\section{Introduction}
The study of supernova remnants lies at the crossroads of astrophysics,
combining the interests in the end products of stellar evolution
with the investigation of the interstellar medium and cosmic ray acceleration.
This is
all the more interesting now that Type Ia supernovae \citep[e.g.][]{tonry03}
have become central to the study of the geometry of the universe and
gamma ray bursts are known to originate from a subset of core
collapse supernovae \citep{stanek03}.

\begin{figure}
\hbox{\psfig{figure=oxygen_ionization_annotated.ps,width=0.9\textwidth}}
\caption{An illustration of the effects non-equilibration ionization.
Although the top and bottom panels look very similar, the top panel
shows the oxygen ionization fraction as a function of electron
temperature for the case of collisional ionization equilibrium (CIE),
whereas the bottom panel shows how in the case of recently heated
plasmas (NEI) the ionization fraction at a fixed temperature of \kte$=1.5$~keV
changes as a function of
ionization age, which is the product
of electron density and time since the plasma was heated (\net).
Note the sturdiness of the He-like ion, which is present over
a wide temperature or \net\ range.
The calculations are based on the simple to use,
but somewhat outdated ionization and recombination rates in \citep{shull82}.
}
\label{ionization}
\end{figure}

\begin{figure}
\psfig{figure=equilibration.eps,angle=-90,width=0.6\textwidth}
\caption{An illustration of the effect of non-equilibration of temperatures,
taking into account ionization effects, but not adiabatic expansion.
The temperature of the electrons (dotted), protons (solid), helium
(dashed-dotted) and oxygen ions (dashed) is proportional to their mass ratio
behind the shock.
Due to Coulomb interactions the temperatures slowly equilibrate.
It is interesting to note that the oxygen-proton equilibration will
eventually be faster than the helium-proton equilibration,
as the oxygen ions acquires a higher charge and the interactions
scale with $Z^2$\
\citep{zeldovich66,nrlplasma}.
}
\end{figure}

However, of more interest for these proceedings
is that supernova remnants are objects in which interesting plasma processes
take place.
Supernova remnant shocks move through a medium with a
typical density of 1~\pcc. At those low densities Coulomb (particle-particle)
interactions are rare, leading to typical collision time scales
of several thousand years.
Shock heating does, however, occur in supernova remnants,
otherwise we would not observe X-ray emission from hot plasmas.
This implies that the heating process takes place through collective effects, involving
plasma waves.
Apart from being collisionless, supernova remnant shocks are also
the most probable source of cosmic rays,
at least for energies up to $\sim 10^{15}$~eV, but possibly even up to
$\sim 10^{18}$~eV \citep[][]{bykov04,vink04b}.

The fact that supernova remnant shocks are collisionless and are
the locations of particle acceleration has two important consequences.
First of all, we can no longer assume that different particles species
(protons and other ions and electrons) are in temperature equilibration.
In an extreme case, and ignoring particle acceleration,
the temperature, $kT_i$, of each plasma component, $i$, is:

\begin{equation
        kT_{i} = \frac{2(\gamma-1)}{(\gamma+1)^2} m_i v_s^2  =
	\frac{3}{16} m_i v_s^2,
       \label{eq-shocks}
\end{equation
where $\gamma$ is the adiabatic index,
$m_i$, is the particle mass, and $v_s$ is the shock velocity.
For full equilibration this is $kT = 3/16 <m>v_s^2$.
In addition,
we have to consider that efficient
cosmic ray acceleration, and possible magnetic field generation, will take
away energy, lowering the plasma temperature(s) of the thermal plasma
components \citep{hughes00b}.

As a result measuring the plasma temperature through X-ray spectroscopy,
which usually will give us the {\em electron} temperature,
will no longer inform us directly about the shock velocity.
This has been known for quite some time \citep[e.g.][]{itoh77,itoh84},
but until recently it was ignored consciously \citep[e.g.][]{jansen88}
or subconsciously, as it was difficult to assess the amount of
temperature equilibration from the observational data.

However, another form of non-equilibration,
namely non-equilibration of ionization (NEI), has received more attention
over the last three decades, because its consequences could be more easily
deduced from X-ray spectra of sufficient quality
\citep[e.g][]{winkler81,
gronenschild82,hughes85,jansen88}.

The concept of  NEI is relatively simple \citep{itoh77,mewe80,liedahl99}.
NEI is important in supernova remnants as
in the relatively recently heated plasmas the ions have not had
sufficient time to ionize up to the ionization
level at which the number of ionizations is compensated
by the number of recombinations
(i.e. collisional ionization equilibration or CIE).
The effect of NEI is illustrated in
Fig.~\ref{ionization}. Observationally NEI gives rise to a mismatch between
the electron temperature derived from line emission ratios,
and the electron temperature derived from the continuum shape, which reflects
the actual electron temperature.
In addition, spectra of NEI plasmas will display lines that are unique
for NEI, and are the result of
inner shell excitations and ionizations.
The Fe-K line emission is most widely used as a diagnostic tool
for NEI conditions, as for a plasma with \kte$ \gtrsim 2$~keV the centroid
of the $n=2\rightarrow1$\ line emission shifts gradually
from $\sim6.4$~keV to $\sim 6.7$~keV for Fe XXV \citep{palmeri03,mendoza04}.

Here I discuss high resolution spectra of a supernova
remnant that is one of the most extreme cases of non-equilibration
of ionization and temperatures, \snts.

\begin{figure}
\hbox{
\vbox{
\psfig{figure=sn1006_acis_pointings.ps,width=0.5\textwidth}
\vskip 4mm
}

\psfig{figure=profile_sn1006nw.eps,width=0.5\textwidth}
}
\caption{\chandra\ mosaic of SN 1006. The circles indicate the \xmm\ pointings discussed here.
The emissivity profile is taking in the dispersion direction of the
RGS
\citep[from][]{vink03b}.}
\label{image}
\end{figure}

\begin{sidewaysfigure}
\hbox{
\psfig{figure=sn1006nw_rgs12_o1_annotated.eps,width=0.9\textheight}
}
\caption{The \xmm-RGS spectrum of the northwestern knot.}
\end{sidewaysfigure}

\section{The \xmm-RGS spectra of SN 1006}
The young historical remnant SN 1006 is thought to be a Type Ia supernova
remnant on account of its high Galactic latitude (+14.6)
and the long duration of the historical light curve \citep{stephenson02}.
The high latitude also explains the relatively low density of the local
medium of
$\sim 0.1$~cm$^{-3}$\
\citep[][]{vink00a,allen01,dyer04,long03}.
This makes \snts\ the ideal candidate for studying the two non-equilibration
effects,
as it has an ionization parameter of \net$\sim 2\times 10^9$~\netunit.
The distance to \snts\ is 2.1~kpc \citep{winkler03}.

\xmm\ observed \snts\ several times, both as a guaranteed time and guest
observer target.
I will discuss here the guest observer observations of August 2002, which consisted
of two pointings.
One pointing was on the bright knot at the northwestern edge of the remnant,
the other was on a narrow feature on the eastern rim (Fig.~\ref{image}).
SN 1006 is a large remnant (30\arcmin), but the pointings were optimized
in order to obtain high resolution spectra of bright, small structures.
Moreover, the pointings were on the rest of the remnant where it is unlikely that
there are bulk plasma motions in our line of sight. This is necessary in order to
allow estimates of the thermal line broadening.

A sufficiently high resolution spectrum was obtained for the
northwestern pointing, but for the eastern rim, only the non-thermal X-ray emission is very narrow, whereas
the thermal emission is much more extended, degrading the spectral resolution.
This is somewhat unfortunate, as the electron and ion temperatures of the plasma may be
influenced by efficient cosmic ray acceleration, and non-thermal emission from the rims of SN 1006
indicate a very efficient electron cosmic ray acceleration at the west and eastern rims, but
not at the northwestern edge \citep{koyama95,rothenflug04}.

\begin{figure}
\hbox{
  \psfig{figure=sn1006nw_epic_brspot.ps,angle=-90,width=0.5\textwidth}
  \psfig{figure=sn1006_nwknot_mos12_SiK.ps,angle=-90,width=0.5\textwidth}
}
\caption{The \xmm\ PN and MOS spectra (left) and a detail of the Si emission as observed
by the MOS detectors (right). The shifted line shows the position of the Si
emission if most of the Si would have been in the He-like state. }
\label{ccdspectra}
\end{figure}

\subsection{The X-ray spectrum of the northwestern knot}

The \xmm\ Reflection Grating Spectrometer \citep[RGS,][]{denherder01}
spectrum of the northwestern knot shows
many spectral details that cannot be seen in CCD spectra.
The most dominant emission is
coming from O VII, in particular from the O VII triplet, but also O VII line
emission
from higher quantum levels up to the series limits.
The line emission around 19~\AA\ is a mix of O VII He$\beta$\ emission and
O VIII Ly$\alpha$
emission. Lines from Fe-L transitions, in particular Fe XVII which has lines
around 15~\AA and 17~\AA\ are conspicuously absent.
This may be surprising given the fact that SN\ 1006 is a likely Type Ia
remnant, which produce $\sim 0.7$~\msun\ of Fe. However, for the low ionization
age of \snts\ Fe is expected to have a peak ionization state of Fe XIII,
so no Fe XVII emission is to be expected.

The CCD spectra were modeled using the SPEX\footnote{
\url{http://www.sron.nl/divisions/hea/spex/version2.0/index.html}}
\citep{kaastra96} spectral code.
The detailed results can
be found in \citep{vink03b} and confirm the low ionization age of the plasma,
\net $= 2.4\times10^9$~\netunit, whereas \kte$=1.5\pm0.2$keV.
There is a relative overabundance of Si, which suggests
that the knot is an ejecta knot, and not a shocked pre-existing density
enhancement
in the interstellar medium. Interestingly, there is a weak enhancement of
X-ray emission
in front of the knot as well, that seems to emit slightly harder X-ray emission.
It is not clear how these relate to each other.

\begin{figure}
\hbox{
 \psfig{figure=centroids.ps,width=0.5\textwidth}
 \psfig{figure=sn1006_net.ps,width=0.5\textwidth}
}
\caption{
Line
energy versus charge for O, Ne, Mg, and Si. The crosses give the expected
line energies for different ionization states (from Dr. M.F. Gu's FAC code)
and the
arrows approximately indicate the observed centroid energies.}
\label{centroids}
\end{figure}

The low ionization age of the plasma can be deduced not only by globally
fitting the spectrum, but
also by measuring the centroid of each individual line, or by modeling the
O~VII triplet.
For instance the EPIC PN and MOS spectra clearly indicate that the centroid of
the Si emission
is $1.80\pm0.01$~keV instead of at 1.85~keV, which is the centroid expected if
Si is in the He-like
state (Fig.~\ref{ccdspectra}). Also the Ne and Mg line centroids are at lower
energies
than the He-like.  Purely combining the information of the centroids,
and assuming
\kte $=1.5$~keV one also finds an ionization age of
\net\ $\sim 2\times10^9$~\netunit\
(Fig.~\ref{centroids}).
The most important inner shell lines are present in the SPEX spectral code,
but in general one should be aware that spectral codes are not always
complete in this sense, as most atomic databases were initially compiled with
CIE plasmas in mind. Moreover, the line parameters may also be less accurate,
as less experimental data exists to measure the line properties.

\subsection{Non-equilibration of temperatures in the northwestern knot}
One of the main goals of the \xmm\ observations was to measure the amount of
non-equilibration
of electron and ion temperatures, by measuring the thermal Doppler broadening
of O VII and O VIII
line emission. So far evidence for non-equilibration in supernova remnants
comes mainly from measuring
the H$\alpha$\ line width and broad to narrow line ratios.
This has been done for several remnants including \snts\
\citep{ghavamian02,rakowski03}.
The amount of non-equilibration
has also been measured in the UV from C IV, N V and O VI lines by
\hut\ \citep{laming96} and very recently by \fuse\ \citep{korreck04}.
The optical and UV results indicate that there is no equilibration at the
shock front, although  the \fuse\ results
suggests that the temperature of different species is not proportional to the
species mass, as implied by eq.~\ref{eq-shocks} and the \hut\ spectrum.

The O VII He$\alpha$\ emission consists of the He triplet, plus a potential
contribution from an O VI line very close
to the O VII forbidden line emission.
In order to measure the Doppler broadening models for the line ratios
for a grid of \net\ values were fitted, using parameters obtained from
the flexible  atomic code (FAC) of Dr. M. F. Gu \citep{gu03}.
The spectrum of the RGS1 O VII spectrum and the best fit models are shown in
Fig.~\ref{broadening}.
The best fit suggests a very significant broadening at a confidence level of
9.8$\sigma$.
Allowing full freedom of the line normalizations still indicates thermal
broadening, at a still high confidence level
of 6.5$\sigma$. The broadening itself is $\sigma_E = 3.4\pm0.4$~eV.
The published results are based on a response matrix that was corrected for
the emissivity  profile shown in Fig.~\ref{image}.
However, the fitting program XSPEC also contains the convolution model
{\em rgsxsrc} made by Dr. A. Rasmussen, which uses an input image to determine
the emissivity profile. Using {\em rgsxsrc} gives very similar results.

The derived value for the broadening implies an oxygen temperature of
$kT = 530\pm150$~keV, which looks incredibly high,
but is in effect consistent with eq.~\ref{eq-shocks} if the gas was
shocked with a velocity of $\sim 5000$~{km\,s$^{-1}$}.
Of course the oxygen temperature is more than a factor 300 higher than the
electron temperature, clearly indicating non-equilibration of temperatures.
The measured O VII line broadening is more than reported for O VI UV emission
from other regions of the northwestern rim
\footnote{Unfortunately in \citep{korreck04} the measured broadening in
$\sigma$ is mistakenly taken to be a FWHM.} \citep{korreck04}.
There are several reasons why there could be a discrepancy.
The knot may be an ejecta knot, as implied by the high Si abundance,
and the heating may have been caused by the reverse shock,
which has a different speed than the shock.
Another possibility is that the plasma of the knot was shocked earlier than
the regions observed in O VII, after all the X-ray
measurements concern more highly ionized material, which is
situated further downstream from the shock.

\begin{figure}
\psfig{figure=o_triplet.ps,angle=-90,width=0.64\textwidth}
\caption{The O\,VII triplet from the northwestern not as observed by the RGS1.
The dotted line shows the best fit spectrum with (solid) and without including
thermal Doppler broadening.}
\label{broadening}
\end{figure}

\subsection{What about the eastern rim?}

The eastern and western rims of SN 1006 are likely sites of efficient cosmic
ray shock acceleration \citep{koyama95}.
If the cosmic ray pressure dominates over the thermal pressure,
eq.~\ref{eq-shocks} is no longer
valid. It will result in an average plasma temperature that is lower. Moreover,
if the cosmic ray is very efficient and the cosmic ray power law
index is equal to or flatter than -2, the energetic cosmic rays that escape
the shock will drain the shock of its energy and the shock will become
(cosmic) radiative, resulting in much higher shock compression ratios
\citep{ellison04}.

Given the interesting physics at the eastern and western rim, it is
unfortunate that the RGS spectrum from the eastern rim
does have a high spectral resolution,
especially since no UV emission lines from the eastern rim have been
detected by \fuse\ \citep{korreck04}.
It is therefore still worthwhile to obtain as much information from
the RGS spectrum as possible. I will therefore discuss here a preliminary
analysis of RGS spectrum.

The effective spectral resolution is mainly determined by the width of the
shell, about 5\arcmin, resulting in an effective resolution for the first
order spectrum of $\sim0.6$~\AA.
This is not sufficient to resolve the O VII triplet, which is also for the
eastern rim the dominant line emission. However, there is still high spectral
resolution information in the spectrum, which results from the sharp increase
of the emission at the edge of the remnant.

