x_{i})=i(n-i)x_{i+1},\quad H(x_{i})=(n-2i+1)x_{i},i=1,\ldots,n.$

It can be directly verified (see [2]) that the following commutation relations
hold

(3) $[D,\widehat{D}]=H,\quad[H,D]=2D,\quad[H,\widehat{D}]=-2\widehat{D},$

that is, $L\simeq\mathfrak{sl}_{2}(\mathbb{K})$.

The derivation $\widehat{D}$ is obviously locally nilpotent;
$\widehat{D}$-order of a polynomial $f\in A$ we will denote by
$\operatorname{ord}(f)$ and call the order of $f$. It is easy to see that each
monomial $x_{1}^{\alpha_{1}}\dots x_{n}^{\alpha_{n}}\in A$ is an eigenvector
for $H$ with the eigenvalue

$\lambda_{(\alpha_{1},\ldots,\alpha_{n})}=n\sum_{i=1}^{n}\alpha_{i}-(\alpha_{1}+3\alpha_{2}+\cdots+(2i-1)\alpha_{i}+\cdots+(2n-1)\alpha_{n}).$

We call this eigenvalue the weight of the monomial $x_{1}^{\alpha_{1}}\dots
x_{n}^{\alpha_{n}}.$ A polynomial $f\in A$ will be called _isobaric_ , if all
its monomials have the same weight. Since all the monomials form a basis of
the linear space $A=\mathbb{K}[x_{1},\dots,x_{n}]$ over $\mathbb{K}$, the
linear operator $H$ is diagonalized in this basis. But then the vector space
$A$ is the direct sum
$A=\bigoplus_{\lambda_{(\alpha_{1},\dots,\alpha_{n})}}A_{\lambda_{(\alpha_{1},\dots,\alpha_{n})}}$
of subspaces, where $A_{\lambda_{(\alpha_{1},\dots,\alpha_{n})}}$ is the
linear span of all monomials of weight
$\lambda_{(\alpha_{1},\dots,\alpha_{n})}$, that is
$A_{\lambda_{(\alpha_{1},\dots,\alpha_{n})}}$ consists of isobaric monomials
of the weight $\lambda_{(\alpha_{1},\dots,\alpha_{n})}$.

The statement of the next technical lemma follows immediately from the main
properties of $sl_{2}(\mathbb{K})$-modules (see, for example, [7]).

######  Lemma 4.

Let $D$ be the basic Weitzenb$\ddot{o}$ck derivation on
$\mathbb{K}[x_{1},\ldots,x_{n}]$ and let $f,g\in A$ be isobaric polynomials
with weights $\lambda$ and $\mu,$ respectively. Then:

(1) $D(f)$ and $\widehat{D}(f)$ are isobaric polynomials of the weights
$\lambda+2$ and $\lambda-2,$ respectively;

(2) $fg$ is an isobaric polynomial with the weight $\lambda+\mu;$

(3) If $f\in{\rm Ker}D$ and $f=f_{\lambda_{1}}+\cdots+f_{\lambda_{k}}$ is a
decomposition of the polynomial $f$ into the sum of isobaric polynomials of
pairwise distinct weights, then $f_{\lambda_{i}}\in{\rm Ker}D,i=1,\ldots,k.$

It follows from the known Weitzenb$\ddot{\rm o}$ck theorem [14] that the
algebra of constants ${\rm Ker}D$ of the Weitzenb$\ddot{\rm o}$ck derivation
on $\mathbb{K}[x_{1},\ldots,x_{n}]$ is finitely generated. Increasing the
number of generators (if needed) we can always assume (by Lemma 4) that ${\rm
Ker}D$ has a finite system of generators consisting of isobaric polynomials.

The following lemma indicates how the derivations $D$ and $\widehat{D}$ act on
the terms of the filtration $0=A_{0}\subset A_{1}\subset\dots\subset
A_{n}\subset\dots$ (see (2)).

