# Stability of Optically Injected Two-State Quantum-Dot Lasers

Stefan Meinecke  Institut für Theoretische Physik, Technische Universität
Berlin, Hardenbergstr. 36, 10623 Berlin, Germany  Benjamin Lingnau  Institut
für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36,
10623 Berlin, Germany  André Röhm  Institut für Theoretische Physik,
Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany  Kathy
Lüdge  Institut für Theoretische Physik, Technische Universität Berlin,
Hardenbergstr. 36, 10623 Berlin, Germany

###### Abstract

Simultaneous two-state lasing is a unique property of semiconductor quantum-
dot (QD) lasers. This not only changes steady-state characteristics of the
laser device but also its dynamic response to perturbations. In this paper we
investigate the dynamic stability of QD lasers in an external optical
injection setup. Compared to conventional single-state laser devices, we find
a strong suppression of dynamical instabilities in two-state lasers.
Furthermore, depending on the frequency and intensity of the injected light,
pronounced areas of bistability between both lasing frequencies appear, which
can be employed for fast optical switching in all-optical photonic computing
applications. These results emphasize the suitability of QD semiconductor
lasers in future integrated optoelectronic systems where a high level of
stability is required.

Quantum Dot Lasers, Two-State Lasing, Optical Injection, Optical Bistability,
All-Optical Switching

This is the pre-peer reviewed version of the following article: Stability of
Optically Injected Two-State Quantum-Dot Lasers, S. Meinecke, B. Lingnau, A.
Röhm, K. Lüdge, ANNALEN DER PHYSIK 2017, 529, 1600279, which has been
published in final form at https://doi.org/10.1002/andp.201600279. This
article may be used for non-commercial purposes in accordance with Wiley Terms
and Conditions for Use of Self-Archived Versions.

##  I Introduction

Quantum-dot (QD) based semiconductor lasers have a variety of beneficial
properties, e.g. low threshold current, high differential gain, small line
width and weak temperature sensitivity, which make them appealing for
innovative applications in optical communication networks [1, 2, 3]. Quantum
dots are most commonly self-assembled nanometer-sized semiconductor structures
embedded in a surrounding quantum well. They provide confinement in all
spatial dimensions, which results in localized electronic states and a
discrete density of states. Relatively slow charge-carrier dynamics between
the localized states [4] then allow for simultaneous lasing from different
electronic states at distinct wavelengths. Lasing has been experimentally
demonstrated for emission at two [5, 6] and even three [7] wavelengths.
Simultaneous lasing from the QD ground (GS) and first excited (ES) state is
called two-state lasing. Additionally, some devices can exhibit a decreasing
emission from the GS wavelength with increasing pump current, resulting in the
complete turn-off of the GS emission (quenching) and a switching to higher-
energy emission. Since both wavelengths also have different threshold pump
currents, there are typically three distinct regimes of operation: ground-
state lasing, two-state lasing, and excited-state lasing. The modeling of two-
state lasing has been the topic of previous research[8, 9, 10, 11, 12].
Applications of two-state lasing effects are discussed for free-running all-
optical gating [13], feedback induced switching [14], anti-phase excitability
[15] and self-mixing velocimetry [16].

Our optical injection setup considers an external laser, that drives the QD
laser at a slightly detunend frequency. This method has been widely studied
for quantum well and single-color QD lasers and can be used to stabilize the
emitted frequency [17, 18, 19, 20]. We want to concentrate on the effects that
arise due to the ability of the QD laser to exhibit two-state lasing,
specifically the role of the resulting higher dimensional phase-space, which,
surprisingly, improves the dynamical stability. An increased stability is also
theoretically predicted when the microscopic polarization introduces an
additional degree of freedom into the system dynamics [21], which possibly
suggests a counter-intuitive but general trend towards higher stability with
increasing number of degrees of freedom in semiconductor lasers.

We will discuss the stability boundaries by exploring the parameter space of
the injected frequency and intensity. Our aim is to analyse the influence of
driving current and amplitude-phase coupling upon the bifurcation diagram,
especially on the optical bistability between the two different lasing
frequencies. A bistability between two different locked solutions already
occurs in single-state QD laser with optical injection[20], however, we focus
on the optical bistability between two wavelengths that was recently
demonstrated experimentally for QD lasers in an all-optical switching setup
[22, 23]. Previously, stability boundaries have been derived analytically in
the context of two-polarization-mode lasing with optical injection [24] and
elaborate bifurcation analysis was performed for optically injected Fabry-
Pérot cavities that support two longitudinal modes [25, 26, 27, 28]. Yet, in
the case of two-state QD lasers, the underlying dynamics are different as the
two wavelengths emerge from different localized levels with a large energy
separation of about 70meV. Thus, they do not directly compete for the same
gain medium which on the one hand requires a modeling with more dynamic
variables and on the other hand gives rise to new dynamics as for example
ground state quenching [12]. For our investigations, we consider a 1mm long
edge-emitting laser based on InAs/InGaAs QDs, which consists of 15 QD layers,
each $2\mu m$ wide and $4nm$ tall. The GS emits at $\hbar\omega^{GS}=0.952eV$
and the ES at $\hbar\omega^{ES}=1.022eV$ ($1300$nm and $1210$nm,
respectively.).

Theoretical results for two-state lasers with optical injection suggest the
inhomogeneous broadening of QD states to be crucial for the observation of the
hysteresis [23]. The authors also emphasize the need for a model that goes
beyond the $\alpha$-factor approximation to describe the amplitude-phase
coupling of the gain medium. In our results, however, we show that the
bistability is directly caused by the GS amplitude-phase coupling and that the
usual linear $\alpha$-factor approximation can be sufficient, if the driving
current and the strength of the amplitude-phase coupling is tuned correctly.

Figure 1: Sketch of the optically injected two-state QD laser. The injection
$E_{inj}$ is at the ground state wavelength, while the resulting dynamics at
the ground state energy ($\hbar\omega^{GS}$) and excited state energy
($\hbar\omega^{ES}$) are investigated.

##  II Model & Equations

The model used within this paper is a rate-equation model that is based upon
semiconductor-Bloch-equations with microscopically calculated charge-carrier
scattering rates and builds upon previous work [29, 30, 12]. By comparison
with the full semiconductor-Bloch-equation approach we extracted the frequency
shift $\delta\omega$ of the GS electric field induced by changes in the entire
carrier distribution [30], whereas the conventional $\alpha$-factor approach
only takes into account the influence of the active carrier population on the
refractive index. Thus, the phase dynamics of the field inside the QD-laser
cavity, i.e., the instantaneous frequency shift, is decoupled from the gain
dynamics in our model, which makes the description more accurate than using a
linear $\alpha$-factor [31]. The equation for the ground and excited state
field amplitudes ($E^{GS},E^{ES}$) are given in Eq. (1)-(2).

$\displaystyle\dot{E}^{GS}=$
$\displaystyle\left[g_{GS}(\rho_{e,\textit{act}}^{GS}+\rho_{h,\textit{act}}^{GS}-1)-i\gamma\delta\omega-\kappa^{GS}\right]E^{GS}$
$\displaystyle+K\kappa^{GS}E_{0}e^{-i2\pi\Delta\nu_{inj}t}$ (1)
$\displaystyle\dot{E}^{ES}=$
$\displaystyle\left[g_{ES}(\rho_{e,\textit{act}}^{ES}+\rho_{h,\textit{act}}^{ES}-1)-\kappa^{ES}\right]E^{ES}$
(2) $\displaystyle\dot{\rho}_{b,\textit{act}}^{m}=$
$\displaystyle-\frac{g_{m}}{\nu_{m}Z^{QD}f^{act}}(\rho_{e,\textit{act}}^{m}+\rho_{h,\textit{act}}^{m}-1)\frac{|E^{m}|^{2}}{\eta_{m}^{2}}$
$\displaystyle-
W_{m}\rho_{e,\textit{act}}^{m}\rho_{h,\textit{act}}^{m}+S^{m,cap}_{b,\textit{act}}+S^{m,rel}_{b,\textit{act}}$
(3) $\displaystyle\dot{\rho}_{b,\textit{inact}}^{m}=$ $\displaystyle-
W_{m}\rho_{e,\textit{inact}}^{m}\rho_{h,\textit{inact}}^{m}+S^{m,cap}_{b,\textit{inact}}+S^{m,rel}_{b,\textit{inact}}$
(4) $\displaystyle\dot{w_{b}}=$ $\displaystyle+J-R_{loss}^{W}w_{e}w_{h}$
$\displaystyle-2N^{QD}\left(f^{\textit{act}}S^{GS,cap}_{b,\textit{act}}+f^{\textit{inact}}S^{GS,cap}_{b,\textit{inact}}\right)$
$\displaystyle-4N^{QD}\left(f^{\textit{act}}S^{ES,cap}_{b,\textit{act}}+f^{\textit{inact}}S^{ES,cap}_{b,\textit{inact}}\right)$
(5)

We consider optical injection into the ground state (see sketch of the setup
in Fig.1) and therefore the phase dynamics of the ES can be neglected. The
relative injection strength $K$ is normalized to a combined electric field
amplitude $E_{0}$, which is defined via the the total out-coupled power

$\displaystyle
2\kappa^{GS}|E_{0}|^{2}:=\sum_{m\in\\{GS,ES\\}}2\kappa^{m}|E^{m}|^{2}\bigg{|}_{K=0}\propto
P_{out}$ (6)

This normalization ensures that the injection strength does not depend upon
the distribution of the GS and ES to the total power. $K=1.0$ then corresponds
to injecting as much electric field intensity into to the GS as is being
coupled out from the GS and the ES. The relative frequency detuning of the
injected signal from the solitary GS frequency is given by
$\Delta\nu_{inj}=(\omega_{inj}-\omega_{las}^{GS})/(2\pi)$.

Figure 2: Energy band structure across one QD within the self-organized layer
of heterogeneously distributed QDs. Green arrows indicate carrier scattering
transitions while blue and red straight arrows indicate optical transmissions.
Blue and red wavy arrows sketch stimulated emission of photons. The green
symbols on the right denote the dynamic charge-carrier variables described
with Eq. (3)-(5).

The charge-carrier variables inside the active medium and their interaction
processes needed for their dynamic description are shown in Fig. 2. There are
six different carrier types: electrons and holes inside the ground state
$(\rho_{e}^{GS},\rho_{h}^{GS})$, the excited state
$(\rho_{e}^{ES},\rho_{h}^{ES})$ and inside the surrounding quantum-well
reservoir $(w_{e},w_{h})$. The reservoir is pumped with an electric current
density $J$. Due to the inhomogeneous broadening not all QD transitions are
resonant with the optical mode and we therefore introduce a separation into
resonant active carriers $\rho_{\textit{act}}$ and non-resonant carriers
$\rho_{\textit{inact}}$ that do not directly interact with the light mode but
indirectly via the common carrier reservoir. The resulting equations are given
in Eg.(3)-(5). We use $b\in\\{e,h\\}$ and
$x\in\\{\textit{act},\textit{inact}\\}$ to indicate the carrier type and
$m\in\\{GS,ES\\}$ to indicate the state. Short explanations of the parameters
and the values used for numerical calculations are given in Tab. 1.

