# On the half-quantized Hall conductance of massive surface electrons in
magnetic topological insulator films

Rui Chen  Department of Physics, The University of Hong Kong, Pokfulam Road,
Hong Kong 999077, China  Department of Physics, Hubei University, Wuhan
430062, China  Shun-Qing Shen<EMAIL_ADDRESS>Department of Physics, The
University of Hong Kong, Pokfulam Road, Hong Kong 999077, China

(August 31, 2024)

###### Abstract

In topological insulators, massive surface bands resulting from local symmetry
breaking are believed to exhibit a half-quantized Hall conductance. However,
such scenarios are obviously inconsistent with the Thouless-Kohmoto-
Nightingale-Nijs theorem, which states that a single band in a lattice with a
finite Brillouin zone can only have an integer-quantized Hall conductance. To
explore this, we investigate the band structures of a lattice model describing
the magnetic topological insulator film that supports the axion insulator,
Chern insulator, and semi-magnetic topological insulator phases. We reveal
that the gapped and gapless surface bands in the three phases are
characterized by an integer-quantized Hall conductance and a half-quantized
Hall conductance, respectively. This result is distinct from the previous
consensus that the gapped surface band is responsible for the half-quantized
Hall conductance and the gapless band should exhibit zero Hall response. We
propose an effective model to describe the three phases and show that the low-
energy dispersion of the surface bands inherits from the surface Dirac
fermions. The gapped surface band manifests a nearly half-quantized Hall
conductance at low energy near the center of Brillouin zone, but is
compensated by another nearly half-quantized Hall conductance at high energy
near the boundary of Brillouin zone because a single band can only have an
integer-quantized Hall conductance. The gapless state hosts a zero Hall
conductance at low energy but is compensated by another half-quantized Hall
conductance at high energy, and thus the half-quantized Hall conductance can
only originate from the gapless band. Moreover, we calculate the layer-
resolved Hall conductance of the system. The conclusion suggests that the
individual gapped surface band alone does not support the half-quantized
surface Hall effect in a lattice model.

##  I Introduction

The half-quantized Hall effect occurs on the topological insulator surface
once the surface Dirac fermion acquires a mass due to local symmetry breaking
(Qi  _et al._ , 2008; Nomura and Nagaosa, 2011; Qi and Zhang, 2011; Haldane,
1988; Fu  _et al._ , 2007; Niemi and Semenoff, 1983; Jackiw, 1984; Semenoff,
1984) for three reasons: (i) The bulk value of the axion angle $\theta=\pi$ in
topological insulators allows either gapless or gapped surface states (Qi  _et
al._ , 2008; Qi and Zhang, 2011; Mong  _et al._ , 2010; Wilczek, 1987; Essin
_et al._ , 2009; Hasan and Kane, 2010). In the gapped case, there must be a
half-quantized Hall effect; (ii) The half-quantized Hall effect can be
captured by calculating the layer-resolved Hall conductance (Mong  _et al._ ,
2010; Wang  _et al._ , 2015; Varnava and Vanderbilt, 2018; Essin  _et al._ ,
2009; Fu  _et al._ , 2021); and (iii) The gapped surface state could be
described by the massive Dirac equation, which hosts a half-quantized Hall
conductance (Chu  _et al._ , 2011; Liu  _et al._ , 2021; Sun  _et al._ , 2020;
Shen, 2017; Shan  _et al._ , 2010; Lu  _et al._ , 2013). Therefore, it is
widely believed that the half-quantized Hall conductance originates from the
gapped surface state in magnetic topological insulators (Mogi  _et al._ ,
2022; Gao  _et al._ , 2021; Mogi  _et al._ , 2017a; Hao  _et al._ , 2019).
However, there are still concerns for the following reasons: (i) The axion
term and (ii) layer-resolved Hall conductance do not guarantee that the half-
quantized Hall effect originates from the gapped surface state; (iii) The half
quantization in the massive Dirac equation contradicts the common belief that
a single band on a two-dimensional finite Brillouin zone can only have an
integer-quantized Hall conductance. Moreover, our recent works focus on
“parity anomalous semimetals”, in which massive and massless Dirac fermions
coexist (Zou  _et al._ , 2022, 2023; Fu  _et al._ , 2022). The half-quantized
Hall conductance in the parity anomalous semimetal is attributed to the
massless Dirac fermions, rather than the previous consensus that it should be
attributed to the massive Dirac fermions. Therefore, how the half quantization
is manifested in magnetic topological insulator films is still an open
question.