The O VII triplet emission is displayed in Fig.~\ref{eastern}.
It shows that a model that does only contain the triplet lines
and an instrumental response that only takes into account the spatial
emission profile does not give an adequate fit.
The long wavelength side seems to fit reasonably well,
but there is a dip in the data around 21.5~\AA, and the model underpredicts
the emission around 21.3~\AA. Note that here the interior of remnant is
dispersed in an opposite direction compared to
the spectrum of the northwestern knot. So the edge of the remnant determines
the shape of the long wavelength side. This suggests that discrepancy is
caused not so much by line broadening, but by line shifts.
The plasma motions at the  edge of the remnant, will be mostly in the plane of
the sky, whereas going to the interior of the remnant radial
motions will give rise to Doppler shifts.
As we are observing both the front and backside of the shell,
both blueshifts and redshifts should be taken into account.
The redshifts are not directly noticeable
(they blend in with the rest of the triplet emission),
but the blueshifts can be seen as the mismatch between model and spectrum
around 22.3~\AA.

I have attempted to fit a simple model for the O VII triplet,
taking into account the
increase in blueshifts and redshifts moving interior to the remnant
(corresponding to shorter wavelength).
As it is a mixing of spectral and spatial information,
the method consisted in making a grid of response
matrices in which the spatial and redshift and blueshift were incorporated.
The input to the model consisted of the projected plasma velocity,
which increased toward the interior,
and a parameter that determined how much of the emission was coming from the
near and far side of the remnant.
This is a simplification, as velocity gradients are presents
and the morphology of
the shell may be complicated. It was nevertheless possible to obtain
acceptable fits to
the O VII triplet emission (Fig.~\ref{eastern}).

It turns out there are two good fitting models.
One with no, or only a small additional (thermal) broadening
($\sigma_E = 0.4\pm0.3$~eV), and
a shell with
velocity $6500\pm830$~{km\,s$^{-1}$}\ (68\% confidence range, Fig.~\ref{eastern}).
The other solution is for a much higher shell velocity of $> 9000$~{km\,s$^{-1}$},
and quite a substantial additional broadening of $\sigma_E = 3\pm1$~eV.
In both cases the best solution suggests that about 70\% emission
comes from material moving toward us ($>55\%$ at a 3$\sigma$ confidence).
The solution with a large broadening gives a better fit (Fig.~\ref{eastern}),
but is less plausible. The reason is that a bulk velocity of $>9000$~{km\,s$^{-1}$}
would imply that the shock heated plasma is still in free expansion,
whereas a solution with $v\sim6500$~{km\,s$^{-1}$}\ fits in much better
with radio proper motions \cite{moffet93}.
The average shock velocity of \snts, given its age and distance is
$<v>=9375$~{km\,s$^{-1}$}.
The radio expansion measurements suggest that
$v_s = (0.48\pm0.13)<v> = (4500\pm1220)$~{km\,s$^{-1}$}.
Both solutions exclude the possibility that the plasma velocity is
similar to what has been measured for the northeastern H$\alpha$\
filaments $v_s = 2800$~{km\,s$^{-1}$}\ \citep{ghavamian02}. This is consistent with
the smaller radius in the northwest and the finding of an increased
density as indicated by
21cm line measurements \citep{dubner02}.

The shape of the O VII triplet suggests that more material
is blueshifted than redshifted. This asymmetry is
reminiscent of the difference in forward shock and reverse shock
velocity found in absorption toward a  UV star projected on the center
of \snts\ \citep{hamilton97}. The reverse shock appears to have penetrated
more into the innerejecta layers of the front side than on the backside.
Such an asymmetry is likely to
be the result of a higher density at the front side,
which will also result in more shocked heated plasma at the front side.

\begin{figure}
\vbox{
\hbox{
\vbox{
 \psfig{figure=sn1006e1_0kms.ps,angle=-90,width=0.5\textwidth}
\vskip 2mm
}
 \psfig{figure=sn1006velocity.ps,angle=-90,width=0.5\textwidth}
}
\vskip 2mm
\hbox{
  \psfig{figure=sn1006e_6500_f0.7_combined.ps,angle=-90,width=0.5\textwidth}
  \psfig{figure=sn1006e_9500_f0.7_combined.ps,angle=-90,width=0.5\textwidth}
}}
\caption{
The O VII triplet from the eastern rim as observed by RGS1.
Top left: The best fit model with only taking into account
the spatial profile.
Top right: the likelihood statistic as a function of shell velocity,
for a front back asymmetry of 70\%. The arrows indicate other velocity
measurements from respectively the northwest, radio and the average velocity.
Bottom: The left panel shows the best fit solution
for a shell velocity of 6500~{km\,s$^{-1}$} with no broadening. The right panel
shows a solution with a velocity of 9500~{km\,s$^{-1}$}\ including substantial
thermal broadening. }
\label{eastern}
\end{figure}

\section{Concluding remarks}

\snts\ is from many points of view an interesting supernova remnants.
It is one of the few historical remnants, and it was the first remnant
for which it was established that the particle acceleration by the shock
was efficient enough to produce X-ray synchrotron emission.
The fact that \snts\ expands into a low density medium makes it
also an ideal laboratory to study the effects of
non-equilibrium ionization and non-equilibration of temperatures,
both of which depend on the value of \net, which is around
$2\times10^9$~\netunit\ for \snts, one of the lowest values found
in a supernova remnant.

The measurements presented here show the diagnostic power of high resolution
spectroscopy when it comes to characterizing the ionization
state, temperature and velocities of hot plasma.
The results on a remnant like \snts\
raises the expectation for the results from micro-calorimeters carried
by \astroe, and {\it Constellation-X/Xeus},
which will make it much more straightforward to measure Doppler motions
from extended sources.

I thank the organizers of the XDAP 2004 workshop for their invitation and
financial support.


\section{Introduction}

The formation and evolution of elliptical galaxies continues to be an
outstanding problem in astrophysics (e.g. Meza et al. 2003, Wright et
al. 2003). In the current CDM models of hierarchical structure formation,
mergers are the common process through which larger objects form from
small ones (Frenk et al. 1988, Kauffmann \& White 1993, Lacey \& Cole 1993).
In this scenario, ellipticals are a `natural' outcome (Toomre 1977) and
accumulating observational evidence seems to support this
(Schweizer 1998 and references therein). However, an important challenge
to the merger hypothesis is to explain how this seemingly random merging
process can lead to the tight correlations found in ellipticals (White 1997).

One of this correlations found in ellipticals is the so-called Fundamental
Plane (FP); a linear relation, in the logarithmic space, among their
effective radius, $R_{\mathrm e}$, their effective surface
brightness, $I_{\mathrm e}$, and their one-dimensional central
velocity dispersion, $\sigma_0$ \citep{dd87,Dressler87}:  $R_\mathrm{e}
\!\propto \! \sigma_0^a I_\mathrm{e}^b$. The FP exhibits a
general trend that follows from the application of the virial theorem to
ellipticals: $M \propto V^2 R_\mathrm{g}$,
with $M$ being the total mass, $V^2$ the 3-dimensional mean square
velocity and $R_\mathrm{g}$ the gravitational radius of the system
($V^2=2K/M$ and $R_\mathrm{g}=G M^2/|W|$, where $K$ and $W$ are
the total kinetic
and potential energy, respectively). Assuming that ellipticals are
homologous systems in equilibrium
($ V^2 \! \propto \! \sigma_0^2$, $R_\mathrm{g} \! \propto \!
R_\mathrm{e}$) with a constant mass-to-light ratio,
the virial theorem leads to
\begin{equation}\label{eq:virial}
R_\mathrm{e} \propto \sigma_0^2 I_\mathrm{e}^{-1}\, \to \,
\log R_\mathrm{e} \approx 2[\log \sigma_0 + 0.2  \mu_\mathrm{e}]
\end{equation}
where $\mu_\mathrm{e}=-2.5 \log I_\mathrm{e}$. Deviations of an observed
FP from the theoretical expectation are attributed to variations of
the mass-to-light ratio with
luminosity and/or to the breaking of
homology in galaxies (e.g. Trujillo, Burkert \& Bell 2004).

The observed FP presents some differences depending on the
wave-band spectrum and on the environment where galaxies reside
\citep{jfk96,Pahre98,mobasher,Kelson2000,Bernardi03}.  Recently,
from a sample of about $9000$ elliptical galaxies taken from the
Sloan Digital Sky Survey (SDSS), Bernardi et al. (2003) obtained a FP
relation given by $R_\mathrm{e}\! \propto \! \sigma_0^{1.49 \pm 0.05}
I_\mathrm{e}^{-0.75\pm0.01}$; where the $R^{1/4}$--profile was assumed.

On the theoretical side, simulations of spiral mergers are numerous
and have been extensively explored to study:
the morphology of merger remnants (e.g. Mihos \& Hernquist 1996),
the line-of-sight velocity distribution (e.g. Bendo
\& Barnes 2000), and the effect of the orientation of the angular momentum
of discs (e.g. Naab \& Burkert 2003). Other works have considered the
effect of mergers on the properties of the FP (e.g. Capelato et al. 1995,
Levine \& Aguilar 1996, Bekki 1998,
Dantas et al. 2003, Nipoti et al. 2003, G\'onzalez-Garc\'{\i}a
\& van Albada 2003). A common feature of previous studies is that they
have used models with properties chosen rather arbitrary resembling
present day spiral galaxies. In particular, those works addressing the FP
of merger remnants have not considered self-consistent models of disc galaxies.

Disc galaxies also define a tight correlation between the maximum of their
rotation velocity  $V_\mathrm{m}$ and their luminosity, termed the
 Tully-Fisher
(TF) relation, that can be written as
$L=\mathcal{A} V_\mathrm{m}^{\alpha}$;
where $\alpha$ and $\mathcal{A}$ are the `slope' and the zero-point,
respectively.
The observed values for $\alpha$ lie between $\approx\,$2.5-4, depending
on the waveband (e.g. Strauss \& Willick 1995). In particular, the
$I$-band TF relation is given by
\begin{equation}\label{eq:tfrel}
M_{I} - 5 \log h = -21.00 -7.68(\log W -2.5)
\end{equation}
where $M_I$ is the absolute magnitude of the disc, and
$W\approx 2\,V_\mathrm{m}$ is the 21-cm hydrogen line width resulting in
a slope of $\alpha \approx 3.1$ (Giovanelli et al. 1997).

In this {\it Letter} we study whether merger remnants of self-consistent
$N$-body spirals satisfying the TF relation can lead to the FP of ellipticals.
Spirals properties are obtained from the Press-Schechter
formalism (Press \& Schechter 1974) in combination with the model of
disc galaxy formation of Mo, Mao \& White (1998, hereafter MMW) and the
TF relation under the $\Lambda$CDM model.
The rest of this work has been structured
as follows: in $\S$\ref{sec:model}
we summarise the model used to set up the theoretical TF
relation, the initial conditions and the orbital
parameters for our simulations. In $\S$\ref{sec:results} the results and a
discussion are given.

\section{Numerical Model}\label{sec:model}
In this section we summarise the method and properties of our models to
study the FP associated to merger remnants.

\subsection{Tully-Fisher relation of spirals}

To construct a theoretical TF we use the Press-Schechter framework of
hierarchical clustering and the model of MMW for the
formation of galaxy discs. In this model five
parameters are required to obtain the radial scale-length of a disc formed
inside a spherical dark halo. These are the circular velocity, $V_\mathrm{c}$,
the spin parameter, $\lambda$, the concentration of the halo, $c$,
the fraction of disc to halo mass, $m_\mathrm{d}$  and the
fraction of angular momentum in the disc, $j_\mathrm{d}$. To obtain
 $(V_\mathrm{c},
\lambda, c, m_\mathrm{d}, j_\mathrm{d})$ we have proceeded as in
Shen, Mo \& Shu (2002). The $\Lambda$CDM cosmology was adopted to obtain
the particular properties of disc galaxies
(with a mass density parameter of $\Omega_0$=0.3, a cosmological constant
of $\Omega_\Lambda$=0.7, the Hubble's constant $h$=0.7, and the perturbation
power-spectrum normalization $\sigma_8$=1).

We have used the theoretical TF at $z$=1, an epoch thought to be close
to the one where formation of the discs occurred (Peebles 1993),
as our fiducial redshift to construct the numerical disc galaxies.
This allowed us to consider remnants resulting from binary encounters and
to compare them with ellipticals that might have formed through these
encounters. Present day massive ellipticals are probably the result of
more than a binary merger, but their study is out of the scope of this work.

We have selected only disc galaxies with a stability criterion
$\varepsilon_\mathrm{m}$$\,\ge\,$$0.9$, where
$\varepsilon_\mathrm{m}$=$V_\mathrm{m}
(G M_\mathrm{d} / R_\mathrm{d})^{-1/2}$, and $V_\mathrm{m}$ is the maximum
 rotation velocity (Efstathiou, Lake \& Negroponte 1982; Syer, Mao \& Mo 1997).
  Galaxies were evolved in isolation during 2 Gyr, about twice
    the time needed for the first pericentric passage in the
    encounter. Despite the fact that our galaxy models match the previous
    criterion some form a well defined bar (B) while in other
    cases they develop a transient bar (T) or remain stable (S)
    (see last column of Table~\ref{tab:glxmodels}).

 In Figure~\ref{fig:tfzetas} the TF relation,
at redshift $z$=$1$, is shown.
The straight solid line indicates the observed slope of the TF in
 the $I$-band of
equation (2) at the present epoch. A sample of twenty random points from this
TF relation were selected in order to set up the self-consistent
$N$-body disc galaxies.

\begin{figure}
\centering
\includegraphics[width=8.5cm]{mymmw6.eps}
\vspace{-0.7cm}
\caption{Theoretical Tully-Fisher relation for an ensemble of
spirals (dots) at $z$=1. The dashed and solid lines correspond to the
linear-square fit with a slope of $\alpha=-8.0$ and to the observed
relation (2), respectively. The properties of discs of galaxies to be
merged were selected from this ensemble. As an illustration, progenitors of
mergers $M01$, $M02$ and $M03$ of Table~1 are shown.}
\label{fig:tfzetas}

