# COMETARY ACTIVITY BEYOND THE PLANETS

Naceur Bouziani Center of Research in Astronomy, Astrophysics and Geophysics
Route de l’Observatoire, 16340 Bouzareah, Algiers, Algeria  David Jewitt
Department of Earth, Planetary and Space Sciences, UCLA
595 Charles Young Drive East, Los Angeles, CA 90095-1567, USA  Department of
Physics and Astronomy, UCLA
430 Portola Plaza, Box 951547, Los Angeles, CA 90095-1547, USA
<EMAIL_ADDRESS>
(Revised 2021 Oct 18)

###### Abstract

Recent observations show activity in long-period comet C/2017 K2 at
heliocentric distances beyond the orbit of Uranus. With this as motivation, we
constructed a simple model that takes a detailed account of gas transport
modes and simulates the time-dependent sublimation of super-volatile ice from
beneath a porous mantle on an incoming cometary nucleus. The model reveals a
localized increase in carbon monoxide (CO) sublimation close to heliocentric
distance $r_{H}$ = $150$ AU (local blackbody temperature $\sim$23 K), followed
by a plateau and then a slow increase in activity towards smaller distances.
This localized increase occurs as heat transport in the nucleus transitions
between two regimes characterized by the rising temperature of the CO front at
larger distances and nearly isothermal CO at smaller distances. As this
transition is a general property of sublimation through a porous mantle, we
predict that future observations of sufficient sensitivity will show that in-
bound comets (and interstellar interlopers) will exhibit activity at distances
far beyond the planetary region of the solar system.

comets: general comets: individual (C/2017 K2 (PANSTARRS), Oort Cloud)

##  1 Introduction

It has long been known that some comets are active far beyond the range of
heliocentric distances over which water ice can sublimate, the latter
corresponding roughly to the region interior to Jupiter’s orbit (semimajor
axis 5 AU). The interpretation of activity in distant comets is complicated.
Ices more volatile than water (e.g. CO, CO2) may be involved, as might the
exothermic crystallization of amorphous water ice, at least at distances
$r_{H}\lesssim$ 12 AU (Jewitt, 2009; Guilbert-Lepoutre et al., 2012). In
comets observed after perihelion, the origin of distant activity is further
complicated by the slow conduction into the interior of heat acquired at
perihelion, resulting in the delayed activation of sub-surface volatiles and
driving distant outbursts (e.g. Prialnik & Bar-Nun (1992)).

Significantly, in long-period comets that are active at large distances while
still inbound to perihelion, the conduction of perihelion heat can play no
role. The long-period comet C/2017 K2 (PANSTARRS), which was discovered active
pre-perihelion at heliocentric distance $r_{H}=16$ AU, falls into this
category. Observations reveal a spherical coma of large ($\sim$0.1 - 1 mm)
size grains, slowly ejected (speeds $\sim$ 4 m s-1) close to steady-state from
a nucleus no more than 9 km in radius, Jewitt et al. (2017, 2019a). Coma was
present in prediscovery observations at 23.8 AU, a linear extrapolation of
photometry indicates that activity began at $r_{H}\sim$ 26 AU, (Jewitt et al.,
2019a) while more detailed considerations indicate that C/2017 K2 was active
even at $\sim$35 AU (Jewitt et al., 2021). Comet C/2010 U3 (Boattini) likewise
possessed a coma at 25.8 AU pre-perihelion (Hui et al., 2019), while C/2014 B1
(Schwartz) was also active far beyond the water sublimation zone with a coma
of large, slowly-moving grains, but observed only starting at 11.9 AU, pre-
perihelion (Jewitt et al., 2019b). Recently, comet C/2014 UN271
(Bernardinelli-Bernstein) was reported to be active at 23.8 AU (Farnham, 2021)
and may have been active at much larger distances.

Free sublimation of exposed super-volatile ices provides an initially
attractive but ultimately unsatisfactory explanation for the distant activity.
Gas drag forces produced by free-sublimation of CO in equilibrium with
sunlight at distances $>$ 15 AU are orders of magnitude too small to eject 100
$\mu$m sized particles against the gravity of the nucleus (Jewitt et al.,
2019a). Neither could smaller particles be ejected owing to inter-particle
(van der Waals) cohesive forces naturally between grains. Taken at face value,
this “cohesion bottleneck” prevents the ejection of particles of any size in
distant comets, as noted by Gundlach et al. (2015). However, sublimation in a
confined space, for example beneath a porous refractory mantle, offers the
possibility of building the pressure to values much higher than reached in
free sublimation to space. Fulle et al. (2020) have suggested that sublimation
occurs within centimeter-sized “pebbles” whose small but finite strength
allows gas pressure to build up to values sufficient to eject dust by gas
drag. Activity driven by the crystallization of amorphous water ice has also
been proposed as a mechanism but, at $\geq$ 15 AU, temperatures are too low
for this process to occur.

We here explore solutions to the bottleneck problem involving the build-up of
pressure by the sublimation of sub-surface ice, taking careful account of the
processes by which heat and gas transport occur. In so doing, we find that
cometary activity is possible at extraordinarily large distances, even far
beyond the planetary region of the solar system. We present the model in
Section (2) and the results in Section (3).

##  2 The Model

The physical model we present is a refinement of Whipple’s (1950) icy
conglomerate model, enhanced to include the presence of a non-volatile, low
conductivity porous mantle (see Figure 1). In active comets, such a “rubble
mantle” is produced by fallback of suborbital particles and by volatile loss
from the strongly heated surface. In inactive comets, for example those in the
Oort cloud and Kuiper belt storage reservoirs, an irradiation mantle can grow
as pure ices are rendered non-volatile by prolonged exposure to cosmic rays
(Kaiser & Roessler (1997), Abplanalp, Frigge, & Kaiser (2019), Gautier et al.
(2019)). In our model, we assume that a non-volatile mantle of variable
thickness and porosity overlies CO ice.

In equilibrium with sunlight, the energy balance averaged over the surface of
a spherical, isothermal nucleus is written as

$\frac{F_{\textsubscript{$\odot$}}\left(1-A_{B}\right)}{r_{H}^{2}}+4\sigma\epsilon_{IR}T_{OC}^{4}=4\sigma\epsilon_{IR}T^{4}+4k(T)\frac{\partial{T}}{\partial{x}}$
(1)

The first term on the left of Equation (1), in which $F_{\odot}$ is the solar
constant, $r_{H}$ is the heliocentric distance in AU and $A_{B}$ is the bond
albedo, represents the absorbed solar power. The second term on the left, only
important at the largest heliocentric distances, accounts for heating of the
nucleus from other external sources. These include the cosmic microwave
radiation background and the integrated light of the stars, as represented by
an effective temperature $T_{OC}$ = 10 K. The emissivity of the nucleus is
$\epsilon_{IR}$ while $\sigma$ is the Stefan-Boltzmann constant. On the right,
the first term is the flux of radiation from the nucleus surface at
temperature, $T$, while the second term is the conducted flux. There is no
contribution from sublimation since the surface layers are assumed to be
volatile-free.

The conductivity in Equation (1), $k(T)$, is written as in Mendis & Brin
(1977)

$k(T)=A+B(4\sigma\epsilon_{IR}T^{3})$ (2)

where the first term, $A$, is the effective thermal conductivity of the porous
mantle and second term accounts for radiative conductivity. $B$ is an
empirical attenuation factor for radiative conduction of void and adjacent
grain surfaces within the porous medium.

As shown schematically in Figure (1), the ice is not on the surface as in
Whipple’s model but hidden under a mantle. Energy propagated through the
mantle to deeper layers can trigger the sublimation of buried ices. At the
ice-mantle interface, the conducted energy flux sublimates ice at rate

$k(T)\frac{\partial{T}}{\partial{x}}=E_{CO}f_{CO}$ (3)

in which we ignore heat conducted into the “cold core” beneath the sublimation
front. A justification of this neglect is given in the Appendix.

Here, the sublimation is entirely controlled by the conducted heat; a non-
equilibrium process. The generated gas is also subject to another non-
equilibrium process caused by the pressure gradient. The heat is driven down
the temperature gradient towards the ice front, $B$ in Figure (1), while the
gas is driven up towards the surface of the comet, $A$, by the pressure
gradient established in the porous medium. In this simple comet model the
reverse flow of molecules is not favored, so condensation is somewhat unlikely
and is neglected for simplicity. The sublimation flux that has been calculated
in Watson & al. (1961) and in Delsemme, & Miller (1971) approximates the
sublimation flux in our case too. Otherwise, we should model the condensation
process and calculate the net flux

$f_{CO}=P(T)\sqrt{\frac{1}{2\pi k_{B}Tm}}$ (4)

Practically, $f_{CO}$ is an upper limit to the sublimated flux because of the
possibility of back-flow and sticking of molecules reflected from particles
above the ice front. This is the necessary Neumann boundary condition. At all
levels above the ice front, we calculate the net flux, $J_{CO}$, formed as
molecules diffuse through the porous medium.

In equation (4), $P(T)$ is the equilibrium pressure of the $CO$ gas over the
$CO$ ice as function of temperature. $P(T)$ contains valuable information
about both the gas and the ice structure. We consider sublimation and
condensation as facets of a single process which is a first order phase
transition phenomenon. It is worthwhile to invoke the chemical reaction
formalism to address it. Sublimation is considered as the direct process while
condensation is the reverse process; analogous to the direct and reverse
arrows between reactants and products in chemical reaction theory. Usually,
experiments are conducted in closed thermodynamic equilibrium to characterize
the chemical reaction and find the equilibrium constant called $K(T)$. We
found that for pure ice, the temperature dependence of $P(T)$ varies just like
$K(T)$ in the language of chemical reactions. The resulting pressure relation
resembles the Arrhenius equation (5). Physically, the similarity arises
because in a chemical reaction we consider the breakage of bonds within
molecules while, in sublimation, the breaking bonds are those between
molecules. For this reason, the activation energies $E_{i}$ in the sublimation
case are small compared to the energies involved in chemical reactions.

We fitted the sublimation data from Fray & Schmitt (2009) using

$P(T)=A_{CO}e^{-{E_{CO}}/{RT}}$ (5)

where $R$ is the molar gas constant, $E_{CO}$ is the activation energy of $CO$
sublimation, and $A_{CO}$ is a constant, using a non-linear Levenberg
Marquardt technique. The data and the fits are shown in Figure (2). Equation
(5) provides a better fit to the data than the polynomials used by Fray &
Schmitt (2009). We succeeded in inferring the activation energy of $CO$ with
an exceptional accuracy, see the energies actually measured using various
dedicated experimental methods and we did this independently and in a
temperature range not included in the data (Luna & al., 2014).

Our model is governed by two coupled partial differential equations, one for
heat transport

$\rho c_{p}(T)\frac{\partial{T}}{\partial{t}}=\frac{\partial}{\partial
x}\left(k(T)\frac{\partial{T}}{\partial{x}}\right)\\\ $ (6)

and one for mass transport

$\frac{\partial n}{\partial
t}~{}~{}~{}~{}~{}=~{}~{}~{}\epsilon\frac{\partial}{\partial
x}\left(D_{K}\frac{\partial n}{\partial x}+\frac{\kappa n}{\mu}\frac{\partial
p}{\partial x}\right).$ (7)

In Equation (6) $\rho$ is the density and $k(T)$ is the conductivity at
temperature, $T$, while $t$ is the time and $x$ is the depth in the nucleus,
and $c_{p}(T)$ (see equations 8 and 9) is the heat capacity of the mantle. In
Equation (7), $D_{K}=\frac{r_{p}}{3}V_{th}$ is the Knudsen diffusion
coefficient, $V_{th}=\sqrt{\frac{8k_{B}T}{\pi m}}$ is the thermal speed,
$\kappa=\frac{r_{p}^{2}}{32}$ the permeability of the porous medium (Cussler,
2009), and $\mu=\frac{\alpha}{\pi^{3/2}}\frac{\sqrt{k_{B}mT}}{\sigma^{2}}$ is
the dynamic viscosity. Other quantities are $k_{B}$, the Boltzmann constant,
$\alpha$, a numerical constant on the order of 1, $T$, the temperature,
$\sigma$ the CO molecule’s collision diameter and $m$ the CO molecular mass,
$p$, the CO gas pressure $p(n,T)=nk_{B}T$, $n$ is the molecular density of CO;
$r_{p}$ the pore radius, $\epsilon$ is the porosity.