The instantaneous frequency shift $\delta\omega$ given in Eq. (1) of the GS
lasing mode is given by

$\displaystyle\delta\omega=\delta\omega_{ES}(\rho_{e}^{ES}+\rho_{h}^{ES})+\delta\omega_{QW}^{e}w_{e}+\delta\omega_{QW}^{h}w_{h}$
(7)

with parameters
$\delta\omega_{ES},\delta\omega_{QW}^{e},\delta\omega_{QW}^{h}$ as given in
Tab. 1. We additionally include the artificial scaling parameter $\gamma$ to
investigate the consequences of stronger amplitude-phase coupling (the default
value is $1$). The microscopic treatment of the charge-carrier dynamics is
important for an accurate modelling of the two-mode lasing characteristics of
our QD laser device [12]. We found the carrier scattering to only weakly
depend on small variations of, e.g., the energy structure of the QD or their
inhomogeneous broadening. On the other hand, the carrier-induced refractive
index change and, thus, the amplitude-phase coupling depends sensitively on
those parameters. While an isolated rescaling of our dynamic amplitude-phase
coupling with $\gamma$ is not a rigorous microscopic treatment of the QD laser
carrier dynamics, it nevertheless gives reliable results on the dynamic
changes expected for QD laser structures with different amplitude-phase
coupling strength.

Figure 3: Input-output characteristics of the QD-laser without optical
injection. Total outcoupled power (dashed green) and GS (red) and ES (blue)
outcoupled powers are plotted as a function of the pump current $J$ normalized
to the threshold current of the GS ($J_{th}^{GS}$). With increasing pump
current three dynamic regimes can be distinguished. They are indicated by the
vertical lines and are labeled "GS lasing", "two-state lasing" and "ES
lasing". The bended arrow shows the operating point for Figs. 4-7.

The scattering rates for the charge-carrier exchange processes between QD
states and the quantum well reservoir are implemented via nonlinear fits to
microscopically calculated rates (at $T=300K$) (see [30] for details) and
enter the carrier equations according to Eq. (8)-(9).

$\displaystyle S_{b,x}^{m,cap}$
$\displaystyle=S_{b}^{m,in}(1-\rho_{b,x}^{m})-S_{b}^{m,cap,out}(\rho_{b,x}^{m})$
(8) $\displaystyle S_{b,x}^{m,rel}$
$\displaystyle=\pm\frac{1}{\nu_{m}}\left[S_{b,in}^{rel}(1-\rho_{b,x}^{GS})\rho_{b,x}^{ES}-S_{b,out}^{rel}(1-\rho_{b,x}^{ES})\rho_{b,x}^{GS}\right]$
(9)

Table 1: Parameters used for our numerical calculations unless otherwise noted. Symbol | Value | Meaning
---|---|---
$g_{GS}$ | $230ns^{-1}$ | GS linear gain
$g_{ES}$ | $460ns^{-1}$ | ES linear gain
$\kappa^{GS}$ | $50ns^{-1}$ | GS optical losses
$\kappa^{ES}$ | $10ns^{-1}$ | ES optical losses
$Z^{QD}$ | $3.0\times 10^{7}$ | number of QDs
$N^{QD}$ | $10^{11}cm^{-2}$ | area density of QDs
$f^{act}$ | $0.5$ | fraction of active QDs
$W_{GS}$ | $0.44ns^{-1}$ | GS spont. emission rate
$W_{ES}$ | $0.55ns^{-1}$ | ES spont. emission rate
$R_{loss}^{W}$ | $0.54\times 10^{11}cm^{2}ns^{-1}$ | quantum well loss rate
$\eta_{GS}^{2}$ | $101.1V^{2}cm^{-2}$ | conversion factor
$\eta_{ES}^{2}$ | $108.5V^{2}cm^{-2}$ | conversion factor
$\nu_{m}$ | 1(GS),2(ES) | QD state degeneracy
$\gamma$ | $1$ | scaling factor
$\delta\omega_{ES}$ | $125ns^{-1}$ | frequency shift coefficient
$\delta\omega_{QW}^{e}$ | $11.3\times 10^{-11}cm^{2}ns^{-1}$ | frequency shift coefficient
$\delta\omega_{QW}^{h}$ | $5.5\times 10^{-11}cm^{2}ns^{-1}$ | frequency shift coefficient

The GS and ES electric field amplitudes are only indirectly coupled via
charge-carrier scattering processes between the GS and ES QD states. These
processes are slow when compared to quantum well lasers [4] and thereby allow
for an incomplete gain clamping of the ES inversion [32], which enables two-
state lasing.

The simulated input-output diagram (LI-curve) of the two-state QD laser is
shown in Fig. 3. The total out-coupled power is shown with a dashed green line
while GS and ES out-coupled powers are indicated with solid red and blue
lines, respectively. The ground state shows GS quenching (decreasing red line)
as soon as the ES turns on. At high currents the GS is off and only ES lasing
remains. We thus have three different operating regimes that we indicate by:
"GS", "two-state" and "ES" -lasing in Fig. 3. This behavior mimics
experimental results [5]. In a model with separate GS and ES carrier
populations, it can only be reproduced, if an asymmetry in the electron and
hole dynamics is included [12]. Simpler two-mode laser models, which take into
account only one charge-carrier type, have to include nonlinear gain terms
with self- and cross-compression contributions to reproduce a similar
qualitative shape of the input-output curve [33]. Such an approach, however,
fails to describe the asynchronous refractive index dynamic of QD lasers [34],
which requires multiple carrier types to be modeled. Therefore, a relatively
complex model with in total 10 different carrier types is chosen.

Note that without optical injection the QD laser always emits stable light and
does not exhibit bistability. For the numerical studies of the stability an
operating current within the "GS-lasing" regime close to the onset of two-
state lasing is chosen (see arrow in Fig. 3). The regimes and the respective
threshold currents are characteristic for each device and among other
parameters depend upon the optical loss rates $\kappa^{GS}$ and $\kappa^{ES}$
[35, 12]. Note that the ES losses $\kappa^{ES}$ are chosen to be
$0.2\kappa^{GS}$ to allow for ES lasing at reasonable pump currents.

##  III Injection Dynamics

Figure 4: Dynamics under optical injection at the GS wavelength as a function
of the injection strength $K$ and the injected frequency (detuning
$\Delta\nu_{inj}$ from the GS wavelength). Color coded in blue/green is the
intensity of the GS within the locking region. Red/orange colors indicate the
intensity of the injection induced pulsations of the GS. Solid white lines
mark the region where injection induces two-state lasing. Parameters:
$J\approx 7.0J_{th}^{GS}$, other parameters as given in Tab. 1.

We first characterize the dynamics of the optically injected two-state QD
laser that is operated below the ES threshold. Thus, without injection only
one-color GS emisison is observed. The control parameters that are varied are
the injection strength $K$ and the relative frequency detuning
$\Delta\nu_{inj}$. The results are plotted in Fig. 4. The blue-green color
code shows the intensity of the GS within the locking region, i.e. where the
GS still emits stable cw light even with injection. Outside this locking
region pulsations are observed and their pulsation amplitudes are indicated by
the red/orange colors. For $K\lesssim 0.3$, the laser unlocks in a saddle-
node-infinite-period (SNIPER) bifurcation [36], where two fixed points on a
periodic orbit collide and annihilate each other and only a limit cycle
solution remains. As a result spiking dynamics are observed close to the
bifurcation [19]. For $K\gtrsim 0.3$, the cw-solution loses its stability in a
Hopf bifurcation [37], where a stable limit cycle solution is born. Hence,
undamped relaxation oscillations are found close to the bifurcation. The
signature of the bifurcations at the edge of the locking cone can be seen in
the pulsation amplitude in Fig. 4: The Hopf bifurcation shows a square root
like increase of the pulsation amplitudes while at a SNIPER the amplitude is
about constant at the bifurcation line.

Moreover, injection into the GS can induce two-state lasing even though the
laser is electrically pumped below the ES threshold. For large enough
detunings, the injection acts destructively on the GS and thus increases the
effective losses for the GS. This, in turn, increases the carrier densities in
the QD, allowing the ES to reach a sufficient inversion for lasing. We observe
regions where GS and ES simultaneously emit stable cw-lasing as well as
regions where both perform pulsations. Note that due to their indirect
coupling via the charge-carriers, pulsations of the GS electric field will
always cause the ES electric field to also perform pulsations. The regions
where ES lasing occurs are indicated with solid white lines in Fig. 4. At the
boundary of the two-state lasing region (solid white lines), the laser either
crosses a transcritical bifurcation of fixed points (white line inside blue
region) or a transcritical bifurcation of limit-cycles (white lines inside
orange region). In both cases the two-state solution and the single-color
solution exchange stability. These transcritical bifurcations are also found
for the conventional optically injected two-mode laser by Osborne et al. [27,
28], however, the shape of the bifurcation lines in the
($K-\Delta\nu_{inj}$)-parameter plane differs, as those lasers already emit
two modes without injection and crucial parameters like carrier lifetimes and
amplitude-phase coupling are different.

In general, the locking cone in the ($K-\Delta\nu_{inj}$)-parameter plane
observed for the GS emission of the optically injected QD laser that is
operated at a pump current below the ES lasing threshold is similar to what is
known from normal single color lasers. However, we only observe period-one
pulsations outside of the locking cone and neither complex dynamics nor chaos,
which are usually found in optically injected single-mode [19, 38, 25] and
dual-mode [25] semiconductor lasers. We will explore the reasons for this
increase of dynamic stability in the next section.

##  IV Stability

Figure 5: Comparison between a single-color laser (a) and a two-state laser
(b) under optical injection. The color code denotes the number of different
maxima found in the time series of the emitted light as a function of
injection strength $K$ and optical detuning $\Delta\nu_{inj}$ (blue indicates
stable lasing and orange to yellow periodic pulsations with increasingly
complicated pulse trains). Hatched regions indicate frequency locking. The
vertical lines indicate the parameter scans presented in Fig. 6. Parameters:
$J\approx 7.0J_{th}^{GS}$, other parameters as given in Tab. 1.

To compare the impact of two-state lasing on the dynamic stability, we compare
two different QD lasers under optical injection: first, the two-state QD laser
on the edge of two-state lasing as discussed in the previous section and,
second, an identical QD laser with increased ES losses $\kappa_{ES}$. This
way, ES emission of the QD laser is suppressed in the second case, and, thus,
the typical behaviour of an optically injected single-color QD laser that has
been studied before [19, 30] is recovered. The observed dynamics of this
single-color QD laser are shown in Fig. 5a. Note, the color code now indicates
the number of maxima found in the time series. Blue corresponds to steady
state (cw)-lasing and the areas plotted in beige correspond to chaotic and
high-order limit-cycle dynamics. The well known period-doubling cascade into
chaos can be found at the negative detuning side in Fig. 5a. There, regions of
dynamic instability e.g. higher period oscillations, irregular oscillation and
chaos, appear. The hatching indicates the regions with frequency locking
(locking of the time-averaged frequency) that also occur if the laser is
pulsing.