\end{figure}

\subsection{Galaxy models}\label{sec:galmodels}

\begin{table*}
 \centering
 \begin{minipage}{7in}
  \caption{Properties of merging galaxies.}\label{tab:glxmodels}

  \begin{tabular}{l|crccrr|ccccrrr}
  \hline
   Merger     &  \multicolumn{6}{c}{Halo Properties} & \multicolumn{7}{c}{Disc Properties}           \\
  & $M_\mathrm{h}$ & $r_\mathrm{200}$ & $\lambda$ & $j$
   & $c$ & $N_{\rm h}$ &$f_\mathrm{d}$ & $R_\mathrm{d}$ & $z_\mathrm{d}$ & $Q$ & $R_\mathrm{Q}$ & $N_{\rm d}$  &  \\
& [M$_\odot$] & [kpc] &  &  & & &  & [kpc] & [kpc] & & [kpc] & &\\
 \hline
 $M01$  & $4.02\times 10^{11}$ & 104.3 & 0.063 & 0.041 & 7.63 & 57126 & 0.0498 & 2.4 & 0.39 & 0.402 & 5.8 &  12000 & T\\
      & $1.25\times 10^{12}$ & 152.3 & 0.056 & 0.068 & 3.84 &  177571 & 0.0586 & 5.7 & 0.47 & 0.321 & 13.8 & 44267 & T\\
\hline
 $M02$  & $1.39\times 10^{11}$ & 73.3 & 0.021 & 0.052 & 8.94 & 172403 & 0.0312 & 1.2 & 0.12 & 0.651 & 3.0 & 57436  & T \\
      & $6.45\times 10^{10}$ & 56.7 & 0.028 & 0.033 & 6.61 &  80000 & 0.0236 & 1.3 & 0.13 & 1.302 & 3.1 & 20000  & S \\
\hline
 $M03$  & $8.65\times 10^{10}$ & 62.6 & 0.078 & 0.033 & 11.12 & 149138 & 0.0402 & 1.7 & 0.20 & 0.995 & 4.1 & 42093  & S \\
      & $4.64\times 10^{10}$ & 50.8 & 0.031 & 0.043 & 12.38 & 80000 & 0.0357 & 0.7 & 0.12 & 0.789 & 1.8 & 20000 & B \\
\hline
 $M04$  & $2.36\times 10^{11}$ & 87.4 & 0.036 & 0.029 & 9.17 & 80000 & 0.0393 & 1.1 & 0.19 & 0.387 & 2.6 & 20000  & T \\
      & $1.75\times 10^{11}$ & 79.2 & 0.048 & 0.054 & 5.29 & 59428 & 0.0372 & 3.3 & 0.58 & 1.064 & 8.0 & 15671  & S \\
\hline
 $M05$  & $5.42\times 10^{10}$ & 53.5 & 0.053 & 0.114 & 15.95 & 74829 & 0.0817 & 1.2 & 0.19 & 0.561 & 2.8 & 20000  & B  \\
      & $6.02\times 10^{10}$ & 55.4 & 0.034 & 0.077 & 12.05 & 83124 & 0.0541 & 1.0 & 0.14 & 0.610 & 2.3 & 14245 & S  \\
\hline
 $M06$  & $7.49\times 10^{10}$ & 59.6 & 0.098 & 0.059 & 7.91 & 91546 & 0.0638 & 2.5 & 0.35 & 0.943 & 6.1 & 25000 & B \\
      & $6.68\times 10^{10}$ & 57.4 & 0.078 & 0.043 & 13.32 & 81742 & 0.0420 & 1.9 & 0.23 & 1.253 & 4.5 & 14327 & S \\
\hline
 $M07$  & $4.88\times 10^{10}$ & 51.7 & 0.074 & 0.064 & 11.02 & 73643 & 0.0635 & 1.5 & 0.25 & 0.841 & 3.6 & 20000 & B \\
      & $9.48\times 10^{10}$ & 64.5 & 0.047 & 0.103 & 11.79 & 143062 & 0.0656 & 1.6 & 0.31 & 0.579 & 3.9 & 42119 & B  \\
\hline
 $M08$  & $9.77\times 10^{10}$ & 65.2 & 0.032 & 0.034 & 6.56  & 245462 & 0.0239 & 1.7 & 0.20 & 1.261 & 4.1 & 36000  & S \\
      & $1.02\times 10^{11}$ & 66.1 & 0.023 & 0.012 & 7.80  & 188124 & 0.0127 & 0.9 & 0.12 & 1.217 & 2.2 & 19725 & S \\
\hline
 $M09$ & $8.11\times 10^{10}$ & 61.2 & 0.099 & 0.142 & 10.87 & 150000 & 0.0818 & 4.6 & 0.73 & 1.217 & 11.1 & 30000 & S \\
  & $8.33\times 10^{10}$ & 61.8 & 0.122 & 0.095 & 10.01 & 154050 & 0.0791 & 3.9 & 0.56 & 1.089 & 9.6 & 29686 & S\\
\hline
 $M10$ & $4.74\times 10^{11}$ & 110.3 & 0.112 & 0.106 & 11.39 & 240000 & 0.0807 & 6.6 & 1.29 & 0.576 & 15.9 & 60000 & B \\
  & $6.99\times 10^{10}$ & 58.3 & 0.071 & 0.035 & 9.94 &165992 & 0.0396 & 1.6 & 0.19 & 1.092 & 3.9 & 48689  & S \\
\hline
\end{tabular}
\end{minipage}
\end{table*}

Our $N$-body disc galaxy models consist of a disc and a spherical dark halo
component. Following Hernquist (1993), disc particle positions are
randomly drawn from
the axisymmetric density profile:
\begin{equation}
\rho_\mathrm{d} (R,z) = \frac{M_\mathrm{d}}{4 \pi R^2_\mathrm{d}
z_\mathrm{d}} \exp( -
R/R_\mathrm{d})  \, \mathrm{sech}^{2} (z/z_\mathrm{d}) \;,
\end{equation}
being $R_\mathrm{d}$ and $z_\mathrm{d}$ the radial and vertical scale
length of the
disc, respectively; $z_\mathrm{d}$ was chosen randomly  in the range
$(0.1-0.2)R_\mathrm{d}$. For the halo we have adopted a modified version of
the NFW--profile (Navarro, Frenk \& White 1997) with an exponential
cut-off given by:
\begin{equation}
\rho_\mathrm{h} (r) = \frac{M_\mathrm{h}\, \alpha_\mathrm{h} }{4 \pi r
(r + r_\mathrm{s})^2}
\ \exp\left[ - \left(\frac{r}{r_\mathrm{200}} + q \right)^2  \right]  \;,
\end{equation}
with
$$
\alpha_\mathrm{h} = \frac{\exp( q^2) }{ \sqrt{\pi} q \exp(q^2)
\mathrm{Erfc}(q) +
\frac{1}{2} \exp(q^2) \mathrm{E}_\mathrm{1}(q^2) - 1 } \,,
$$
being $\mathrm{Erfc}(x)$ the complementary error function and
$\mathrm{E}_\mathrm{1}(x)$ the
exponential integral, $r_\mathrm{s}$ and $r_\mathrm{200}$ the scale and
virial radii of
the halo respectively,
$c \!= \! q^{-1}\!= \!(r_\mathrm{s}/r_\mathrm{200})^{-1}$ the
halo  concentration, and $M_\mathrm{h}$ is the halo total mass. Finally,
velocities are derived from Jeans equations.

The parameters characterizing our disc galaxy models are listed in
Table~\ref{tab:glxmodels}; where $f_{\rm d}\!=\!M_{\rm d}/(M_{\rm h}+M_{\rm d})$. We have also included the radius $R_Q$ at which
Toomre's $Q$ parameter is normalized and the number of particles in the halo, $N_{\rm h}$, and the disc, $N_{\rm d}$. Notice the range of concentrations of
haloes and galaxy disc sizes.

Our disc galaxy models do not include a central bulge component since, in the
framework of MMW's galaxy formation, they tend to produce smaller disc sizes
which require a better spatial resolution and smaller timesteps and, hence,
more demanding computational resources.  It is not clear how this central component will affect the FP of the remnants; it will be addressed in a future work.

\subsection{Initial conditions and experiments}

We have considered only parabolic encounters where the initial separation
between galaxy centres is
$
R_\mathrm{s} = 1.25  ( r_\mathrm{200,1} + r_\mathrm{200,2} ) .
$
The pericentre for each encounter is randomly selected in the interval
$R_\mathrm{p}$$\in\,$(5,20)$\,$kpc, consistent with the energies and
eccentricities of cosmological $N$-body simulations (Navarro, Frenk \&
White 1995). The relative orientation between galaxy spins is also taken
randomly.

To evolve the numerical simulations we use the parallel version of {\sc
gadget}, an N-body/SPH code with individual timesteps
(Springel, Yoshida \& White 2001). Softening parameters of
$\epsilon_\mathrm{d}$=35$\,$pc and $\epsilon_\mathrm{h}$=350$\,$pc were
used for
the disc and the halo particles, respectively. {\sc Gadget} uses a spline kernel for the softening, so that the gravitational interaction between two particles is fully Newtonian for separations larger than twice the softening parameter (Power et al. 2003).

The typical time of arrival at pericentre is about
$1\,$Gyr, and all simulations were run for a total time of about
$8\,$Gyr; this corresponds to about the time span
from $z$=1 to $z$=0 in a $\Lambda$CDM scenario. Remnants are already in a
virial state by this time.
All of our runs were performed on a cluster consisting of 32 Pentium
processors running at 450 MHz (Vel\'azquez \& Aguilar 2003).
Energy conservation was better than $0.25$\% in all cases.

\section{Results and discussion}\label{sec:results}

To analyse the merger remnants a constant mass-to-light ratio, $\Upsilon$,
is assumed; hence $I_\mathrm{e}$ is
proportional to the effective surface luminous-mass density
$\Sigma_\mathrm{e}$.
We first compute their `photometric' properties ($R_\mathrm{e},
\Sigma_\mathrm{e}$) and then its central velocity dispersion
$\sigma_\mathrm{0}$ inside
an aperture of radius $R_\mathrm{e}/8$. The surface density profiles
$\Sigma (R)$ were fitted using a S\'ersic profile \citep{Sersic68,Ciotti99,Caon93} and the
$R^{1/4}$ profile, we also fitted the corresponding growth curves of the
luminous part of the remnants: $M_\mathrm{L}(R)=2 \pi \int \Sigma(R)
R \,\mathrm{d}R$.

To check how the photometric properties are affected by the adopted region to
be fitted, we have considered an aperture with outer
radius of $17.5\,$kpc as in Wright et al (2003) and for the inner
radius of the fit, $\xi$, we use two different values: (1) the resolution
of the luminous component in
our simulations,
$\xi \!\approx \! 2.8 \epsilon_\mathrm{d}\!\approx \!100\,$pc,
and (2) $\xi$=300$\,$pc. This also allows us to test the effects of the
boundaries of the fitting region on the values of the exponents $(a,b)$ of the
FP ($R_\mathrm{e} \propto \sigma_\mathrm{0}^a \Sigma_\mathrm{e}^b$).
Each remnant was `observed' from 100 random line-of-sights. A
$\chi^2$--minimisation by Levenberg-Marquart method \citep{NR} was used to
obtain $R_\mathrm{e}$ and $\Sigma_\mathrm{e}$ for each projection.
The exponents
$(a,b)$ of the FP were fitted to the complete set of projected values of
the remnants,
using both orthogonal and direct fits.

The resulting  values for these
exponents are plotted in Figure~\ref{fig:fptheor} ({\sl left panel}).
It can be seen how the values of parameters $(a,b)$ are
sensitive to the fitting function (profile or growth curve), to the adopted
profile (S\'ersic or $R^{1/4}$), to the fitting method (direct or
orthogonal) and to the value of the radius $\xi$.  The
observed values in the $r^*-$band of the SDSS have also been included
(star symbols). In particular, the dotted lines refer to the exponents of
the FP of Bernardi et al. (2003): $R_\mathrm{e} \propto
 \sigma_\mathrm{0}^{1.49} I_\mathrm{e}^{-0.75}$.

This last
relation is quite well reproduced by the model with $\xi$=100$\,$pc, a
S\'ersic growth curve and an orthogonal fitting method (hereafter our fiducial
model, denoted by FP$_\mathrm{f}$) which is expressed by the relation
$R_\mathrm{e} \propto \sigma_\mathrm{0}^{1.48\pm 0.01} \Sigma_\mathrm{e}^{-0.75 \pm 0.01}$ with a standard
deviation orthogonal to the plane of $0.06$. Uncertainties in the exponents were estimated using the bootstrap technique \citep{Efron}.
Table~2 lists the mean projected values of the quantities used to compute our FP$_{\rm f}$ and the mean S\'ersic index.

 However, the exponents $(a,b)$ obtained by
fitting the merger remnants with the $R^{1/4}$-profile have problems
matching the observed ones, and this is independent of the fitting method
(orthogonal or direct). For example, for $\xi=100\,$pc and an orthogonal
fitting, the R$^{1/4}$-profile
leads to a FP described by $R_\mathrm{e} \propto \sigma_\mathrm{0}^{1.79 \pm 0.01}
\Sigma_\mathrm{e}^{-0.60\pm 0.01}$ (which will be denoted by FP$_\mathrm{dV}$), while
for $\xi=300\,$pc it is given by
$R_\mathrm{e} \propto \sigma_\mathrm{0}^{1.65\pm 0.01}
\Sigma_\mathrm{e}^{-0.65\pm 0.01}$; notice that increasing $\xi$ tends to provide
a slight better fit. An edge-view of our FP$_\mathrm{f}$ for the merger
remnants (filled squares)
is shown in Figure~\ref{fig:fptheor} ({\sl right panel}). As a reference,
the long-dashed line indicates the slope, $\alpha_\mathrm{V}=2$, for
virialised homologous systems with a constant mass-to-light ratio while
the solid and dotted lines refer to the
SDSS and to our FP$_\mathrm{f}$, respectively. Our merger remnants were
artificially displaced to coincide with the zero-point of the
FP of the SDSS (Bernardi et al. 2003).

\begin{table}
 \centering
 \begin{minipage}{8cm}
  \caption{Mean of projected values used to determine our FP$_{\rm f}$.}\label{tab:remnants}

  \begin{tabular}{lccrc}
  \hline
   Merger     &  \multicolumn{4}{c}{Parameters$^*$}        \\
 &  $\langle n \rangle $ & $\langle  R_\mathrm{e}\rangle $ & $ \langle \sigma_\mathrm{0}\rangle $ &  $\langle  \mu_\mathrm{e}\rangle $ \\
 &  & [kpc] & [km/s] & \\
 \hline
 $M01$  & 4.3  & 9.0 & 139 & -19.4 \\
 $M02$  & 2.2 & 2.5 & 75 & -19.4\\
 $M03$  & 1.8 & 2.2 & 72 & -19.4 \\
 $M04$  & 2.6 & 3.7 & 95  & -19.4 \\
 $M05$  & 3.8  & 1.7 & 71 &-20.3\\
 $M06$  & 2.3 & 3.7 & 67 &-18.8  \\
 $M07$  & 3.2 & 2.5& 66& -19.8  \\
 $M08$  & 2.8 & 2.1 & 45 & -19.1 \\
 $M09$  & 2.1 & 6.1 & 72 & -18.3 \\
 $M10$  & 2.6 & 5.2& 71& -18.7  \\
\hline
  \multicolumn{5}{l}{$^*\,$  Here $n$ is the index of the S\'ersic profile [$\propto \exp(-R^{1/n})$], } \\
  \multicolumn{5}{l}{ and $\mu_\mathrm{e}=-2.5\log \Sigma_\mathrm{e}$ with $\Sigma_\mathrm{e}$ in M$_\odot/$kpc$^2$.}  \\
\end{tabular}
\end{minipage}
\end{table}

The overall agreement of the parameter values $(a,b)$ for the S\'ersic profile
case with those of the SDSS relies on  a combination of homology breaking
($V^2 \propto \sigma_0^\nu$, $R_g\propto R_\mathrm{e}^\eta$) and
 a mass scaling relation between the total and luminous mass as
$M\propto M_\mathrm{L}^{\gamma}$. For instance, for our
FP$_\mathrm{f}$ non-homology is found to be
$V^2 \propto \sigma_\mathrm{0}^{1.74}$ and
$R_{\rm g} \propto R_\mathrm{e}^{0.51}$,
with a mass scaling relation given by $M\propto M_\mathrm{L}^{0.85}$ and
a negligible dependence of S\'ersic's index with the effective radius
($n \propto R_\mathrm{e}^{0.07}$),
 while  for the FP$_\mathrm{dV}$ case,
these relations are described by $V^2 \propto \sigma_\mathrm{0}^{1.77}$,
$R_\mathrm{g} \propto R_\mathrm{e}^{0.21}$ and $M\propto M_\mathrm{L}^{0.80}$.
Note that a direct fitting method shows larger differences between
our fitted FP and that of the SDSS (see Figure~\ref{fig:fptheor}-{\sl right panel}).  Despite that S\'ersic's profile fitting leads to a better match with observations; a consistent comparison will require an observational FP determined using this type of profile on the large data base of the SDSS.