The half-quantized Hall effect is proposed to be realized on magnetically
doped topological insulators (Yu  _et al._ , 2010; Chang  _et al._ , 2013,
2015; Mogi  _et al._ , 2015, 2017b, 2017a; Xiao  _et al._ , 2018; Tokura  _et
al._ , 2019; Zhao  _et al._ , 2013; Fijalkowski  _et al._ , 2021; Li  _et al._
, 2021a; Liu and Wang, 2020; Varnava and Vanderbilt, 2018; Mogi  _et al._ ,
2022; Zhou  _et al._ , 2022; Zou  _et al._ , 2022; Liu and Hesjedal, 2021) and
intrinsic antiferromagnetic topological insulators (Deng  _et al._ , 2020;
Zhang  _et al._ , 2020; Sun  _et al._ , 2020; Shi  _et al._ , 2019; Rienks
_et al._ , 2019; Otrokov  _et al._ , 2019a; Lin  _et al._ , 2022; Li  _et al._
, 2021b, 2019a, 2019b, 2019c, 2019d; Lee  _et al._ , 2019; Chen  _et al._ ,
2019a, b; Hao  _et al._ , 2019; Ding  _et al._ , 2020; Mong  _et al._ , 2010;
Otrokov  _et al._ , 2019b; Chen  _et al._ , 2021; Gu  _et al._ , 2021; Liu
_et al._ , 2020; Gao  _et al._ , 2021; Chen  _et al._ , 2022; Lei and
MacDonald, 2023; Mei  _et al._ , 2023). According to the electronic band
structures, the magnetic topological insulator can be divided into three
different phases: the semi-magnetic topological insulator phase (Mogi  _et
al._ , 2022; Zhou  _et al._ , 2022; Zou  _et al._ , 2022) consists of a gapped
and a gapless Dirac cone on its top and bottom surfaces [Fig. 1(b)]; the axion
(Liu  _et al._ , 2020; Mogi  _et al._ , 2017b, a; Xiao  _et al._ , 2018; Chen
_et al._ , 2022; Gao  _et al._ , 2021) and Chern (Chu  _et al._ , 2011; Chen
_et al._ , 2021; Chang  _et al._ , 2013; Yu  _et al._ , 2010; Gu  _et al._ ,
2021; Liu  _et al._ , 2020) insulator phases host two gapped Dirac cones on
the top and bottom surfaces [Figs. 1(a) and 1(c)]. Specifically, the gapped
Dirac cones on the axion (Chern) insulator phase were believed to be
characterized by a half-quantized Hall conductance with the opposite (same)
sign on the opposite surface, and they combine to yield a zero (quantized)
Hall conductance. However, such scenarios are obviously inconsistent with the
Thouless-Kohmoto-Nightingale-Nijs theorem, which states that a single band in
a lattice with a finite Brillouin zone can only have an integer-quantized Hall
conductance. In the semi-magnetic topological insulator phase, the magnetic
gap is characterized by a half-quantized Hall conductance contributed by the
gapped surface. The gapless surface is believed to exhibit no effect on the
total Hall conductance. In the past years, the three phases have received
enormous attention. The axion insulator possesses a unique electromagnetic
response from the massive Dirac fermions, which leads to the quantized
topological magnetoelectric effect (Wang  _et al._ , 2015; Morimoto  _et al._
, 2015; Qi  _et al._ , 2009; Maciejko  _et al._ , 2010; Tse and MacDonald,
2010; Yu  _et al._ , 2019). The semi-magnetic topological insulator provides a
condensed-matter realization of the parity anomaly and opens the door for
experimental realization of the single Dirac fermion (Mogi  _et al._ , 2022;
Zhou  _et al._ , 2022; Zou  _et al._ , 2022). The realization of the Chern
insulator phase can lead to the development of novel spintronics devices (Yu
_et al._ , 2010; Chang  _et al._ , 2013).

In this work, we systematically study the spectrum, probability distribution,
and the corresponding Berry curvature distribution in three phases in magnetic
topological insulator films, including the axion insulator, Chern insulator,
and semi-magnetic topological insulator phases. We explore how the band
structures manifest the half-quantized Hall conductance in a lattice model. In
the axion and Chern insulator phases, each gapped surface state contributes a
nearly half-quantized Hall conductance at low energy near the center of
Brillouin zone, but must be compensated by another nearly half-quantized Hall
conductance at high energy near the boundary of Brillouin zone, because a
single band can only have an integer-quantized Hall conductance in a lattice
model according to Thouless-Khomoto-Nightingale-Nijs theorem (Thouless  _et
al._ , 1982). In the semi-magnetic topological insulator phase, the gapless
state hosts a zero Hall conductance at low energy but must be compensated by
another half-quantized Hall conductance at high energy, because the half-
quantized Hall conductance can only originate from the gapless state.
Moreover, we propose an effective four-band model to depict the three phases.
We confirm that the low-energy surface states in these phases inherit from the
Dirac fermions. The states at low and high energies always couple in energy
scale and they cannot be treated separately. Due to the coupling effect, the
exact half quantization is only revealed in the gapless surface state in the
semi-magnetic topological insulator phase, but it is not found in the gapped
surface state in the axion and Chern insulator phases from the band
calculation.