In contrast to the single-color case, the two-state QD laser shown in Fig. 5b
only exhibits stable cw-lasing (blue) and period-one oscillations (red) and
lacks any regions of dynamic instabilities. For parameter values where the
single-color QD laser exhibits complex dynamics, we find that the ES turns on
instead for the two-state QD laser case and stable two-wavelength emission is
found. Consequently the locking cone becomes wider. This is somewhat
surprising and counter-intuitive. The added degree of freedom in the two-state
laser leads to less complicated dynamics. A similar trend was also observed
for conventional optically injected Fabry Pérot lasers that support two
longitudinal modes. Starting from a much more unstable behaviour of the one-
color laser the ability for two-mode emission also lead to suppressed chaotic
regions and favoured two-mode emission [25].

To elaborate the stabilizing effect of ES lasing further, Fig. 6a shows line
scans of the dynamic response of the GS emission to optical injection (dots
correspond to extrema found in the time series) for the case of the two-state
laser (violet) and the single-color laser (green). The scans are performed for
varying $\Delta\nu_{inj}$ along the vertical green and purple lines in Fig. 5.
Figure 6b shows the corresponding dynamics of the ES emission from the two-
state laser. For detunings between $-7\text{GHz}$ and $-5\text{GHz}$ the
single color laser exhibits chaotic pulsing whereas the two-state laser shows
stable cw-emission. Thus, the locking range of the the two-state laser,
extending up to $\Delta\nu_{inj}\approx-7\text{GHz}$, is larger than the
locking range the single-color laser, which only extends to
$\Delta\nu_{inj}\approx-5\text{GHz}$. Although no chaos is observed on the
positive detuning side in the single-color laser, the two-state laser there
also remains locked up to greater detunings than the single-color laser
($\Delta\nu_{inj}\approx 7.6\text{GHz}$ vs. $\Delta\nu_{inj}\approx
6.8\text{GHz}$).

Inspecting the two-state laser ES dynamics in Fig. 6b illustrates how ES
lasing corresponds to the stabilization of the GS. We observe stable ES cw-
emission between $5\text{GHz}\lesssim|\Delta\nu_{inj}|\lesssim 7\text{GHz}$,
period-one pulsations between $7\text{GHz}\lesssim|\Delta\nu_{inj}|\lesssim
10\text{GHz}$ and no ES lasing otherwise. The parameter values where the ES
turns on with $cw$ emission are the signatures of the transcritical
bifurcation (denoted with ’TC’) while it turns on with periodic oscillations
in a transcritical bifurcation of limit cycles (denoted with ’TCLC’). The
stable two-state emission that is born at TC loses its stability in a Hopf
bifurcation (denoted with ’H’). The bifurcations of the single-color laser
(green dots in Fig.6a) involve a period doubling route to chaos at the
negative detuning side as e.g. described in [38], while the single mode lasing
loses stability in a Hopf bifurcation for positive detunings. For increasing
negative detunings we find that the turn on of the ES (at TC) coincides with
the emergence of chaos in the single-color QD laser case (seen as the densely
packed green regions in Fig. 6a), while the Hopf bifurcation of the ES
($\Delta\nu_{inj}\approx-7\text{GHz}$) marks the point where the single-color
laser restabilizes to periodic oscillations.

Figure 6:  Bifurcation diagram: Linescan along increasing detuning (indicated
by the vertical line in Fig. 5). Violet and green dots show extrema found in
the time series of the two-color and single-color laser, respectively. (a)
Extrema found in the GS intensity of both lasers. (b) Extrema found in the ES
intensity of the two-state laser. ’H’ indicates a Hopf-bifurcation, ’TC’ a
transcritical bifurcation and ’TCLC’ a transcritical bifurcation of a limit
cycle. An increase of the dynamic stability is found if two-state lasing is
possible. Parameters: $K=0.62$, $J\approx 7.0J_{th}^{GS}$, other parameters as
given in Tab. 1.

So far our numeric results did not show bistability between the different
sweep directions. However, an optically induced hysteresis has been observed
experimentally in a two-state QD laser [23]. In the following, we therefore
look for bistability in our two-state QD laser model.

##  V Bistability

Figure 7: Impact of amplitude-phase coupling: Same as Fig. 4 but with an
increased $\gamma=4$ to yield a higher effective $\alpha$-factor of $2.08$.The
hatched area indicates bistability between ES on and ES off. The injection
strength $K$ is swept and the up direction is shown. Along the horizontal
line, bifurcation diagrams are evaluated in Fig. 8a. Parameters: $J\approx
7.0J_{th}^{GS}$, other parameters as given in Tab. 1.

Studies of simpler semiconductor-laser models have linked the emergence of
bistability under injection to a sufficiently strong amplitude-phase coupling
($\alpha$-factor) [39]. The strength of the amplitude-phase coupling in QD
devices sensitively depends on the energy level spacings between the GS, the
ES and the quantum well, which are determined by the material system and
physical size of the QDs [40, 41]. The scaling factor $\gamma$ introduced in
Eq. (1) allows us to phenomenologically model those effects, while keeping the
microscopically derived dependencies on the carrier dynamics. Our numerical
investigations then show, that a lower pump current or an increase of the
amplitude-phase coupling leads to the emergence of bistability between ES and
GS lasing.

Calculating an effective $\alpha$-factor allows for comparisons of our
microscopically based amplitude-phase coupling strength with more conventional
models, that use the constant $\alpha$-factor approximation. Generally, the
$\alpha$-factor is defined via changes of the optical susceptibility with
respect to the total carrier number. This approach however, fails for QD
lasers [34] and thus the effective $\alpha$-factor is defined via the response
of the complex gain $g$ of the laser to an injected signal [42]:

$\displaystyle\alpha_{\text{\text{eff}}}=-\frac{\frac{\partial}{\partial
K}\operatorname{Im}(g)}{\frac{\partial}{\partial
K}\operatorname{Re}(g)}\Bigg{|}_{K=0}=-\frac{\frac{\partial}{\partial
K}\gamma\delta\omega}{\frac{\partial}{\partial
K}g_{GS}(\rho_{e,act}^{GS}+\rho_{h,act}^{GS}-1)}\Bigg{|}_{K=0}$ (10)

$\gamma=1$, which was used so far, is the microscopically calculate value and
yields an effective $\alpha\approx 0.5$. In the following, we investigate the
dynamics found for a higher effective $\alpha$-factor $\alpha_{eff}\approx
2.0$, corresponding to the higher end of commonly measured $\alpha$-factors in
QD lasers [43, 44]. We implement this by choosing $\gamma=4$.

Figure 7 shows the dynamics with an amplitude-phase coupling increased by a
factor of four. As a result the instantaneous lasing frequency responds
stronger to optical injection and a large region of bistability appears at the
lower edge of the locking cone ($\Delta\nu_{inj}\lesssim-3.5GHz$ and $K\gtrsim
0.2$, denoted by the black hatching, which will be discussed in further
detail.

Within the large region of bistability the device shows hysteresis. The laser
remains on the cw GS-only solution when decreasing the injection strength.
Contrary, the lasers remains much longer in the two-state lasing regime before
the ES finally turns off, when performing an up-sweep in $K$ . This is
emphasized by a linescan shown in Fig. 8a (taken along the green horizontal
line indicated in Fig. 7 at $\Delta\nu_{inj}=-7.5GHz$). There, solid lines
indicate the up-sweep and dashed lines indicate the down-sweep. The hysteresis
loop can clearly be seen between $K=0.56$ and $K=1.00$ in Fig 8a.

Figure 8:  Hysteresis within the region of bistability for different pump
currents: (a) GS lasing $J\approx 7.0J_{th}^{GS}$, (b) two-state lasing
$J\approx 8.4J_{th}^{GS}$ and (c) ES lasing regime $J\approx 10.6J_{th}^{GS}$.
The solid lines show up-sweeps, the dashed lines down-sweeps. The size of the
bistable region in the injection dimension is denoted with $\Delta K$.

By changing the pump current we can navigate between the different operating
regimes of the solitary QD laser as was shown in Fig. 3. So far the laser was
operated in the GS lasing regime. Figure 8b and c now show that the same kind
of bistability also occurs in the two-state (between $K=0.62$ and $K=0.87$)
and ES lasing regime (between $K=0.89$ and $K=0.95$). The bistability in the
two-state regime Figure 8b is very similar to the one reported in [28] and the
bistability in the ES regime reproduces the behavior that was observed in
[23]. Hence two-state QD lasers combine a variety of different bistability
scenarios of a similar mechanism, among which can be chosen be selecting an
appropriate operating current.

Figure 9:  Parameter dependence of the size of the region of bistability. The
colored regions in (a) indicate the size of the bistable region for different
pump currents $J$ for $\gamma=4$ ($\alpha_{eff}\approx 2.0$) and in (b) the
colored regions indicate different effective $\alpha$-factors for
$J=7J_{th}^{GS}$. The hatched region indicates bistability with a constant
$\alpha$-factor approximation. For $\alpha_{eff}\lesssim 0.75$ and $J\gtrsim
12J_{th}^{GS}$ no bistability is found.

Nevertheless, it needs to be noted that the size of the bistable region
decreases with pump current $J$. This is presented in Fig. 9a, showing the
bistable region for different pump currents in the $K,\Delta\nu_{inj}$ plane
for $\gamma=4$ ($\alpha_{eff}\approx 2.0$). For the lower pump currents
$J=7J_{th}^{GS}$ and $J=8J_{th}^{GS}$ the laser is in the GS lasing regime and
exhibits the largest regions of bistability. For $J=9J_{th}^{GS}$ and
$J=10J_{th}^{GS}$ the laser is in the two-state lasing regime where the
bistable regions are considerably smaller and for $J=11J_{th}^{GS}$ and
$J=12J_{th}^{GS}$ the laser is in the ES lasing regime, where the bistable
regions are the smallest. Above $J=12J_{th}^{GS}$ no bistability is found. The
bistability between GS and ES is therefore predominantly found in the GS
lasing regime.

On the other hand, Fig. 9b plots the size of the bistable region for different
amplitude-phase coupling strengths (effective $\alpha$-factor) for
$J=7J_{th}^{GS}$ (GS lasing regime). Here, the size of the bistable region
monotonously increases with the strength of amplitude-phase coupling. Below
$\alpha_{eff}\approx 0.75$ ($\gamma=1.5$) no bistability is found.
Additionally, the hatching indicates a region of bistability that was found
using a constant $\alpha$-factor approximation instead of the carrier
dependent amplitude-phase coupling. It is similar in shape but shifted towards
stronger injection and detuning when compared to the $\alpha_{eff}\approx 1.0$
region of bistability. This contrasts the results of [23], where bistability
could not be achieved with a constant $\alpha$-factor approximation.

In conclusion, a strong parameter sensitivity of the existence and size of the
bistable region on the amplitude-phase coupling strength and the pump current
is found. To achieve bistability in two-state QD lasers, one must therefore
engineer their physical sizes such that a sufficient amplitude-phase coupling
exists and operate them at a pump current below the onset of solitary two-
state lasing.