######  Lemma 5.

Let $b\in A_{k}\setminus A_{k-1}$ be an isobaric polynomial, $k\geq 2$. Then

  1. (1)

$\widehat{D}(D(b))\in A_{k}\setminus A_{k-1}$;

  2. (2)

$D^{s-1}(\widehat{D}D(b))=\widehat{D}D^{s}(b)+\lambda D^{s-1}(b)$ for any
$s\geq 2$ and some $\lambda=\lambda(s),\;\lambda\in\mathbb{K}$.

###### Proof.

1\. Let $\deg b=t$ and let $V$ be the set of all polynomials of $A$ of degree
at most $t$. Since $D,\widehat{D},H=[D,\widehat{D}]$ are linear derivations on
$A$, the subspace $V$ is invariant under the natural action of the Lie algebra
$L=\mathbb{K}\left<D,\widehat{D},H\right>\simeq\mathfrak{sl}_{2}(\mathbb{K})$.
Obviously $\dim_{\mathbb{K}}V<\infty$ and $b\in V$ Then the $L$-module $V$ can
be decomposed into direct sum $V=V_{1}\oplus\dots\oplus V_{m}$ of irreducible
$L$-modules and $b=b_{1}+\dots+b_{m},\;b_{i}\in V_{i}$. Since $D^{k-1}(b)\neq
0,\;D^{k}(b)=0$, at least one of the summands, for example $b_{i},$ has the
same property, that is $D^{k-1}(b_{i})\neq 0,\;D^{k}(b_{i})=0$. Let us denote
$e=D,f=\widehat{D},h=H$ and choose in the irreducible $L$-module $V_{i}$ the
standard basis $v_{0},\dots,v_{n_{i}}$ (see, for example, [7]). Then the
element $b_{i}$ can be written in the form
$b_{i}=\alpha_{0}v_{0}+\dots+\alpha_{k-1}v_{k-1}.$ One can easily verify in
these standard notations that $D^{k-1}(\widehat{D}D(b_{i}))\neq 0$, but
$D^{k}(\widehat{D}D(b_{i}))=0$. Taking into account our choice of the element
$b_{i}$ we have $D^{k}(\widehat{D}D(b))=0$, $D^{k-1}(\widehat{D}D(b))\neq 0$,
that is $\widehat{D}D(b)\in A_{k}\setminus A_{k-1}$.

2\. Induction by $s$. If $s=2$, then

$D\widehat{D}D(b)=\widehat{D}D^{2}(b)+HD(b)=\widehat{D}D^{2}(b)+\lambda D(b),\
\lambda\in\mathbb{K},$

since $D(b)$ is isobaric by Lemma 4 (recall that by conditions of the lemma
$b$ is an isobaric polynomial). Assume the second statement of the lemma is
true for $s-1$; we will prove it for $s$. We have

$D^{s-1}(\widehat{D}D(b))=D(D^{s-2}\widehat{D}D(b))=D(\widehat{D}D^{s-1}(b)+\mu
D^{s-2}(b))$

for some $\mu\in\mathbb{K}$ by the inductive hypothesis. Then

$\displaystyle D^{s-1}(\widehat{D}D(b))=D\widehat{D}D^{s-1}(b)+\mu
D^{s-1}(b)=(\widehat{D}D+H)D^{s-1}(b)+\mu D^{s-1}(b)$ $\displaystyle=$
$\displaystyle\widehat{D}D^{s}(b)+HD^{s-1}(b)+\mu
D^{s-1}(b)=\widehat{D}D^{s}(b)+\alpha D^{s-1}(b)+\mu D^{s-1}(b)$
$\displaystyle=$ $\displaystyle=\widehat{D}D^{s}(b)+(\alpha+\mu)D^{s-1}(b)$

(here we have used the fact that $D^{s-1}(b)$ is an isobaric polynomial with
some eigenvalue $\alpha$ for $H$). This completes the proof. ∎