 The above results suggest that, ignoring any contribution from
breaking-homology
relations, it does not appear feasible to scale the mass-to-light ratio,
$(M/L)$, as a simple power-law of the luminosity, $L$, (or equivalently of
the mass $M$) alone; see also Bernardi et al. (2003).
 Perhaps, a more general relation of the form
$(M/L) \propto R_\mathrm{e}^x \sigma_0^y L_{\rm e}^z$ need to be explored.
A more extensive study of the scaling relations of merger
remnants, their non-homology, the distribution of luminous and dark matter,
and  their kinematics is under way and it will be presented in a future work.

\begin{figure*}
\centering
\includegraphics[width=8.4cm]{fptheor1a.eps}
\includegraphics[width=8.4cm]{fptheor1b.eps}
\vspace{-0.2cm}
\caption{({\it Left}) Parameters $(a,b)$ for the FP
($R_{\rm e} \propto \sigma_0^a I_{\rm e}^b$) of our galaxy merger remnants.
For comparison purposes, the values of the SDSS (star symbols)
in the $r^*$-band have been included. Horizontal dotted lines indicate the FP
relation, $R_\mathrm{e} \propto \sigma_\mathrm{0}^{1.49}
\Sigma_\mathrm{e}^{-0.75}$, quoted by Bernardi et al. (2003). To the left,
these parameters were obtained using an orthogonal fitting method while to the
right using a direct one. Here, S=S\'ersic profile,
S$_{\rm GC}$=S\'ersic's growth curve, $R^{1/4}$=de Vaucouleurs profile,
and $R^{1/4}_{\rm GC}$ its growth curve. Also, the effect introduced by
two values of the inner radii $\xi$ is illustrated.
({\it Right}) Edge-on view of the FP. Remnants are indicated by
filled squares using the mean values listed in Table~2.
The solid and dotted lines
correspond to the SDSS FP ($R_\mathrm{e} \propto \sigma_\mathrm{0}^{1.49}
\Sigma_\mathrm{e}^{-0.75}$) and to our FP$_\mathrm{f}$ model. As a reference,
the dashed line with slope $\alpha_\mathrm{V}=2$ corresponding to
virialised homologous systems has been plotted. Remnant mergers have been
artificially displaced to coincide with the `zero-point' of the FP of the
SDSS.
}
\label{fig:fptheor}
\end{figure*}

Our results support the idea that mergers of spirals, with properties
obtained from the
 Tully-Fisher relation within a $\Lambda$CDM cosmogony, lead to the tight
correlation expressed by the FP.

 In dissipational numerical simulations of mergers, it is found that a gas component can yield important differences in the stellar component of the remnants in comparison with their collisionless counterparts (e. g., Barnes \& Hernquist 1996). For instance, a gas component deepens the potential well, affecting more the remnant kinematics than its luminous profile. However, it is not clear from these results how the FP is going to be altered; in particular, if its slope is going to be changed or just the dispersion of values around it. This remains an open issue.

\section*{Acknowledgments}

We thank Luis Aguilar and Simon White for their helpful comments on this work.
This research was funded by CONACyT-M\'exico Project 37506-E. An anonymous referee is thanked for useful comments that helped to clarify some points of this work.


\section*{Introduction}
${\mathbb N}$-graded Lie algebras are closely related to
nilpotent Lie algebras, for instance, a finite-dimensional
${\mathbb N}$-graded Lie algebra $\mathfrak{g}$ must to be nilpotent.
Infinite-dimensional ones are also called
residual nilpotent Lie algebras.
M.~Vergne studied in \cite{V} nilpotent
Lie algebras with the maximal possible nilindex
$s(\mathfrak{g})=\dim \mathfrak{g}{-}1$
(by nilindex $s(\mathfrak{g})$ we mean the
length of the descending central series
$\{C^i\mathfrak{g}\}$ of $\mathfrak{g}$). M.~Vergne called them
{\it filiform} Lie algebras. In her study the
${\mathbb N}$-graded filiform Lie algebra $\mathfrak{m}_0(n)$
played a special role.
This Lie algebra is defined by its basis $e_1,\dots,e_n$
and non-trivial commutator relations:
$[e_1,e_i]=e_{i+1}, i=2,\dots,n{-}1$.
Evidently, $\mathfrak{m}_0(n)$ is generated by $e_1$ and $e_2$.
Another example of ${\mathbb N}$-graded two-generated
filiform Lie algebra is $\mathfrak{m}_2(n)$:
$[e_1,e_i]=e_{i+1}, i=2,\dots,n{-}1, \:[e_2,e_i]=e_{i+2}, i=3,\dots,n{-}2$.

A.~Fialowski classified in \cite{Fial} all infinite-dimensional
${\mathbb N}$-graded two-generated Lie algebras
$\mathfrak{g}=\oplus_i\mathfrak{g}_i$
with one-dimensional homogeneous components $\mathfrak{g}_i$. In particular,
there are only three algebras in her list satisfying the "filiform
property": $[\mathfrak{g}_1,\mathfrak{g}_i]=\mathfrak{g}_{i+1}, \forall i$.
They are $\mathfrak{m}_0, \mathfrak{m}_2, L_1$, where
$\mathfrak{m}_0, \mathfrak{m}_2$ denote infinite-dimensional analogues
of $\mathfrak{m}_0(n), \mathfrak{m}_2(n)$, respectively and
$L_1$ is the "positive" part of the Witt or Virasoro algebra. The
classification of finite-dimensional ${\mathbb N}$-graded filiform
Lie algebras over a field of zero characteristic was done in \cite{Mill2}.

A.~Shalev and E.~Zelmanov defined in \cite{ShZ} the {\it coclass}
(which might be infinity) of a finitely generated and
residually nilpotent Lie algebra $\mathfrak{g}$, in analogy with the case
of (pro-)$p$-groups, as
$cc(\mathfrak{g})=
\sum_{i\ge 1}(\dim (C^i\mathfrak{g}/C^{i+1}\mathfrak{g})-1)$.
Obviously the coclass of a filiform algebra is equal to one
and the same is true
for the infinite-dimensional algebras $\mathfrak{m}_0, \mathfrak{m}_2, L_1$.
Algebras of coclass $1$ are also called algebras
of {\it maximal class}. They are also {\it narrow} or {\it thin} Lie algebras
(A.~Shalev, A.~Caranti, M.~Newman, et al.).
Part of Fialowski's classification in \cite{Fial} can be reformulated
in the following way: {\it Up to an isomorphism there are only
three ${\mathbb N}$-graded Lie algebras of maximal class with one-dimensional
homogeneous components:
$\mathfrak{m}_0, \mathfrak{m}_2, L_1$}. The last statement was
rediscovered in \cite{ShZ}.

Algebras of maximal class are in the center
of attention these days both in zero and positive characteristic.
There are many open questions related to them.
One natural question is their cohomology which is the subject of the present
paper.

L.~Goncharova calculated in $1973$ \cite{G}
the Betti numbers $b^q(L_1)=\dim H^q(L_1)$. Her result implies,
as a corollary,
the celebrated Euler identity in combinatorics:
$$\prod_{j=1}^{\infty}(1{-}t^j)
=\sum_{k=0}^{\infty}({-}1)^k(t^{\frac{3k^2{-}k}{2}}{+}
t^{\frac{3k^2{+}k}{2}})
$$
(see \cite{Fu}).
The Betti numbers $\dim H^q(\mathfrak{m}_0(n))$ (finite-dimensional case)
were calculated in \cite{AS} (see also \cite{B}), however, there are no
explicit formulas for basic cocycles and no
description of the multiplicative structure of $H^*(\mathfrak{m}_0(n))$
was obtained.

We give a complete description of the cohomology
algebras $H^*(\mathfrak{m}_0)$ and $H^*(\mathfrak{m}_2)$.
The method we use is based on Dixmier's exact sequence in
Lie algebra cohomology \cite{D}. In our considerations we use combinatorics:
partitions and generating functions.

The paper is organized as follows. In Sections 1--2 we review
all necessary definitions and facts concerning Lie algebra cohomology
and ${\mathbb N}$-graded Lie algebras, in particular we recall
Dixmier's exact sequence in the cohomology \cite{D}.
We start our computations in Section 3 with the algebra
$H^*(\mathfrak{m}_0)$ (Theorem \ref{main_H_m_0}).
In Section 4 we discuss the relations of our results with
representations theory. It turns out that
the basic cocycles representing $H^q(\mathfrak{m}_0)$
are at the same time the highest weight vectors (primitive vectors)
of the $q$-th exterior power $\Lambda^q(V(\lambda))$ of the irreducible
one-dimensional $\mathfrak{sl}(2,{\mathbb K})$-module $V(\lambda)$
for some $\lambda$ (Theorem \ref{sl_2-mod}).
In Section 5 we apply Dixmier's exact sequence
and the results of Section 3 to compute
$H^*(\mathfrak{m}_2)$ (Theorem \ref{main_H_m_2}).
Section 6 is devoted to finite-dimensional analogs of
the algebras considered above. Recall that all of them are filiform Lie
algebras. The Betti numbers $\dim H^q(\mathfrak{m}_0(n))$
are known \cite{AS}, \cite{B}. Some of $\dim H^q(L_1/L_{n+1})$
was found in \cite{Mill1}. The questions on the explicit formulas
for representing cocycles and the multiplicative structure are still
open for these algebras. At the end of the paper we consider
the characteristic $p$ analog of the algebras $\mathfrak{m}_0$
and $\mathfrak{m}_2$.
We briefly remark that other computational tools such as spectral
sequences
or the Hodge Laplacian of the differential $d$ lead to the same
algebraic and combinatorial problems that are solved in the present paper.

\section{Lie algebra cohomology and Dixmier's exact sequence}
Let us consider the cochain complex of a Lie algebra $\mathfrak{g}$
over a field ${\mathbb K}$ of zero characteristic:
$$
\begin{CD}
\mathbb K @>{d_0{=}0}>>
\mathfrak{g}^*=C^1(\mathfrak{g}) @>{d_1}>>
C^2 (\mathfrak{g}) @>{d_2}>>
\dots @>{d_{p-1}}>>C^{p} (\mathfrak{g}) @>{d_p}>> \dots
\end{CD}
$$
where $C^p(\mathfrak{g})$ denotes the vector space of continuous
skew-symmetric $p$-linear forms on $\mathfrak{g}$, and
the differential $d_p$ is defined by:
$$
d_pc(X_1, \dots, X_{p{+}1})=
\sum_{1{\le}i<j{\le}p{+}1}({-}1)^{i{+}j{-}1}
c([X_i,X_j], X_1,\dots, \hat X_j,\dots, \hat X_j, \dots, X_{p{+}1}).
$$
The differential
$d_1: \mathfrak{g}^* \rightarrow \Lambda^2 (\mathfrak{g}^*)$
coincides with the dual mapping of the Lie bracket
$[ \, , ]: \Lambda^2 \mathfrak{g} \to \mathfrak{g}$ and
$$
d(\rho \wedge \eta)=d\rho \wedge \eta+(-1)^{deg\rho} \rho \wedge d\eta,
\; \forall \rho, \eta \in \Lambda^{*} (\mathfrak{g}^*).
$$

\begin{definition}
\label{cohomology}
The cohomology of $(C^{*}(\mathfrak{g}), d)$ is
called the cohomology (with trivial coefficients) of the Lie algebra
$\mathfrak{g}$ and is denoted by $H^*(\mathfrak{g})$.
\end{definition}

Let $\mathfrak{b}$ be an
ideal of codimension $1$ in $\mathfrak{g}$.
One can choose an element $X\in \mathfrak{g}$ such that
$X \notin \mathfrak{b}$.
This element determines a $1$-form $\omega$ by the following condition:
$$\omega(X)=1,\;\: \omega(Y)=0\;\;{\rm for}\;\;  Y \in \mathfrak{b}.$$
As $\mathfrak{b}$ is an ideal in $\mathfrak{g}$, the $1$-form $\omega
\in {\mathfrak{g}}^*$ is closed and $\mathfrak{b}$ is an invariant
subspace of $adX$.

\begin{theorem}[Dixmier ~\cite{D}]
There exists a long exact sequence of Lie algebra cohomology:
\begin{equation}
\dots
\stackrel{(adX^*)_{i{-}1}}{\longrightarrow}
H^{i{-}1}(\mathfrak{b})
\stackrel{\omega \wedge}{\longrightarrow}
H^{i}(\mathfrak{g})
\stackrel{r_i}{\rightarrow} H^{i}(\mathfrak{b})
\stackrel{(adX^*)_i}{\longrightarrow} H^{i}(\mathfrak{b})
\to \dots \end{equation}
where

1) the homomorphism
$r_i: H^{i}(\mathfrak{g}) \to H^{i}(\mathfrak{b})$
is the restriction homomorphism;

2) $\omega \wedge: H^{*{-}1}(\mathfrak{b})
\to H^{*}(\mathfrak{g})$
is induced by the multiplication
$\omega \wedge: \Lambda^{*{-}1}(\mathfrak{b}^*)
\to \Lambda^*(\mathfrak{g}^*)$;

3) the homomorphisms
$(adX^*)_i: H^{i}(\mathfrak{b}) \to H^{i}(\mathfrak{b})$
are induced by the derivation $adX^*_i$ of degree
zero of $\Lambda^*(\mathfrak{b}^*)$ ($adX^*(a\wedge b)=
adX^*a\wedge b+ a\wedge adX^*b$ for all $a, b \in
\Lambda^*(\mathfrak{b}^*)$) that extends the dual mapping
$adX^* : \mathfrak{b}^* \to \mathfrak{b}^*$.
The $0$-derivation $adX^*$ commutes
with $d$ and we will denote the corresponding mapping in the cohomology
by the same symbol, more precisely, $(adX^*)_i$ denotes the mapping
$H^{i}(\mathfrak{b}) \to H^{i}(\mathfrak{b})$.
\end{theorem}
Dixmier's theorem \cite{D} is very important for our computations,
so it appears to be useful to recall his proof.
In fact, Dixmier considered the
cohomology of a nilpotent Lie algebra $\mathfrak{g}$ with coefficients
in arbitrary $\mathfrak{g}$-module $V$, but we restrict ourselves
to the case of scalar coefficients.
\begin{proof}
Each form $f \in \Lambda^*(\mathfrak{g})$
can be decomposed
as $f=\omega \wedge f' + f''$,
where $ f' \in \Lambda^{*{-}1}(\mathfrak{b}^*)$ and
$ f'' \in \Lambda^{*}(\mathfrak{b}^*)$.
One can write a short exact sequence of algebraic complexes
$$
0 \longrightarrow \Lambda^{*{-}1}(\mathfrak{b}^*)
\stackrel{\omega \wedge}{\longrightarrow} \Lambda^{*}(\mathfrak{g}^*)
\longrightarrow \Lambda^{*}(\mathfrak{b}^*)
\longrightarrow 0
$$
where $\Lambda^{*{-}1}(\mathfrak{b}^*)$
is taken with the differential $-d$ as
$d(\omega \wedge c)=d_{\lambda \omega}(\omega \wedge c)=
-\omega \wedge dc$.

This short exact sequence of algebraic complexes gives us
a long exact sequence in the cohomology.
To see this, we have to define the homomorphism
$H^{q}(\mathfrak{b}) \to H^{q}(\mathfrak{b})$
in the long exact sequence.

First of all let us introduce a new mapping
$$
\Lambda^{*}(\mathfrak{g}^*)
\to \Lambda^{*-1}(\mathfrak{b}^*),
f \in \Lambda^{*}(\mathfrak{g}^*)\to f_X \in
\Lambda^{*-1}(\mathfrak{b}^*),$$ where
$f_X(X_1,\dots,X_q)=f(X,X_1,\dots,X_q)$.
That means $f_X= f'$ if $f=\omega \wedge f' +f''$.