The equations are coupled by the transport coefficients. Water ice acts as a
mineral at the low temperatures found in the outer solar system, so we neglect
the H2O gas flux and focus only on the CO flux using a simplified version of
the model described in Bouziani & Fanale (1998). Gas produced by sublimation
passes through the porous mantle towards the comet surface. Figure (3)
illustrates the various modeled mechanisms of gas transport inside the porous
mantle. This model takes into account both microscopic and macroscopic
behavior and also treats the interaction between the gas and the walls. It
provides a continuous physical modeling of the coupling between mechanisms -
from discontinuous collisions to the collective macroscopic behavior of
continuous matter - and thus for the prediction of the mass flow rate and
pressure distribution with reasonable accuracy over the widest range of gas
pressures. The model depends only on molecular data and fundamental parameters
such as temperature, composition and total pressure. This approach allows us
to address the thermal and dynamical coupling of comets in a continuous and
physical way. Table (1) summarizes parameters used in the numerical
simulations.

We assume that the dust mantle is composed of a mixture of refractory minerals
and water ice. To estimate the heat capacity of the ice, we use the heat
capacity $c_{p}$ (J kg-1 K-1)deduced from the data provided in Shulman (2004).

$\begin{split}c_{p}(ice)=7.73T\left(1-e^{-1.263\times
10^{-3}T^{2}}\right)\left(1+e^{-3\sqrt{T}}\times 8.47T^{6}+2.0825\times
10^{-7}T^{4}e^{-4.97\times 10^{2}T}\right)\end{split}$ (8)

For the refractory part of the mantle we use the data for Enstatite found in
Krupka & al (1985) and we find that the Shulman equation gives an excellent
representation of the data at both high and low temperatures;

$\begin{split}c_{p}(dust)=2.03T\left(1-e^{-2.5\times
10^{-4}T^{2}}\right)\left(1+e^{-8.53\times 10^{-1}\sqrt{T}}\times 1.24\times
10^{-09}T^{6}-3.69\times 10^{6}T^{4}e^{-4.76\times
10^{5}T}\right).\end{split}$ (9)

We assume a water ice/dust ratio = 1, and thus take the specific heat capacity
of the mixture as the average of equations (9) and (8). Figure (4) shows how
the heat capacity of the mantle depends on the temperature.

We solved Equations (6 & 7) as a function of time, following the motion of
comet C/2017 K2 upon its approach to the Sun from large heliocentric
distances. For this purpose, we used the barycentric orbital elements from
Królikowska & Dybczyński (2018) while noting that the current Sun-centered
orbit from NASA’s Horizons ephemeris
software111http://ssd.jpl.nasa.gov/horizons.cgi is slightly hyperbolic
(eccentricity $e$ = 1.0004, vs. $e$ = 0.9999 from Królikowska & Dybczyński
(2018)). This difference between the barycentric and Sun-centered elements is
negligible for our purposes. The model solves the heat and gas transport
equations in a coupled manner. As it approaches the Sun, the surface
temperature of C/2017 K2 increases from the limiting interstellar value of 10
K to higher temperatures, creating a temperature gradient within the mantle,
and driving conduction towards the buried ice layer. Heating of the ice
triggers sublimation, producing a pressure gradient and resulting in a
diffusive flow of gas towards the comet’s surface, eventually leading to
escape and the expulsion of embedded dust particles through gas drag. The
downward conduction of heat and the upward transport of sublimated gas
molecules are characterized by distinct and different timescales, each
dependent on the thermal and micro-physical properties of the material, as we
will discuss in Section (3.2). The boundary condition (Equation 1) is time-
dependent because the orbital position is a function of time.

##  3 Results & Discussion

###  3.1 The Activity Mechanism

Figure (5) shows that the surface temperature increases, as expected from
radiative equilibrium, in proportion to $r_{H}^{-1/2}$. On the other hand, the
temperature at the ice-mantle boundary at first rises as $r_{H}$ decreases (we
refer to this region as Regime 1) but becomes nearly constant at
$r_{H}\lesssim$ 150 AU (Regime 2). This flattening occurs because the heat
conducted to the ice-mantle boundary in Region 2 is almost entirely used to
break intermolecular bonds in the ice, leaving little to raise the
temperature. These qualitatively different temperature vs. distance trends
were first noted for H2O by Mendis & Brin (1977) for Comet Kohoutek (1973f)
and later by Prialnik & Bar-Nun (1988). The model confirms this behavior and
extends it to much larger heliocentric distances in more volatile ices. For
convenience, in the remainder of this paper we refer to the junction between
Region 1 and Region 2 as the “Mendis point”. The CO flux coming out from the
comet mantle is given by Equation (7). In order to better understand the
orbital behavior of this flux as the comet approaches the sun, we solve the
coupled equations (6 & 7) and focus on the averaged flux $J_{CO}$ by
estimating the difference in the value of a variables between the surface of
the comet $A$ and the base of the mantle $B$, this approximation is emphasized
next in section (3.2); doing so, we deliberately ignore the local variation of
this flux within the mantle. The flux $J_{CO}$ is

$J_{CO}~{}~{}=~{}~{}\epsilon\left(D_{K}\frac{dn}{dx}+\frac{\kappa
n}{\mu}\frac{dp(n,T)}{dx}\right).$ (10)

Equation (10) shows how Knudsen free diffusion combines with collective
viscous flow (also called Hagen-Poiseuille flow in fluid dynamics). Although
these two flows are additive, they are not independent of each other. They are
coupled through the pre-gradient coefficients, thanks to the microscopic
approach of the model which integrates the walls in the collision from the
very beginning.

Figure (6) shows the CO gas pressure just below the surface, $p_{CO}$, as a
function of $r_{H}$. We find a prominent local bump in $p_{CO}$ at
$r_{H}\approx~{}150$ AU, corresponding to the Mendis point. Further
exploration shows that the location and amplitude of the $J_{CO}$ bump depend
on several physical parameters, including the assumed active surface fraction
of the mantle (the comet surface structure) and the mantle thickness. For
example, Figure (7) shows that the amplitude of the Mendis point bump
increases as the mantle active surface fraction decreases, because a lower
surface pore radius, $r_{s}$, increases the gas speed and allows the build up
of higher pressures that in turn speed up even more the flux (viscous mode is
more rapid). Figure (8) shows as expected that flux $J_{CO}$ increases as the
mantle thickness decreases; while thinner mantles provide less impedance to
the flow of gas, they also allow higher temperatures, and so higher pressures,
at the ice sublimation front.

The steady increase in $J_{CO}$ after the 150 AU Mendis point becomes an
abrupt rise as the comet approaches the Sun (c.f. Figure 9). Part of this
strong increase at small $r_{H}$ is due to the rising contribution of the
radiative term to the effective thermal conductivity of the mantle. To
demonstrate this behavior, we show in Figure (9) the effect of arbitrarily
removing the radiative term by setting $B$ = 0 in Equation (2). In this model
with a 5m thick mantle, the CO flux $J_{CO}$ at 4 AU falls from $6.4\times
10^{-7}$ kg m-2 s-1 (blue curve, including radiative term) to $\approx$
$1.2\times 10^{-7}$ kg m-2 s-1 (red curve, no radiative term). Figures (6, 7,
8 & 9) reveal that, depending on assumed parameters of the mantle, radiative
conduction begins to be significant around $50$ AU, reaches its maximum near
perihelion but has no effect at distances corresponding to the CO Mendis
point.

Why is there a local pressure bump near $r_{H}$ = 150 AU (c.f. Figure 6), and
not a continuous rise towards smaller heliocentric distances? Our models give
several clues. First, consider the unlikely case of a totally sealed comet
surface (the “steam cooker” approximation). Figure (10) shows that, as
expected, the calculated pressure closely follows the temperature of the ice
with no pressure bump and independent of the adopted physical parameters of
the mantle. This numerical experiment gives the total cumulative CO gas
production on the whole path, mainly produced after the CO Mendis point. The
steam cooker approximation proves that the bump and changes inward of the
Mendis point are products of gas transport processes in the mantle because
these fall to zero when the surface is sealed. Second, apart from this
pathological case, which we do not expect to find in nature, the pressure bump
exists in all models, albeit with widely different amplitudes depending on the
model parameters, even in the case of an almost totally (99%) permeable
mantle.

Prialnik & Sierks (2017) noted that the uppermost surface layer of the mantle
may be less permeable to flow than the deeper layers. We model this by
reducing the comet surface pore radii, $r_{s}$, compared to the mantle pore
radii, $r_{p}$. Figure (7) shows that the flux $J_{CO}$ amplitude and behavior
is strongly controlled by the assumed structural properties of the comet
mantle. The more the flux is inhibited, the higher is the flux bump. The
location of the Mendis point at the junction between the two thermal regimes
was noted above. It is at this confluence point that the lower flow entering
the mantle, ($x=B$ in Figure 1) stops increasing over time and slowly becomes
constant in response to the flattening of the temperature dependence caused by
strong CO sublimation in Regime 2. This happens while, at the surface of the
comet ($x=A$ in Figure 1), and also in the mantle depending on the pore
radius, the gas pressure remains high. The transition from the rising ice
front temperature in Regime 1 to the nearly constant ice front temperature in
Regime 2 leads to a temporary under-supply of CO at the base of the mantle,
relative to the loss rate from the surface, creating a local maximum as in
Figure (6) or less visible but there in the orange curve of Figure (7). The
width of the bump is a measure of the mantle’s holding or response time. The
under-supply of CO starting around 150 AU, relative to the rate of loss from
the surface, is responsible for the peak as the mantle switches from regime 1
to regime 2.

###  3.2 Retention Time Scale

The porous mantle of the comet acts as a buffer to the gas flow produced by
sublimation at the bottom. In order to understand this buffer and explore the
physical parameters that affect the gas retention time, we estimate in this
section the time scale corresponding to this buffer.

We first define the Knudsen Number $K_{n}$ by

$K_{n}~{}=~{}\frac{\lambda}{r_{p}}$ (11)

where $r_{p}$ is the pore radius and the mean free path is given by

$\lambda=\frac{1}{\sqrt{2}\pi{\sigma}^{2}n}.$ (12)

Here, $\sigma$ is the effective particle radius and $n$ is the number density
in the gas. The Knudsen number gives a measure of the importance of gas - gas
collisions relative to gas - particle collisions with the solid material
making up the porous mantle.

We define an effective flow velocity of the CO gas through the mantle, $V$,
from $V=J_{CO}/n$, where $n$ is the gas number density. Then, if $\Delta L$ is
the mantle thickness, Equation (10) can be rewritten as

$V~{}~{}=~{}~{}\frac{\epsilon D_{K}}{n}\frac{\Delta n}{\Delta
L}+\frac{\epsilon\kappa}{\mu}\frac{\Delta p}{\Delta L}$ (13)

The time taken for the flow to cross $\Delta L$ defines a holding or retention
timescale, $\tau=\frac{\Delta L}{V}$. Equation (13) gives

$\tau=\frac{{\mu n\Delta L}^{2}}{\epsilon\kappa n\Delta p(n,T)+\epsilon
D_{K}\mu\Delta n}$ (14)

Here $\Delta p=(p_{A}-p_{B})$ and $\Delta n=(n_{A}-n_{B})$ are the differences
in pressure and gas density, respectively, between the surface of the comet
(level A in Figure 1) and the bottom of the mantle (level B). For simplicity,
we calculate the intermediate coefficients $D_{K}$, $\mu$ and $\lambda$ of
this equation, using the averaged value of temperature, molecular density and
pressure (i.e. $T_{m}=\frac{T_{A}+T_{B}}{2}$).