##  VI All-Optical Switching

In this section we want to address possibilities to exploit the observed
dynamics of QD lasers for all-optical switching applications. For two closely
spaced longitudinal modes of a conventional laser different setups have been
studied and experimentally realized [28, 26]. Recently, also the switching
between GS and ES emission of a QD laser was experimentally demonstrated via
injection into the GS [22]. Our model nicely reproduces these results. A time
series of the switching is shown in Fig. 10 where the laser is operated in the
ES lasing regime and driven into GS lasing by optical injection. The injection
is smoothly switched on at $t=2$ns and off at $t=4$ns with a switching time of
$\approx 100$ps. Fig 10a shows the electric field intensities. The timescales
for the GS switch-on of about $300$ps and the GS switch-off of about $900$ps
compare very well to findings in Ref. [22].

Figure 10:  Switching induced by continuous injection within the ES lasing
regime: (a) shows the GS (red) and ES (blue) intensity and (b) the
corresponding carrier inversions. The injection strength is increased to
$K=0.5$ within the time interval indicated in (c). The switching between ES
and GS occurs on a timescale of about $300ps$ and between GS and ES on a
timescale of about $900ps$. The GS to ES transition is governed by strongly
damped relaxation oscillations.

The time series of the population inversion shown in Fig. 10b gives insights
into the switching mechanism. Without the injection, the ground state is
quenched, while the excited state is lasing and the excited state inversion is
gain clamped. Injection into the ground state lowers its effective losses,
thus enabling GS lasing and decreasing the GS population. As a direct
consequence, the excited state inversion, which is coupled to the ground state
inversion via charge-carrier scattering processes, decreases as well and falls
below the lasing threshold, switching it off. As soon as the injection is
turned off, the system returns to its initial state of ES lasing and GS
quenching.

Furthermore, Fig. 11 presents a different method of switching between the two
states using the bistability discussed in the previous section. Now, the laser
with an increased amplitude-phase coupling ($\gamma=4$) is operated in the GS
lasing regime ($J\approx 7.0_{th}^{GS}$) and injected such that it is in a
bistable region ($K=1.0$, $\Delta\nu_{inj}=-8.7GHz$). Short pulses that
increase or decrease the injection strength can then be used to switch between
the two bistable solutions. Fig. 11b shows the pulses while Fig. 11a shows
time traces of the GS and ES electric field intensities. Switching from GS
lasing to two-state lasing occurs in less then $1$ns and is slowed down by
relaxation oscillations while switching back does not exhibit relaxation
oscillations and is considerably faster on a timescale of about $400$ps. This
switching mechanism and operation is similar to what has been reported in
[28]. However, the strong damping of the relaxation oscillations in QD lasers
compared to quantum well lasers [2] enables much faster switching times.

Figure 11:  Switching induced by short injection pulses while the injected QD
laser is operated in the bistability region: (a) ES and GS intensities (blue
and red lines) during switching via short pulses in the injection strength
(black line in (b)). Switching the ES on is slowed down by relaxation
oscillations but occurs in less then $1$ns while switching the ES off is
faster on a timescale of about $400$ps.

##  VII Conclusion

Using a microscopically motivated rate equation model, we have numerically
investigated the dynamic response of a two-state QD laser to optical
injection. We have compared QD lasers with and without the ability to perform
two-state emission and found that the possibility of two wavelength emission
drastically improves the dynamic stability when subjected to optical
injection. Our results show that the added degree of freedom in two-state QD
lasers leads to a high level of stability and suppresses complex dynamics.
Moreover, our model reproduces an experimentally observed bistability in
optically injected QD lasers and allows for numerical investigations of the
parameter dependence. We have found strong dependencies upon the pump current
and the amplitude-phase coupling. The size of the bistability increases with
the amplitude-phase coupling and first increases and then decreases again with
the pump current. Both for too small amplitude-phase couplings and too high
pump currents, the bistability vanishes. Lastly, we have demonstrated two
techniques how injection can be used for all-optical switching between
different operating regimes. The first technique requires a constant change in
injection strength to achieve switching and excellently matches experimental
results, while the second technique exploits the bistability and only requires
short trigger pulses to achieve switching. This setup could be used to
implement all-optical buffers for use in future all-optical computing
applications.

##  VIII Acknowledgment

This work was supported by DFG within the frameworks of SFB787 and SFB910. K.
Lüdge acknowledges the Humboldt Foundation for their support in the framework
of a Feodor Lynen Scholarship for Experienced Researchers.

## Appendix

Table 2: Fit parameters for charge-carrier scattering processes, extracted from microscopic calculations for 64(35)meV GS confinement energy and 50(20)meV GS-ES separation for electrons(holes) at $T=300$K. b: | electrons |  | holes
---|---|---|---
m: | GS | ES |  | GS | ES
A ($10^{-11}$cm2ns-1) | 18.5 | 48.3 |  | 10.5 | 21.4
B ($10^{11}$cm-2) | 1.9 | 0.48 |  | 5.3 | 1.8
C (ns-1) | 1014 |  | 2272
D ($10^{11}$cm-2) | 1.4 |  | 2.3

For numerical efficiency, the nonlinear charge-carrier in-scattering rates are
implemented via Eq. (11) and Eq. (12), where the parameters $A,B,C$ and $D$
are fitted to the fully microscopically calculated rates [29, 30]. The fit
parameters are presented in Tab 2.

$\displaystyle S_{b}^{m,cap,in}(w_{b})$
$\displaystyle=\frac{A_{m,b}w_{b}^{2}}{B_{m,b}+w_{b}}$ (11) $\displaystyle
S_{b}^{rel,in}(w_{b})$ $\displaystyle=\frac{C_{b}w_{b}}{D_{b}+w_{b}}$ (12)

The corresponding out-scattering rates are then calculated via the detailed
balance conditions [2]

$\displaystyle S_{b}^{m,cap,out}$
$\displaystyle=S_{b}^{m,cap,in}\exp\left(-\frac{E_{F,b}^{eq}-\epsilon_{m,b}}{k_{b}T}\right)$
(13) $\displaystyle S_{b}^{rel,out}$
$\displaystyle=S_{b}^{rel,in}\exp\left(-\frac{\epsilon_{ES,b}-\epsilon_{GS,b}}{k_{b}T}\right)$
(14)

where $E_{F,b}^{eq}$ is the quasi-Fermi level of the respective quantum well
band, $\epsilon_{m,b}$ the confinement energy relative to the QW band edge of
the corresponding QD state, $k_{B}$ the Boltzmann constant and $T$ the charge-
carrier temperature. The quasi-Fermi levels are dynamically calculated via

$\displaystyle
E_{F,b}^{eq}=E^{QW}_{b,0}+k_{b}T\ln\left[\exp\left(\frac{w_{b}}{\mathcal{D}_{b}k_{b}T}\right)-1\right]$
(15)

where $E^{QW}_{b,0}$ is the QW band edge and $\mathcal{D}_{b}$ the density of
states. The in and out-scattering rates then enter Eq. (8) and Eq. (9) and
thereby determine the internal QD charge-carrier dynamics.

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# Extraction Algorithm of Hom-Lie Algebras Based on Solvable and Nilpotent
Groups

Shadi Shagagha and Nadeen Kdaisat  Department of Mathematics, Yarmouk
University, Irbid, Jordan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In the theory of groups, investigations of the properties of the solvable and
nilpotent groups are well-developed. The purpose of the paper is to establish
a theory of the solvable and nilpotent Hom-Lie algebras analogous to that of
the solvable and nilpotent groups.

###### Key words and phrases:

Hom-Lie algebras, solvable Hom-Lie algebra, nilpotent Hom-Lie algebra,
multiplicative algebra.

##  1\. INTRODUCTION

The hom-Lie algebras which are generalizations of classical Lie algebras was
constructed by by Hartwig, Larsson, and Silvestrov [5] in 2006. Since then,
many mathematicians have been trying to extend known results in the setting of
Lie algebras to the setting of hom-Lie algebras (see e.g. [7, 9, 10, 11]).
Homo-Lie algebras have received a lot of attention lately because of their
close connection to discrete and deformed vector fields and differential
calculus [5, 15, 16].
W. Killing introduced semisimple and nilpotent Lie algebras in 1888. Lie
previously proposed solvable (or integrable) algebras in the context of his
work on (algebraic) differential equations. Lie sought to develop a theory for
continuous transformation groups (of the space $\mathbb{R}^{n}$) that would be
comparable to the Galois theory (in which finite solvable groups play a
crucial role). Any nilpotent Lie algebra is a fortiori solvable, meanwhile,
the reverse is not true. The Levi decomposition reveals that the solvable Lie
algebras and the semisimple Lie algebras constitute two substantial and
generally complementing classes. The solvable Lie algebras can only be
produced by semidirect products, which start at $0$ and gain one dimension at
a time [6]. According to Lie’s Theorem, if $V$ is a finite-dimensional vector
space over an algebraically closed field with characteristic zero, $g$ is a
solvable Lie algebra, and if $\pi$ is an image of $g$ over $V$, then there
exists a simultaneous eigenvector $v\in V$ of the endomorphisms $\pi(X)$ for
all elements in $X\in g$. Each and every Lie subalgebra as well as its
quotient are solvable Lie algebras, $g$ is solvable only if both $h$ and $g/h$
are solvable, given a Lie algebra $g$ and an ideal $h$ in it. For nilpotent
Lie algebras, the corresponding statement is valid as long as $h$ is contained
in the middle. Thus, a central extension of a nilpotent algebra by a nilpotent
algebra is nilpotent, whereas an extension of a solvable algebra by a solvable
algebra is solvable. The final nonzero term in the derived series of a
solvable nonzero Lie algebra is its nonzero abelian ideal. There are various
ways to define a solvable Lie group because the word ”solvable” is also used
to describe solvable groups in group theory. The typical derived series of the
group $G$ for a Lie group terminates (as an abstract group). The closures of
the derived series have reached their conclusion, and the Lie algebra is
solvable.
In the present article, we study solvable and nilpotent hom-Lie algebras,
which can be viewed as an extension of solvable and nilpotent Lie algebras.

##  2\. PRELIMINARIES

Let $F$ be a ground field.

######  Definition 2.1.

([14]) A Hom-Lie algebra over $F$ is a triple $(L,~{}[~{},~{}],~{}\alpha)$
where $L$ is a vector space over $F$, $\alpha:L\rightarrow L$ is a linear map,
and $[~{},~{}]:L\times L\rightarrow L$ is a bilinear map (called a bracket),
satisfying the following two identities:

  * (i)

$[x,~{}y]=-[y,~{}x]$ for all $x,y\in L$ (skew-symmetry property).

  * (ii)

$[\alpha(x),~{}[y,~{}z]]+[\alpha(y),~{}[z,~{}x]]+[\alpha(z),~{}[x,~{}y]]=0$ ,
for all $x,y,z\in L$ (Hom-Jacobi identity).

A Hom-Lie algebra $(L,~{}[~{},~{}],~{}\alpha)$ is called multiplicative if
$\alpha([x,~{}y])=[\alpha(x),~{}\alpha(y)]$ for all $x,y\in L$.