Then an obvious formula holds:
\begin{multline}
(d f)_X(X_1, \dots, X_{q{+}1})=
\sum_{1{\le}j{\le}q{+}1}({-}1)^{j}
f(adX(X_j), X_1,\dots, \hat X_j, \dots, X_{q{+}1})+\\
+\sum_{1{\le}i{<}j{\le}q{+}1}({-}1)^{i{+}j}
f([X_i,X_j],X,X_1, \dots, \hat X_i, \dots, \hat X_j, \dots, X_{q{+}1})=\\
=((adX^*)_q(f)+d(f_X))(X_1, \dots, X_{q{+}1}).
\end{multline}
Hence the homomorphism
$H^{q}(\mathfrak{b}) \to H^{q}(\mathfrak{b})$,
$[f] \to [(df)']$ of the long exact sequence coincides with the
homomorphism induced by $(adX^*)_q$.
\end{proof}
\section{${\mathbb N}$-graded Lie algebras}
\begin{definition}
A Lie algebra $\mathfrak{g}$ is called $\mathbb N$-graded,
if it is decomposed to the direct sum of subspaces such that
$$\mathfrak{g}=\oplus_{i} \mathfrak{g}_{i}, \; i \in
{\mathbb N},
\quad \quad [\mathfrak{g}_{i}, \mathfrak{g}_{j}]
\subset \mathfrak{g}_{i+j}, \;
\forall \: i, j \in {\mathbb N}.
$$
\end{definition}
\begin{example}
The Lie algebra $\mathfrak{m}_0$ is defined
by its infinite basis $e_1, e_2, \dots, e_n, \dots $
with commutator relations:
$$ \label{m_0}
[e_1,e_i]=e_{i+1}, \; \forall \; i \ge 2.$$
\end{example}
\begin{remark}
We always omit the trivial commutator
relations $[e_i,e_j]=0$ in the definitions of Lie algebras.
\end{remark}
\begin{example}
The Lie algebra $\mathfrak{m}_2$ is defined by its
infinite basis $e_1, e_2, \dots, e_n, \dots $
and commutator relations:
$$
[e_1, e_i ]=e_{i+1}, \quad \forall \; i \ge 2; \quad \quad
[e_2, e_j ]=e_{j+2}, \quad \forall \; j \ge 3.
$$
\end{example}
Consider now the algebra of polynomial vector fields
on the real line ${\mathbb R}^1$.

\begin{example}
Let us define the algebra $L_k$ as the Lie algebra
of polynomial vector fields
on the real line ${\mathbb R}^1$ with zero in
$x=0$ of order not less then $k+1$.
\end{example}
The algebra $L_k$ can be defined by its basis and the commutator relations are
given by the natural commutator of vector fields
$$e_i=x^{i+1}\frac{d}{dx}, \; i \in {\mathbb N},\; i \ge k, \quad \quad
[e_i,e_j]= (j-i)e_{i{+}j}, \; \forall \;i,j \in {\mathbb N}.$$
$L_k$ is a subalgebra of the Witt algebra $W$, where $W$ is
spanned by all $e_i=x^{i+1}\frac{d}{dx}, \; i \in {\mathbb Z}$.
Hence $L_1$ can be regarded as a "positive" part of the Witt
algebra.

The algebras $\mathfrak{m}_0, \mathfrak{m}_2, L_1$ are
${\mathbb N}$-graded Lie algebras generated by two elements $e_1, e_2$.
\begin{theorem}[follows from A.~Fialowski \cite{Fial}]
\label{fialowski}
Let $\mathfrak{g}=\bigoplus_{i=1}^{\infty}\mathfrak{g}_{i}$
be a ${\mathbb N}$-graded Lie algebra such that:
\begin{equation}
\dim \mathfrak{g}_{i}= 1, \; i \ge 1; \quad
[\mathfrak{g}_{1}, \mathfrak{g}_{i}]=\mathfrak{g}_{i+1},
\; \forall i \ge 2.
\end{equation}
Then $\mathfrak{g}$ is isomorphic to one (and only one)
Lie algebra from the three given ones:
$$\mathfrak{m}_0, \; \mathfrak{m}_2, \; L_1.$$
\end{theorem}
\begin{remark}
All  these Lie algebras have exactly two defining relations in
degree $5$ and $7$.
\end{remark}
The ideals $C^{k}\mathfrak{g}$ of the descending central series
of a Lie algebra $\mathfrak{g}$ determine
a decreasing filtration $C$ of $\mathfrak{g}$, i.e.
$[C^{k}\mathfrak{g},C^{l}\mathfrak{g}]
\subset C^{k+l}\mathfrak{g}, \: k,l \ge 1$.
One can consider the associated ${\mathbb N}$-graded
Lie algebra ${\rm gr}_C\mathfrak{g}$:
$${\rm gr}_C\mathfrak{g}=\oplus_{i\ge 1} ({\rm gr}_C\mathfrak{g})_i=
\oplus_{i\ge 1} C^i\mathfrak{g}/C^{i+1}\mathfrak{g}.$$
We have the following obvious isomorphisms of ${\mathbb N}$-graded
Lie algebras:
$$ {\rm gr}_C\mathfrak{m}_2 \cong
{\rm gr}_C L_1 \cong {\rm gr}_C\mathfrak{m}_0 \cong
\mathfrak{m}_0.$$
\begin{remark}
The Lie algebra $\mathfrak{m}_0$ has at least two
different ${\mathbb N}$-gradings: one of them we have already
discussed, and the second one (sometimes called natural) is defined by
$$\mathfrak{m}_0=\oplus_{i\ge 1}(\mathfrak{m}_0)_i, \;
(\mathfrak{m}_0)_1=\langle e_1, e_2 \rangle, \:
(\mathfrak{m}_0)_i=\langle e_{i+1} \rangle, \: i \ge 2.$$
The first grading is more convenient in cohomological
computations and we will use only this one.
\end{remark}

\begin{theorem}[Vergne \cite{V}]
\label{V_ne1}
Let $\mathfrak{g}=\oplus_{i\ge 1}\mathfrak{g}_{i}$ be a
${\mathbb N}$-graded Lie algebra such that
$$
\dim \mathfrak{g}_{1}=2, \: \dim \mathfrak{g}_{i}=1, \: i \ge 2; \;
[\mathfrak{g}_1,\mathfrak{g}_i]=\mathfrak{g}_{i+1}, \: i \ge 1.
$$
Then $\mathfrak{g}$ is isomorphic to $\mathfrak{m}_0$.
\end{theorem}

A Lie algebra $\mathfrak{g}$ in Definition \ref{cohomology}
is assumed to be topological
and the spaces $C^{*}(\mathfrak{g})$ can have
rather complicated nature, but here we will consider
only ${\mathbb N}$-graded Lie algebras
$\mathfrak{g}=\oplus_{i}\mathfrak{g}_{i}=
\oplus_{i}\langle e_{i} \rangle$.
Let us denote by $e^i, e^i(e_j)=\delta^i_j$ the corresponding dual $1$-forms.
In this situation we assume $C^{*}(\mathfrak{g})$ to be a vector space
of formal series of elements from $\Lambda^{*}(e^1, e^2, \dots)$
(we will also use the notation $\Lambda^{*}(\mathfrak{g}^*)$
for this space).

One can define a second grading in the
cochain complex $(C^*(\mathfrak{g}),d)=(\Lambda^{*}(\mathfrak{g}^*),d)$:
$\Lambda^{*}(\mathfrak{g}^*) =
\hat {\bigoplus_{k}} \Lambda^*_{k}(\mathfrak{g})$, where
a finite-dimensional subspace $\Lambda^q_{k}(\mathfrak{g})$ is spanned by
$q$-forms $e^{i_1} {\wedge} \dots {\wedge}
e^{i_q}, i_1<\dots <i_q$ such that
$i_1{+}\dots{+}i_q=k$. The symbol $\hat {\bigoplus_k}$
means the completed direct sum.

The second grading is compatible with the differential
$d$ and with the exterior product:
$$d \Lambda^q_{k} (\mathfrak{g})
\subset \Lambda^{q{+}1}_{k} (\mathfrak{g}), \qquad
\Lambda^{q}_{k}(\mathfrak{g}) \wedge
\Lambda^{p}_{l}(\mathfrak{g}) \subset
\Lambda^{q{+}p}_{k+l}(\mathfrak{g}).
$$
The exterior product in $\Lambda^*(\mathfrak{g})$
induces a structure of a
bigraded algebra in the cohomology
$H^*(\mathfrak{g})$:
$$
H^{q}_{k} (\mathfrak{g}) \wedge
H^{p}_{l} (\mathfrak{g}) \to
H^{q{+}p}_{k+l} (\mathfrak{g}).
$$
\begin{theorem}[Goncharova \cite{G}]
The Betti numbers $b^q(L_1)={\rm dim} H^q(L_1)=2$ for every
$q \ge 1$, more precisely
$$ ~~~b_k^q(L_1)={\rm dim} H_k^q(L_1)=
\left\{\begin{array}{r}
   1, \hspace{0.6em}{\rm if}~k=\frac{3q^2 \pm q}{2}, \\
   0,\hspace{1.76em}{\rm otherwise.}\\
   \end{array} \right . \hspace{3.3em} $$
\end{theorem}
The numbers
$\frac{3q^2 \pm q}{2}$ are so called Euler pentagonal numbers.
A sum of two arbitrary pentagonal numbers is not a pentagonal number,
hence the algebra
$H^*(L_1)$ has a trivial multiplication.
One can consider the Euler characteristic
$$\chi_k(L_1)=\sum_{q}({-}1)^q{\rm dim}C_k^q(L_1)=
\sum_{q}({-}1)^qb_k^q(L_1)$$
of each subcomplex $C^*_k(L_1), k \ge 0$, and
associate the corresponding generating function
$\chi_{L_1}(t)=\sum_{k=0}\chi_k(L_1)t^k$:
$$\prod_{j=1}^{\infty}(1{-}t^j){=}\sum_{k,q{\ge}0}({-}1)^q
{\rm dim}C_k^q(L_1)t^k{=}
\sum_{k,q{\ge}0}({-}1)^qb_k^q(L_1)t^k{=}
\sum_{k=0}^{\infty}({-}1)^k(t^{\frac{3k^2{-}k}{2}}{+}
t^{\frac{3k^2{+}k}{2}}).
$$
This equality of two expressions for the Euler characteristic
proves the celebrated Euler identity in combinatorics
(see \cite{Fu}, \cite{H} for details).
\begin{remark}
The ${\mathbb N}$-graded Lie algebras
$L_1$, $\mathfrak{m}_0$, $\mathfrak{m}_2$
have the same cochain spaces (but different differentials), hence
the cohomology $H^*(\mathfrak{m}_0)$ and $H^*(\mathfrak{m}_2)$
(that we are going to compute in the present article) also satisfy
the "Euler property":
$$\sum_{k=0}^{\infty}\sum_{q\ge 0}({-}1)^qb_k^q(\mathfrak{m}_0)t^k=
\sum_{k=0}^{\infty}\sum_{q\ge 0}({-}1)^qb_k^q(\mathfrak{m}_2)t^k=
\sum_{q=0}^{\infty}({-}1)^q(t^{\frac{3q^2{-}q}{2}}{+}
t^{\frac{3q^2{+}q}{2}}).
$$
\end{remark}

\section{The cohomology $H^*(\mathfrak{m}_0)$}
\label{H^*(m_0)}
Now we are going to apply Dixmier's exact sequence
for the situation
$$\mathfrak{g}=\mathfrak{m}_0, \; \mathfrak{b}= {\rm Span}
(e_2, e_3, \dots, e_k,\dots), \; X=e_1.$$

Obviously the ideal $\mathfrak{b}= {\rm Span}(e_2, e_3, \dots)$
is an abelian Lie algebra and its cohomology is the exterior algebra
$\Lambda^*(e_2, e_3, \dots)$.

Let us denote by $D_1$ the operator $ade_1^*: \Lambda^*(e_2, e_3, \dots)
\to \Lambda^*(e_2, e_3, \dots)$.
It can be defined explicitly by
\begin{equation}
\begin{split}
D_1(e^2)=0, \; D_1(e^i)= e^{i-1}, \; \forall i\ge 3,\\
D_1(\xi\wedge \eta)=D_1(\xi)\wedge \eta +\xi\wedge D_1(\eta),
\; \: \forall \xi, \eta \in \Lambda^*(e_2, e_3, \dots).
\end{split}
\end{equation}

\begin{lemma}
\label{surj}
The operator $D_1$ is surjective.
\end{lemma}
\begin{proof}
Let us define the operator $D_{-1}: \Lambda^*(e^2,e^3,\dots) \to
\Lambda^*(e^2,e^3,\dots)$,
\begin{equation}
\begin{split}
\label{D_{-1}}
D_{-1}e^i=e^{i+1}, \;
D_{-1}(\xi{\wedge} e^i)=
\sum_{l\ge 0}(-1)^l D_{1}^{l}(\xi){\wedge} e^{i+1+l},
\end{split}
\end{equation}
where $i\ge 2$ and $\xi$ is an arbitrary form in
$\Lambda^*(e^2,\dots,e^{i-1})$.
The sum in the definition (\ref{D_{-1}}) of $D_{-1}$ is always finite
because $D_1^l$ decreases the second grading by $l$.
For instance,
$D_{-1}(e^i\wedge e^k)=\sum_{l=0}^{i-2} ({-}1)^l e^{i-l}{\wedge} e^{k+l+1}$.

The operator $D_{-1}$ is right inverse to $D_1$, as
$D_1D_{-1}=Id$ on $\Lambda^*(e^2,e^3,\dots)$.

In fact
$D_1D_{-1}(e^i)=e^i, \; i \ge 2$ and for arbitrary
$\xi \in \Lambda^*(e^2,\dots,e^{i-1})$ we have
$$D_1D_{-1}(\xi\wedge e^i)=\sum_{l\ge 0}
(-1)^l D_{1}^{l+1}(\xi)\wedge e^{i+l+1}+ \sum_{l\ge 0}
(-1)^l D_{1}^{l}(\xi)\wedge e^{i+l}=\xi\wedge e^{i}.$$
\end{proof}
One can write the formula for
$D_{-1}(e^{i_1} {\wedge} \dots {\wedge} e^{i_q}{\wedge} e^{i_q})=D_{-1}(0)$:
\begin{equation}
\label{m_0cocycles}
\omega(e^{i_1}{\wedge}\dots {\wedge} e^{i_q} {\wedge}
e^{i_q{+}1})=
\sum\limits_{l\ge 0}(-1)^l D_1^l(e^{i_1}\wedge \dots
\wedge e^{i_q})\wedge e^{i_q+1+l}.
\end{equation}
This sum is also always finite and determines
a homogeneous closed $(q{+}1)$-form
of the second grading $i_1{+}{\dots}{+}i_{q{-}1}{+}2i_{q}{+}1$.