Moreover, we adopt the layer-resolved Hall conductance to study the three
phases. The three phases are characterized by distinct patterns of half-
quantized surface Hall effect. The results indicate that the half-quantized
surface Hall effect is contributed by all the occupied bands, rather than the
individual gapped surface bands.

##  II Model

To carry out numerical studies on the magnetic topological insulator, we
consider a tight-binding Hamiltonian on a cubic lattice for an isotropic
three-dimensional topological insulator (Chu  _et al._ , 2011; Shan  _et al._
, 2010; Lu  _et al._ , 2010)

$\mathcal{H}=\sum_{i}c_{i}^{\dagger}\mathcal{M}_{0}c_{i}+\sum_{i,\alpha=x,y,z}\left(c_{i}^{\dagger}\mathcal{T}_{\alpha}c_{i+\alpha}+c_{i+\alpha}^{\dagger}\mathcal{T}_{\alpha}^{\dagger}c_{i}\right),$
(1)

where
$\mathcal{T}_{\alpha}=B\sigma_{z}\tau_{0}-i\frac{A}{2}\sigma_{x}\tau_{\alpha}$
and
$\mathcal{M}_{0}=(M-6B)\sigma_{z}\tau_{0}+\Delta\left(z\right)\sigma_{0}\tau_{z}$
with the lattice space is taken to be unity. Near the $\bm{k}=0$ point in the
momentum space (i.e., the low-energy regime), this model is reduced to a
Dirac-like model in the absence of the magnetic effect. $\sigma$ and $\tau$
are Pauli matrices. The magnetic effect is represented by layer-dependent
Zeeman splitting $\Delta\left(z\right)$. We take
$\Delta\left(z\right)=\Delta_{b}$ for the bottom surface with $z=1,2$,
$\Delta\left(z\right)=\Delta_{t}$ for the top surface with $z=n_{z}-1,n_{z},$
and $\Delta\left(z\right)=0$ elsewhere. The axion and Chern insulator phases
are characterized by $\Delta_{t}\Delta_{b}<0$ and $\Delta_{t}\Delta_{b}>0$,
respectively. The semi-magnetic topological insulator phase is characterized
by either $\Delta_{b}\neq 0$ and $\Delta_{t}=0$ or $\Delta_{b}=0$ and
$\Delta_{t}\neq 0$. In the subsequent calculations, we fix the parameters as
$A=0.5$, $B=0.25$, and $M=0.4$.

Figure 1: (a) Schematic illustration of the axion insulator phase. The arrows
indicate the propagating direction of the surface chiral currents. Blue and
red correspond to the top and bottom surfaces, respectively. (d) Energy
spectra of the axion insulator phase. Here the color scheme of the bands
indicates the wave function distribution. (g) Numerically calculated Hall
conductance $\sigma_{1}^{c,v}$ (blue), $\sigma_{2}^{c,v}$ (red),
$\sigma_{3}^{c,v}$ (green), and $\sigma_{t}$ (orange) as functions of $E_{F}$.
Here, $\sigma_{t}$ depicts the conductance contributed from all the bands,
$\sigma_{1}^{c,v}$ depict the conductance contributed from the lowest
conduction and highest valence bands, $\sigma_{2}^{c,v}$ depict the
conductance contributed from the second lowest conduction and second highest
valence bands, and so on. The dashed black line corresponds to
$\sigma=-e^{2}/2h.$ In (d) and (g), the yellow and light blue regions
correspond to the magnetic gap on the top and bottom surfaces, respectively.
(b, e, h) and (c, f, i) are the same as (a, d, g), except that they depict the
semi-magnetic topological insulator phase and the Chern insulator phase,
respectively. We take $\Delta_{t}=0.05$ for the axion insulator phase,
$\Delta_{t}=0$ for the semi-magnetic topological insulator phase, and
$\Delta_{t}=-0.05$ for the Chern insulator phase. The Zeeman splitting term
for the bottom surface is $\Delta_{b}=-0.2$ for all three phases. The film
thickness is taken as $n_{z}=10$.