Next, let $a_{1},\ldots,a_{k}$ be a system of generators of the kernel
$A_{1}={\rm Ker}D$ (as a subalgebra of $A$). Without loss of generality one
can assume that the polynomials $a_{1},\ldots,a_{k}$ are isobaric. For
convenience, we write ${\widehat{D}}^{0}(a_{i})=1,i=1,\ldots k.$ Consider a
system of subsets of $A$ of the form

$S_{i}=\\{\widehat{D}^{k_{1}}(a_{i_{1}})\times\cdots\times\widehat{D}^{k_{t}}(a_{i_{t}})\
|\ a_{i_{j}}\in\\{a_{1},\dots,a_{k}\\},\;k_{j}\geq 0,\ \sum_{j=1}^{t}k_{j}\leq
i-1\\},\ i=1,2,\ldots.$

By our convention, we have $S_{1}=\\{1\\},$
$S_{2}=\\{1,{\widehat{D}}(a_{1}),\ldots,{\widehat{D}}(a_{k})\\},$ and by Lemma
4 all the elements of the set $S_{i},i\geq 1$ are isobaric polynomials.

######  Theorem 2.

Let $D$ be the basic Weitzenb$\ddot{o}$ck derivation on
$A=\mathbb{K}[x_{1},\dots,x_{n}]$. Let us choose a system of isobaric
generators $a_{1},\dots,a_{k}$ of the kernel $A_{1}=\operatorname{Ker}D$ (as a
subalgebra in $A$) and denote $A_{i}=\operatorname{Ker}D^{i},\;i\geq 1$. Then
$A_{i},\;i\geq 1$ is an $A_{1}$-module with the set of generators (as a
module) of the form

$S_{i}=\\{\widehat{D}^{k_{1}}(a_{i_{1}})\times\cdots\times\widehat{D}^{k_{t}}(a_{i_{t}})\
|\ a_{i_{j}}\in\\{a_{1},\dots,a_{k}\\},\;\sum_{j=1}^{t}k_{j}\leq i-1\\}.$

###### Proof.

Induction by $i$. For $i=1$ everything is clear because $A_{1}$ as
$A_{1}$-module is generated by the element $1.$ Assuming the statement of the
theorem to hold for $i-1$, we prove it for $i$. Let $b\in A_{i}\setminus
A_{i-1}$ be an arbitrary element. Then the element $b$ can be written in the
form $b=b_{1}+\dots+b_{t}$, where $b_{s}$ are isobaric polynomials of pairwise
different weights, $b_{s}\in A_{i},\;s=1,\dots,t,$ and at least one of the
polynomials $b_{s}$ belongs to the set $A_{i}\setminus A_{i-1}$. Therefore,
without loss of generality we can assume that $b$ is isobaric.

By Lemma 5, it holds $\widehat{D}D(b)\in A_{i}\setminus A_{i-1}$ and by the
same Lemma

(4) $D^{i-1}(\widehat{D}D(b))=\widehat{D}D^{i}(b)+\lambda D^{i-1}(b)$

for some $\lambda=\lambda(i)\in\mathbb{K}$. Since $b\in A_{i}$, we have
$D^{i}(b)=0$; thus from the equality (4) we obtain that
$D^{i-1}(\widehat{D}D(b))=\lambda D^{i-1}(b)$ for some $\lambda\in\mathbb{K}.$
Additionally, by Lemma 5, $\lambda\neq 0$. Hence, we have the equality
$D^{i-1}(\widehat{D}D(b)-\lambda b)=0$. The latter means that the element
$d:=\widehat{D}D(b)-\lambda b$ belongs to $A_{i-1}$ and
$b=\lambda^{-1}(\widehat{D}D(b)-d).$ By the inductive hypothesis $d$ is a
linear combination of elements of the set $S_{i-1}$ with coefficients in
$A_{1}=\operatorname{Ker}D$. Besides, the element $D(b)\in A_{i-1}$ and
consequently by inductive hypothesis