Let $(L_{1},~{}[~{},~{}]_{1},~{}\alpha_{1})$ and
$(L_{2},~{}[~{},~{}]_{2},~{}\alpha_{2})$ be Hom-Lie algebras. A linear map
$\varphi:L_{1}\rightarrow L_{2}$ is called a morphism of Hom-Lie algebras if
the following two identities are satisfied:

  * (i)

$\varphi([x,~{}y]_{1})=[\varphi(x),~{}\varphi(y)]_{2}$ for all $x,~{}y\in
L_{1}$.

  * (ii)

$\varphi\circ\alpha_{1}=\alpha_{2}\circ\varphi$.

A linear map $\varphi:L_{1}\rightarrow L_{2}$ is called an isomorphism of Hom-
Lie algebras if $\varphi$ is a bijection morphism of Hom-Lie algebras. We say
$L_{1}$ and $L_{2}$ are isomorphic and write $L_{1}\cong L_{2}$.
A subspace $H$ of $L$ is a Hom-Lie subalgebra if $\alpha(H)\subseteq H$ and
$[x,~{}y]\in H$ for all $x,y\in H$. A Hom-Lie subalgebra $H$ is said to be a
Hom-Lie ideal if $[x,~{}y]\in H$ for all $x\in H$ and $y\in L$.

######  Example 2.1.

([14]) Every Lie algebra is a Hom-Lie algebra by setting $\alpha=id_{L}$ (The
identity map). $\blacksquare$

######  Example 2.2.

Let $L$ be a vector space over $F$, $[~{},~{}]:L\times L\rightarrow L$ be
arbitrary skew-symmetric bilinear map, and $\alpha:L\rightarrow L$ be the zero
map. Then one can easily see that $(L,[~{},~{}],\alpha)$ is a multiplicative
Hom-Lie algebra. $\blacksquare$

######  Example 2.3.

([14]) Let $L$ be a vector space and $\alpha:L\rightarrow L$ be any linear
operator. Then $(L,~{}[~{},~{}],~{}\alpha)$ is a Hom-Lie algebra, where
$[x,~{}y]=0$ for all $x,y\in L$. Such Hom-Lie algebras are called abelian or
commutative Hom-Lie algebras. $\blacksquare$

######  Example 2.4.

([13]) Let
$(L_{1},~{}[~{},~{}]_{1},~{}\alpha_{1}),(L_{2},~{}[~{},~{}]_{2},\alpha_{2}),\ldots,(L_{n},~{}[~{},~{}]_{n},\alpha_{n})$
be Hom-Lie algebras. Then $(L_{1}\oplus L_{2}\oplus\cdots\oplus
L_{n},~{}[~{},~{}],~{}\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n})$ becomes a Hom-
Lie algebra by setting

$\displaystyle[~{},~{}]~{}:~{}(L_{1}\oplus L_{2}\oplus\cdots\oplus
L_{n})\times(L_{1}\oplus L_{2}\oplus\cdots\oplus L_{n})$
$\displaystyle\rightarrow$ $\displaystyle(L_{1}\oplus L_{2}\oplus\cdots\oplus
L_{n})$ $\displaystyle((x_{1},\ldots,~{}x_{n}),~{}(y_{1},\ldots,~{}y_{n}))$
$\displaystyle\mapsto$
$\displaystyle([x_{1},~{}y_{1}]_{1},\ldots,~{}[x_{n},~{}y_{n}]_{n}),$

and

$\displaystyle(\alpha_{1}+\alpha_{2}+\ldots+\alpha_{n})~{}:~{}(L_{1}\oplus
L_{2}\oplus\cdots\oplus L_{n})$ $\displaystyle\rightarrow$
$\displaystyle(L_{1}\oplus L_{2}\oplus\cdots\oplus L_{n})$
$\displaystyle(x_{1},~{}x_{2},\ldots,~{}x_{n})$ $\displaystyle\mapsto$
$\displaystyle(\alpha_{1}(x_{1}),~{}\alpha_{2}(x_{2}),\ldots,~{}\alpha_{n}(x_{n})).$

$\blacksquare$

######  Example 2.5.

([7]) Let $F=\mathbb{C}$ be the field of complex numbers. Consider the vector
space $\mathbb{C}^{2}$ and define the linear map

$\alpha_{*}:\mathbb{C}^{2}\rightarrow\mathbb{C}^{2};~{}(x,~{}y)\mapsto(-y,~{}-x).$

We define the bilinear map
$[~{},~{}]_{*}:\mathbb{C}^{2}\times\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$,
where

$[(x_{1},~{}x_{2}),~{}(y_{1},~{}y_{2})]_{*}=(i(x_{1}y_{2}-x_{2}y_{1}),~{}i(x_{1}y_{2}-x_{2}y_{1})).$

Then $(\mathbb{C}^{2},~{}[~{},~{}]_{*},~{}\alpha_{*})$ is a multiplicative
Hom-Lie algebra. $\blacksquare$

######  Example 2.6.

([7]) Consider the set

$L=\left\\{\begin{bmatrix}\frac{i(x+y)}{2}&x\\\
y&\frac{-i(x+y)}{2}\end{bmatrix}~{}|~{}x,y\in\mathbb{C}\right\\}$

with the linear map

$\alpha:L\rightarrow L;~{}A\mapsto-A^{T},$

and the skew-symmetric bilinear map

$[~{},~{}]~{}:~{}L\times L\rightarrow L;~{}(A,~{}B)\mapsto[A,~{}B],$

where $[A,~{}B]=A^{T}B^{T}-B^{T}A^{T}$. For any $x,y,z,w\in\mathbb{C}$. Then
$(L,~{}[~{},~{}],~{}\alpha)$ is a multiplicative Hom-Lie algebra.
$\blacksquare$

######  Example 2.7.

Consider the vector space $\mathbb{R}[x]$ of polynomials with coefficients in
$\mathbb{R}$. Define the linear map

$\alpha:\mathbb{R}[x]\rightarrow\mathbb{R}[x];~{}p(x)\mapsto p(0).$

Then $(\mathbb{R}[x],~{}[~{},~{}],~{}\alpha)$ becomes a multiplicative Hom-Lie
algebra by setting

$[p(x),~{}q(x)]=p^{\prime\prime}(x)q^{\prime}(x)-q^{\prime\prime}(x)p^{\prime}(x)-p^{\prime\prime}(0)q^{\prime}(0)+q^{\prime\prime}(0)p^{\prime}(0),$

for any $p(x),q(x)\in\mathbb{R}[x]$. Indeed if $p(x),q(x)\in\mathbb{R}[x]$,
then

$\displaystyle[p(x),q(x)]$ $\displaystyle=$ $\displaystyle
p^{\prime\prime}(x)q^{\prime}(x)-q^{\prime\prime}(x)p^{\prime}(x)-p^{\prime\prime}(0)q^{\prime}(0)+q^{\prime\prime}(0)p^{\prime}(0)$
$\displaystyle=$
$\displaystyle-(q^{\prime\prime}(x)p^{\prime}(x)-p^{\prime\prime}(x)q^{\prime}(x)-q^{\prime\prime}(0)p^{\prime}(0)+p^{\prime\prime}(0)q^{\prime}(0))$
$\displaystyle=$ $\displaystyle-[q(x),p(x)].$

For $p(x),q(x),h(x)\in\mathbb{R}[x]$, then one can easily see that

$[\alpha(h(x)),~{}[p(x),~{}q(x)]]=[h(0),~{}p^{\prime\prime}(x)q^{\prime}(x)-q^{\prime\prime}(x)p^{\prime}(x)-p^{\prime\prime}(0)q^{\prime}(0)+q^{\prime\prime}(0)p^{\prime}(0)]=0.$

Thus, for each $p(x),q(x),h(x)\in\mathbb{R}[x]$ we have

$[\alpha(h(x)),~{}[p(x),~{}q(x)]]+[\alpha(p(x)),~{}[q(x),~{}h(x)]]+[\alpha(q(x)),~{}[h(x),~{}p(x)]]=0.$

Also,

$\displaystyle\alpha([p(x),~{}q(x)])$ $\displaystyle=$ $\displaystyle
p^{\prime\prime}(0)q^{\prime}(0)-q^{\prime\prime}(0)p^{\prime}(0)-p^{\prime\prime}(0)q^{\prime}(0)+q^{\prime\prime}(0)p^{\prime}(0)$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle[p(0),~{}q(0)]$ $\displaystyle=$
$\displaystyle[\alpha(p(x)),~{}\alpha(q(x))].$

Note that, $(\mathbb{R}[x],~{}[~{},~{}])$ is not a Lie algebra, since

$[x^{2},~{}[x^{3},~{}x^{4}]]+[x^{3},~{}[x^{4},~{}x^{2}]]+[x^{4},~{}[x^{2},~{}x^{3}]]=96x^{3}\neq
0$

$\blacksquare$

######  Example 2.8.

([3]) Let $(L,~{}[~{},~{}],~{}\alpha)$ be a Hom-Lie algebra and let $H$ be a
Hom Lie ideal. Then the quotient space
$(L/H,~{}\overline{[~{},~{}]},~{}\overline{\alpha})$ is a Hom-Lie algebra
where

$\overline{[~{},~{}]}:L/H\times L/H\rightarrow
L/H;~{}(x+H,~{}y+H)\mapsto[x,~{}y]+H,$

and

$\overline{\alpha}:L/H\rightarrow L/H;~{}x+H\mapsto\alpha(x)+H.$

Let $H$ and $K$ be ideals of a Hom-Lie algebra $L$. The sum and multiplication
of $H$ and $K$ are defined as follows:

$H+K=\\{h+k~{}|~{}h\in H\mathrm{~{}and}~{}k\in K\\}$

and

$[H,~{}K]=\mathrm{Span}(\\{[h,~{}k]~{}|~{}h\in H~{}\mathrm{and~{}}k\in K\\}).$

######  Theorem 2.1.

([subalgebra]) Let $H$ and $K$ be Hom-Lie ideals of a multiplicative Hom-Lie
algebra $(L,~{}[~{},~{}],~{}\alpha)$. Then,

  * (i)

$[H,~{}K]$ is a Hom-Lie subalgebra of $L$.

  * (ii)

$[H,~{}K]$ is a Hom-Lie ideal of $H$ and $K$, respectively.

  * (iii)

$[H,~{}K]$ is a Hom-Lie ideal of $L$ when $\alpha$ is onto.

######  Example 2.9.

([7]) Let $(L,~{}[~{},~{}],~{}\alpha)$ be a Hom-Lie algebra and H be a Hom-Lie
ideal. Then $(L/H,~{}\overline{[~{},~{}]},~{}\overline{\alpha})$ is a Hom-Lie
algebra and the linear map

$\pi:L\rightarrow L/H;$ $x\mapsto x+H$

is a morphism of Hom-Lie algebras. $\blacksquare$

##  3\. Solvable Hom-Lie Algebra

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a Hom-Lie algebra. The sequence of Hom-Lie
subalgebras $L_{1},L_{2},\ldots,L_{n}\ldots$ such that

$L=L_{0}\supseteq L_{1}\supseteq\cdots\supseteq L_{n}\supseteq\cdots$

is called a descending series.