When magnetization is introduced to a certain surface, it is believed that the
gapless Dirac cone will open an energy gap characterized by a half-quantized
Hall conductance, with its sign depending on the magnetization direction. For
both Chern and axion insulator phases, the Zeeman effect is introduced to the
top and bottom surfaces. Therefore the surface states open energy gaps at the
$\Gamma$ point on both the top and bottom surfaces [Figs. 1(a) and 1(c)]. When
the two surfaces have antiparallel magnetization alignment [Fig. 1(a)], the
system is characterized by half-quantized Hall conductance with the opposite
sign on the opposite surface, leading to the emergence of the axion insulator
phase. Because the top and bottom surfaces have parallel magnetization
alignment in the Chern insulator phase, the system is characterized by the
half-quantized Hall conductance with the same sign [Fig. 1(c)], and they
combine to yield a quantized Hall conductance. The semi-magnetic topological
insulator phase is characterized by a half-quantized Hall conductance,
contributed by the gapped Dirac cone on the magnetic bottom surface [Fig.
1(b)].

However, the above scenarios seem to contradict the common belief that a
single band in a lattice model can only host an integer-quantized Hall
conductance in units of $e^{2}/h.$ Next, we study the spectrum and the
corresponding probability distribution of the three phases in the lattice
model in Eq. (1), and explore how the band structures reconcile the
contradiction and how the half quantization is manifested.

##  III Spectrum and Hall conductance

The second and third rows in Fig. 1 show the numerically calculated energy
spectra and the corresponding Hall conductance for the three phases,
respectively. We take $\Delta_{b}=0.2$ and $\Delta_{t}=\mp 0.05$ for the axion
and Chern insulator phases and $\Delta_{b}=0.2$ and $\Delta_{t}=0$ for the
semi-magnetic topological insulator phase, respectively. The results are as
expected. In the axion insulator phase, the spectrum opens a gap [Fig. 1(d)]
characterized by a zero Hall conductance [the orange line in Fig. 1(g)]. The
spectrum of the Chern insulator phase [Fig. 1(f)] is the same as the axion
insulator phase, except that the gap is characterized by a quantized Hall
conductance [the orange line in Fig. 1(i)]. The spectrum of the semi-magnetic
topological insulator phase is gapless [Fig. 1(e)], and the system is
characterized by a half-quantized Hall conductance when the Fermi energy is
located at the magnetic gap of the bottom surface [the orange line in Fig.
1(h)]. Here, the conductance for the $m$-band is calculated by (Shen, 2017)

$\displaystyle\sigma_{m}\left(E_{F}\right)$ $\displaystyle=-i\hbar
e^{2}\sum_{n\neq
m}\intop\frac{d^{2}k}{\left(2\text{$\pi$}\right)^{2}}\frac{\left\langle
m\right|v_{x}\left|n\right\rangle\left\langle
n\right|v_{y}\left|m\right\rangle}{E_{m}-E_{n}}$
$\displaystyle\times\frac{f\left(E_{F}-E_{n}\right)-f\left(E_{F}-E_{m}\right)}{E_{m}-E_{n}+i\eta},$
(2)

wth $\left|m\right\rangle$ is the eigensate of energy $E_{m}$ for the quasi-2D
system confined along the $z$ direction. $v_{x}$ and $v_{y}$ are the velocity
operators. $f\left(x\right)$ is the Fermi distribution and $\eta$ is an
infinitesimal quantity. The total conductance contributed from all the bands
has the form
$\sigma_{t}\left(E_{F}\right)=\sum_{m}\sigma_{m}\left(E_{F}\right).$

Figure 2: (a) Probability distribution
$\left|\psi_{1}^{v}\left(k_{x},z\right)\right|^{2}$ as a function of layer
index $z$ for the highest valence band of the axion insulator phase for
different $k_{x}.$ (b) and (c) are the same as (a), except that they depict
the second and third highest valence bands. The probability distributions of
the semi-magnetic topological insulator phase and the Chern insulator phase
are similar to that of the axion insulator phase. (d) The logarithm of the
energy gap between the first and second valence bands as a function of
$\Delta_{t}$. Each point is obtained by searching the minimum energy
difference between the first and second valence bands by scanning in the two-
dimensional Brillouin zone. The $k$ points used in the calculation is
$N_{k}\times N_{k}$. The Zeeman splitting for the bottom surface is taken as
$\Delta_{b}=-0.2$. The film thickness is $n_{z}=10$. In (a-c), we take
$\Delta_{t}=0.05$.

When the Fermi energy is located at the magnetic surface gaps [the yellow
region in Figs. 1(d-f)], the topology nature of the three phases is dominated
by the lowest conduction and highest valence bands [see the blue and orange
lines in Fig. 1(g-i)]. The higher conduction and lower valence bands exhibit
no contribution to the total Hall conductance [the red and green lines in Fig.
1(g-i)].