Let us consider the restriction
$(D_1)_k^{q+1}$ of $D_1$ on $(q+1)$-forms of the second grading $k$.
\begin{lemma} \label{KerD_1} Let $q\ge 1$.
Then ${\rm Ker}(D_1)_k^{q+1}$ is spanned by
$$
\omega(e^{i_1}{\wedge}\dots {\wedge} e^{i_q} {\wedge}
e^{i_q{+}1}), \:2\le i_1{<}{\dots}{<}i_q,
\:i_1{+}{\dots}{+}i_{q-1}{+}2i_q{+}1=k.
$$
\end{lemma}
\begin{proof}
Obviously, if these forms exist, they are linearly independent.
How many they are? Let us consider
the inclusion $A^{q+1}_{k-1}:\Lambda_{k-1}^{q+1}(e^2,e^3,\dots)
\to \Lambda_{k}^{q+1}(e^2,e^3,\dots)$ defined on basic monomials
by the shift of the last superscript:
$$
A^{q+1}_{k-1}(\xi{\wedge}e^i{\wedge}e^j)=
\xi{\wedge}e^i{\wedge}e^{j{+}1}.
$$
It follows immediately that ${\rm Im}A^{q+1}_{k-1}$ is spanned by monomials
$e^{j_1}{\wedge}\dots{\wedge}e^{j_{q}}{\wedge}e^{j_{q+1}}$,
such that $j_{q+1}-j_q\ge 2$. Hence $\dim {\rm Im} A^{q+1}_{k-1}=
\dim\Lambda_{k-1}^{q+1}(e^2,e^3,\dots)$ and
$$
{\rm Im}A^{q+1}_{k-1}\oplus
{\rm Span}
(\{\omega(e^{i_1}{\wedge}\dots{\wedge}e^{i_q}{\wedge}e^{i_q+1}),
i_1{+}{\dots}{+}2i_q{+}1=k \})=
\Lambda_{k}^{q+1}(e^2,e^3,\dots).
$$
On the another hand, $(D_1)_k^{q+1}$ is surjective
and hence
$$\dim {\rm Ker}(D_1)_k^{q+1}=\dim \Lambda_{k}^{q+1}(e^2,e^3,\dots)-
\dim \Lambda_{k-1}^{q+1}(e^2,e^3,\dots)$$
which completes the proof.
\end{proof}

We will think of $\omega$ as a linear map defined on
the subspace of $\Lambda^{q+1}(e^2,e^3,\dots)$, spanned by the monomials
$e^{i_1}{\wedge}\dots{\wedge}e^{i_q}{\wedge}e^{i_q+1}$:
$$
\omega(\sum_{i_1\dots i_{q}} \alpha_{i_1\dots i_{q}}
e^{i_1}{\wedge}\dots{\wedge}e^{i_q}{\wedge}e^{i_q+1}):=
\sum_{i_1\dots i_{q+1}} \alpha_{i_1\dots i_{q+1}}
\omega(e^{i_1}{\wedge}\dots{\wedge}e^{i_q}{\wedge}e^{i_q+1}).
$$
Let us denote by $P_q(k)$ the number of (unordered) partitions
of a positive integer $k$ into $q$ parts, i.e.
(see \cite{H}) $P_q(k)$ is the number
of solutions in positive integers $x_i$ of
$$
k=x_1+\dots+x_q, \; 1\le x_1 \le x_2 \le \dots \le x_q.
$$
Denote by $V_q(k)$ the number of partitions of a positive integer
$k$ into $q$ {\it distinct} parts, i.e. $V_q(k)$ is the number
of solutions in positive integers $y_i$ of
$$
k=y_1+\dots+y_q, \; 1< y_1 < y_2 < \dots < y_q.
$$
\begin{lemma}
\label{partitions}
${\rm dim} \Lambda_k^q(e^2, e^3, \dots)=
V_q\left(k{-}q\right)=
P_q\left(k{-}\frac{q(q{+}1)}{2}\right)$.
\end{lemma}
\begin{proof}
It consists of two standard tricks in combinatorics.
Let us consider

1) a one-to-one correspondence between the set
of partitions of a positive integer $N$ exactly into
$q$ parts and the standard basis of
$\Lambda_{N{+}\frac{q(q{+}1)}{2}}^q(e^2, e^3, \dots)$:
\begin{multline*}
(x_1, x_2, x_3, \dots, x_q) \to e^{x_1{+}1}{\wedge}
e^{x_2{+}2}{\wedge}e^{x_3{+}3}{\wedge}\dots {\wedge}e^{x_q{+}q},\\
N=x_1+\dots+x_q, \quad 1\le x_1 \le x_2 \le \dots \le x_q;
\end{multline*}

2) an analogous bijection between the partitions of $N$ into
$q$ distinct parts and the standard basis of
$\Lambda_{N{+}q}^q(e^2, e^3, \dots)$:
\begin{multline*}
(y_1, y_2, y_3, \dots, y_q) \to e^{y_1{+}1}{\wedge}
e^{y_2{+}1}{\wedge}e^{y_3{+}1}{\wedge}\dots {\wedge}e^{y_q{+}1},\\
N=y_1+\dots+y_q, \quad 1\le y_1 < y_2 < \dots < y_q.
\end{multline*}
\end{proof}
\begin{theorem}
\label{main_H_m_0}
The bigraded cohomology algebra
$H^*(\mathfrak{m}_0)=\oplus_{k,q} H^q_k(\mathfrak{m}_0)$
is spanned by the cohomology classes of the following homogeneous cocycles:
\begin{equation}
\label{sumHm_0}
\begin{split}
e^1, \: e^2, \:
\omega(e^{i_1}{\wedge}\dots
{\wedge} e^{i_q}{\wedge} e^{i_q{+}1})=
\sum\limits_{l\ge 0}({-}1)^l (ad e_1^*)^l
(e^{i_1}{\wedge} e^{i_2}{\wedge} \dots
{\wedge} e^{i_q}){\wedge} e^{i_q{+}1{+}l}, \\
\end{split}
\end{equation}
where
$q \ge 1, \; 2\le i_1 {<}i_2{<}{\dots} {<}i_q$,
in particular
$${\rm dim} H_{k{+}\frac{q(q{+}1)}{2}}^q(\mathfrak{m}_0)=
P_q(k)-P_q(k{-}1).$$
The multiplicative structure is defined by
\begin{multline}
\label{Hm_0multiplication}
[e^1] {\wedge} \omega(\xi{\wedge}e^i{\wedge}e^{i+1})=0,\;
[e^2] {\wedge} \omega(\xi{\wedge}e^i{\wedge}e^{i+1})
=\omega(e^2{\wedge}\xi{\wedge}e^i{\wedge}e^{i+1}),\\
\omega(\xi{\wedge}e^i{\wedge}e^{i+1})
{\wedge} \omega(\eta{\wedge}e^j{\wedge}e^{j+1})
=\sum_{l=0}^{j-i+1}(-1)^l\omega((ade_1^*)^l(\xi{\wedge}e^i){\wedge}e^{i+1+l}
{\wedge}\eta{\wedge}e^j{\wedge}e^{j+1})+\\
+(-1)^{j{-}i{+}deg\eta}\sum_{s \ge 1}
\omega((ade_1^*)^{j{-}i{-}1{+}s}(\xi{\wedge}e^i){\wedge}
(ade_1^*)^s(\eta{\wedge}e^j){\wedge}e^{j{+}s}
{\wedge}e^{j{+}s{+}1})+\\
+(-1)^{j{-}i{+}deg\eta {+}1}\sum_{s \ge 1}
\omega((ade_1^*)^{j{-}i{+}1{+}s}(\xi{\wedge}e^i){\wedge}
(ade_1^*)^s(\eta{\wedge}e^j){\wedge}e^{j{+}s{+}1}
{\wedge}e^{j{+}s{+}2}),
\end{multline}
where $i \le j$, $\xi$ and $\eta$ are arbitrary homogeneous forms in
$\Lambda^*(e^2,\dots,e^{i{-}1})$ and $\Lambda^*(e^2,\dots,e^{j{-}1})$,
respectively.
\end{theorem}
\begin{proof}
It follows from Lemma \ref{surj} that
in our case Dixmier's sequence is equivalent to the following exact
sequences:
\begin{equation*}
\begin{split}
0\stackrel{}{\longrightarrow}
H^{0}(\mathfrak{b})={\mathbb K}
\stackrel{e^1 {\wedge}}{\longrightarrow}
H^1(\mathfrak{m}_0)
\stackrel{r_1}{\longrightarrow} {\langle} e^2 \rangle
\to 0;\\
0\stackrel{}{\longrightarrow}
H^q(\mathfrak{m}_0)
\stackrel{r_q}{\longrightarrow} {\rm Ker}(ade_1^*)_q
\to 0, \; q \ge 2.
\end{split}
\end{equation*}
Here ${\rm Ker}(ade_1^*)_q={\rm Ker}(D_1)^q$ and the dimensions
$\dim {\rm Ker}(D_1)^q_k$ were found in Lemma \ref{KerD_1}
and the final formula for
$\dim H^q_{k{+}\frac{q(q{+}1)}{2}}(\mathfrak{m}_0)$
follows from Lemma \ref{partitions}.

Let us remark that the sum (\ref{sumHm_0}) is always finite. Namely, the
maximal value of the superscript $l$ is equal
to $l_{max}=i_1{+}\dots{+}i_q-\frac{q(q+3)}{2}$ and the corresponding
summand in
(\ref{sumHm_0}) is
$(-1)^{l_{max}}\alpha(i_1,\dots, i_q)e^{2}{\wedge}e^{3}{\wedge}\dots
{\wedge} e^{q{+}2}{\wedge} e^{i_q{+}1{+}l_{max}}$
(see Example \ref{3cocycle} below).

We will obtain the formulas for the multiplicative structure of
 $H^*(\mathfrak{m}_0)$ by using the explicit expressions (\ref{m_0cocycles})
for basic cocycles.
An arbitrary homogeneous $q$-cocycle
$\omega(e^{i_1}{\wedge}\dots {\wedge}
e^{i_{q{-}2}}{\wedge}e^r{\wedge}e^{r{+}1})$ is completely determined
by its leading term
$e^{i_1}{\wedge}\dots {\wedge}e^{i_{q{-}2}}{\wedge}e^r{\wedge}e^{r{+}1}$
 -- the unique monomial
$\alpha e^{i_1}{\wedge}{\dots}e^{i_{q{-}1}}{\wedge}e^{i_q},
\:i_1<\dots < i_{q{-}1}<i_q$ in its decomposition
such that $i_q-i_{q{-}1}=1$.
Hence we only have to consider the summands with this property in the
right part of the formula
$$
\omega(\xi{\wedge}e^i{\wedge}e^{i{+}1}){\wedge}
\omega(\eta{\wedge}e^j{\wedge}e^{j{+}1}){=}
\sum_{l,k\ge 0}({-}1)^{k{+}l}D_1^l(\xi{\wedge}e^i){\wedge}
e^{i{+}1{+}l}{\wedge}D_1^k(\eta{\wedge}e^j){\wedge}e^{j{+}1{+}k}.
$$
They are of the following three kinds:

{\parindent 0pt
\medskip
1) \ $({-}1)^{l}D_1^l(\xi{\wedge}e^i){\wedge}
e^{i{+}l{+}1}{\wedge}\eta{\wedge}e^j{\wedge}e^{j{+}1}\quad
{\rm for} \quad i+1+l \le j+2;$
\smallskip

2) \  $({-}1)^{j{-}i{-}1}D_1^{j{-}i{-}1{+}k}
(\xi{\wedge}e^i){\wedge}
e^{j{+}k}{\wedge}D_1^k(\eta{\wedge}e^j){\wedge}e^{j{+}1{+}k} \:
\quad{\rm for}\quad k\ge 1 ;$
\smallskip

3) \ $({-}1)^{j{-}i{+}1}D_1^{j{-}i{+}1{+}k}
(\xi{\wedge}e^i){\wedge}
e^{j{+}2{+}k}{\wedge}D_1^k(\eta{\wedge}e^j){\wedge}e^{j{+}1{+}k} \:
\quad{\rm for}\quad k\ge 1.$

\medskip

Taking $\omega$ of their sum we get formula (\ref{Hm_0multiplication})
for the multiplication in $H^*(\mathfrak{m}_0)$ (recall that $D_1$ denotes
the operator $ade_1^*$).
}
\end{proof}

\begin{example}
We choose the following basis of $H^2(\mathfrak{m}_0)$:
$$
e^2{\wedge} e^3, e^3{\wedge} e^4 -e^2{\wedge} e^5, \dots,
\omega(e^j{\wedge}e^{j+1})=
\sum_{l=0}^{j-2} ({-}1)^l e^{j-l} {\wedge} e^{j+1+l}, \dots
$$
>From this it is clear that
$$  ~~~{\rm dim} H_k^2(\mathfrak{m}_0)= \left\{\begin{array}{r}
   1, \; k=2j+1 \ge 5, \\
   0, \; {\rm otherwise} \quad \quad \; \\
   \end{array} \right.$$
and for $2 \le i < j$ we have
\begin{multline*}
\omega(e^2{\wedge}e^{3}){\wedge}\omega(e^j{\wedge}e^{j{+}1})=
\omega(e^2{\wedge}e^{3}{\wedge}e^j{\wedge}e^{j{+}1}), \\
\omega(e^i{\wedge}e^{i+1})
{\wedge} \omega(e^j{\wedge}e^{j+1})
{=}{\sum_{l{=}0}^{j{-}i{+}1}{(}{-}1{)}^l}\omega{(}e^{i{-}l}{\wedge}e^{i+1+l}
{\wedge}e^j{\wedge}e^{j+1}{)}+\\
+({-}1)^{j{-}i}\sum_{s=1}^{2i{-}j{-}1}\omega(e^{2i{-}j{+}1{-}s}{\wedge}
e^{j{-}s}{\wedge}e^{j{+}s}{\wedge}e^{j{+}s{+}1})+\\
+({-}1)^{j{-}i{+}1}
\sum_{s=1}^{2i{-}j{-}3}
\omega (e^{2i{-}j{-}1{-}s}{\wedge}
e^{j{-}s}{\wedge}e^{j{+}s{+}1}
{\wedge}e^{j{+}s{+}2}).
\end{multline*}
\end{example}
\begin{corollary} $H_k^3(\mathfrak{m}_0)=0$ if $k < 9$,

and for $k\ge 9$ we
have
$$  ~~~{\rm dim} H_k^3(\mathfrak{m}_0)= \left\{\begin{array}{r}
   l-1, \; k=6l{+}r, \: r=0{,}1{,}2{,}4,\\
   l, \quad \quad \quad k=6l{+}r, \: r=3{,}5.\\
   \end{array} \right . \hspace{3.3em} $$
\end{corollary}
This follows from the formulas for $P_3(k)$ given in \cite{H}.
Using other remarks on $P_q(k)$ in \cite{H},
one can show that ${\rm dim} H_k^q(\mathfrak{m}_0)$
is a polynomial of degree $q-2$ in $k$ with  leading term
$\frac{k^{q{-}2}}{(q{-}2)!q!}$ and the other coefficients depend
on residue $k$ modulo $q!$.
\begin{example}
\label{3cocycle}
\begin{multline*}
\omega(e^5{\wedge} e^6 {\wedge} e^7)=
e^5{\wedge} e^6 {\wedge}e^7-e^4{\wedge} e^6 {\wedge} e^8+(e^3{\wedge} e^6+
e^4 {\wedge} e^5){\wedge} e^9-\\
-(e^2{\wedge}e^6+2e^3 {\wedge}e^5){\wedge}e^{10}
+(3e^2{\wedge} e^5+2e^3{\wedge} e^4){\wedge} e^{11}-
5e^2{\wedge} e^4{\wedge} e^{12}+5e^2{\wedge} e^3{\wedge} e^{13}.
\end{multline*}
\end{example}

Now we can finish this section with the formula for the generating
function for Betti numbers $b^q_k(\mathfrak{m}_0)$:
$$
\sum_{k=0}^{\infty}\sum_{q=0}^k b^q_k(\mathfrak{m}_0)t^kx^q=
t(1+x)+(1-t)\prod_{j=2}^{\infty}(1+xt^j),
$$
and verify the Euler property:
\begin{multline*}
\sum_{k=0}^{\infty}\sum_{q\ge 0}(-1)b^q_k(\mathfrak{m}_0)t^k=
1-t-t^2+\sum_{k\ge 3}t^k\sum_{q\ge 2}({-}1)^q(V_q(k{-}q){-}V_q(k{-}q{-}1))=\\
=1{-}t{-}t^2{+}(\prod_{j=2}^{\infty}(1{-}t^j)-1+t^2)-
t(\prod_{j=2}^{\infty}(1{-}t^j)-1)=(1{-}t)\prod_{j=2}^{\infty}(1{-}t^j)=
\prod_{j=1}^{\infty}(1{-}t^j).
\end{multline*}

\section{The cohomology $H(\mathfrak{m}_0)$ and highest weight representations of
$\mathfrak{sl}(2,{\mathbb K})$}
This section was influenced by Appendix B in
Bordemann's article \cite{B}.