Equation (14) shows that $\tau$ is governed by the two main modes of gas
transport, namely 1) Knudsen diffusion in the second denominator term and 2)
viscous flow (first denominator term). Figure (11) illustrates these two modes
as well as the intermediate transition and slip stages that are intermediate
steps between them over the wide range of pressures and pore sizes. The
Knudsen number approximately indicates the ranges of the different modes.

At what heliocentric distance does the Knudsen-Viscous transition occur? The
answer will depend on the structure of the dust mantle, in particular the pore
radius and the average free path, $\lambda$, which varies with pressure.
Figure (12) shows how the retention time changes with the heliocentric
distance, $r_{H}$, focusing on the coupling between the Knudsen flow and the
viscous flow and the different steps between the two flows. It reveals two
very distinct configurations. At large $r_{H}$ we find an almost constant time
scale of one century, leading to a slow and almost uniform motion of the gas
leaving the comet surface at a global bulk flow velocity of V =
$\frac{1}{3}~{}10^{-9}$ (m $s^{-1}$). The retention time scale decreases
dramatically as the comet approaches the sun, falling to about an hour at 50
AU. This indicates an acceleration of the gas, which we approximate by
estimating the slope of the Figure (12), given the assumed physical
parameters. By examining the behavior of the retention time Figure (13), we
can easily identify the four known modes for each independently computed
Knudsen number value and this is then a check of the model. The model was able
to capture the complete physics of the flow without any artificial
intervention by using the Knudsen number to switch from one mode to the other.

Knudsen diffusion predominates mainly at low pressure and the model allows a
viscous flow to develop continuously when the pressure becomes appreciable. As
we produce gas by sublimation, Figures (14) show that depending on the
transport mode, the duration of this retention time can vary widely.

###  3.3 Largest ejected dust particle

To be ejected, a dust particle must first overcome cohesive forces, $S_{0}$,
exerted on it by the surrounding dust material (Gundlach et al., 2015). The
minimum size of grains that could escape is (Jewitt et al., 2019a):

$a_{s}(r_{H})=\frac{S_{0}}{C_{D}f_{s}(r_{H})V_{s}(r_{H})},$ (15)

where $C_{D}$ is a dimensionless constant of order unity, $f_{s}$ (kg m-2 s-1)
is the mass sublimation flux and $V_{s}$ is the speed of the sublimated gas.
Once detached, the dust grain needs a force to accelerate above escape
velocity. This force can be provided by gas drag. For particles that are
spherical and of uniform density, the maximum size of grains that can be
ejected is approximated by (Jewitt et al., 2019a):

$a_{c}(r_{H})=\frac{9C_{D}f_{s}(r_{H})V_{s}(r_{H})}{16\pi
G\rho\rho_{n}r_{n}}.$ (16)

Here $a_{s}(r_{H})$ & $a_{c}(r_{H})$ are the dust grain sizes escaping the
comet surface against cohesive forces and against cometary gravity,
respectively. The model outputs : $f_{s}(CO)$ and
$V_{s}(CO)=\sqrt{\frac{8k_{B}T_{s}}{\pi\mu m_{H}}}$ are respectively the CO
gas outflow and its thermal speed, taken at the comet surface, where $T_{s}$
is the comet surface temperature, $\mu$ is the molecular weight, $m_{H}$ the
mass of hydrogen, $G$ is the gravitational constant, $\rho=\rho_{n}$
respectively dust and nucleus densities, $C_{D}$ is the dimensionless drag
efficiency coefficient, assumed to be equal to 1, cohesive strength constant
$S_{0}\approx 1.5\times 10^{-4}$ N m-1 (Sánchez & Scheeres, 2014), and $r_{n}$
is the comet nucleus radius.

Figure (15) displays Equations (15) and (16) as a function of $r_{H}$.
Particle radii must lie above the red curve in order for gas drag to exceed
inter-grain cohesive forces and below the blue curve for gas drag to exceed
the gravitational pull of the nucleus. The Figure shows that these conditions
can be simultaneously satisfied at all distances inside the $r_{H}\sim$ 150 AU
CO Mendis point, thus presenting a resolution of the cohesion bottleneck
problem for all active comets observed to-date.

##  4 Summary

We present a detailed gas-physics model of sublimation of cometary CO ice
buried beneath a permeable, refractory dust mantle. The model is solved as a
function of time and distance along the orbit of the distantly active long-
period comet C/2017 K2.

  1. 1.

Our main result is that we find an unexpected local peak in CO production at
very large heliocentric distances ($r_{H}\approx~{}150$ AU), caused by the
build up of pressure beneath a permeable mantle of modest thickness.

  2. 2.

This peak, whose magnitude is a function of several mantle microphysical
parameters, corresponds to the Mendis point, where the buried CO ice front
transitions between distinct temperature regimes.

  3. 3.

While modeled on C/2017 K2 as a specific case, our conclusions are general.
Comets entering the solar system from Oort cloud and interstellar distances
may become active far beyond the planets, at heliocentric distances
$r_{h}\sim$ 150 AU. Observational attempts to detect such ultra-distant
activity are encouraged.

We thank the anonymous referee for their valuable comments. We thank B.
Schmitt and N. Fray for providing the empirical pressure-temperature data for
cometary ices. To justify our neglect of conduction into the cold core beneath
the sublimation front (Equation 3), we can show that sublimation dominates the
energy budget over conduction down into the cold core. If $l$ is the thermal
skin depth to the cold core, then,

$l\sim\left(\frac{k\tau}{\rho c_{p}}\right)^{1/2}.$ (1)

Here, $k$ is the thermal conductivity of the material, $\rho$ and $c_{p}$ are
respectively the density and heat capacity. Quantity $\tau$ is the heating
timescale.

The conducted heat flux into the cold core is $F_{C}=k(dT/dx)\approx k\Delta
T/l$, where $\Delta T=T_{CO}-T_{C}$ is the temperature difference between the
top and bottom of the heated layer of thickness $l$. Then

$F_{C}=\left(\frac{k\rho c_{p}}{\tau}\right)^{1/2}\Delta T$ (2)

For comparison, the energy flux used by sublimation is

$F_{sub}=f_{CO}L_{CO}$ (3)

where $f_{CO}$ is the mass flux of CO in kg m-2 s-1 and $L=2.66\times 10^{5}$
J kg-1 is the latent heat of CO from Table (1).

Conduction into the cold core is smaller than sublimation when
$F_{C}<F_{sub}$, leading to

$f_{CO}>\left(\frac{k\rho c_{p}}{\tau}\right)^{1/2}\frac{\Delta T}{L_{CO}}.$
(4)

Setting $\Delta T=(26-10)=16$ K, $k=10^{-3}$ W m-1 K-1, $\rho$ = 533 kg m-3,
$c_{p}$ = 119 J kg-1 K-1 and $\tau\approx 5\times 10^{9}$ s as the travel time
from the Mendis point to 10 AU, we find that sublimation dominates provided
$f_{CO}>6.7\times 10^{-9}$ kg m-2 s-1. For comparison, Eq. (4) gives
$f_{CO}=1.4\times 10^{-7}$ kg m-2 s-1 at $T$ = 26 K confirming that the
sublimation energy flux is much larger than the conducted energy flux.

## References

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Table 1: Physical Parameters  Parameters | Symbols | Values | References
---|---|---|---
Oort Cloud Temperature (K) | $T_{OC}$ | $10$ | Baxter et al. (2018)
Gravitational Constant ($m^{3}kg^{-1}s^{-2}$ ) | $G$ | $6.67\times 10^{-11}$ |
Comet Nucleus Surface Albedo | $A_{B}$ | $0.04$ |
Comet Nucleus Radius (m) | $r_{n}$ | $9\times 10^{3}$ | Jewitt et al. (2017)
Comet Nucleus Density (kg m-3) | $\rho_{n}$ | $533$ |
Dust Grains Density (kg m-3) | $\rho=\rho_{n}$ | $533$ |
Porous Crust Thickness (m) | $L$ |  $10^{-2}$ – $10^{2}$ |
Main Pore Radius (m) | $r_{p}$ |  $10^{-8}$ – $10^{-2}$ |
Porosity | $\epsilon$ | 0.80 |
Emissivity | $\epsilon_{IR}$ | $0.95$ |
Stefan-Boltzmann Constant (W m-2 K-4 ) | $\sigma$ | $5.67\times 10^{-8}$ |
Solar Constant (W m-2 ) | $F_{\odot}$ | $1360.8$ |
Boltzmann Constant (J K-1 ) | $k_{B}$ | $1.380649\times 10^{-23}$ |
Ideal Gas Constant (J K-1 mol-1 ) | $R$ | $8.314$ |
Latent Heat–CO (J mol-1) | $E_{CO}$ | $7.45\times 10^{3}$ | Data & Fits Fig. (2)
Sublimation Pressure of CO (N m-2) | $P(T)$ | $8\times 10^{9}e^{-7450/RT}$ | Data & Fits Fig. (2)
Thermal Conductivity (W m-1 K-1) | $k(T)=A+B(4\sigma\epsilon_{IR}T^{3})$ |  $A$ & $B=10^{-4}$-$10^{-1}$ | Mendis & Brin (1977)

Figure 1: Schematic of the model physics. The photo on the right is from the
ESA Rosetta Mission to 67P/Churyumov-Gerasimenko (Osiris Team), the one on the
left is sugar powder from J. C. M.
https://stock.adobe.com/contributor/265923/jcm

Figure 2: Sublimation pressures from Fray & Schmitt (2009) fitted by the
relation $P_{i}(T)=A_{i}e^{-E_{i}/RT}$ to determine the activation energy,
$E_{i}$ (J mol-1), and the frequency factor, $A_{i}$. The resulting numerical
values are listed in Table (1).

Figure 3: Schematic showing different molecular transport modes. In the porous
mantle, gas transport is hindered by the walls encountered by the molecules
during their journey. Panels (1), (2) and (3) show the three gas transport
mechanisms within the porous mantle. This figure is adapted from
https://tinyurl.com/y3xyj9ss.

Figure 4: Heat capacity of the mantle as a function of temperature, $c_{p}(T)$
in (J kg-1 K-1)

Figure 5: Nucleus surface (red line) & buried ice front (blue line)
temperatures as a function of heliocentric distance, for in-falling comet
C/2017 K2. In this illustrative example the effective thermal conductivity is
$k(T)=10^{-2}$ W m-1 K-1, pore radius is $r_{p}=10^{-4}$ m, 30% of the flux is
allowed to escape the mantle, and the mantle thickness is 1 m.

Figure 6:  CO ice front temperature (blue line) and pressure just bellow the
surface, $p_{CO}$ (red line), the latter showing the Mendis point at about
150AU. This simulation assumes thermal conductivity $k(T)=10^{-4}$ W m-1 K-1
and a mantle thickness of 1 m. The comet surface pores are 10% smaller than
the mantle pores which have $r_{p}$ = 1.15$\times 10^{-4}$ m.

Figure 7:  CO flux $J_{CO}$ as a function of heliocentric distance for two
surface pore radii, $r_{s}$. Mantle pore radius is $r_{p}$= 1.15$\times
10^{-4}$ m. Assumed thermal conductivity is $k(T)$= 10-3 W m-1 K-1; the mantle
thickness is 1 m.

Figure 8: Dependence of the CO flux $J_{CO}$ on heliocentric distance and the
assumed mantle thickness.

Figure 9:  Effective of radiative conduction. Red curve shows the $J_{CO}$
flux when thermal conductivity includes a radiative contribution as described
in the text. Blue curve shows the surface pressure calculated without a
radiative contribution. Here mantle thickness is 5m. All other parameters are
held fixed between the models.