Figures 2(a-c) show the probability distributions
$\lvert\psi_{i=1,2,3}^{v}\left(k_{x},z\right)\rvert^{2}$ of the valence bands
as functions of $z$ for different $k_{x}$ in the axion insulator phase, where
$\psi_{i}^{v}$ is the wave function distribution of the $i$-th highest valence
band. It is noted that the probability distributions of the semi-magnetic
topological insulator phase and Chern insulator phase are similar to that of
the axion insulator phase [as shown in Figs.1(d-f)]. The probability
distribution of the highest valence band
$\lvert\psi_{1}^{v}\left(k_{x},z\right)\rvert^{2}$ is mainly distributed at
the top surface for the low-energy states near $k_{x}=0$ [the blue lines in
Fig. 2(a)], and is mainly distributed at the bottom surface for the high-
energy states near $k_{x}=\pi$ [the red lines in Fig. 2(a)].

The second and third highest valence bands exhibit no contribution to the
total Hall conductance when the Fermi energy resides in the magnetic gap.
Their probability distributions
$\lvert\psi_{2,3}^{v}\left(k_{x},z\right)\rvert^{2}$ are mainly distributed at
the bottom surfaces for the low-energy state near $k_{x}=0$ [the blue lines in
Figs. 2(b) and 2(c)], and are mainly distributed at the bulk for the high-
energy state near $k_{x}=\pi$ [the red lines in Figs. 2(b) and 2(c)].
Moreover, to confirm that the first and second valence bands are well
separated in energy scale, we plot $\log_{10}\left(E_{g}\right)$ as a function
of $\Delta_{t}$ in Fig. 2(d), where $E_{g}$ is the minimum energy difference
between the first and second valence bands by scanning the whole two-
dimensional Brillouin zone. $E_{g}$ converges with the increasing density of
the $k$ points. The results indicate that the first and second valence bands
are well separated in energy scale, except at two special points with
$\Delta_{t}=\pm\Delta_{b}$.

The above scenario indicates that the highest valence band, that dominates the
topology of the three phases when Fermi energy resides in the magnetic gap, is
localized at one surface at low energy near the Dirac point and at the other
surface at high energy near the Brillouin boundary. In contrast to the
previous consensus (Li  _et al._ , 2019a; Wang  _et al._ , 2015; Morimoto  _et
al._ , 2015), the topology natures of the three phases are dominated by the
two surface states near the $\Gamma$ point.

##  IV Effective model

Figure 3: (a, d) Energy spectrum of the axion insulator phase with
$\Delta_{t}=0.05$ and $\Delta_{b}=-0.2$. The red dashed lines are obtained by
numerical calculations (where only the two lowest conductance and two highest
valence bands are illustrated). The solid lines in (a) and (d) are obtained
from the effective Hamiltonians in Eqs. (3) and (4), respectively. (b, e) and
(c, f) are the same as (a, d), except that they depict (b, e) the semi-
magnetic topological insulator phase with $\Delta_{t}=0$ and (c, f) the Chern
insulator phase with $\Delta_{t}=0.05$, respectively.

The lowest four bands of the magnetic topological insulator thin film can be
effectively described by a $4\times 4$ Hamiltonian in the two-dimensional
Brillouin zone (Lu  _et al._ , 2010; Zou  _et al._ , 2022, 2023; Shan  _et
al._ , 2010):

$H=\left(\begin{array}[]{cc}h\left(\mathbf{{k}}\right)+V_{t}\tau_{z}&j\left[m_{0}\left(\mathbf{{k}}\right)/T\right]m_{0}\left(\mathbf{{k}}\right)\\\
j\left[m_{0}\left(\mathbf{{k}}\right)/T\right]m_{0}\left(\mathbf{{k}}\right)&-h\left(\mathbf{{k}}\right)+V_{b}\tau_{z}\end{array}\right),$
(3)

where
$m_{0}\left(\mathbf{{k}}\right)=M-4B\left(\sin^{2}k_{x}/2+\sin^{2}k_{y}/2\right)$,
$V_{t/b}\left(k\right)=\Delta_{t/b}j\left[-m_{0}\left(\mathbf{{k}}\right)/T\right]+\Delta^{\prime}_{t/b}j\left[m_{0}\left(\mathbf{{k}}\right)/T\right]$,
and $h\left(\mathbf{{k}}\right)=A\left(\tau_{y}\sin k_{x}-\tau_{x}\sin
k_{y}\right)$ describes the massless Dirac fermion. The Fermi-Dirac-
distribution like factor $j(x)=\left[\exp\left(x\right)+1\right]^{-1}$
describes the process that the surface states merge into the bulk states.
$\Delta^{\prime}_{t/b}/\Delta_{t/b}\simeq n_{z}^{\textrm{Mag}}/n_{z}\ll 1$
where $n_{z}^{\textrm{Mag}}$ is the thickness of the magnetically doped film.
For thick films with a large $n_{z}$, we have
$\Delta^{\prime}_{t/b}\rightarrow 0.$ The coefficient $T^{*}=0.05$ is a model-
specific parameter. The first row in Fig. 3 compares the spectra obtained from
the numerical tight-binding calculations and the analytical effective models
in Eq. (3). The effective Hamiltonian captures the band structures of the
three phases in the magnetic topological insulator. Moreover, in the following
content, we will show that this effective Hamiltonian also captures the band
topologies of the three topological phases.