######  Definition 3.1.

([8]) Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra. We
define ,$\\{L^{(i)}\\},i\geq 0$, the derived series of $L$ by

$\displaystyle L^{(0)}$ $\displaystyle=$ $\displaystyle L,$ $\displaystyle
L^{(1)}$ $\displaystyle=$ $\displaystyle[L,~{}L],$ $\displaystyle L^{(i)}$
$\displaystyle=$ $\displaystyle[L^{(i-1)},~{}L^{(i-1)}],i\geq 2.$

Note that $L^{(i)}=[L^{(i-1)},~{}L^{(i-1)}]$ is a Hom-Lie ideal of $L^{(i-1)}$
(by induction and Theorem 2.1(ii)).

$L=L^{(0)}\supseteq L^{(1)}\supseteq\cdots\supseteq L^{(i-1)}\supseteq
L^{(i)}\cdots$

Thus the derived series is a descending series.

######  Definition 3.2.

([8]) A multiplicative Hom-Lie algebra $(L,~{}[~{},~{}],~{}\alpha)$ is said to
be solvable if there exists $n\in\mathbb{N}$ such that $L^{(n)}=\\{0\\}$. We
say $L$ is solvable of class $k$ if $L^{(k)}=\\{0\\}$ and
$L^{(k-1)}\neq\\{0\\}$.

Clearly a multiplicative Hom-Lie algebra is solvable of class $\leq k$ iff
$L^{(k)}=\\{0\\}$. Metabelian Hom-Lie algebras is the same as in the case of
Lie algebras ([12]) are the solvable Hom-Lie algebras of class at most $2$.

######  Example 3.1.

Let $L$ be the space spanned by a basis $\\{e_{1},e_{2},\ldots,e_{n}\\}$
($n\geq 5$) over $F$. Consider the multiplicative Hom-Lie algebra
$(L,~{}[~{},~{}],\alpha)$ where $\alpha$ is the zero map and $[~{},~{}]$ is
the skew-symmetric bilinear map such that $[e_{i},e_{j}]=0$ if $i=j$ or $i=1$
and $[e_{i},e_{j}]=e_{i-1}$ if $1<i<j\leq n$. Note that
$[e_{2},~{}[e_{4},~{}e_{5}]]+[e_{4},~{}[e_{5},~{}e_{2}]]+[e_{5},~{}[e_{2},~{}e_{4}]]=e_{1}\neq
0$. Thus $L$ is not a Lie algebra.
Now, $L^{(1)}=[L,~{}L]=\\{e_{1},~{}e_{2},\ldots,~{}e_{n-2}\\}$
$L^{(2)}=[L^{(1)},~{}L^{(1)}]=\\{e_{1},~{}e_{2},\ldots,~{}e_{n-4}\\}$
$\vdots$
$L^{(i)}=\\{e_{1},~{}e_{2},\ldots,~{}e_{n-2i}\\},i<{\frac{n}{2}}$
If $n$ is even then $L^{({\frac{n}{2}-1})}=\\{e_{1},~{}e_{2}\\}$ and
$L^{({\frac{n}{2}})}=\\{0\\}$. Thus $L$ is a solvable Hom-Lie algebras of
class ${\frac{n}{2}}$. If $n$ is odd then $L^{({\frac{n-1}{2}})}=\\{e_{1}\\}$
and $L^{({\frac{n+1}{2}})}=\\{0\\}$. Thus $L$ is a solvable Hom-Lie algebras
of class ${\frac{n+1}{2}}$. $\blacksquare$

######  Definition 3.3.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra. Then the
descending series $L=L_{0}\supseteq L_{1}\supseteq...\supseteq L_{n}=\\{0\\}$
called a solvable series if for each i, we have $L_{i+1}$ is a Hom-Lie ideal
of $L_{i}$ and $L_{i}/L_{i+1}$ is an abelian Hom-lie algebra.

######  Lemma 3.1.

Let $H$ be a Hom-Lie subalgebra of the Hom-Lie algebra
$(L,~{}[~{},~{}],~{}\alpha)$. Then $H$ is a Hom-Lie ideal of $L$ and $L/H$ is
abelian Hom-Lie algebra if and only if $[L,~{}L]\subseteq H$

Proof. If $L/H$ is an abelian Hom-Lie algebra, then for any
$[x,~{}y]\in[L,~{}L]$ we find $H=\overline{[x+H,~{}y+H]}=[x,~{}y]+H$.
Therefore $[x,~{}y]\in H$. Conversely, if $[L,~{}L]\subseteq H$ then
$[x,~{}y]\in H$ for all $x\in H~{}(\subseteq L)$ and $y\in L$, which implies
$H$ is a Hom-Lie ideal of $L$. Also, for any $x,y\in L$ we have
$\overline{[x+H,~{}y+H]}=[x,~{}y]+H=H$ (because $[x,~{}y]\in[L,~{}L]\subseteq
H$). $\Box$

######  Corollary 3.1.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a Hom-Lie algebra. Then the descending
series $L=L_{0}\supseteq L_{1}\supseteq...\supseteq L_{n}=\\{0\\}$ is solvable
if and only if $[L_{i},~{}L_{i}]\subseteq L_{i+1}$ for each
$i=0,1,\ldots,n-1$.

Proof.  This follows directly from Definition 3.3 and Lemma 3.1. $\Box$

######  Theorem 3.1.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra. Then
$(L,~{}[~{},~{}],~{}\alpha)$ is a solvable Hom-Lie algebra of class $\leq k$
iff $L=L^{(0)}\supseteq L^{(1)}\supseteq\cdots\supseteq L^{(k)}=\\{0\\}$ is a
solvable series.

Proof.  This follows directly from the definition of the derived series and
the corollary above. $\Box$

######  Theorem 3.2.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra. If
$L=L_{0}\supseteq L_{1}\supseteq\cdots\supseteq L_{n}=\\{0\\}$ is a solvable
series, then for each i, $L^{(i)}\subseteq L_{i}$.

Proof.  We use induction. For $k=0$, we have $L^{(0)}=L=L_{0}$. For $k>0$ and
because the induction assumption we have
$L^{(k+1)}=[L^{(k)},L^{(k)}]\subseteq[L_{k},L_{k}]\subseteq L_{k+1}$. Now
according to Corollary 3.1, we have $[L_{k},L_{k}]\subseteq L_{k+1}$.
Therefore $L^{(k+1)}\subseteq L_{k+1}$. $\Box$

######  Theorem 3.3.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra. Then,
$L$ is solvable of class $\leq k$ if and only if there exists a solvable
series of length $k$.

Proof.  If $L$ is a solvable Hom-Lie algebra of class $\leq k$, then, using
Theorem 3.1, the series $L=L^{(0)}\supseteq L^{(1)}\supseteq...\supseteq
L^{(k)}=\\{0\\}$ is solvable. Conversely suppose that

$L=L_{0}\supseteq L_{1}\supseteq\cdots\supseteq L_{k}=\\{0\\}$

is a solvalbe series. Then, using $L^{(k)}\subseteq L_{k}=\\{0\\}$, we find
$L^{(k)}=\\{0\\}$. $\Box$

######  Corollary 3.2.

Solvable Hom-Lie algebras of class $1$ are the abelian Hom-Lie algebras.

Proof.  $L$ is solvable of class 1 iff there exists a solvable series
$L=L_{0}\supseteq L_{1}=\\{0\\}$ of length $1$ iff $L/\\{0\\}$ is abelian Hom-
Lie algebra iff $L$ is abelian Hom-Lie algebra. $\Box$

######  Theorem 3.4.

Let
$\varphi:(L_{1},~{}[~{},~{}]_{1},~{}\alpha_{1})\longrightarrow(L_{2},~{}[~{},~{}]_{2},~{}\alpha_{2})$
be a morphism of multiplicative Hom-Lie algebras. Then

  * (i)

$(\varphi(L_{1}))^{(i)}=\varphi(L_{1}^{(i)})$,

  * (ii)

if $L_{1}$ is solvable of class $k$, then $\varphi(L_{1})$ is solvable of
class $\leq k$,

  * (iii)

if $\varphi$ is an isomorphism of Hom-Lie algebras, then $L_{1}$ is solvable
of class $k$ if and only if $L_{2}$ is solvable of class $k$.

Proof.

  * (i)

By applying induction we find
$(\varphi(L_{1}))^{(0)}=\varphi(L_{1})=\varphi(L_{1}^{(0)})$.
Also, if $(\varphi(L_{1}))^{(i)}=\varphi(L_{1}^{(i)})$ then
$(\varphi(L_{1}))^{(i+1)}=[(\varphi(L_{1}))^{(i)},~{}(\varphi(L_{1}))^{(i)}]=[\varphi(L_{1}^{(i)}),~{}\varphi(L_{1}^{(i)})]=\varphi([L_{1}^{(i)},~{}L_{1}^{(i)}])=\varphi(L_{1}^{(i+1)})$.

  * (ii)

Since $L_{1}$ is solvable of class $k$, it follows $L_{1}^{(k)}=\\{0\\}$. So,
$(\varphi(L_{1}))^{(k)}=\varphi(L_{1}^{(k)})=\varphi(0)=0$. Thus,
$\varphi(L_{1})$ is solvable of class $\leq k$.

  * (iii)

Using $(ii)$ we may assume that $L_{1}$ and $L_{2}$ are solvable of classes
$k$ and $m$, respectively. Again by (ii) we have $m\leq k$. Also, since
$\varphi^{-1}$ is an isomorphism of Hom-Lie algebras, then
$L_{1}=\varphi^{-1}(L_{2})$ is solvable of class $\leq m$; that is $k\leq m$.
Thus, $k=m$. $\Box$

######  Example 3.2.

The Hom-Lie algebras $(L,~{}[~{},~{}],~{}\alpha)$ (Example2.6) and
$(\mathbb{C}^{2},~{}[~{},~{}]_{*},~{}\alpha_{*})$ ( Example 2.5) are
isomorphic Hom-Lie algebras. Since

$\varphi:L\rightarrow\mathbb{C}^{2};~{}\begin{bmatrix}\frac{i(x+y)}{2}&x\\\
y&\frac{-i(x+y)}{2}\end{bmatrix}\mapsto(x,~{}y).$

is an isomorphism Hom-Lie algebra ([7]). Now,
$(\mathbb{C}^{2})^{(1)}=[\mathbb{C}^{2},~{}\mathbb{C}^{2}]_{*}=\\{(x,x);x\in\mathbb{C}\\}$,
and
$(\mathbb{C}^{2})^{(2)}=[(\mathbb{C}^{2})^{(1)},~{}(\mathbb{C}^{2})^{(1)}]_{*}=\\{0\\}$.
Thus $\mathbb{C}^{2}$ is a solvable Hom-Lie algebra of class 2. And so $L$ is
a solvable Hom-Lie algebra of class 2

######  Lemma 3.2.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra and $H$
be a Hom-Lie subalgebra of $L$. Then $H^{(i)}\subseteq L^{(i)}$, for each
$i\in\mathbb{N}$.