Figure 10: Extreme case of a totally sealed surface (“steam cooker”
approximation). This shows the cumulative CO gas mass production in (kg m-2).
For this numerical experiment we used $k(T)=10^{-2}$ W m-1 K-1 and the mantle
thickness is 1 m. The curve is independent of the mantle structure.

Figure 11:  The diagram shows how the different gas transport processes should
evolve as the comet moves from the Oort cloud to the inner solar system,
starting from high $Kn$ numbers ($Kn=\lambda/d$, where $\lambda$ is the mean
free path and $d$ is the pore size) to low $Kn$ numbers and changing
continuously over a wide range. Flow regimes evolve from Knudsen free
diffusion (a), followed by a transition regime (b) that turns into a slip
regime (c) that leads to a viscous continuum regime (d). Figure modified from
(Moghaddam & Jamiolahmady, 2016) & from https://tinyurl.com/y3xyj9ss.

Figure 12:  Retention time scale $\tau$ (Eq. 14) vs. the heliocentric distance
$r_{H}$ for two thermal conductivities (in W m-1 K-1); (1)
$k_{1}(T)=10^{-2}+10^{-2}(4{\sigma}_{IR}T^{3})$, (2)
$k_{2}(T)=10^{-4}+10^{-2}(4{\sigma}_{IR}T^{3})$

Figure 13:  This figure shows retention time ($\tau_{Model}$) (thick line) in
various transport modes. The Knudsen number (thin line) is shown on the right-
hand axis. (a) is the Knudsen Diffusion region ($10\leq Kn$), (b) the
Transition regime ($0.1\leq Kn\leq 10$) and, (c) the Slip regime region where
($0.01\leq Kn\leq 0.1$). We have assumed for purposes of illustration a
thermal conductivity $k(T)=10^{-2}+10^{-2}(4{\sigma}_{IR}T^{3})$ (W m-1 K-1),
mantle pore radius $r_{p}=10^{-4}$ m and, comet surface pores are taken equal
to 95% $r_{p}$.

Figure 14:  The effect of pore radius on retention time (Equation 14). The
curves show that the behavior of $\tau$ changes depending on whether the
transition (Knudsen-Viscous) in the gas transport occurs before (the blue
curves) or after (the red curves) the 150 AU Mendis point. It is shown here
that the sublimation temperature transition has an important retention effect
and this effect scales with $r^{2}$. We have assumed that
$k(T)=10^{-2}+10^{-2}(4{\sigma}_{IR}T^{3})$ (W m-1 K-1), with mantle thickness
1 m and surface pore radii $r_{s}=0.95r_{m}$.

Figure 15:  The largest dust particle escaping the comet surface. The
forbidden and the allowed zones for dust particle production are separated by
the bottleneck intersection point. The critical sizes $a_{c}$ & $a_{s}$ are
labeled (1) & (2) to show the effect of mantle pore sizes (1) $r_{p}$ is
$10^{-4}$ m and (2) $r_{p}$ is $10^{-3}$ m, here thermal conductivity is
$k(T)=10^{-4}$ (W m-1 K-1). We observe on this figure that for $r_{p}=10^{-3}$
(m), we can’t overcome cohesion and detach 0.1 mm dust particles (see the
inner curves $a_{c}(2)$ & $a_{s}(2)$) unless we reach a heliocentric distance
of $\sim$150 AU. The assumed thickness of the mantle is 1 m and the surface
pores have radii $r_{s}=0.97r_{p}$.


4.2cm4.2cm3.6cm3cm

# Generators of simple Lie superalgebras in characteristic zero

Wende Liu1,2111Supported by the NSF for Distinguished Young Scholars, HLJ
Province (JC201004) and the NSF of China (10871057) and Liming
Tang1,2222Correspondence<EMAIL_ADDRESS>(W. Liu<EMAIL_ADDRESS>(L. Tang)
1Department of Mathematics, Harbin Institute of Technology
Harbin 150006, China
2School of Mathematical Sciences, Harbin Normal University
Harbin 150025, China

> Abstract: It is shown that any finite dimensional simple Lie superalgebra
> over an algebraically closed field of characteristic 0 is generated by 2
> elements.
>
> Keywords: Classical Lie superalgebra; Cartan Lie superalgebra; generator
>
> Mathematics Subject Classification 2000: 17B05, 17B20, 17B70

##  0\. Introduction

Our principal aim is to determine the minimal number of generators for a
finite-dimensional simple Lie superalgebra over an algebraically closed field
of characteristic 0. The present work is dependent on the classification
theorem due to Kac [4], which states that a simple Lie superalgebra (excluding
simple Lie algebras) is isomorphic to either a classical Lie superalgebra or a
Cartan Lie superalgebra (see also [6]). In 2009, Bois [1] proved that a simple
Lie algebra in arbitrary characteristic $p\neq 2,3$ is generated by 2
elements. In 1976, Ionescu [3] proved that a simple Lie algebra $L$ over the
field of complex numbers is generated by $1.5$ elements, that is, given any
nonzero $x,$ there exists $y\in L$ such that the pair $(x,y)$ generates $L.$
In 1951, Kuranashi [5] proved that a semi-simple Lie algebra in characteristic
0 is generated by 2 elements.

As mentioned above, all the simple Lie superalgebras split into two series:
Classical Lie superalgebras and Cartan Lie superalgebras. The Lie algebra
(even part) of a classical Lie superalgebra is reductive and meanwhile there
exists a similarity in the structure side between the Cartan Lie superalgebras
in characteristic 0 and the simple graded Lie algebra of Cartan type in
characteristic $p$. Thus, motivated by Bois’s paper [1] and in view of the
observation above, we began this work in 2009. In the process we benefit in
addition much from the literatures above, especially from [1], which contains
a considerable amount of information in characteristic 0 and characteristic
$p$. We also use certain information about classical Lie superalgebras from
[7].

Throughout we work over an algebraically closed field $\mathbb{F}$ of
characteristic 0 and all the vector spaces and algebras are finite
dimensional. The main result is that any simple Lie superalgebra is generated
by 2 elements.

##  1\. Classical Lie superalgebras

###  1.1. Basics

A classical Lie superalgebra by definition is a simple Lie superalgebra for
which the representation of its Lie algebra (its even part) on the odd part is
completely reducible [4, 6]. Throughout this section, we always write
$L=L_{\bar{0}}\oplus L_{\bar{1}}$ for a classical Lie superalgebra. Our aim is
to determine the minimal number of generators for a classical Lie superalgebra
$L$. The strategy is as follows. First, using the results in Lie algebras [1,
3], we show that the Lie algebra $L_{\bar{0}}$ is generated by 2 elements.
Then, from the structure of semi-simple Lie algebras and their simple modules,
we prove that each classical Lie superalgebra is generated by 2 elements.

A classical Lie superalgebra is determined by its Lie algebra in a sense.

######  Proposition 1.1.

[6, p.101, Theorem 1] A simple Lie superalgebra is classical if and only if
its Lie algebra is reductive.

The following facts including Table 1.1 may be found in [4, 6]. The odd part
$L_{\bar{1}}$ as $L_{\bar{0}}$-module is completely reducible and
$L_{\bar{1}}$ decomposes into at most two irreducible components. By
Proposition 1.1, $L_{\bar{0}}=C(L_{\bar{0}})\oplus[L_{\bar{0}},L_{\bar{0}}].$
If the center $C(L_{\bar{0}})$ is nonzero, then $\dim C(L_{\bar{0}})=1$ and
$L_{\bar{1}}=L_{\bar{1}}^{1}\oplus L_{\bar{1}}^{2}$ is a direct sum of two
irreducible $L_{\bar{0}}$-submodules. For further information the reader is
refereed to [6, 4].

Classical Lie superalgebras (Table 1.1)
---
$L$ | $L_{\bar{0}}$ |  $L_{\bar{1}}$ as $L_{\bar{0}}$-module
${\rm{A}}(m,n),\;m,n\geq 0,n\neq m$ | ${\rm{A}}_{m}\oplus{\rm{A}}_{n}\oplus\mathbb{F}$ | $~{}~{}\mathfrak{sl}_{m+1}\otimes\mathfrak{sl}_{n+1}\otimes\mathbb{F}\oplus(\mbox{its dual})$
${\rm{A}}(n,n),\;n>0$ | ${\rm{A}}_{n}\oplus{\rm{A}}_{n}$ | $~{}~{}\mathfrak{sl}_{n+1}\otimes\mathfrak{sl}_{n+1}\oplus(\mbox{its dual})$
${\rm{B}}(m,n),\;m\geq 0,n>0$ | ${\rm{B}}_{m}\oplus{\rm{C}}_{n}$ | $~{}~{}\mathfrak{so}_{2m+1}\otimes\mathfrak{sp}_{2n}$
${\rm{D}}(m,n),\;m\geq 2,n>0$ | ${\rm{D}}_{m}\oplus{\rm{C}}_{n}$ | $\mathfrak{so}_{2m}\otimes\mathfrak{sp}_{2n}$
${\rm{C}}(n),\;n\geq 2$ | ${\rm{C}}_{n-1}\oplus\mathbb{F}$ | $~{}~{}\mathfrak{csp}_{2n-2}\oplus(\mbox{its dual})$
${\rm{P}}(n),\;n\geq 2$ | ${\rm{A}}_{n}$ | $~{}~{}\Lambda^{2}\mathfrak{sl}^{*}_{n+1}\oplus{\rm{S}}^{2}\mathfrak{sl}_{n+1}$
${\rm{Q}}(n),\;n\geq 2$ | ${\rm{A}}_{n}$ | $~{}~{}{\rm{ad}}\mathfrak{sl}_{n+1}$
${\rm{D}}(2,1;\alpha),\;\alpha\in\mathbb{F}\setminus\\{-1,0\\}$ | ${\rm{A}}_{1}\oplus{\rm{A}}_{1}\oplus{\rm{A}}_{1}$ | $~{}~{}\mathfrak{sl}_{2}\otimes\mathfrak{sl}_{2}\otimes\mathfrak{sl}_{2}$
${\rm{G}}(3)$ | $\mathfrak{G}_{2}\oplus{\rm{A}}_{1}$ | $\mathfrak{G}_{2}\otimes\mathfrak{sl}_{2}$
${\rm{F}}(4)$ | ${\rm{B}}_{3}\oplus{\rm{A}}_{1}$ | $\mathfrak{spin}_{7}\otimes\mathfrak{sl}_{2}$

###  1.2. Even parts

Let $\mathfrak{g}$ be a semi-simple Lie algebra. Consider the root
decomposition relative to a Cartan subalgebra $\mathfrak{h}$:
$\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}^{\alpha}.$
For $x\in\mathfrak{g}$ we write
$x=x_{\mathfrak{h}}+\sum_{\alpha\in\Phi}x^{\alpha}$ for the corresponding root
space decomposition. It is well-known that [2]

$\displaystyle{\rm{dim}}\mathfrak{g}^{\alpha}=1\quad\mbox{for
all}\;\alpha\in\Phi,$ (1.1)
$\displaystyle\mathfrak{h}=\sum_{\alpha\in\Phi}[\mathfrak{g}^{\alpha},\mathfrak{g}^{-\alpha}],$
(1.2)
$\displaystyle[\mathfrak{g}^{\alpha},\mathfrak{g}^{\beta}]=\mathfrak{g}^{\alpha+\beta}\quad\mbox{whenever}\;\alpha,\beta,\alpha+\beta\in\Phi.$
(1.3)

Let $V$ be a vector space and $\mathfrak{F}:=\\{f_{1},\ldots,f_{n}\\}$ a
finite set of non-zero linear functions on $V$. Write

$\Omega_{\mathfrak{F}}:=\\{v\in V\mid\Pi_{1\leq i\neq j\leq
n}(f_{i}-f_{j})(v)\neq 0\\}.$

######  Lemma 1.2.

Suppose $\mathfrak{F}$ is a finite set of non-zero functions in $V^{*}$. Then
$\Omega_{\mathfrak{F}}\neq\emptyset.$ If $\mathfrak{G}\subset\mathfrak{F}$
then $\Omega_{\mathfrak{F}}\subset\Omega_{\mathfrak{G}}.$

###### Proof.

The first statement is from [1, Lemma 2.2.1] and the second is
straightforward. ∎

This lemma will be usually used in the special situation when $V$ is a Cartan
subalgebra of a simple Lie superalgebra.