$D(b)=\sum_{s_{j}\in S_{i-1}}c_{j}s_{j},\;c_{j}\in A_{1}.$

Applying $\widehat{D}$ to the both sides of the last equality we get

(5) $\widehat{D}D(b)=\sum_{s_{j}\in
S_{i-1}}(\widehat{D}(c_{j})s_{j}+c_{j}\widehat{D}(s_{j})).$

Note that $\widehat{D}(s_{j})$ is a linear combination of elements of $S_{i}$
with integral coefficients. In addition, for $c_{j}\in
A_{1}=\operatorname{Ker}D$ we have that $c_{j}=c_{j}(a_{1},\dots,a_{k})$ is a
polynomial in the generators $a_{1},\dots,a_{k}$ of the kernel
$\operatorname{Ker}D=A_{1}$. It is easy to see that

$\widehat{D}(c_{j})=\sum_{s=1}^{k}\frac{\partial c_{j}}{\partial
a_{s}}(a_{1},\dots,a_{k})\widehat{D}(a_{s}),$

and hence $\widehat{D}(c_{j})$ is a linear combination of elements of $S_{2}$
with the coefficients in $A_{1}$. Therefore, from the equality (5) we see that
$\widehat{D}D(b)$ is a linear combination of elements of $S_{i}$ with the
coefficients in $A_{1}=\operatorname{Ker}D$. The latter means that $S_{i}$ is
a system of generators of $A_{1}$-module $A_{i}.$ The proof is completed. ∎

######  Theorem 3.

Let $D$ be the basic Weitzenb$\ddot{o}$ck derivation on
$A=\mathbb{K}[x_{1},\dots,x_{n}]$, let $a_{1},\dots,a_{k}$ be a system of
generators of the kernel $A_{1}=\operatorname{Ker}D$ (as a subalgebra in $A$.)
If

$S_{n}=\\{\widehat{D}^{k_{1}}(a_{i_{1}})\times\cdots\times\widehat{D}^{k_{t}}(a_{i_{t}})\
|\ a_{i_{j}}\in\\{a_{1},\dots,a_{k}\\},\;\sum_{j=1}^{t}k_{j}\leq n-1\\},$

then $C_{W_{n}(\mathbb{K})}(D)$ has a system of generators $\\{D_{s}\ |\ s\in
S_{n}\\}$ (as a module over $\operatorname{Ker}D$), where
$D_{s}=\sum_{i=1}^{n}D^{n-i}(s)\frac{\partial}{\partial x_{i}}.$

###### Proof.

For any element $s\in S_{n}$ we have $D^{n}(s)=0$. Let $T$ be an arbitrary
element of the centralizer $C_{W_{n}}(D)$. Then, by Proposition 1,
$T=D_{b},\;b\in A_{n}.$ By Theorem 2, the element $b$ is of the form
$b=\sum_{s\in S_{n}}r_{i}s_{i}$, where $r_{i}\in\operatorname{Ker}D$.
Therefore $D_{b}=\sum_{s\in S_{n}}r_{i}D_{s_{i}}$. But then
$\\{D_{s_{i}},\;s_{i}\in S_{n}\\}$ is a system of generators of the
$A_{1}$-module $C_{W_{n}}(D)$. ∎

######  Example 1.

Let $D_{3}$ be the basic Weitzenb$\ddot{o}$ck derivation $D_{3}$ on the
polynomial ring $\mathbb{K}[x_{1},x_{2},x_{3}].$ Let us build a system of
generators for the centralizer $C_{W_{3}(\mathbb{K})}(D_{3})$ as a module over
${\rm Ker}D_{3}.$

The kernel $D_{3}$ is generated by isobaric polynomials $a_{1}=x_{1},\
a_{2}=x_{1}x_{3}-(1/2)x_{2}^{2}$ (see, for example, [2]). Note that

${\widehat{D}}(x_{1})=x_{2},{\widehat{D}}(x_{1}x_{3}-(1/2)x_{2}^{2})=0,{\widehat{D}}^{2}(x_{1})=x_{3},{\widehat{D}}^{2}(x_{1}x_{3}-(1/2)x_{2}^{2})=0.$