Let $X, Y, H$ denote the standard basis of
$\mathfrak{sl}(2,{\mathbb K})$:
$$[X,Y]=H, \; [H,X]=2X, \; [H,Y]=-2Y.$$
One can define an infinite-dimensional
$\mathfrak{sl}(2,{\mathbb K})$-module $V(\lambda)$ by its basis
$\{f_i, \: i \ge 0\}$ and the well-known classical formulas (\cite{S}):
\begin{equation}
\begin{split}
Hf_i=&(\lambda-2i)f_i,\\
Yf_i=&(i+1)f_{i+1},\\
Xf_i=&(\lambda-i+1)f_{i-1},
\end{split}
\end{equation}
where we set $f_{-1}=0$ and $\lambda \in {\mathbb K}$.
The $\mathfrak{sl}(2,{\mathbb K})$-module $V(\lambda)$
is generated by its highest weight vector (primitive vector \cite{S})
$f_0$:
$Hf_0=\lambda f_0$,
$Xf_0=0$, $f_i=Y^if_0/i!$. The module $V(\lambda)$ is
irreducible if and only if $\lambda \notin {\mathbb N}, \lambda \ne 0$.
Sometimes the module $V(\lambda)$ is called the standard cyclic module or
the Verma module.

Consider now the $q$-th exterior power $\Lambda^q(V(\lambda))$.
It is decomposed to the direct sum of its weight subspaces
$(\Lambda^q(V(\lambda)))^{\alpha}=\{v \in \Lambda^q(V(\lambda)),
H(v)=\alpha v\}$:
$$
\Lambda^q(V(\lambda))=\oplus_k (\Lambda^q(V(\lambda)))^{\lambda q-2k}=
\oplus_k \Lambda^q_k(V(\lambda)),
$$
where  $\Lambda^q_k(V(\lambda))$ is spanned by monomials
$f_{i_1}{\wedge}\dots{\wedge}f_{i_q}$ such that
$i_1+\dots+i_q=k$. Obviously
$H(f_{i_1}{\wedge}\dots{\wedge}f_{i_q})=(\lambda q -2\sum_{l=1}^qi_l)
f_{i_1}{\wedge}\dots{\wedge}f_{i_q}$.

Now we take $\lambda\notin {\mathbb N}, \lambda \ne 0$ and consider
a new basis of $V(\lambda)$:
$$
\label{tilde_f}
{\tilde f}_0=f_0,\: {\tilde f}_i=f_i/\prod_{l=1}^i(\lambda-l+1),\:
i\ge 1, \; X{\tilde f}_i={\tilde f}_{i-1}, \: i \ge 0.
$$
In fact, we can reformulate Theorem \ref{main_H_m_0} as follows.
\begin{theorem} \label{sl_2-mod}
Let $q \ge 2$ and $V(\lambda)$ be an infinite-dimensional
irreducible $\mathfrak{sl}(2,{\mathbb K})$-module, then
its $q$-th exterior power $\Lambda^q(V(\lambda))$ is reducible.
The subspace of its primitive vectors of weight $\lambda q - 2k$ has
dimension $P_q(k-\frac{(q-3)q}{2})-P_q(k-\frac{(q-3)q}{2}-1)$ and
we can take the following basis:
\begin{multline}
\omega({\tilde f}_{i_1}{\wedge}\dots
{\wedge}{\tilde f}_{i_{q-2}}{\wedge}{\tilde f}_{i}{\wedge}{\tilde f}_{i{+}1})
=\sum\limits_{l\ge 0}({-}1)^l X^l
({\tilde f}_{i_1}{\wedge}\dots{\wedge}{\tilde f}_{i_{q-2}}{\wedge}
{\tilde f}_i){\wedge} {\tilde f}_{i{+}1{+}l}, \\
0 \le i_1 < i_2 < \dots < i_{q-2}< i, \; \sum_{l=1}^{q-2}i_l+2i+1=k.
\end{multline}
That means the module
$\Lambda^q(V(\lambda))$ is decomposed to the direct sum of its highest weight
submodules $\Lambda^q(V(\lambda))=\oplus V_{i_1\dots i_{q-2},i}$,
where the highest weight vector of $V_{i_1\dots i_{q-2},i}$ is
$\omega({\tilde f}_{i_1}{\wedge}\dots{\wedge}{\tilde f}_{i_{q-2}}
{\wedge}{\tilde f}_{i}{\wedge}{\tilde f}_{i{+}1})$.
\end{theorem}
\begin{corollary}
Let $\lambda \notin {\mathbb Q}$ and $q \ge 2$, then each
submodule $V_{i_1\dots i_{q-2},i}$
in the decomposition of $\Lambda^q(V)$ considered above
is irreducible (and hence isomorphic to $V(\lambda)$).
\end{corollary}
\begin{proof}
The operator $D_1: {\rm Span}(e^2,e^3,\dots) \to
{\rm Span}(e^2,e^3,\dots)$ considered above
plays the role of the operator $X$
in an infinite-dimensional irreducible
$\mathfrak{sl}(2,{\mathbb K})$-module
$V(\lambda)={\rm Span}(e^2,e^3,\dots)$. We just rescale
${\tilde f}_i=e^{i+2},i \ge 0$, where $\{{\tilde f}_i, \: i \ge 0\}$
is the basis (\ref{tilde_f}) of $V(\lambda)$.
The subspace of primitive elements of weight $\lambda q - k$
coincides by definition with $({\rm Ker}D_1)^q_{k+2q}$ and
the theorem follows from Lemma~\ref{KerD_1}.
\end{proof}

\section{The cohomology $H^*(\mathfrak{m}_2)$}

Let us set in Dixmier's exact sequence
$$\mathfrak{g}=\mathfrak{m}_2, \; \mathfrak{b}= {\rm Span}
(e_1,e_3,\dots), \; X=e_2.$$
The ideal $\mathfrak{b}$ is isomorphic to
$\mathfrak{m}_0$, by shifting $e_1=e_1',\: e_i=e_{i-1}', i\ge3$.
Hence it follows from the results of the previous section
that the cohomology $H^*(\mathfrak{b})$
is spanned by the cohomology classes of
$$e^1, \:e^3, \:
\omega_{\mathfrak{b}}(e^{i_1}{\wedge}\dots {\wedge} e^{i_p} {\wedge}
e^{i_p{+}1}), \; p\ge 1,\: 3\le i_1 <\dots < i_p,$$
where in the formula
$\omega_{\mathfrak{b}}(\xi{\wedge}e^{i} {\wedge}
e^{i+1})=\sum_{l\ge 0}D_1^l(\xi{\wedge}e^{i}){\wedge}e^{i+1+l},
\xi \in \Lambda^*(e^2,\dots,e^{i-1})$ and
we also assume that $D_1(e^3)=0$, for instance
$\omega_{\mathfrak{b}}(e^{3}{\wedge}e^4)=e^{3}{\wedge}e^4$.

The operator $D_2=ade_2^*: \Lambda^*(\mathfrak{b})
\to \Lambda^*(\mathfrak{b})$
inducing ${\mathcal D}_2: H^*(\mathfrak{b}) \to H^*(\mathfrak{b})$
can be defined by
\begin{equation}
\begin{split}
D_2(e^1)=D_2(e^4)=0,\; D_2(e^3)=e^1,\; D_2(e^i)= e^{i-2},
\: i\ge 5,\\
D_2(\xi\wedge \eta)=D_2(\xi)\wedge \eta +\xi\wedge D_2(\eta), \;
\forall \xi, \eta \in \Lambda^*(\mathfrak{b}).
\end{split}
\end{equation}

It is immediate that
\begin{equation}
\label{2-surj}
\begin{split}
{\mathcal D}_2(e^3)=e^1&, \quad {\mathcal D}_2(e^1)=0,\\
{\mathcal D}_2(\omega_{\mathfrak{b}}(e^3{\wedge} e^4))=0,& \;
{\mathcal D}_2(\omega_{\mathfrak{b}}(e^k{\wedge} e^{k+1}))=
-2\omega_{\mathfrak{b}}(e^{k-1}{\wedge} e^k), \;
k \ge 4.
\end{split}
\end{equation}
\begin{proposition} Let $3\le i_1<\dots<i_{p-1}<i$ and
$\xi=e^{i_1}{\wedge}\dots{\wedge}e^{i_{p-1}}$, then
$$
\label{mathcal D_2}
{\mathcal D}_2(\omega_{\mathfrak{b}}(\xi{\wedge}e^{i}{\wedge}e^{i+1}))=
\omega_{\mathfrak{b}}((D_2+D_1^2)(\xi){\wedge}e^{i}{\wedge}e^{i+1})
-2\omega_{\mathfrak{b}}(\xi{\wedge}e^{i-1}{\wedge}e^{i}).$$
\end{proposition}
\begin{proof}
First of all let us remark that in general $D_2$ and $D_1^2$ coincide
only on $\Lambda^1(\mathfrak{b})$:
$$D_1^2(e^i)=D_2(e^i)=e^{i-2}, \:i\ge 5, \;
D_1^2(\xi{\wedge}\eta)=D_1^2(\xi){\wedge}\eta+
2D_1(\xi){\wedge}D_1(\eta)+\xi{\wedge}D_1^2(\eta).$$
An arbitrary cocycle is completely determined by its terms
$\alpha e^{j_1}{\wedge}\dots{\wedge}e^{j_{p}}{\wedge}
e^{j_p+1}$. On the another hand, the operator $D_2$ decreases the
difference between the two last superscripts of some
monomial $e^{j_1}{\wedge}\dots{\wedge}e^{j_{p}}$ by two.
Denoting all "non-interesting" terms by dots, we obtain
an expression for
$D_2(\omega_{\mathfrak{b}}(\xi{\wedge}e^{i}{\wedge}e^{i+1}))$:
\begin{multline*}
D_2(\xi{\wedge}e^{i}{\wedge}e^{i+1}-
D_1(\xi{\wedge}e^{i}){\wedge}e^{i+2}+
D_1^2(\xi{\wedge}e^{i}){\wedge}e^{i+3}+\dots)=\\
=D_2(\xi){\wedge}e^{i}{\wedge}e^{i+1}+
\xi{\wedge}e^{i}{\wedge}D_2(e^{i+1})-\\
-\xi{\wedge}D_1(e^{i}){\wedge}D_2(e^{i+2})+
D_1^2(\xi){\wedge}e^{i}{\wedge}D_2(e^{i+3})+\dots=\\
=D_2(\xi){\wedge}e^{i}{\wedge}e^{i+1}+
\xi{\wedge}e^{i}{\wedge}e^{i-1}
-\xi{\wedge}e^{i-1}{\wedge}e^{i}+
D_1^2(\xi){\wedge}e^{i}{\wedge}e^{i+1}+\dots.
\end{multline*}
\end{proof}
\begin{lemma}
$
H^*(\mathfrak{b})={\rm Im}{\mathcal D}_2 \oplus \langle e^3 \rangle.
$
\end{lemma}
\begin{proof}
We have already seen (\ref{2-surj}) that ${\mathcal D}_2$ is surjective
on $H^2(\mathfrak{b})$. Now
for an arbitrary $\xi=e^{i_1}{\wedge}\dots{\wedge}e^{i_{q-1}}$
with $q \ge 2$ and $3\le i_1<\dots<i_{q-1}<i$, define an operator
${\mathcal D}_{-2}:H^2(\mathfrak{b}) \to H^2(\mathfrak{b})$ by the formula:
$$
\label{DD_{-2}}
{\mathcal D}_{-2}(\omega_{\mathfrak{b}}(\xi {\wedge}e^{i}{\wedge} e^{i+1}))=
\sum_{l\ge 0}\frac{1}{2^{l+1}}\omega_{\mathfrak{b}}\left((D_2+D_1^2)^l
(\xi) {\wedge}
e^{i+l}{\wedge} e^{i+1+l}\right).
$$
Like in the definition of $D_{-1}$, remark that the sum (\ref{DD_{-2}})
is always finite: the operator $(D_2+D_1^2)^l$ decreases the second
grading of an arbitrary homogeneous $\xi$ by $2l$.
Now using formula (\ref{mathcal D_2}) we obtain
${\mathcal D}_{2}{\mathcal D}_{-2}=-Id$.
\end{proof}
Let us consider the  restriction
$({\mathcal D}_2)_k^{q+1}={\mathcal D}_2:H^{q+1}_k(\mathfrak{b}) \to
H^{q+1}_{k-2}(\mathfrak{b})$.
\begin{corollary} \label{dimH_m_2}
Let $q\ge 1$. Then
$\dim {\rm Ker}({\mathcal D}_2)_k^{q+1}=
\dim H_{k}^{q+1}(\mathfrak{b})-
\dim H_{k-2}^{q+1}(\mathfrak{b})$.
\end{corollary}

\begin{lemma} \label{KerD_2} The space
${\rm Ker}{\mathcal D}_2$ is spanned by
$$e^1, \; \omega_{\mathfrak{b}}(e^3{\wedge}e^4)=e^3{\wedge}e^4, \;
\sum_{l\ge 1}\frac{1}{2^{l}}\omega_{\mathfrak{b}}\left((D_2+D_1^2)^l
(e^{i_1}{\wedge} {\dots} {\wedge} e^{i_q}) {\wedge}
e^{i_q{+}1{+}l}{\wedge} e^{i_q{+}2{+}l}\right),
$$
where $1\le q, \; 3\le i_1 <i_2<\dots <i_q$.
\end{lemma}
\begin{proof}
A mimic of the proof of Lemma \ref{KerD_1}.
\end{proof}

\begin{theorem}
\label{main_H_m_2}
The bigraded cohomology algebra
$H^*(\mathfrak{m}_2)=\oplus_{q,k} H^q_k(\mathfrak{m}_2)$ is spanned
by cohomology classes of the following homogeneous cocycles:
\begin{multline}
e^1, \; e^2, \;  e^2 \wedge e^3, \; e^3\wedge e^4-e^2\wedge e^5, \\
w_{i_1{,} {\dots}{,} i_q{,} i_q{+}1{,} i_q{+}2}=
\sum_{l{\ge}1}\frac{1}{2^l}\omega\left({(}ade_2^*{+}{(}ade_1^*{)}^2{)}^l
{(}e^{i_1}{\wedge}{\dots} {\wedge} e^{i_q}{)} {\wedge}
e^{i_q{+}1{+}l}{\wedge} e^{i_q{+}2{+}l}\right),
\end{multline}
where $1\le q, \; 3\le i_1 <i_2<\dots <i_q$, in particular
for $q \ge 3$,
$$
{\rm dim} H_{k{+}\frac{q(q{+}1)}{2}}^q(\mathfrak{m}_2)=
P_q(k)-P_q(k{-}1)-P_q(k{-}2)+
P_q(k{-}3).
$$
\end{theorem}
\begin{proof}
Dixmier's sequence is equivalent to the following exact
sequences:
\begin{multline*}
0\stackrel{}{\longrightarrow}
H^{0}(\mathfrak{b})={\mathbb K}
\stackrel{e^2 {\wedge}}{\longrightarrow}
H^1(\mathfrak{m}_0)
\stackrel{r_1}{\longrightarrow} {\langle} e^1 \rangle
\longrightarrow 0;\\
0\stackrel{}{\longrightarrow}\langle e^3 \rangle
\stackrel{e^2 {\wedge}}{\longrightarrow}
H^2(\mathfrak{m}_2)
\stackrel{r_2}{\longrightarrow}
\langle \omega_{\mathfrak{b}}(e^3{\wedge}e^4) \rangle
\longrightarrow 0,\\
0\stackrel{}{\longrightarrow}
H^q(\mathfrak{m}_2)
\stackrel{r_q}{\longrightarrow} {\rm Ker}(ade_2^*)_q
\longrightarrow 0, \; q \ge 3.
\end{multline*}
The equality ${\rm Ker}(ade_2^*)_q={\rm Ker}({\mathcal D}_2)^q$
was found in Lemma \ref{KerD_2}.
formulas for $\dim H^q_k(\mathfrak{m}_2)$
follows from Corollary \ref{dimH_m_2} and Theorem \ref{main_H_m_0}.
The last remark is that $\omega(\xi)$ represents the inverse image of
$\omega_{\mathfrak{b}}(\xi) \in {\rm Ker}(ade_2^*)_q$ with respect
to $r_q:H^q(\mathfrak{m}_2) \to H^q(\mathfrak{b})$, for instance
$r_q^{-1}(\omega_{\mathfrak{b}}(e^3{\wedge}e^4))=
r_q^{-1}(e^3{\wedge}e^4)=\omega(e^3{\wedge}e^4)=
e^3{\wedge}e^4-e^2{\wedge}e^5.$
\end{proof}
\begin{example}
\begin{multline*}
w_{5,6,7}=\omega(e^5{\wedge} e^6 {\wedge} e^7) +
\omega(e^3{\wedge} e^7{\wedge}e^8)
=e^5{\wedge} e^6 {\wedge}e^7+(e^3{\wedge} e^7{-}e^4{\wedge} e^6){\wedge} e^8+\\
{+}(e^4 {\wedge} e^5{-}e^2{\wedge} e^7){\wedge} e^9+
(e^2{\wedge}e^6{-}e^3 {\wedge}e^5){\wedge}e^{10}
+e^3{\wedge} e^4{\wedge} e^{11}-
e^2{\wedge} e^4{\wedge} e^{12}+e^2{\wedge} e^3{\wedge} e^{13}.
\end{multline*}
\end{example}
\begin{corollary}