An element $x$ in a semi-simple Lie algebra $\mathfrak{g}$ is called balanced
if it has no zero components with respect to the standard decomposition of
simple Lie algebras. If $\mathfrak{h}$ is a Cartan subalgebra of
$\mathfrak{g}$, $x\in\mathfrak{g}$ is called $\mathfrak{h}$-balanced provided
that $x^{\alpha}\neq 0$ for all $\alpha\in\Phi.$

######  Lemma 1.3.

[1] An element of a semi-simple Lie algebra $\mathfrak{g}$ is balanced if and
only if it is $\mathfrak{h}$-balanced for some Cartan subalgebra
$\mathfrak{h}$.

###### Proof.

One direction is obvious. Suppose $x\in\mathfrak{g}$ is balanced and let
$\mathfrak{h}^{\prime}$ is a Cartan subalgebra of $\mathfrak{g}$. From the
proof of [1, Theorem 2.2.3], there exists $\varphi\in\mathfrak{g}$ such that
$\varphi(x)$ is $\mathfrak{h}^{\prime}$-balanced. Letting
$\mathfrak{h}=\varphi^{-1}(\mathfrak{h}^{\prime})$, one sees that
$\mathfrak{h}$ is a Cartan subalgebra and $x$ is $\mathfrak{h}$-balanced. ∎

For an algebra $\mathfrak{A}$ and $x,y\in\mathfrak{A}$, we write $\langle
x,y\rangle$ for the subalgebra generated by $x$ and $y$. We should notice that
for a Lie superalgebra $\langle x,y\rangle$ is not necessarily a
$\mathbb{Z}_{2}$-graded subalgebra (hence not necessarily a sub-Lie
superalgebra). The following technical lemma will be frequently used.

######  Lemma 1.4.

Let $\mathfrak{A}$ be an algebra. For $a\in\mathfrak{A}$ write $L_{a}$ for the
left-multiplication operator given by $a$. Suppose
$x=x_{1}+x_{2}+\cdots+x_{n}$ is a sum of eigenvectors of $L_{a}$ associated
with mutually distinct eigenvalues. Then all $x_{i}$’s lie in the subalgebra
generated $\langle a,x\rangle$.

###### Proof.

Let $\lambda_{i}$ be the eigenvalues of $L_{a}$ corresponding to $x_{i}$.
Suppose for a moment that all the $\lambda_{i}$’s are nonzero. Then

$(L_{a})^{k}(x)=\lambda_{1}^{k}x_{1}+\lambda_{2}^{k}x_{2}+\cdots+\lambda_{n}^{k}x_{n}\quad\mbox{for}\;k\geq
1.$

Our conclusion in this case follows from the fact that the Vandermonde
determinate given by $\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$ is nonzero
and thereby the general situation is clear. ∎

We write down a lemma from [1, Theorem B and Corollory 2.2.5] and the
references therein, which is also a consequence of Lemmas 1.2, 1.3 and 1.4.

######  Lemma 1.5.

Let $\mathfrak{g}$ be a semi-simple Lie algebra. If $x\in\mathfrak{g}$ is
balanced then for a suitable Cartan subalgebra $\mathfrak{h}$ and the
corresponding root system $\Phi$ we have $\mathfrak{g}=\langle x,h\rangle$ for
all $h\in\Omega_{\Phi}$.

Denote by $\Pi:=\\{\alpha_{1},\ldots,\alpha_{n}\\}$ the system of simple roots
of a semi-simple Lie algebra $\mathfrak{g}$ relative to a Cartan subalgebra
$\mathfrak{h}$. As above, $x\in\mathfrak{g}$ is refereed to as $\Pi$-balanced
if $x$ is a sum of all the simple-root vectors, that is,
$x=\sum_{\alpha\in\Pi}x^{\alpha},$ where $x^{\alpha}$ is a root vector of
$\alpha$. Recall that $\Omega_{\Pi}\neq\emptyset$ by Lemma 1.2.

######  Corollary 1.6.

A semi-simple Lie algebra $\mathfrak{g}$ is generated by a $\Pi$-balanced
element and an element in $\Omega_{\Pi}$.

###### Proof.

This is a consequence of Lemma 1.4 and the facts (1.1), (1.2) and (1.3). ∎

######  Proposition 1.7.

The Lie algebra of a classical Lie superalgebra is generated by 2 elements.

###### Proof.

Let $L=L_{\bar{0}}\oplus L_{\bar{1}}$ be a classical Lie superalgebra. By
Proposition 1.1, $L_{\bar{0}}$ is reductive, that is,
$[L_{\bar{0}},L_{\bar{0}}]$ is semi-simple and

$L_{\bar{0}}=C(L_{\bar{0}})\oplus[L_{\bar{0}},L_{\bar{0}}].$ (1.4)

If $C(L_{\bar{0}})=0$, the conclusion follows immediately from Lemma 1.5. If
$C(L_{\bar{0}})$ is nonzero, then $C(L_{\bar{0}})=\mathbb{F}z$ is
1-dimensional. Choose a balanced element $x\in[L_{\bar{0}},L_{\bar{0}}]$. By
Lemma 1.5, there exists $h\in[L_{\bar{0}},L_{\bar{0}}]$ such that
$[L_{\bar{0}},L_{\bar{0}}]=\langle x,h\rangle.$ Claim that

$L_{\bar{0}}=\langle x,h+z\rangle.$

Indeed, considering the projection of $L_{\bar{0}}$ onto
$[L_{\bar{0}},L_{\bar{0}}]$ with respect to the decomposition (1.4), denoted
by $\pi$, which is a homomorphism of Lie algebras, we have

$\pi(\langle x,h+z\rangle)=\langle\pi(x),\pi(h+z)\rangle=\langle
x,h\rangle=[L_{\bar{0}},L_{\bar{0}}].$

Hence only two possibilities might occur: $\langle x,h+z\rangle=L_{\bar{0}}$
or ${\rm{dim}}\langle x,h+z\rangle={\rm{dim}}[L_{\bar{0}},L_{\bar{0}}].$ The
first case is the desired. Let us show that the second does not occur. Assume
the contrary. Then $\pi$ restricting to $\langle x,h+z\rangle$ is an
isomorphism and thereby $\langle x,h+z\rangle$ is semi-simple. Thus

$\langle x,h+z\rangle=[\langle x,h+z\rangle,\langle x,h+z\rangle]=[\langle
x,h\rangle,\langle x,h\rangle]=\langle x,h\rangle.$

Hence $h\in\langle x,h+z\rangle$. It follows that

$z\in\langle x,h+z\rangle=\langle x,h\rangle=[L_{\bar{0}},L_{\bar{0}}],$

contradicting (1.4). ∎

######  Remark 1.8.

By Corollary 1.6, $\mathfrak{sl}(n)$ is generated by a $\Pi$-balanced element
$x$ and an element $y$ in $\Omega_{\Pi}$. As in the proof of Proposition 1.7,
one may prove that $\mathfrak{gl}(n)$ is generated by $h$ and $x+z$, where $z$
is a nonzero central element in $\mathfrak{gl}(n)$.

###  1.3. Classical Lie superalgebras

Suppose $L$ is a classical Lie superalgebra with the standard Cartan
subalgebra $H$. The corresponding weight (root) space decompositions are

$\displaystyle
L_{\bar{0}}=H\oplus\bigoplus_{\alpha\in\Delta_{\bar{0}}}L_{\bar{0}}^{\alpha},\qquad
L_{\bar{1}}=\bigoplus_{\beta\in\Delta_{\bar{1}}}L_{\bar{1}}^{\beta};$
$\displaystyle
L=H\oplus\bigoplus_{\alpha\in\Delta_{\bar{0}}}L_{\bar{0}}^{\alpha}\oplus\bigoplus_{\beta\in\Delta_{\bar{1}}}L_{\bar{1}}^{\beta}.$
(1.5)

Every $x\in L$ has a unique decomposition with respect to (1.5):

$x=x_{H}+\sum_{\alpha\in\Delta_{\bar{0}}}x_{\bar{0}}^{\alpha}+\sum_{\beta\in\Delta_{\bar{1}}}x_{\bar{1}}^{\beta},$
(1.6)

where $x_{H}\in H,$ $x_{\bar{0}}^{\alpha}\in L_{\bar{0}}^{\alpha}$,
$x_{\bar{1}}^{\beta}\in L_{\bar{1}}^{\beta}$. Write

$\Delta:=\Delta_{\bar{0}}\cup\Delta_{\bar{1}}\quad\mbox{and}\quad
L^{\gamma}:=L_{\bar{0}}^{\gamma}\oplus
L_{\bar{1}}^{\gamma}\quad\mbox{for}\quad\gamma\in\Delta.$

Note that the standard Cartan subalgebra of a classical Lie superalgebra is
diagonal:

$\mathrm{ad}h(x)=\gamma(h)x\quad\mbox{for all}\;\;h\in H,\;x\in
L^{\gamma},\;\gamma\in\Delta.$ (1.7)

For $x\in L$, write

$\mathbf{supp}(x):=\\{\gamma\in\Delta\mid x_{\gamma}\neq 0\\}.$ (1.8)

For $x=x_{\bar{0}}+x_{\bar{1}}\in L,$

$\mathbf{supp}(x)=\mathbf{supp}(x_{\bar{0}})\cup\mathbf{supp}(x_{\bar{1}}).$

######  Lemma 1.9.

  * $\mathrm{(1)}$

If $L\neq{\rm{Q}}(n)$ then $0\notin\Delta_{\bar{1}}$ and
$\Delta_{\bar{0}}\cap\Delta_{\bar{1}}=\emptyset.$

  * $\mathrm{(2)}$

If $L={\rm{Q}}(n)$ then $\Delta_{\bar{1}}=\\{0\\}\cup\Delta_{\bar{0}}.$

  * $\mathrm{(3)}$

If $L\neq\mathrm{A}(1,1)$, $\mathrm{Q}(n)$ or $\mathrm{P}(3)$ then
$\mathrm{dim}L^{\gamma}=1$ for every $\gamma\in\Delta.$

  * $\mathrm{(4)}$

Suppose $L={\rm{A}}(m,n)$, ${\rm{A}}(n,n),{\rm{C}}(n)$ or ${\rm{P}}(n),$ where
$m\neq n$.

    * $\mathrm{(a)}$

$L_{\bar{1}}=L_{\bar{1}}^{1}\oplus L_{\bar{1}}^{2}$ is a direct sum of two
irreducible $L_{\bar{0}}$-submodules.

    * $\mathrm{(b)}$

Let $\Delta_{\bar{1}}^{i}$ be the weight set of $L_{\bar{1}}^{i}$ relative to
$H,$ $i=1,2$. Then there exist $\alpha_{\bar{1}}^{i}\in\Delta_{\bar{1}}^{i}$
such that $\alpha_{\bar{1}}^{1}\neq\alpha_{\bar{1}}^{2}.$

###### Proof.

(1), (2) and (3) follow from [6, Proposition 1, p.137]. (4)(a) follows from
Table 1.1. Let us consider (4)(b). For $L={\rm{A}}(m,n),{\rm{A}}(n,n)$ or
${\rm{C}}(n),$ it follows from the fact that $L_{0}$-modules $L_{-1}$ and
$L_{1}$ are contragradient. For $L={\rm{P}}(n),$ a direct computation shows
that $-\varepsilon_{1}-\varepsilon_{2}\in\Delta_{\bar{1}}^{1}$ and
$2\varepsilon_{1}\in\Delta_{\bar{1}}^{2}.$ ∎

######  Theorem 1.10.

A classical Lie superalgebra is generated by 2 elements.