The Brillouin zone is divided into two regimes by the sign of
$m_{0}(\mathbf{k})$. In the low-energy regime near the center of Brillouin
zone, i.e., $k^{2}<k_{c}^{2}$, we have
$j\left[m_{0}\left(\mathbf{{k}}\right)/T\right]\rightarrow 0$ and
$j\left[-m_{0}\left(\mathbf{{k}}\right)/T\right]\rightarrow 1$, where the
value of $k_{c}$ is given by $m_{0}(\mathbf{k}_{c})=0$. The effective
Hamiltonian in Eq. (3) reduces to

$H=\left(\begin{array}[]{cc}h^{\prime}\left(\mathbf{{k}}\right)+\Delta_{t}\tau_{z}&0\\\
0&-h^{\prime}\left(\mathbf{{k}}\right)+\Delta_{b}\tau_{z}\end{array}\right),$
(4)

with
$h^{\prime}\left(\mathbf{{k}}\right)=A\left(\tau_{y}k_{x}-\tau_{x}k_{y}\right)$.
The low-energy effective model describes two decoupled Dirac fermions with the
Dirac mass determined by $V_{t}$ and $V_{b}$, respectively. The effective low-
energy effective Hamiltonian reproduces the key features of the band
structures, that the spectra are dominated by the two decoupled Dirac fermions
near the $\Gamma$ point [the second row in Fig. 3].

In the high-energy regime near the boundary of Brillouin zone, i.e.,
$k^{2}>k_{c}^{2}$, we have
$j\left[m_{0}\left(\mathbf{{k}}\right)/T\right]\rightarrow 1$ and
$j\left[-m_{0}\left(\mathbf{{k}}\right)/T\right]\rightarrow 0.$ The effective
Hamiltonian in Eq. (3) reduces to

$H=\left(\begin{array}[]{cc}h\left(\mathbf{{k}}\right)+\Delta^{\prime}_{t}\tau_{z}&m_{0}\left(\mathbf{{k}}\right)\\\
m_{0}\left(\mathbf{{k}}\right)&-h\left(\mathbf{{k}}\right)+\Delta^{\prime}_{b}\tau_{z}\end{array}\right).$
(5)

The top and bottom surface states are coupled via the term $m_{0}(\mathbf{k})$
in the high-energy regime.

Figure 4: (a) $\Omega_{1}^{v}(k_{x},k_{y})$ as a function of $k_{x}$ and
$k_{y}$, where $\Omega_{1}^{v}(k_{x},k_{y})$ corresponds to the Berry
curvature of the highest valence band in the axion insulator phase. (f)
$\sigma\left(k_{r}\right)$ as a function of $k_{r}$. (b, g) and (c, h) are the
same as (a, f), except that they depict the semi-magnetic topological
insulator and the Chern insulator phase, respectively. The solid red line is
obtained by numerical calculations. The blue circle and green triangle points
are obtained from the effective Hamiltonians in Eqs. (3) and (4),
respectively. The black dashed lines correspond to the half-quantized value
with $\sigma_{1}^{v}\left(k_{r}\right)=\pm e^{2}/2h$. The red dashed line
corresponds to $k_{r}=k_{c}.$ (d, i) and (e, j) are the same as (a, f), except
that they depict the second and third highest valence bands for the axion
insulator phase. The results for the semi-magnetic topological insulator phase
and the Chern insulator phase are similar to the axion insulator phase. The
parameters are $\Delta_{b}=-0.2$ and $\Delta_{t}=0.05$ for the axion insulator
phase, $\Delta_{t}=0$ for the semi-magnetic topological insulator phase, and
$\Delta_{t}=-0.05$ for the Chern insulator phase. The film thickness is taken
as $n_{z}=10$.