Then the $A_{1}$-modules $A_{1}$, $A_{2}$, $A_{3}$ have the systems of
generators

$S_{1}=\\{1\\},\ S_{2}=\\{1,\ x_{2}\\},\ S_{3}=\\{1,\ x_{2},\ 2x_{3},\
4x_{2}^{2}\\},\ {\rm respectively.}$

By Theorem 3, we obnain a system of generators of the centralizer
$C_{W_{3}(\mathbb{K})}(D_{3})$ as a $A_{1}$-module

$\partial_{3},\ x_{1}\partial_{2}+x_{2}\partial_{3},\
2x_{1}\partial_{1}+2x_{2}\partial_{2}+2x_{3}\partial_{3},\
8x_{1}^{2}\partial_{1}+8x_{1}x_{2}\partial_{2}+4x_{2}^{2}\partial_{3}.$

######  Remark 2.

The systems of generators, constructed in Theorems 2 and 3 are not necessarily
minimal.

##  4\. Centralizers of locally nilpotent derivations

In this section, we assume that $A$ is a finitely generated domain over an
arbitrary field $\mathbb{K}$ of characteristic zero. The next lemma contains
some known properties of locally nilpotent derivations on such domains.

######  Lemma 6 (see, for example, [5]).

Let $D\neq 0$ be a locally nilpotent derivation on a finitely generated domain
$A$ over a field $\mathbb{K}$ of characteristic zero and $C={\rm Ker}_{A}(D).$
Then

  1. (1)

$C$ is a subalgebra of $A$ that is invariant under any
$T\in\operatorname{Der}_{\mathbb{K}}A$ commuting with $D$;

  2. (2)

if $D$ has a slice $s$ in $A$, i.e. there exists $s\in A$ such that $D(s)=1$,
then $A$ is the polynomial ring in $s$ over $C$, that is $A\simeq C[s]$;

  3. (3)

if $D$ does not have any slices in $A$, then there exist $h_{0}\in
C,\;h_{1}\in A$ such that the element $s=h_{1}/h_{0}$ is a slice for the
extension $\overline{D}$ of $D$ on the ring $A[h_{0}^{-1}]$ and the kernel
$\overline{D}$ in $A[h_{0}^{-1}]$ coincides with $C[h_{0}^{-1}]$.

######  Proposition 2.

Let $D\neq 0$ be a locally nilpotent derivation of a domain $A$ over
$\mathbb{K}$ and $C={\rm Ker}D$ in $A.$ If $D$ has a slice in $A,$ then
$C_{\operatorname{Der}_{\mathbb{K}}A}(D)\simeq
C\rightthreetimes\operatorname{Der}_{\mathbb{K}}C,$ where $C$ is considered as
a Lie algebra over $\mathbb{K}$ with zero multiplication and the action of
$\operatorname{Der}_{\mathbb{K}}C$ on $C$ is natural.

###### Proof.

It follows from Lemma 6 that $A\simeq C[s].$ Let $T$ be an arbitrary element
of $C_{\operatorname{Der}_{\mathbb{K}}A}(D).$ Then the equality $[T,D]=0$
implies $T(s)\in C.$ Denote $c_{0}=T(s).$ It is easy to see that $T-c_{0}D\in
C_{\operatorname{Der}_{\mathbb{K}}A}(D)$ and $(T-c_{0}D)(s)=0$. Consider two
subspaces $L$ and $M$ of $C_{\operatorname{Der}_{\mathbb{K}}A}(D)$ of the form

$L=\\{T\in C_{\operatorname{Der}_{\mathbb{K}}A}(D)\ |\ T(s)=0\\},\ M=\\{T\in
C_{\operatorname{Der}_{\mathbb{K}}A}(D)\ |\ T(C)=0\\},{\rm respectively}.$