1) The space $H^2(\mathfrak{m}_2)$ is two-dimensional
and it is spanned by the cohomology classes represented by
cocycles $e^2\wedge e^3$ and $e^3\wedge e^4-e^2\wedge e^5$
with second grading $5$ and $7$ respectively;\\
2) The space $H^3(\mathfrak{m}_2)$ is
infinite-dimensional and is spanned by
$$w_{k,k{+}1,k{+}2}=
\sum_{l\ge 0}\omega\left(e^{k-2l}{\wedge}
e^{k+1+l}{\wedge} e^{k+2+l}\right), \; k\ge 3.$$
Hence
$  ~~{\rm dim} H_q^3(\mathfrak{m}_2)= \left\{\begin{array}{r}
   1, \; q=3k{+}3 \ge 12, \\
   0, \; {\rm otherwise.} \quad \quad \; \\
   \end{array} \right . \hspace{3.3em} $
\end{corollary}

\begin{remark}
Again (see \cite{H}), one can show that $b^q_k(\mathfrak{m}_2)$
for $q \ge 3$ is a polynomial of degree $q-3$ with leading term
$\frac{2k^{q{-}3}}{(q{-}3)!q!}$ and other coefficients depending
on residue $k$ mod $q!$. We have the following
identity for the corresponding generating function:
$$
\sum_{k=0}^{\infty}\sum_{q=0}^k b^q_k(\mathfrak{m}_2)t^kx^q=
(1+x)(t+t^2-t^3+xt^5)+(1-t-t^2+t^3)\prod_{j=3}^{\infty}(1+xt^j).
$$
It is easy to verify that the Euler property of $b^q_k(\mathfrak{m}_2)$
is equivalent to the following obvious equality:
$(1{-}t{-}t^2{+}t^3)\prod_{j=3}^{\infty}(1{-}t^j)=
\prod_{j=1}^{\infty}(1{-}t^j)$.
\end{remark}

\section{Finite-dimensional analogs and their cohomology}
Let $\mathfrak{g}$ be a ${\mathbb N}$-graded Lie algebra,
then $\oplus_{i=n{+}1} \mathfrak{g}_i$ is an ideal and we
can consider the corresponding quotient Lie algebra
$\mathfrak{g}/\oplus_{i=n{+}1} \mathfrak{g}_i$ and denote it
by $\mathfrak{g}(n)$.
The quotient Lie algebras $\mathfrak{m}_0(n)$, $\mathfrak{m}_2(n)$ and
${\mathcal V}_n=L_1/L_{n+1}$ are $n$-dimensional nilpotent Lie algebras
with the same length $s(\mathfrak{g})=n-1$
of the descending central series $\{C^i\mathfrak{g}\}$.
In fact, $n-1$ is the maximum of $s(\mathfrak{g})$ in the
set of $n$-dimensional nilpotent Lie algebras.
Studying this class of nilpotent Lie algebras was initiated
by Vergne in \cite{V}.

Recall that
a nilpotent $n$-dimensional Lie algebra $\mathfrak{g}$ is  called
filiform Lie algebra, if its descending central series $\{C^i\mathfrak{g}\}$
has the length (nil-index) $s(\mathfrak{g})=n-1$.

The filiform Lie algebra $\mathfrak{m}_0(n)$ plays a special role in the
theory of filiform Lie algebras: an arbitrary $n$-dimensional
filiform Lie algebra can be obtained as a nilpotent deformation
of $\mathfrak{m}_0(n)$ (\cite{V}).
Vergne's explicit formulas for
basic cocycles $\Psi_{k,r}$ of the second cohomology
$H^2(\mathfrak{m}_0, \mathfrak{m}_0)$
with coefficients in the adjoint representation
are one of the main tools in the study of filiform Lie algebras
via the deformation theory. We can mention that
the cohomology groups $H^2(\mathfrak{m}_0)$ and
$H^2(\mathfrak{m}_2)$ were also found by M.~Vergne (\cite{V}).

Later all the Betti numbers of $\mathfrak{m}_0(n)$ were calculated
by M.~Bordemann in \cite{B}(Appendix B). We would like to discuss
his very elegant approach, in fact similar results are well known
in combinatorics (R.~Stanley's  and his school, see \cite{AS} for
references). By arguments equivalent to Dixmier's exact sequence,
he reduced the computation of $\dim H^q(\mathfrak{m}_0(n))$ to the
problem of finding ${\rm Ker}(D_1)_q$:
$$
\dim H^q(\mathfrak{m}_0(n))=\dim{\rm Ker}(D_1(n))_q+
\dim{\rm Ker}(D_1(n))_{q-1}.
$$
Let us take $\lambda=n-2$ and consider the irreducible
$(n-1)$-dimensional $\mathfrak{sl}(2,{\mathbb K})$-module $V(n-2)=
{\rm Span}(e^2,\dots,e^n)$, where $X=D_1$.
The dimension of ${\rm Ker}(D_1(n))_q$ is equal to
the number of irreducible $\mathfrak{sl}(2,{\mathbb K})$-modules
in the decomposition of $\Lambda^q(V(n-2))$ and the last number
is equal in its
turn to the dimension of the zero-eigenspace plus
the dimension of one-eigenspace of
$H:\Lambda^q(V(n{-}2)) {\to} \Lambda^q(V(n{-}2))$.
Now we rescale as before ${\tilde f}_i=e^{i+2}, \: i=0,\dots,n-2$.
$$H({\tilde f}_{i_1}{\wedge}\dots{\wedge}{\tilde f}_{i_q})=
(q\lambda-2\sum_{l=1}^qi_l)
{\tilde f}_{i_1}{\wedge}\dots{\wedge}{\tilde f}_{i_q}, \; \lambda=n-2.
$$
Hence $\dim{\rm Ker}(D_1(n))_q$ is equal to the number of solutions
of the equation
$$
\sum_{l=1}^qi_l=\left[\frac{q(n-2)}{2}\right], \;
0 \le i_1 <\dots <i_q\le n-2.
$$
In other words,
$\dim{\rm Ker}(D_1(n))_q=V_{q,n-1}(\left[\frac{qn}{2}\right])$,
where we denoted by $V_{q,n-1}(N)$ the number of partitions
of a positive integer $N$ into $q$ distinct summands
$i_1,\dots,i_q$ such that $1\le i_1 <\dots <i_q\le n{-}1$ and
$[x]$ stands for the integer part of $x \in {\mathbb Q}$.
We conclude that
\begin{equation}
\dim H^q(\mathfrak{m}_0(n))=V_{q,n-1}\left(\left[qn/2\right]\right)+
V_{q-1,n-1}\left(\left[(q-1)n/2\right]\right).
\end{equation}
It is possible to write some of the first formulas in terms of more
convenient
combinatorial functions:
\begin{equation*}
\begin{split}
\dim H^2(\mathfrak{m}_0(n))=\left[ \frac{n+1}{2}\right],\;
\dim H^3(\mathfrak{m}_0(n))=\left[ \binom{\frac{n+1}{2}}{2}+
\frac{1}{8}\right],\\
\dim H^4(\mathfrak{m}_0(n))=\left[ \frac{4}{3}\binom{\frac{n+1}{2}}{3}+
\frac{4n+13}{36}\right].
\end{split}
\end{equation*}
\begin{remark}

The last formulas give no information about the bigraded structure
of $H^*(\mathfrak{m}_0(n))$. Also we have no explicit formulas for
basic cocycles. Later Bordemann's results were generalized in
\cite{AS} to the case of an arbitrary  finite-dimensional
nilpotent Lie algebra with an abelian ideal of codimension one
(the preprint version of \cite{B} appeared earlier than
\cite{AS}).
\end{remark}

It was shown in \cite{Mill1} that
the case $\mathfrak{g}(n)$ can be more complicated then
its infinite-dimensional analog $\mathfrak{g}$.
The Betti numbers of $L_1/L_{n+1}$ stabilize as
$n \to \infty$  and $\dim H^q(L_1/L_{n+1})$
are equal to the Fibonacci numbers $F_{q+2}$ for sufficiently large $n>>q$.
For instance,
$$\dim H^2(L_1/L_{n+1})=3,\:\dim H^3(L_1/L_{n+1})=5,\:
\dim H^4(L_1/L_{n+1})=8,\:\dots.$$
However, the full description of $H^*(L_1/L_{n+1})$ is still
an open question. The same situation is with the cohomology
$H^*(\mathfrak{m}_2)$.

The classification of finite-dimensional ${\mathbb N}$-graded filiform Lie
algebras $\mathfrak{g}=\oplus\langle e_i \rangle$
with one-dimensional homogeneous components
(there is the one-parametric family $\mathfrak{g}_{\alpha}$
of non-isomorphic algebras
in each dimension $7\le \dim \mathfrak{g} \le 11$, see \cite{Mill2})
also shows that one can expect the difficulties in the
finite-dimensional case.

\section{Final remarks}
1) We can consider the Lie algebras $\mathfrak{m}_0$ and
$\mathfrak{m}_2$ not only over a field of zero characteristic but
over an arbitrary field of positive characteristic.
>From the proof of Theorem \ref{main_H_m_0} it follows that the statement
is also valid over an arbitrary field and Theorem \ref{main_H_m_2}
is valid over any field of non-even characteristic.
This remark appears important
because of applications of Lie algebras of maximal class in the
theory of (pro-)$p$-groups (see \cite{ShZ}.

2) Let $\mathfrak{g}=\oplus_{i\ge 1}\langle e_i \rangle$
be an ${\mathbb N}$-graded Lie algebra with one-dimensional
homogeneous components.
We can equip the (finite-dimensional)
space $\Lambda^q_k(\mathfrak{g}^*)$
with an euclidean scalar product $(,)$, such that the basic
monomials $e^{i_1}{\wedge}\dots{\wedge}e^{i_q}$ form
an orthonormal basis of $\Lambda^q_k(\mathfrak{g}^*)$:
$$
(e^{i_1}{\wedge}\dots{\wedge}e^{i_q},
e^{j_1}{\wedge}\dots{\wedge}e^{j_q})=
\delta^{i_1}_{j_1}\dots\delta^{i_q}_{j_q}.
$$
One of the main methods to compute the cohomology $H^*(\mathfrak{g})$
of the cochain complex $(\Lambda^*(\mathfrak{g}),d)$ is
to find the zero-eigenspace of the Hodge Laplacian $dd^*+d^*d$
(see \cite{Fu}).

\begin{proposition}
The Hodge Laplacian $dd^*+d^*d$ of the Lie algebra $\mathfrak{m}_0$
satisfies the following properties:
\begin{equation}
\begin{split}
(dd^*+d^*d)(e^1)=0,& \: \
(dd^*+d^*d)(e^1{\wedge}\xi)=e^1{\wedge}D_1D_1^*(\xi),\\
(dd^*+d^*d)(\eta)&=D_1^*D_1(\eta),\;
\quad\eta \notin e^1{\wedge}\Lambda^*(\mathfrak{g}^*)
\end{split}
\end{equation}
where $D_1^*=ade_1$ is the $0$-derivation of the exterior algebra
$\Lambda^*(\mathfrak{g})$ that extends the operator $ade_1:\mathfrak{m}_0 \to
\mathfrak{m}_0$.
\end{proposition}
Let us study the kernel of the Hodge Laplacian.
We have proved in Lemma \ref{surj} that
$D_1: \Lambda^*(e^2,e^3,\dots)\to \Lambda^*(e^2,e^3,\dots)$
is surjective, hence ${\rm Ker}D_1^*=0$, moreover it is easy to see that
$e^1 \notin {\rm Im} D_1$ and on the another hand,
${\rm Ker}D_1D_1^*=0$. Hence
${\rm Ker}(dd^*+d^*d) \subset e^1\oplus \Lambda^*(e^2,e^3,\dots)$
and the problem of determining the zero-eigenspace
of $dd^*+d^*d$ reduces to the computation
of ${\rm Ker}D_1^*D_1={\rm Ker}D_1$. The basic cocycles
$\omega(e^{i_1}{\wedge}\dots{\wedge} e^{i_q}{\wedge} e^{i_q{+}1})$
from Theorem \ref{main_H_m_0} together with
$e^1, e^2$ form the basis of the harmonic forms.

3) One can compute the cohomology of $\mathfrak{m}_0$ directly
without Dixmier's exact sequence, but for the Lie algebra
$\mathfrak{m}_2$ such a computation is not so easy. We chose
Dixmier's method (see also \cite{AS}) to demonstrate a general
principle which is applicable for other finite or infinite
dimensional Lie algebras as well. Dixmier remarked in \cite{D}
that his exact sequence in the cohomology is equivalent to some
cohomological spectral sequence.

Let $\mathfrak{g}=\oplus_i \mathfrak{g}_i$
be an arbitrary ${\mathbb N}$-graded Lie algebra
of the filiform type, i.e. $\dim \mathfrak{g}_i =1$ for all $i$ and
$[\mathfrak{g}_1,\mathfrak{g}_i]=\mathfrak{g}_{i+1}, \: i \ge 2$.
We showed earlier
that it is filtered by the ideals $\{C^i\mathfrak{g}\}$
of the descending central series and the corresponding
associated graded Lie algebra ${\rm gr}_C \mathfrak{g}$ is isomorphic
to $\mathfrak{m}_0$. It is easy to see that this filtration induces
a filtration in the cochain complex
$\Lambda^*(\mathfrak{g})$ and the first term $E_1$
of the corresponding spectral sequence $(E_r, d_r)$ is isomorphic to the
cohomology $H^*({\rm gr}_C \mathfrak{g})=H^*(\mathfrak{m}_0)$.
Moreover, the spectral sequence $(E_r, d_r)$ degenerates at the first term
$E_1$ in the case $\mathfrak{g}=\mathfrak{m}_0$,
and at the second term $E_2$ in the case $\mathfrak{g}=\mathfrak{m}_2$,
respectively. In both cases the differential $d_0$
is equal to $e^1{\wedge}D_1$, and $d_1=0$ for $\mathfrak{g}=\mathfrak{m}_0$,
while $d_1=e^2{\wedge}D_2$ for $\mathfrak{g}=\mathfrak{m}_2$.
The case of $\mathfrak{g}=L_1$ is much more complicated. We have infinite
number of non-trivial differentials $d_r$ of the spectral sequence $E_r$
and computing all of them would give a new proof of
Goncharova's theorem. This approach might be helpful to obtain
explicit formulas for Goncharova's cocycles
which have not yet been found so far.