Proof.  Clearly $H^{(0)}\subseteq L^{(0)}$. Using induction, if
$H^{(i)}\subseteq L^{(i)}$ then
$H^{(i+1)}=[H^{(i)},~{}H^{(i)}]\subseteq[L^{(i)},~{}L^{(i)}]=L^{(i+1)}$.
$\Box$

######  Theorem 3.5.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a solvable Hom-Lie algebra of class $k$.

  * (i)

Any Hom-Lie subalgebra is solvable of class $\leq k$.

  * (ii)

Any quotient Hom-Lie algebra of $L$ is solvable of class $\leq k$.

Proof.

  * (i)

Let $H$ be a Hom-Lie subalgebra. Then $H$ is multiplicative (because $L$ is
multiplicative ). Also, according to the lemma above, we have
$H^{(k)}\subseteq L^{(k)}=\\{0\\}$. Thus $H^{(k)}=\\{0\\}$.

  * (ii)

Let $I$ be a Hom-Lie ideal of $L$. Then so is $L/I$ (because $L$ is
multiplicative). Consider the natural map $\pi:L\rightarrow L/I$ in Example
2.9. According to Theorem 3.4(i), $\pi(L^{(K)})=(\pi(L))^{(k)}$, which implies

$(L/I)^{(k)}=(\pi(L))^{(k)}=\pi(L^{(k)})=\pi(\\{0\\})=\\{0+I\\}.$

$\Box$

######  Theorem 3.6.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra. If H is
a solvable Hom-Lie ideal of class $k$ and $L/H$ is solvable of class $m$, then
$L$ is solvable of class $\leq k+m$.

Proof.  According to Theorem 3.3, we have the following two solvable series,

$H=H_{0}\supseteq H_{1}\supseteq\cdots\supseteq H_{k}=\\{0\\},$
$L/H=(L/H)_{0}\supseteq(L/H)_{1}\supseteq\cdots\supseteq(L/H)_{m}=\\{0+H\\}.$

Consider the natural map $\pi$, and let
$L_{i}=\pi^{-1}((L/H)_{i}),i=1,2,...,m$. Hence $L_{i}$ is a Hom-Lie subalgebra
of $L$ and $L_{i+1}\subseteq L_{i}$. Therefore,

$L=L_{0}\supseteq L_{1}\supseteq\cdots\supseteq
L_{m}=\pi^{-1}(\\{0+H\\})=H=H_{0}\supseteq H_{1}\supseteq\cdots\supseteq
H_{k}=\\{0\\}$

is a descending series. Now it suffices to prove that
$[L_{i},~{}L_{i}]\subseteq L_{i+1}$ ($i=0,2,\ldots,m-1$). If $x,y\in
L_{i}=\pi^{-1}((L/H)_{i})$, then $\pi(x),\pi(y)\in(L/H)_{i}$ and so
$\pi([x,~{}y])=\overline{[\pi(x),~{}\pi(y)]}\in\overline{[(L/H)_{i},~{}(L/H)_{i}]}\subseteq(L/H)_{i+1}$
(Corollary 3.1). Therefore, $[x,~{}y]\in\pi^{-1}((L/H)_{i+1})=L_{i+1}$ for
each $x,y\in L_{i}$. This shows that $[L_{i},~{}L_{i}]\subseteq L_{i+1}$.
Therefore

$L=L_{0}\supseteq L_{1}\supseteq\cdots\supseteq L_{m}=H=H_{0}\supseteq
H_{1}\supseteq\cdots\supseteq H_{k}=\\{0\\}$

is a solvable series of length $k+m$. By Theorem 3.3, we have $L$ is solvable
of class $\leq k+m$. $\Box$

In [10], we proved that if $(L_{1},~{}[~{},~{}]_{1},~{}\alpha_{1})$ and
$(L_{2},~{}[~{},~{}]_{2},~{}\alpha_{2})$ are Hom-Lie algebras and $H_{i}$ is a
Hom-Lie ideal of $L_{i}$, $i=1,2.$, then $H_{1}\times H_{2}$ is a Hom-Lie
ideal of $L_{1}\times L_{2}$ and

$(L_{1}\times L_{2})/(H_{1}\times H_{2})\equiv L_{1}/H_{1}\times L_{2}/H_{2}$

.

######  Theorem 3.7.

Let $(L_{1},~{}[~{},~{}]_{1},~{}\alpha_{1})$ and
$(L_{2},~{}[~{},~{}]_{2},~{}\alpha_{2})$ be solvable Hom-Lie algebras of class
$k$ and $m$, respectively. Then $(L_{1}\times L_{2},~{}[~{},~{}],~{}\alpha)$
is a solvable Hom-Lie algebra of class $\leq k+m$.

Proof.  Note that, $L_{1}\times L_{2}$ is a multiplicative Hom-Lie algebra
because $L_{1}$ and $L_{2}$ are multiplicative Hom-Lie algebras. Since
$L_{1}\times\\{0\\}\equiv L_{1}$, so $L_{1}\times\\{0\\}$ is a solvable Hom-
Lie ideal (of class $k$) of $L_{1}\times L_{2}$. Also, $(L_{1}\times
L_{2})/(L_{1}\times\\{0\\})\equiv L_{1}/L_{1}\times L_{2}/\\{0\\}\equiv
L_{2}$, so $(L_{1}\times L_{2})/(L_{1}\times\\{0\\})$ is a solvable Hom-Lie
algebra of class $m$. According to Theorem 3.6, $L_{1}\times L_{2}$ is a
solvable Hom-Lie algebra of class $\leq k+m$. $\Box$

##  4\. Nilpotent Hom-Lie algebra

######  Definition 4.1.

([8]) Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra. We
define , $\\{L^{i}\\},i\geq 0$ , the lower central series of $L$ by $L^{0}=L$,
$L^{1}=[L,~{}L]$, and $L^{i}=[L,~{}L^{i-1}]$.

Note that $L^{i+1}=[L,~{}L^{i}]$ is a Hom-Lie ideal of $L^{i}$ (by Theorem
2.1(ii) and induction).

$L=L^{0}\supseteq L^{1}\supseteq\cdots\supseteq L^{i}\supseteq
L^{i+1}\supseteq\cdots$

Thus the lower central series is a descending series.

######  Definition 4.2.

([8]) Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra. We
say that $L$ is nilpotent if there exists $n\in\mathbb{N}$ such that
$L^{n}={0}$. It is nilpotent of class $k$ if $L^{k}=\\{0\\}$ and
$L^{k-1}\neq\\{0\\}$

It is clear now that $L$ is nilpotent of class $\leq k$ iff $L^{k}=\\{0\\}$.

######  Example 4.1.

Consider the multiplicative Hom-Lie algebra $(L,~{}[~{},~{}],\alpha)$ in
Example 3.1 where $\alpha$ is the zero map and $[~{},~{}]$ is the skew-
symmetric bilinear map such that $[e_{i},e_{j}]=0$ if $i=j$ or $i=1$ and
$[e_{i},e_{j}]=e_{i-1}$ if $1<i<j\leq n$.
Now, $L^{1}=[L,~{}L]=\\{e_{1},~{}e_{2},\ldots,~{}e_{n-2}\\}$
$L^{2}=[L,~{}L^{1}]=\\{e_{1},~{}e_{2},\ldots,~{}e_{n-3}\\}$
$\vdots$
$L^{i}=\\{e_{1},~{}e_{2},\ldots,~{}e_{n-(i+1)}\\},i<{n-3}$
$\vdots$
$L^{n-3}=\\{e_{1},~{}e_{2}\\}$
$L^{n-2}=\\{0\\}$.
Thus $L$ is a nilpotent Hom-Lie algebras of class ${n-2}$. $\blacksquare$

######  Definition 4.3.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra. Then a
descending series $L=L_{0}\supseteq L_{1}\supseteq L_{2}\supseteq\cdots$ is
said to be central if for each $i\in\mathbb{N}$, $[L,~{}L_{i}]\subseteq
L_{i+1}$. It has a length $k\in\mathbb{N}$ if $L_{k}=\\{0\\}$ but
$L_{k-1}\neq\\{0\\}$.

######  Theorem 4.1.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra. Then
$(L,~{}[~{},~{}],~{}\alpha)$ is a nilpotent Hom-Lie algebra of class $\leq k$
iff $L=L^{0}\supseteq L^{1}\supseteq\cdots\supseteq L^{k}=\\{0\\}$ is a
central series.

Proof.  It follows directly from the definition of $L^{i}$. $\Box$

######  Theorem 4.2.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra. If
$L=L_{0}\supseteq L_{1}\supseteq L_{2}\supseteq\cdots$ is a central series,
then for each $i\in\mathbb{N}$, $L^{i}\subseteq L_{i}$.

Proof.  Applying induction we see $L^{0}=L=L_{0}$. Also, if $L^{i}\subseteq
L_{i}$ then $L^{i+1}=[L,~{}L^{i}]\subseteq[L,~{}L_{i}]\subseteq L_{i+1}$.
$\Box$

######  Theorem 4.3.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra. Then $L$
is nilpotent of class $\leq k$ iff there exists a central series of length
$k$.

Proof.  If $L$ is a nilpotent Hom-Lie algebra of class $\leq k$. Then

$L=L^{0}\supseteq L^{1}\supseteq\cdots\supseteq L^{k}=\\{0\\}$

is a central series. The converse is true, since $L^{k}\subseteq
L_{k}=\\{0\\}$ so $L^{k}=\\{0\\}$. $\Box$

######  Corollary 4.1.

Nilpotent Hom-Lie algebras of class $1$ are the abelian Hom-Lie algebras.

Proof.  A Hom-Lie algebra $L$ is nilpotent of class $1$ iff there exists a
central series of length 1,$L=L_{0}\supseteq L_{1}=\\{0\\}$ iff
$[L,~{}L_{0}]\subseteq L_{1}$ iff $[L,~{}L]=\\{0\\}$ iff $L$ is an abelian
Hom-Lie algebra. $\Box$

######  Theorem 4.4.

Let
$\varphi:(L_{1},~{}[~{},~{}]_{1},~{}\alpha_{1})\longrightarrow(L_{2},~{}[~{},~{}]_{2},~{}\alpha_{2})$
be a morphism of multiplicative Hom-Lie algebras. Then

  * (i)

$(\varphi(L_{1}))^{i}=\varphi(L_{1}^{i})$,

  * (ii)

if $L_{1}$ is nilpotent of class $k$, then $\varphi(L_{1})$ is nilpotent of
class $\leq k$,

  * (iii)

if $\varphi$ is an isomorphism of Hom-Lie algebras, then $L_{1}$ is nilpotent
of class $k$ if and only if $L_{2}$ is nilpotent of class $k$.

Proof.

  * (i)

We note that $(\varphi(L_{1}))^{0}=\varphi(L_{1})=\varphi(L_{1}^{0})$. Also if
$(\varphi(L_{1}))^{i}=\varphi(L_{1}^{i})$, then
$(\varphi(L_{1}))^{i+1}=[(\varphi(L_{1})),~{}(\varphi(L_{1}))^{i}]=[\varphi(L_{1}),~{}\varphi(L_{1}^{i})]=\varphi([L_{1},~{}L_{1}^{i}])=\varphi(L_{1}^{i+1})$.