It is easy to check that $L$ is a subalgebra of
$C_{\operatorname{Der}_{\mathbb{K}}A}(D),$ $M$ is an abelian ideal from
$C_{\operatorname{Der}_{\mathbb{K}}A}(D)$ and $M=C\cdot D.$ One can easily
show that $C_{\operatorname{Der}_{\mathbb{K}}A}(D)\simeq M\rightthreetimes L$
is a semidirect sum. Note also that $L\simeq\operatorname{Der}_{\mathbb{K}}C$
because any derivation of the algebra $C$ over $\mathbb{K}$ can be uniquely
extended to a derivation $T$ of the algebra $A$ with $T\in
C_{\operatorname{Der}_{\mathbb{K}}A}(D)$ (it suffices to put $T(s)=0$). ∎

######  Theorem 4.

Let $A$ be a finitely generated domain over a field $\mathbb{K}$ of
characteristic zero and $D\neq 0$ be a locally nilpotent derivaton on $A$.
Then the centralizer $C_{\operatorname{Der}_{\mathbb{K}}A}(D)$ has rank $n$
over $A,$ where $n=tr.\deg_{\mathbb{K}}A.$

###### Proof.

First, let the derivation $D$ have a slice $s$ in $A$. It follows from Lemma 6
that ${\rm tr.deg}_{\mathbb{K}}C=n-1.$ Choose any transcendence basis
$c_{1},\dots c_{n-1}$ of $C$ (over the field $\mathbb{K}$) and consider
partial derivations $\frac{\partial}{\partial c_{i}},\;i=1,\dots,n-1$ on the
subalgebra $\mathbb{K}[c_{1},\dots,c_{n-1}]\subseteq C.$ It is known that
these derivations can be uniquely extended on the field of fractions
$\mathrm{Frac}(C)$ (we use the same symbols for extended
$\frac{\partial}{\partial c_{i}}$ when no confusion can arise). Let
$S=C\setminus\\{0\\}$ and let $A[S^{-1}]$ be the ring of fractions. Since
$A=C[s],$ we have $A[S^{-1}]=\mathrm{Frac}(C)[s].$

Further, let us extend the derivations $\frac{\partial}{\partial
c_{i}},\;i=1,\dots,n-1$ on $A[S^{-1}]$, putting $\frac{\partial}{\partial
c_{i}}(s)=0,\;i=1,\dots,n-1$. Let $a_{1},\dots,a_{k}$ be a system of
generators of the algebra $A.$ Denote $\frac{\partial}{\partial
c_{i}}(a_{j})=e_{ij}/d_{ij}$ for some $e_{ij}\in A,\,d_{ij}\in
S=C\setminus\\{0\\},$ and put $d=\prod_{i,j}d_{ij}.$ Then we obviously have
$d\in C$ and $d\frac{\partial}{\partial c_{i}}(a_{j})\in
A,\;i=1,\dots,n-1,\,j=1,\dots,k.$. The latter means that
$d\frac{\partial}{\partial c_{i}}(A)\subseteq A,\;i=1,\dots,n-1.$ It is easy
to see that the derivations $d\frac{\partial}{\partial c_{i}}$ commute with
the derivation $D$ on $A$ (recall that we extended $\frac{\partial}{\partial
c_{i}}$ on $A[S^{-1}]$). One can easily show that the derivations
$D,d\frac{\partial}{\partial c_{i}},i=1,\ldots,n-1$ on the domain $A$ are
linearly independent over $R$ and form a basis of the vector space
$RC_{\operatorname{Der}A}(D)$ over the ring $R=\mathrm{Frac}(A)$. The latter
means that $C_{\operatorname{Der}A}(D)$ has rank $n$ over $A.$

Now let $D$ do not have any slices in $A.$ Then, by Lemma 6, there exists a
local slice $h_{1}\in A$ such that $D(h_{1})=h_{0}\neq 0$ and $s=h_{1}/h_{0}$
is a slice for the extension $\overline{D}$ of the derivation $D$ on the ring
of fractions $A[h_{0}^{-1}].$