###### Proof.

Let $L=L_{\bar{0}}\oplus L_{\bar{1}}$ be a classical Lie superalgebra.

Case 1. Suppose ${\rm{dim}}C(L_{\bar{0}})=1.$ In this case $L={\rm{C}}(n)$ or
${\rm{A}}(m,n)$ with $m\neq n$ (see Table 1.1). Then
$L_{\bar{1}}=L_{\bar{1}}^{1}\oplus L_{\bar{1}}^{2}$ is a direct sum of two
irreducible $L_{\bar{0}}$-submodules and $[L_{\bar{0}},L_{\bar{0}}]$ is simple
or a direct sum of two simple Lie algebras. Let $x_{\bar{0}}$ be a balanced
element in $[L_{\bar{0}},L_{\bar{0}}].$ From Lemma 1.3, there exists a Cartan
subalgebra $\mathfrak{h}$ of $[L_{\bar{0}},L_{\bar{0}}]$ such that
${\bf{supp}}(x_{\bar{0}})=\Delta_{\bar{0}},$ the latter is viewed as the root
system relative to $\mathfrak{h}.$ By Lemma 1.5, we have
$[L_{\bar{0}},L_{\bar{0}}]=\langle x_{\bar{0}},h\rangle$ for all
$h\in\Omega_{\Delta_{\bar{0}}}$. Furthermore, from the proof of Proposition
1.7 it follows that $L_{\bar{0}}=\langle x_{\bar{0}},h+z\rangle$ for $0\neq
z\in C(L_{\bar{0}}).$ By Lemma 1.9(1) and (4), there exist
$\alpha_{\bar{1}}^{1}\in\Delta_{\bar{1}}^{1}$ and
$\alpha_{\bar{1}}^{2}\in\Delta_{\bar{1}}^{2}$ such that
$\alpha_{\bar{1}}^{1}\neq\alpha_{\bar{1}}^{2}$ and
$\alpha_{\bar{1}}^{1},\alpha_{\bar{1}}^{2}\notin\Delta_{\bar{0}}.$ Set
$x:=x_{\bar{0}}+x_{\bar{1}}^{\alpha_{\bar{1}}^{1}}+x_{\bar{1}}^{\alpha_{\bar{1}}^{2}}+z$
for some weight vectors $x_{\bar{1}}^{\alpha_{\bar{1}}^{i}}\in
L_{\bar{1}}^{\alpha_{\bar{1}}^{i}},\;i=1,2.$ Then

$x=(x_{\mathfrak{h}}+z)+\sum_{\alpha\in\Delta_{\bar{0}}}x_{\bar{0}}^{\alpha}+x_{\bar{1}}^{\alpha_{\bar{1}}^{1}}+x_{\bar{1}}^{\alpha_{\bar{1}}^{2}}.$

Write
$\Phi:=\Delta_{\bar{0}}\cup\\{\alpha_{\bar{1}}^{1}\\}\cup\\{\alpha_{\bar{1}}^{2}\\}$
and choose an element $h^{\prime}\in\Omega_{\Phi}.$ Assert $\langle
x,h^{\prime}\rangle=L.$ To show that, write $L^{\prime}:=\langle
x,h^{\prime}\rangle.$ Lemma 1.4 implies all components $x_{\bar{0}}^{\alpha},$
$x_{\bar{1}}^{\alpha_{\bar{1}}^{1}}$, $x_{\bar{1}}^{\alpha_{\bar{1}}^{2}}$ and
$x_{H}+z$ belong to $L^{\prime}$. Since $x_{\bar{0}}^{\alpha}\in L^{\prime}$
for all $\alpha\in\Delta_{\bar{0}},$ from (1.2) we have $x_{H}\in L^{\prime}$
and then $z\in L^{\prime}.$ As
$h^{\prime}\in\Omega_{\Phi}\subset\Omega_{\Delta_{\bar{0}}},$ we obtain
$\langle x_{\bar{0}},h^{\prime}+z\rangle=L_{\bar{0}}\subset L^{\prime}.$ Since
$x_{\bar{1}}^{\alpha_{\bar{1}}^{i}}\in L^{\prime}$ and
$L_{\bar{1}}^{\alpha_{i}}$ is an irreducible $L_{\bar{0}}$-module, we have
$L_{\bar{1}}^{i}\subset L^{\prime},$ where $i=1,2.$ Therefore, $L=L^{\prime}.$

Case 2. Suppose $C(L_{\bar{0}})=0.$ Then $L_{\bar{0}}$ is a semi-simple Lie
algebra and $L_{\bar{1}}$ decomposes into at most two irreducible components
(see Table 1.1).

Subcase 2.1. Suppose $L_{\bar{1}}$ is an irreducible $L_{\bar{0}}$-module.
Note that in this subcase, $L$ is of type $\mathrm{B}(m,n)$,
$\mathrm{D}(m,n)$, ${\rm{D}}(2,1;\alpha),$ $\mathrm{Q}(n),$ $\mathrm{G}(3)$ or
$\mathrm{F}(4).$ We choose a weight vector $x_{\bar{1}}^{\alpha_{\bar{1}}}\in
L_{\bar{1}}^{\alpha_{\bar{1}}}$ ($\alpha_{\bar{1}}\neq 0$) and any balanced
element $x_{\bar{0}}$ in $L_{\bar{0}}.$ By Lemma 1.3, we may assume that
${\bf{supp}}(x_{\bar{0}})=\Delta_{\bar{0}}.$

If $L\neq{\rm{Q}}(n),$ according to Lemma 1.9(1),
$\alpha_{\bar{1}}\notin\Delta_{\bar{0}}.$ Let
$x=x_{\bar{0}}+x_{\bar{1}}^{\alpha_{\bar{1}}}.$ Then

$x=x_{H}+\sum_{\alpha\in{\Delta_{\bar{0}}}}x_{\bar{0}}^{\alpha}+x_{\bar{1}}^{\alpha_{\bar{1}}}$

is the root-vector decomposition. Let
$\Phi=\Delta_{\bar{0}}\cup\\{\alpha_{\bar{1}}\\}$. By Lemmas 1.2 and 1.4, all
components $x_{H}$, $x_{\bar{0}}^{\alpha}$ and
$x_{\bar{1}}^{\alpha_{\bar{1}}}$ belong to $\langle x,h\rangle$ for
$h\in\Omega_{\Phi}\subset H$. By (1.1) and (1.2), this yields
$L_{\bar{0}}=\langle x_{\bar{0}},h\rangle\subset\langle x,h\rangle.$ Since
$x_{\bar{1}}^{\alpha_{\bar{1}}}\in\langle x,h\rangle$ and $L_{\bar{1}}$ is
irreducible as $L_{\bar{0}}$-module, we have $L=\langle x,h\rangle.$

Suppose $L={\rm{Q}}(n).$ Denote by
$\Pi:=\\{\delta_{1},\delta_{2},\ldots,\delta_{n}\\}$ the set of simple roots
of $L_{\bar{0}}$ relative to the Cartan subalgebra $H.$ According to Lemma
1.9(2), without loss of generality we may assume that
$\alpha_{\bar{1}}:=\delta_{1}+\delta_{2}.$ Let
$x=x_{\bar{0}}+x_{\bar{1}}^{\alpha_{\bar{1}}}.$ Then

$x=x_{H}+\sum_{\alpha\in{\Delta_{\bar{0}}}\setminus\\{\alpha_{\bar{1}}\\}}x_{\bar{0}}^{\alpha}+(x_{\bar{0}}^{\alpha_{\bar{1}}}+x_{\bar{1}}^{\alpha_{\bar{1}}}).$

By Lemma 1.4, all components $x_{H}$, $x_{\bar{0}}^{\alpha}$
($\alpha\in\Delta_{\bar{0}}\setminus\\{\alpha_{\bar{1}}\\}$), and
$x_{\bar{0}}^{\alpha_{\bar{1}}}+x_{\bar{1}}^{\alpha_{\bar{1}}}$ belong to
$\langle x,h\rangle$, where $h\in\Omega_{\Delta_{\bar{0}}}\subset H.$ From
(1.3) and (1.1) we conclude that
$x_{\bar{0}}^{\alpha_{\bar{1}}}\in\mathbb{F}[x_{\bar{0}}^{\delta_{1}},x_{\bar{0}}^{\delta_{2}}]\subset\langle
x,h\rangle$ and then $x_{\bar{1}}^{\alpha_{\bar{1}}}\in\langle x,h\rangle.$ As
above, the irreducibility of $L_{{\bar{1}}}$ yields $L=\langle x,h\rangle.$

Subcase 2.2. Suppose $L_{\bar{1}}=L_{\bar{1}}^{1}\oplus L_{\bar{1}}^{2}$ is a
direct sum of two irreducible $L_{\bar{0}}$-submodules. In this case,
$L={\rm{A}}(n,n)$ or ${\rm{P}}(n).$ Choose any balanced element
$x_{\bar{0}}\in L_{\bar{0}}$ and weight vectors
$x_{\bar{1}}^{\alpha_{\bar{1}}^{i}}\in L_{\bar{1}}^{\alpha_{\bar{1}}^{i}},$
where ${\alpha_{\bar{1}}^{1}}$ and ${\alpha_{\bar{1}}^{2}}$ are different
nonzero weights and $\alpha_{\bar{1}}^{i}\notin\Delta_{\bar{0}}$ (Lemma 1.9(1)
and (4)). Lemma 1.3 allows us to assume that
${\bf{supp}}(x_{\bar{0}})=\Delta_{\bar{0}}.$ Let
$x:=x_{\bar{0}}+x_{\bar{1}}^{\alpha_{\bar{1}}^{1}}+x_{\bar{1}}^{\alpha_{\bar{1}}^{2}}$
and
$\Phi:=\Delta_{\bar{0}}\cup\\{\alpha_{\bar{1}}^{1}\\}\cup\\{\alpha_{\bar{1}}^{2}\\}.$
As before, we are able to deduce that $L_{\bar{0}}\subset\langle x,h\rangle$
and
$x_{\bar{1}}^{\alpha_{\bar{1}}^{1}},x_{\bar{1}}^{\alpha_{\bar{1}}^{2}}\in\langle
x,h\rangle$ for $h\in\Omega_{\Phi}\subset\Omega_{\Delta_{\bar{0}}}\subset H.$
Thanks to the irreducibility of $L_{\bar{1}}^{1}$ and $L_{\bar{1}}^{2}$, we
have $L=\langle x,h\rangle$. The proof is complete. ∎

######  Remark 1.11.

In view of the proof of Theorem 1.10, starting from any balanced element in
the semi-simple part of the Lie algebra of a classical Lie superalgebra $L$ we
are able to find two elements generating $L.$

By Theorem 1.10, as in the proof of Proposition 1.7, one is able to prove the
following

######  Corollary 1.12.

The general linear Lie superalgebra $\mathfrak{gl}(m,n)$ is generated by 2
elements.

As a subsidiary result, let us show that a classical Lie superalgebra, except
for $\mathrm{A}(1,1)$, $\mathrm{Q}(n)$ or $\mathrm{P}(3)$, is generated by 2
homogeneous elements. By Lemma 1.9(3), for such a classical Lie superalgebra,
all the odd-weight subspaces are 1-dimensional. Here we give a more general
description in Remark 1.13. As before, an element $x\in L$ is called
$\Delta_{\bar{1}}$-balanced if $x$ is a sum of all the odd-weight vectors,
namely, $x=\sum_{\gamma\in\Delta_{\bar{1}}}x_{\bar{1}}^{\gamma},$ where
$x_{\bar{1}}^{\gamma}$ is a weight vector of $\gamma$.

######  Remark 1.13.

A finite dimensional simple Lie superalgeba (not necessarily classical) for
which all the odd-weight is $1$-dimensional is generated by 2 homogeneous
elements.