  * (ii)

Since $L_{1}$ is nilpotent of class $k$, then $L_{1}^{k}=\\{0\\}$. So,
$(\varphi(L_{1}))^{k}=\varphi(L_{1}^{k})=\varphi(\\{0\\})=\\{0\\}$. Thus,
$\varphi(L_{1})$ is nilpotent of class $\leq k$.

  * (iii)

Let $L_{1}$ be nilpotent of class $k$. By (ii) $L_{2}=\varphi(L_{1})$ is
nilpotent of class $\leq k$. Let $L_{2}$ be nilpotent of class $m$. Since
$\varphi^{-1}$ is an isomorphism of Hom-Lie algebras, it follows
$L_{1}=\varphi^{-1}(L_{2})$ is solvable of class $\leq m$. Thus, $k=m$. $\Box$

######  Example 4.2.

Consider Example3.2. Since
$(\mathbb{C}^{2})^{1}=[\mathbb{C}^{2},~{}\mathbb{C}^{2}]_{*}=\\{(x,x);x\in\mathbb{C}\\}$,
$(\mathbb{C}^{2})^{2}=[\mathbb{C}^{2},~{}(\mathbb{C}^{2})^{1}]_{*}=(\mathbb{C}^{2})^{1}$
and $(\mathbb{C}^{2})^{i}=(\mathbb{C}^{2})^{1}$, $i>1$. Thus $\mathbb{C}^{2}$
is not a nilpotent Hom-Lie algebra. And so $L$ is not a nilpotent Hom-Lie
algebra.

######  Lemma 4.1.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra and $H$
be a Hom-Lie subalgebra of $L$. Then $H^{i}\subseteq L^{i}$, for each
$i\in\mathbb{N}$.

Proof.  $H^{0}=H\subseteq L=L^{0}$, and by using induction, if $H^{i}\subseteq
L^{i}$ then $H^{i+1}=[H,~{}H^{i}]\subseteq[L,~{}L^{i}]=L^{i+1}$. $\Box$

######  Theorem 4.5.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a nilpotent Hom-Lie algebra of class k.

  * (i)

Any Hom-Lie subalgebra is nilpotent of class $\leq k$.

  * (ii)

Any quotient Hom-Lie algebra of $L$ is nilpotent of class $\leq k$.

Proof.

  * (i)

A Hom-Lie subalgebra $H$ of a multiplicative Hom-Lie algebra is
multiplicative. By the lemma obove, we have $H^{k}\subseteq L^{k}=\\{0\\}$.
Thus $H^{k}=\\{0\\}$.

  * (ii)

Let $I$ be a Hom-Lie ideal of $L$. The Hom-Lie algebra $L/I$ is
multiplicative. Consider the natural morphism $\pi:L\rightarrow L/I$.
According to Theorem 4.4(i), $\pi(L^{k})=(\pi(L))^{k}$, which implies
$(L/I)^{k}=(\pi(L))^{k}=\pi(L^{k})=\pi(\\{0\\})=\\{0\\}$. $\Box$

######  Remark 1.

Let $(L,~{}[~{},~{}],~{}\alpha)$ be a multiplicative Hom-Lie algebra. If H is
a nilpotent Hom-Lie ideal and $L/H$ is a nilpotent Hom-Lie algebra, then $L$
need not be a nilpotent Hom-Lie algebra.

######  Example 4.3.

Let $L$ be the space spanned by a basis $\\{e_{1},e_{2}\\}$ over $F$. Consider
the multiplicative Hom-Lie algebra $(L,~{}[~{},~{}],\alpha)$ where $\alpha$ is
the zero map and $[~{},~{}]$ is the skew-symmetric bilinear map such that
$[e_{1},e_{1}]=[e_{2},e_{2}]=0$ and $[e_{1},e_{2}]=e_{1}$. Let
$H=\mathrm{Span(\\{e_{1}\\})}$. Then $H$ is a nilpotent Hom-Lie ideal, because
$H^{1}=[H,~{}H]=\\{0\\}$. Also, $L/H$ is a nilpotent Hom-Lie algebra, because
$\overline{[L/H,~{}L/H]}=H$. But $L$ not a nilpotent Hom-Lie algebra, since
$L^{1}=[L,~{}L]=\mathrm{Span(\\{e_{1}\\})}=H$, and
$L^{i}=[L,~{}L^{i-1}]=[L,~{}H]=\mathrm{Span(\\{e_{1}\\})}\neq\\{0\\}$ for all
$i\in\mathbb{N}$.

######  Theorem 4.6.

Let $(L_{1},~{}[~{},~{}]_{1},~{}\alpha_{1})$ and
$(L_{2},~{}[~{},~{}]_{2},~{}\alpha_{2})$ be nilpotent Hom-Lie algebras of
class $k$ and $m$, respectively. Then $(L_{1}\times
L_{2},~{}[~{},~{}],~{}\alpha)$ is a nilpotent Hom-Lie algebra of class
$M=Max\\{m,~{}k\\}$.

Proof. We use induction to show that $(L_{1}\times L_{2})^{i}=L_{1}^{i}\times
L_{2}^{i}$. For $i=0$, we have $(L_{1}\times L_{2})^{0}=L_{1}\times
L_{2}=L_{1}^{0}\times L_{2}^{0}$. For $i>0$ and because the induction
assumption we have $(L_{1}\times L_{2})^{i+1}=[L_{1}\times
L_{2},~{}(L_{1}\times L_{2})^{i}]=[L_{1}\times L_{2},~{}L_{1}^{i}\times
L_{2}^{i}]=[L_{1},~{}L_{1}^{i}]_{1}\times[L_{2},~{}L_{2}^{i}]_{2}=L_{1}^{i+1}\times
L_{2}^{i+1}$.
We may assume that $m\leq k$. Since $L_{1}$ and $L_{2}$ are nilpotent Hom-Lie
algebras of class $k$ and $m$, respectively, then $L_{1}^{k}=\\{0\\}$ and
$L_{1}^{k-1}\neq\\{0\\}$ and $L_{2}^{k}=\\{0\\}$.
Now, $(L_{1}\times L_{2})^{k}=L_{1}^{k}\times
L_{2}^{k}=\\{0\\}\times\\{0\\}=\\{(0,~{}0)\\}$ and $(L_{1}\times
L_{2})^{k-1}=L_{1}^{k-1}\times L_{2}^{k-1}\neq\\{(0,~{}0)\\}$. Thus
$L_{1}\times L_{2}$ is a nilpotent Hom-Lie algebras of class
$k=Max\\{m,~{}k\\}=M$ $\Box$

######  Theorem 4.7.

Every central series is a solvable series.

Proof.  Let $(L,~{}[~{},~{}],~{}\alpha)$ be a Hom-Lie algebra and
$L=L_{0}\supseteq L_{1}\supseteq L_{2}\supseteq\cdots L_{k}$ be a central
series. Then for each $i=0,1,\ldots,k-1$, $[L,~{}L_{i}]\subseteq L_{i+1}$.
Since $[L_{i},~{}L_{i}]\subseteq[L,~{}L_{i}]\subseteq L_{i+1}$ so
$L=L_{0}\supseteq L_{1}\supseteq L_{2}\supseteq\cdots L_{k}$ is a solvable
series (Theorem 3.1). $\Box$

######  Corollary 4.2.

Every nilpotent Hom-Lie algebra is a solvable Hom-Lie algebra.

Proof.  If $(L,~{}[~{},~{}],~{}\alpha)$ is a nilpotent Hom-Lie algebra, then
there exists a central series $L=L_{0}\supseteq L_{1}\supseteq
L_{2}\supseteq\cdots L_{k}$(Theorem 4.3). From the theorem above,
$L=L_{0}\supseteq L_{1}\supseteq L_{2}\supseteq\cdots L_{k}$ is a solvable
series. Thus $(L,~{}[~{},~{}],~{}\alpha)$ is a solvable Hom-Lie algebra
(Theorem 3.3). $\Box$
The converse not true, as in the following Example.

######  Example 4.4.

Consider the Hom-Lie algebra $(\mathbb{R}[x],~{}[~{},~{}],~{}\alpha)$ in
Example 2.7. It is easy to show that $H=\\{p(x)\in\mathbb{R}[x]:deg(p)\leq
3\\}$ is a Hom-Lie subalgebra of $\mathbb{R}[x]$. For any
$p_{i}(x)=a_{i}x^{3}+b_{i}x^{2}+c_{i}x+d_{i}\in H$,
$[p_{1},p_{2}]=(6a_{1}b_{2}-6a_{2}b_{1})x^{2}+(6a_{1}c_{2}-6a_{2}c_{1})x$
where $a_{i},b_{i},c_{i},d_{i}\in\mathbb{R}$, and so
$H^{(1)}=[H,~{}H]=\\{Ax^{2}+Bx:A,B\in\mathbb{R}\\}$ and
$H^{(2)}=[H^{(1)},~{}H^{(1)}]=\\{0\\}$. Thus $H$ is a solvable Hom-Lie algebra
of class 2. But $H$ is not a nilpotent Hom-Lie algebra, since
$H^{1}=[H,~{}H]=\\{Ax^{2}+Bx:A,B\in\mathbb{R}\\}$, $H^{2}=[H,~{}H^{1}]=H^{1}$
and $H^{i}=H^{1}\neq\\{0\\}$ for all $i>1$. $\blacksquare$

######  Example 4.5.

Consider the Hom-Lie subalgebra $I=\\{p(x)\in\mathbb{R}[x]:deg(p)\leq 4\\}$ of
$(\mathbb{R}[x],~{}[~{},~{}],~{}\alpha)$ in Example 2.7. For any
$p_{i}(x)=a_{i}x^{4}+b_{i}x^{3}+c_{i}x^{2}+d_{i}x+e_{i}\in H$,
$[p_{1},p_{2}]=(12a_{1}b_{2}-12a_{2}b_{1})x^{4}+(16a_{1}c_{2}-16a_{2}c_{1})x^{3}+(12a_{1}d_{2}-12a_{2}d_{1}+6b_{1}c_{2}-6b_{2}c_{1})x^{2}+(6b_{1}d_{2}-6b_{2}d_{1})x$
where $a_{i},b_{i},c_{i},d_{i},e_{i}\in\mathbb{R}$, and so
$H^{(1)}=[H,~{}H]=\\{Ax^{4}+Bx^{3}+Cx^{2}+Dx:A,B,C,D\in\mathbb{R}\\}$,
$H^{(2)}=[H^{(1)},~{}H^{(1)}]=H^{(1)}$ and $H^{(i)}=H^{(1)}\neq\\{0\\}$ for
all $i>1$. Thus $H$ is not a solvable Hom-Lie algebra. Also $H$ is not a
nilpotent Hom-Lie algebra by corollary4.2.
Note that $(\mathbb{R}[x],~{}[~{},~{}],~{}\alpha)$ is not a solvable and not a
nilpotent Hom-Lie algebra because there exists a non-solvable and non-
nilpotent Hom-Lie subalgebra of $\mathbb{R}[x]$ (Theorem 3.5(i) and Theorem
4.5(i)). $\blacksquare$

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