Every derivation $T$ of the ring $C={\rm Ker}D$ can be uniquly lifted to the
derivation $\overline{T}$ of the ring $A[h_{0}^{-1}]$ by putting
$\overline{T}(s)=0$ (because $A[h_{0}^{-1}]=C[h_{0}^{-1}][s]$ and the case of
slice was considered above). It is easy to check that
$[\overline{T},\overline{D}]=0$, where $\overline{D}$ is the extension of $D$
on the ring $A[h_{0}^{-1}]$ (clearly, $\overline{D}$ is a locally nilpotent
derivation and $\operatorname{Ker}\overline{D}=C[h_{0}^{-1}]$). For the system
of generators $a_{1},\dots,a_{k}$ of the algebra $A$ we have
$\overline{T}(a_{i})={b_{i}}/(h_{0}^{k_{i}})$ for some $b_{i}\in A$ and
$k_{i}\geq 0.$ Denote $k=\max k_{i}$. Then $h_{0}^{k}\overline{T}$ commutes
with $\overline{D}$ and the inclusion $h_{0}^{k}\overline{T}(A)\subseteq A$
holds. As the multiplication of derivations from $C_{\operatorname{Der}A}(D)$
by the powers of $h_{0}$ does not change the rank of
$C_{\operatorname{Der}A}(D)$ over $R=\mathrm{Frac}(A)$, we can repeat
considerations from the case when there exists a slice of $D$ in $A$ and
obtain the desired equality $\dim_{R}RC_{\operatorname{Der}A}(D)=n$.

∎

## References

  * [1] V.V. Bavula, Lie algebras of triangular polynomial derivations and an isomorphism criterion for their Lie factor algebras, Izvestiya: Mathematics, (2013), v.77, issue 6, 1067–1104.
  * [2] L.Bedratyuk, Kernels of derivations of polynomial rings and Kazimir elements, Ukrainian Mathematical Journal, vol.62, no.4 (2010), 495–517.
  * [3] Y.Chapovskyi, D.Efimov, A.Petravchuk, Centralizers of elements in Lie algebras of vector fields with polynomial coefficients, Proceedings of the International Geometry Center, (2021), 14 (4), 257–270.
  * [4] D.R. Finston and S.Walcher, Centralizers of locally nilpotent derivations, J. Pure and Appl. Math., 120 (1997), 39–49.
  * [5] G.Freudenburg, Algebraic Theory of Locally Nilpotent Derivations, _Encyclopedia of Mathematical Scieces_ , vol.136, Springer, 2006.
  * [6] F.R. Gantmacher, The theory of matrices (Chelsea, New York, 1959). Vol. I, 1988\.
  * [7] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer Verlag, New York, 1972.
  * [8] M.Miyanishi, Normal affine subalgebras of a polynomial ring, in: Algebraic and Topological Theories, Kinokuniya, Tokyo, 1985, 37–51.
  * [9] Joel Nagloo, Alexey Ovchinnikov, Peter Thompson, Commuting planar polynomial vector fields for conservative newton systems. Communications in Contemporary Mathematics, 22(04):1950025, 2020.
  * [10] Andrzej Nowicki. Polynomial derivations and their rings of constants. Uniwersytet Mikolaja Kopernika Torun, 1994.
  * [11] A. Nowicki A., M. Nagata, Rings of constants for $k$-derivations in $k[x_{1},\ldots,x_{n}]$, J. Math. Kyoto Univ., 28 (1988), 111–118.
  * [12] A.P.Petravchuk, O.G.Iena, On centralizers of elements in the Lie algebra of the special Cremona group ${\rm SA}_{2}(k)$, J. Lie Theory. (2006), v.16, no.3, 561–567.
  * [13] D. Panyushev, Two results on centralisers of nilpotent elements, Journal of Pure and Applied Algebra, (2008), v.212, Issue: 4, 774-779.
  * [14] R.Weitzenb$\ddot{\rm o}$ck, Ueber die Invarianten von linearen Gruppen, Acta Math., (1932) 58, 231–293.