##  V Berry curvature distribution

Now, we investigate the Berry curvature distributions and search for the
underlying half quantization in the three systems. The first row in Fig. 4
shows the Berry curvature distribution $\Omega(k_{x},k_{y})$ as a function of
$k_{x}$ and $k_{y}$. The Berry curvature distributions exhibit distinct
behaviors for $k_{r}<k_{c}$ (the low-energy regime) and $k_{r}>k_{c}$ (the
high-energy regime), where $k_{r}$ is the radius of the circle centered at the
origin in the Brillouin zone and $k_{c}$ corresponds to the critical radius
[see the red circle in Fig. 4(b)]. The second column shows the conductance of
the $m$-th band $\sigma_{m}\left(k_{r}\right)$ as a function of $k_{r}$ at
$E_{F}=0$, where

$\displaystyle\sigma_{m}\left(k_{r}\right)$ $\displaystyle=-i\hbar
e^{2}\times\sum_{n\neq
m}\intop_{k^{2}<k_{r}^{2}}\frac{d^{2}k}{\left(2\pi\right)^{2}}\frac{\left\langle
m\right|v_{x}\left|n\right\rangle\left\langle
n\right|v_{y}\left|m\right\rangle}{E_{m}-E_{n}}$
$\displaystyle\times\frac{f\left(-E_{n}\right)-f\left(-E_{m}\right)}{E_{m}-E_{n}+i\eta},$
(6)

Let us focus on the low-energy regime (i.e., $k_{r}<k_{c}$). The highest
valence band is characterized by
$\sigma_{1}^{v}\left(k_{c}\right)=0.47e^{2}/h$ for the axion insulator phase
[Fig. 4(f)], $\sigma_{1}^{v}\left(k_{c}\right)=0$ for the semi-magnetic
topological insulator [Fig. 4(g)], and
$\sigma_{1}^{v}\left(k_{c}\right)=-0.47e^{2}/h$ for the Chern insulator phase
[Fig. 4(h)]. The second and third highest valence bands are characterized by
$\sigma_{2}^{v}\left(k_{c}\right)=-0.38e^{2}/h$ [Fig. 4(i)] and
$\sigma_{3}^{v}\left(k_{c}\right)=0.35e^{2}/h$ [Fig. 4(j)] for all three
phases. In the low-energy regime, the calculated Hall conductance from the
analytical low-energy effective model in Eq. (4) [the blue circle points in
Figs. 4(f)-4(i)] can reproduce the numerically calculated Hall conductance
from the tight-binding models [the solid red lines in Figs. 4(f)-4(i)]. This
further confirms that, at low-energy regimes, the topologies of the systems
are dominated by the Dirac fermions on the top and bottom surfaces.

When the whole Brillouin zone is considered, each band in the axion and Chern
insulator phases can only host a quantized Hall conductance, i.e.,
$\sigma_{1}^{v}\left(\pi\right)=0$ for the axion insulator phase [Fig. 4(f)]
and $\sigma_{1}^{v}\left(\pi\right)=e^{2}/h$ for the Chern insulator phase
[Fig. 4(h)]. The gapless band in the semi-magnetic topological insulator is
characterized by an exact half-quantized Hall conductance with
$\sigma_{1}^{v}\left(\pi\right)=e^{2}/2h$ [Fig. 4(g)]. The second and third
highest valence bands are characterized by
$\sigma_{2,3}^{v}\left(\pi\right)=0$ for all three phases [Figs. 4(i) and
4(j)].

##  VI Half-quantized Hall conductance

In previous studies based on the low-energy effective model, the surface
states are believed to host a half-quantized Hall conductance when local time-
reversal symmetry is broken. However, this is not realizable in a realistic
system where each band can only host an integer-quantized Hall conductance.
The above scenarios explain how this contradiction is reconciled in a lattice
model. In the axion and Chern insulator phases, the highest valence band is
dominated by a nearly half-quantized Hall conductance from one surface at low
energy, and must be compensated by another nearly half-quantized Hall
conductance from the other surface at high energy. In the semi-magnetic
topological insulator phase, the highest valence band is dominated by a zero
Hall conductance from one surface at low energy, and is compensated by another
half-quantized Hall conductance from the other surface at high energy. This is
because the half-quantized Hall conductance can only originate from the
symmetry-protected gapless highest valence band, and all the rest bands can
only contribute a quantized Hall conductance.

Moreover, due to the coupling between the low-energy and high-energy states,
the exact half quantization can not be observed in the axion and Chern
insulator phases. As shown in Fig. 4(f), the contribution to the Hall
conductance $\sigma\left(k_{r}\right)$ from the low-energy massive Dirac cone
(i.e., the blue circle dots) increases with the increasing $k_{r}.$ The exact
half quantization $\sigma\left(k_{r}\right)\rightarrow e^{2}/2h$ shall be
achieved only if $k_{r}\rightarrow\infty$. However, the high-energy state also
contributes to $\sigma\left(k_{r}\right)$ when $k_{r}>k_{c}$, which prohibits
the observation of the exact half quantization. Thus, only a nearly half
quantized Hall conductance $\sigma_{1}^{v}\left(k_{c}\right)=0.47e^{2}/h$ is
observed at $k_{r}=k_{c}$. In contrast, the exact half quantization could only
be observed in the semi-magnetic topological phase, because its low-energy
state does not contribute to the Hall conductance.