###### Proof.

Let $L$ be such a Lie superalgebra. Choose a $\Delta_{\bar{1}}$-balanced
element $x=\sum_{\gamma\in\Delta_{\bar{1}}}x_{\bar{1}}^{\gamma}$ and any
$h\in\Omega_{\Delta_{\bar{1}}}\subset H.$ By Lemmas 1.2 and 1.4, all
components $x_{\bar{1}}^{\gamma}$ belong to $\langle x,h\rangle$ for
$h\in\Omega_{\Delta_{\bar{1}}}\subset H$. Since $\mathrm{dim}L^{\gamma}=1,$ we
conclude that $L^{\gamma}\subset\langle x,h\rangle$ for all
$\gamma\in\Delta_{\bar{1}}.$ By [4, Proposition 1.2.7(1), p.20],
$L_{\overline{0}}=[L_{\overline{0}},L_{\overline{0}}]$ and then $\langle
x,h\rangle=L$. ∎

Finally we give an example to explain how to find the pairs of generators in
Theorem 1.10 and Remark 1.13.

######  Example 1.14.

Let $\mathrm{A}={\rm{A}}(1;0)$. Find the generators of $\mathrm{A}$ as in
Theorem 1.10 and Remark 1.13.

Recall that ${\rm{A}}=\\{x\in\mathfrak{gl}(2;1)\mid{\rm{str}}(x)=0\\}$. Its
Lie algebra is a direct sum of the $1$-dimensional center and the semi-simple
part:

${\rm{A}}_{\bar{0}}=\mathbb{F}(e_{11}+e_{22}+2e_{33})\oplus[{\rm{A}}_{\bar{0}},{\rm{A}}_{\bar{0}}],$

where
$[{\rm{A}}_{\bar{0}},{\rm{A}}_{\bar{0}}]=\mathrm{span}_{\mathbb{F}}\\{e_{11}-e_{22},e_{12},e_{21}\\}.$
The odd part is a direct sum of two irreducible
$\mathrm{A}_{\bar{0}}$-submodules:

$\displaystyle{\rm{A}}_{\bar{1}}={\rm{A}}_{\bar{1}}^{1}\oplus{\rm{A}}_{\bar{1}}^{2}=\mathrm{span}_{\mathbb{F}}\\{e_{13},e_{23}\\}\oplus\mathrm{span}_{\mathbb{F}}\\{e_{31},e_{32}\\}.$

The standard Cartan subalgebra is
$H=\mathrm{span}_{\mathbb{F}}\\{e_{11}-e_{22},e_{11}+e_{22}+2e_{33}\\}.$

Table 1.2 gives all the roots and the corresponding root vectors.

Table 1.2
---
roots | $\varepsilon_{1}-\varepsilon_{2}$ | $\varepsilon_{2}-\varepsilon_{1}$ | $\varepsilon_{1}-2\varepsilon_{3}$ | $\varepsilon_{2}-2\varepsilon_{3}$ | $-\varepsilon_{1}+2\varepsilon_{3}$ | $-\varepsilon_{2}+2\varepsilon_{3}$
vectors | $\hfill e_{12}\hfill$ | $\hfill e_{21}\hfill$ | $\hfill e_{13}\hfill$ | $\hfill e_{23}\hfill$ | $\hfill e_{31}\hfill$ | $\hfill e_{32}\hfill$

  * •

Theorem 1.10-Version. Put
$x:=(e_{12}+e_{21})+e_{13}+e_{31}+(e_{11}+e_{22}+2e_{33})$ and
$h:=3e_{11}+e_{22}+4e_{33}.$ From Table 1.2, the weight values corresponding
to $e_{12},e_{21},e_{13},e_{31}$ are $2,-2,-5,5,$ respectively. As in the
proof of Theorem 1.10, we have

$e_{12},e_{21},e_{13},e_{31},e_{11}+e_{22}+2e_{33}\in\langle x,h\rangle.$

Furthermore,

$\displaystyle\langle
e_{12}+e_{21},h+(e_{11}+e_{22}+2e_{33})\rangle={\rm{A}}_{\bar{0}}\subset\langle
x,h\rangle.$

Since ${\rm{A}}_{\bar{1}}^{i}$ is an irreducible ${\rm{A}}_{\bar{0}}$-module,
${\rm{A}}_{\bar{1}}^{i}\subset\langle x,h\rangle$, $i=1,2.$ Hence
$\mathrm{A}=\langle x,h\rangle$.

  * •

Remark 1.13-Version. Consider the $\Delta_{\overline{1}}$-balanced element
$x:=e_{13}+e_{31}+e_{23}+e_{32}$ and write $h:=e_{11}+e_{33}.$ By Table 1.2,
the weight values corresponding to $e_{13},e_{31},e_{23},e_{32}$ are
$-1,1,-2,2,$ respectively. As in the proof of Remark 1.13, we have
$e_{13},e_{31},e_{23},e_{32}\in\langle x,h\rangle.$ Since
$\mathrm{dimA}_{\bar{1}}^{\lambda}=1$ for $\lambda\in\Delta_{\bar{1}}$ and
$[\mathrm{A}_{\bar{1}},\mathrm{A}_{\bar{1}}]=\mathrm{A}_{\overline{0}},$ we
obtain $\mathrm{A}=\langle x,h\rangle.$

##  2\. Cartan Lie superalgebras

All the Cartan Lie superalgebras are listed below [4, 6]:

  * $W(n)$ ($n\geq 3$), $S(n)$ ($n\geq 4$), $\widetilde{S}(2m)$ ($m\geq 2$), $H(n)$ ($n\geq 5$).

Let $\Lambda(n)$ be the Grassmann superalgebra with $n$ generators
$\xi_{1},\ldots,\xi_{n}$. For a $k$-shuffle $u:=(i_{1},i_{2},\ldots,i_{k})$,
that is, a strictly increasing sequence between $1$ and $n$, we write $|u|:=k$
and $x^{u}:=\xi_{i_{1}}\xi_{i_{2}}\cdots\xi_{i_{k}}.$ Letting
${\rm{deg}}\xi_{i}=1,\;i=1,\ldots,n,$ we obtain the so-called standard
$\mathbb{Z}$-grading of $\Lambda(n).$ Let us briefly describe the Cartan Lie
superalgebras.

  * •

$W(n)={\rm{der}}\Lambda(n)$ is $\mathbb{Z}$-graded,
$W(n)=\oplus_{k=-1}^{n-1}W(n)_{k},$

$W(n)_{k}={\rm{span}}_{\mathbb{F}}\\{x^{u}\partial/\partial\xi_{i}\mid|u|=k+1,\;1\leq
i\leq n\\}.$

  * •

$S(n)=\oplus_{k=-1}^{n-2}S(n)_{k}$ is a $\mathbb{Z}$-graded subalgebra of
$W(n)$,

$S(n)_{k}={\rm{span}}_{\mathbb{F}}\\{\mathrm{D}_{ij}(x^{u})\mid|u|=k+2,\,\
1\leq i,j\leq n\\}.$

Hereafter,
${{\mathrm{D}_{ij}}}(f):=\partial(f)/\partial\xi_{i}\partial/\partial\xi_{j}+\partial(f)/\partial\xi_{j}\partial/\partial\xi_{i}$
for $f\in\Lambda(n).$

  * •

$\widetilde{S}(2m)$ ($m\geq 2$) is a subalgebra of $W(2m)$ and as a
$\mathbb{Z}$-graded subspace,

$\widetilde{S}(2m)=\oplus_{k=-1}^{2m-2}\widetilde{S}(2m)_{k},$

where

$\displaystyle\widetilde{S}(2m)_{-1}={\rm{span}}_{\mathbb{F}}\\{(1+\xi_{1}\cdots\xi_{2m})\partial/\partial\xi_{j}\mid
1\leq j\leq 2m\\},$ $\displaystyle\widetilde{S}(2m)_{k}=S(2m)_{k},\;0\leq
k\leq 2m-2.$

Notice that $\widetilde{S}(2m)$ is not a $\mathbb{Z}$-graded subalgebra of
$W(2m)$.

  * •

$H(n)=\oplus_{k=-1}^{n-3}H(n)_{k}$ is a $\mathbb{Z}$-graded subalgebra of
$W(n)$, where

$H(n)_{k}={\rm{span}}_{\mathbb{F}}\\{{\rm{D_{H}}}(x^{u})\mid|u|=k+2\\}.$

To explain the linear mapping ${\rm{D_{H}}}:\Lambda(n)\longrightarrow W(n)$,
write $n=2m$ $(m\geq 3)$ or $2m+1$ $(m\geq 2).$ By definition,
${\rm{D_{H}}}(x^{u}):=(-1)^{|u|}\sum_{i=1}^{n}\partial(x^{u})/\partial\xi_{i}\partial/\partial\xi_{i^{\prime}}$
for any shuffle $u,$ where ′ is the involution of the index set
$\\{1,\ldots,n\\}$ satisfying that $i^{\prime}=i+m$ for $i\leq m$.

For simplicity we usually write $W,S,\widetilde{S},H$ for
$W(n),S(n),\widetilde{S}(2m),H(n),$ respectively. Throughout this section $L$
denotes one of Cartan Lie superalgebras. Consider its decomposition of
subspaces mentioned above:

$L=L_{-1}\oplus\cdots\oplus L_{s}.$ (2.1)

For $W,S,\widetilde{S}$ and $H$, the height $s$ is $n-1,$ $n-2,$ $2m-2$ or
$n-3,$ respectively. Note that $S$ and $H$ are $\mathbb{Z}$-graded subalgebras
of $W$ with respect to (2.1), but $\widetilde{S}$ is not. The null $L_{0}$ is
isomorphic to
$\mathfrak{gl}(n),\mathfrak{sl}(n),\mathfrak{sl}(2m),\mathfrak{so}(n)$ for
$L=W,S,\widetilde{S},H,$ respectively.

######  Lemma 2.1.

  * $\mathrm{(1)}$

$L_{-1}$ and $L_{s}$ are irreducible as $L_{0}$-modules.

  * $\mathrm{(2)}$

$L_{1}$ is an irreducible $L_{0}$-module for $L=S,\widetilde{S}$ or $H,$
except for $H(6).$ For $L=H(6),$ $L_{1}$ is a direct sum of two irreducible
$L_{0}$-submodules.

  * $\mathrm{(3)}$

$L$ is generated by the local part $L_{-1}\oplus L_{0}\oplus L_{1}.$

  * $\mathrm{(4)}$

$L$ is generated by $L_{-1}$ and $L_{s}$ for $L=W$, $S$ or $H$.

###### Proof.

All the statements are standards (see [4, 6] for example), except for that
$\widetilde{S}_{-1}$ is irreducible as $\widetilde{S}_{0}$-module. Indeed, a
direct verification shows that $\widetilde{S}_{-1}$ is an
$\widetilde{S}_{0}$-module and the irreducibility follows from the canonical
isomorphism of $S_{0}$-modules
$\varphi:S_{-1}\longrightarrow\widetilde{S}_{-1}$ assigning
$\partial/\partial\xi_{i}$ to
$(1+\xi_{1}\cdots\xi_{2m})\partial/\partial\xi_{i}$ for $1\leq i\leq 2m.$ ∎

The following is a list of bases of the standard Cartan subalgebras
$\mathfrak{h}_{L_{0}}$ of $L_{0}.$

Table 2.1
---
$L$ | basis of $\mathfrak{h}_{L_{0}}$
$W(n)$ |  $\xi_{i}\partial/\partial\xi_{i},$ $1\leq i\leq n$
$S(n)$ |  $\xi_{1}\partial/\partial\xi_{1}-\xi_{j}\partial/\partial\xi_{j},$ $2\leq j\leq n$
$\widetilde{S}(2m)$ |  $\xi_{1}\partial/\partial\xi_{1}-\xi_{j}\partial/\partial\xi_{j},$ $2\leq j\leq 2m$
$H(2m)$ |  $\xi_{i}\partial/\partial\xi_{i}-\xi_{m+i}\partial/\partial\xi_{m+i},$ $1\leq i\leq m$
$H(2m+1)$ |  $\xi_{i+1}\partial/\partial\xi_{i+1}-\xi_{m+i}\partial/\partial\xi_{m+i},$ $1\leq i\leq m$

The weight space decomposition of the component $L_{k}$ relative to
$\mathfrak{h}_{L_{0}}$ is:

$L_{k}=\delta_{k,0}\mathfrak{h}_{L_{0}}\oplus_{\alpha\in\Delta_{k}}L_{k}^{\alpha},\,\
\mbox{where}\,\ -1\leq k\leq s.$

By Lemma 2.1(2), $H(6)_{1}$ is a direct sum of two irreducible
$H(6)_{0}$-modules

$H(6)_{1}=H(6)_{1}^{1}\oplus H(6)_{1}^{2}.$

Let $\Delta_{1}^{i}$ be the weight set of $H(6)_{1}^{i},$ $i=1,2.$ Write $\Pi$
for the set of simple roots of $L_{0}$ relative to the Cartan subalgebra
$\mathfrak{h}_{L_{0}}$. We have

######  Lemma 2.2.

  * $\mathrm{(1)}$

If $L=W$ or $S$ then
$\Pi\cap\Delta_{-1}=\Pi\cap\Delta_{s}=\Delta_{-1}\cap\Delta_{s}=\emptyset.$

  * $\mathrm{(2)}$

If $L=\widetilde{S}$ then
$\Pi\cap\Delta_{-1}=\Pi\cap\Delta_{1}=\Delta_{-1}\cap\Delta_{1}=\emptyset.$

  * $\mathrm{(3)}$

If $L=H(2m)$ then $\Pi\cap\Delta_{-1}=\Pi\cap\Delta_{1}=\emptyset$ and
$\Delta_{-1}\neq\Delta_{1}$.

  * $\mathrm{(4)}$

If $L=H(2m+1)$ then $0\in\Delta_{-1},$ $\Pi\neq\Delta_{1}$ and
$\Delta_{-1}\neq\Delta_{1}.$

  * $\mathrm{(5)}$

There exist nonzero weights $\alpha_{1}^{i}\in\Delta_{1}^{i}$ such that
$\alpha_{1}^{1}\neq\alpha_{1}^{2}.$

###### Proof.

We first compute the weight sets of the desired components and the system of
simple roots of $L_{0}.$ For $W(n)$,

$\displaystyle\Delta_{-1}=\\{-\varepsilon_{j}\mid 1\leq j\leq
n\\},\qquad~{}~{}~{}~{}~{}~{}~{}~{}\Delta_{0}=\\{\varepsilon_{i}-\varepsilon_{j}\mid
1\leq i\neq j\leq n\\},$
$\displaystyle\Pi=\\{\varepsilon_{i}-\varepsilon_{i+1}\mid 1\leq i\leq
n-1\\},\qquad\Delta_{s}=\bigg{\\{}\sum_{k=1}^{n}\varepsilon_{k}-\varepsilon_{j}\mid
1\leq j\leq n\bigg{\\}}.$

For $S(n)$ and $\widetilde{S}(n),$

$\displaystyle\Delta_{-1}=\\{-\varepsilon_{j}\mid 1\leq j\leq
n\\},\qquad~{}~{}~{}~{}~{}~{}~{}~{}\Delta_{0}=\\{\varepsilon_{i}-\varepsilon_{j}\mid
1\leq i\neq j\leq n\\},$
$\displaystyle\Pi=\\{\varepsilon_{i}-\varepsilon_{i+1}\mid 1\leq i\leq
n-1\\},\qquad\Delta_{1}=\big{\\{}\varepsilon_{k}+\varepsilon_{l}-\varepsilon_{j}\mid
1\leq k,l,j\leq n\big{\\}},$
$\displaystyle\Delta_{s}=\bigg{\\{}\sum_{i=1}^{n}\varepsilon_{i}-\varepsilon_{j}-\varepsilon_{k}\mid
1\leq j,k\leq n\bigg{\\}}.$

For $H(2m)$,

$\displaystyle\Delta_{-1}=\\{\ \pm\varepsilon_{j}\mid 1\leq j\leq
m\\},\qquad\Delta_{0}=\\{\pm(\varepsilon_{i}+\varepsilon_{j}),\pm(\varepsilon_{i}-\varepsilon_{j})\mid
1\leq i<j\leq m\\},$
$\displaystyle\Pi=\\{\varepsilon_{i}-\varepsilon_{i+1},\varepsilon_{m-1}+\varepsilon_{m}\mid
1\leq i<m\\},$
$\displaystyle\Delta_{1}=\\{\pm(\varepsilon_{i}+\varepsilon_{j})\pm\varepsilon_{k},\pm(\varepsilon_{i}-\varepsilon_{j})\pm\varepsilon_{k}\mid
1\leq i<j<k\leq m\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\cup\\{\pm\varepsilon_{l}\mid 1\leq
l\leq m\\}.$ (2.2)

For $H(2m+1)$, write $\varepsilon_{i}^{\prime}=\varepsilon_{i+1}$ for $1\leq
i\leq m.$ We have

$\displaystyle\Delta_{-1}=\\{0\\}\cup\\{\pm\varepsilon_{i}^{\prime}\mid 1\leq
i\leq m\\},$
$\displaystyle\Delta_{0}=\\{\pm\varepsilon_{k}^{\prime},\pm(\varepsilon_{i}^{\prime}+\varepsilon_{j}^{\prime}),\pm(\varepsilon_{i}^{\prime}-\varepsilon_{j}^{\prime})\mid
1\leq k\leq m,1\leq i<j\leq m\\},$
$\displaystyle\Pi=\\{\varepsilon_{i}^{\prime}-\varepsilon_{i+1}^{\prime},\varepsilon_{m}^{\prime}\mid
1\leq i<m\\},$
$\displaystyle\Delta_{1}=\\{0\\}\cup\\{\pm\varepsilon_{l}^{\prime},\pm(\varepsilon_{i}^{\prime}+\varepsilon_{j}^{\prime}),\pm(\varepsilon_{i}^{\prime}-\varepsilon_{j}^{\prime})\mid
1\leq l\leq m,1\leq i<j\leq m\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\cup\\{\pm(\varepsilon_{i}^{\prime}+\varepsilon_{j}^{\prime})\pm\varepsilon_{k},\pm(\varepsilon_{i}^{\prime}-\varepsilon_{j}^{\prime})\pm\varepsilon_{k}^{\prime})\mid
1\leq i<j<k\leq m\\}.$

All the statements follow directly, except (5) for $L=H(6).$ In this special
case, from (2.2) one sees that $0\notin\Delta_{1}$ and $|\Delta_{1}|>1$.
Consequently, (5) holds. ∎

Recall that an element $x\in\mathfrak{g}$ is refereed to as $\Pi$-balanced if
$x$ is a sum of all the simple-root vectors.

######  Theorem 2.3.

A Cartan Lie superalgebra is generated by 2 elements.

###### Proof.

Recall the null $L_{0}$ is isomorphic to
$\mathfrak{gl}(n),\mathfrak{sl}(n),\mathfrak{sl}(2m)$ or $\mathfrak{so}(n)$.
From Remark 1.8 and Corollary 1.6, for a $\Pi$-balanced element $x_{0}\in
L_{0}$ and $h_{0}\in\Omega_{\Pi}\subset\mathfrak{h}_{L_{0}}$ we have
$L_{0}=\langle x_{0}+\delta_{L,W}z,h_{0}\rangle,$ where $z$ is a central
element in $\mathfrak{gl}(n).$

For simplicity, write $t:=s$ for $L=W$ or $S$ and $t:=1$ for $L=\widetilde{S}$
or $H$. Suppose $L\neq H(6)\;\mbox{and}\;H(2m+1).$ According to Lemma 2.2, we
are able to choose nonzero weights $\alpha_{-1}\in\Delta_{-1}$ and
$\alpha_{t}\in\Delta_{t}$ such that $\alpha_{-1}\neq\alpha_{t}$,
$\alpha_{-1}\notin{\Pi},$ and $\alpha_{t}\notin{\Pi}.$ Put
$x:=x_{-1}+x_{0}+\delta_{L,W}z+x_{t}$ for some weight vectors $x_{-1}\in
L_{-1}^{\alpha_{-1}}$ and $x_{t}\in L_{t}^{\alpha_{t}}.$ Now set
$\Phi:=\Pi\cup\\{\alpha_{-1}\\}\cup\\{\alpha_{t}\\}\subset\mathfrak{h}_{L_{0}}^{*}$
and choose an element $h_{0}\in\Omega_{\Phi}.$ Assert $\langle
x,h_{0}\rangle=L.$ Lemma 1.4 implies all components $x_{-1}$ $x_{0}$,
$\delta_{L,W}z$ and $x_{t}$ belong to $\langle x,h_{0}\rangle.$ As
$h_{0}\in\Omega_{\Phi}\subset\Omega_{\Pi},$ we obtain $L_{0}=\langle
x_{0}+\delta_{L,W}z,h_{0}\rangle\subset\langle x,h_{0}\rangle.$ By Lemma
2.1(1) and (2), since $L_{-1}$ and $L_{t}$ are irreducible $L_{0}$-modules, we
have $L_{-1}+L_{t}\subset\langle x,h_{0}\rangle.$ From Lemma 2.1(3) and (4) it
follows that $L=\langle x,h_{0}\rangle.$

If $L=H(6),$ by Lemma 2.2(3), we are able to choose
$\alpha_{-1}\in\Delta_{-1},$ $\alpha_{1}^{1}\in\Delta_{1}^{1}$ and
$\alpha_{1}^{2}\in\Delta_{1}^{2}$ such that
$\alpha_{-1},\alpha_{1}^{1},\alpha_{1}^{2}$ are pairwise distinct and
$\alpha_{-1}\notin\Pi,$ $\alpha_{1}^{1}\notin\Pi$ and
$\alpha_{1}^{2}\notin\Pi$. Put $x:=x_{-1}+x_{0}+x_{1}^{1}+x_{1}^{2}$ for some
weight vectors $x_{-1}\in L_{-1}^{\alpha_{-1}}$ and $x_{1}^{i}\in
L_{1}^{\alpha_{1}^{i}},$ $i=1,2.$ Write
$\Phi:=\Pi\cup\\{\alpha_{-1}\\}\cup\\{\alpha_{1}^{1}\\}\cup\\{\alpha_{1}^{2}\\}.$
For $h_{0}\in\Omega_{\Phi}\subset\Omega_{\Pi}$, as in the above, one may show
that $L=\langle x,h_{0}\rangle$.

If $L=H(2m+1),$ by Lemma 2.2(4), choose $\alpha_{-1}\in\Delta_{-1},$
$\alpha_{1}\in\Delta_{1}$ such that $\alpha_{-1}=0,$ $\alpha_{1}\notin\Pi.$
Set $x:=x_{-1}+x_{0}+x_{1}$ for some weight vectors $x_{-1}\in
L_{-1}^{\alpha_{-1}}$ and $x_{1}\in L_{t}^{\alpha_{1}}.$ Now put
$\Phi:=\Pi\cup\\{\alpha_{-1}\\}\cup\\{\alpha_{1}\\}\subset\mathfrak{h}_{L_{0}}^{*}.$
Let $h_{0}\in\Omega_{\Phi}\subset\Omega_{\Pi}$ and claim that $L=\langle
x,h_{0}\rangle.$ By Lemma 1.4, $x_{0},$ $x_{-1}$ and $x_{1}\in\langle
x,h_{0}\rangle.$ Consequently, $L_{0}\subset L$. The irreducibility of
$L_{-1}$ and $L_{1}$ ensures $L_{-1}+L_{1}\subset\langle x,h_{0}\rangle.$ By
Lemma 2.1(3), the claim holds. The proof is complete. ∎

Theorems 1.10 and 2.3 combine to the main result of this paper:

######  Theorem 2.4.

Any simple Lie superalgebra is generated by 2 elements.

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