##  VII Layer-resolved Hall conductance

Figure 5: (a) The layer-resolved Hall conductance
$\tilde{\sigma}\left(z\right)$ as a function of layer $z$ for different
$\Delta_{t}$. Green, red, and blue correspond to the axion insulator
($\Delta_{t}=0.05$), semi-magnetic insulator ($\Delta_{t}=0$), and Chern
insulator phases ($\Delta_{t}=-0.05$), respectively. (b)
$\tilde{\sigma}_{t/b}$ as functions of $\Delta_{t}$. Here,
$\tilde{\sigma}_{t/b}$ depicts the layer-resolved Hall conductance contributed
from the top/bottom surface layers. In (a-b), the calculations take into
account the contributions from all the occupied bands. (c-d) and (e-f) are the
same as (a-b), except that the obtained layer-resolved Hall conductance only
takes into account the contributions from the first valence band in (c-d) and
from the second valence band in (e-f), respectively. Here, we take the Fermi
energy $E_{F}=0$, the film thickness $n_{z}=10$, and $\Delta_{b}=-0.2$.

The axion insulator phase can be distinguished from the normal insulator phase
by calculating the layer-resolved Hall conductance. The layer-resolved Hall
conductance of the $m$-th occupied band is given by (Fu  _et al._ , 2021; Wang
_et al._ , 2015; Mong  _et al._ , 2010; Varnava and Vanderbilt, 2018; Essin
_et al._ , 2009)

$\tilde{\sigma}_{m}\left(z\right)=\frac{e^{2}}{2\pi h}\int
d^{2}\mathbf{k}\mathcal{F}_{xy}^{mm}\left(\mathbf{k},z\right),$ (7)

where

$\mathcal{F}_{\alpha\beta}^{mn}\left(\mathbf{k}\right)=\partial_{\alpha}\mathcal{A}_{\beta}^{mn}\left(\mathbf{k}\right)-\partial_{\beta}\mathcal{A}_{\alpha}^{mn}\left(\mathbf{k}\right)+i\left[\mathcal{A}_{\alpha}^{mn}\left(\mathbf{\mathbf{k}}\right),\mathcal{A}_{\beta}^{mn}\left(\mathbf{\mathbf{k}}\right)\right]$
(8)

is the non-Abelian Berry curvature in terms of
$\mathcal{A}_{\alpha}^{mn}\left(\mathbf{\mathbf{k}},z\right)$ with the band
indexes $m$ and $n$. The layer-resolved Hall conductance of all occupied bands
is given by
$\tilde{\sigma}\left(z\right)=\sum_{E_{m}<E_{F}}\tilde{\sigma}_{m}\left(z\right)$.
Moreover, the net Hall conductance of the $m$-th band and the layer-resolved
Hall conductance of the $m$-th band is connected through
$\sigma_{m}=\sum_{z=1}^{n_{z}}\tilde{\sigma}_{m}\left(z\right)$.

Figure 5(a) shows $\tilde{\sigma}\left(z\right)$ as a function of $z$ for the
axion insulator ($\Delta_{t}=0.05$), semi-magnetic topological insulator
($\Delta_{t}=0$), and Chern insulator phases ($\Delta_{t}=-0.05$). The bottom
surface layers have the same magnetization alignments in the three phases,
thus they host the same layer-resolved Hall conductance
[$\tilde{\sigma}\left(z=1,2,3\right)$ in Fig. 5(a)]. The top surface layers
have the opposite magnetization alignments in the axion insulator (green) and
Chern insulator (blue) phases, thus the corresponding layer-resolved Hall
conductances have the opposite signs in the two phases
[$\tilde{\sigma}\left(z=8,9,10\right)$ in Fig. 5(a)]. The top surface layers
are non-mangnetic in the semi-magnetic topological insulator (red) phase, thus
the corresponding layer-resolved Hall conductances is zero. This phenomenon is
observed more clearly in Fig. 5(b), where we show $\tilde{\sigma}_{t/b}$ as
functions of $\Delta_{t}$.
$\tilde{\sigma}_{t}=\sum_{z=8}^{10}\tilde{\sigma}\left(z\right)$ and
$\tilde{\sigma}_{b}=\sum_{z=1}^{3}\tilde{\sigma}\left(z\right)$ depict the
layer-resolved Hall conductance of the top and bottom surface layers,
respectively. With the increasing $\Delta_{t}$, $\tilde{\sigma}_{b}$ remains
unchanged but $\tilde{\sigma}_{t}$ increases from $-0.5$ to $0$ and then to
$0.5$, which corresponds to the phase crossovers from the Chern insulator
phase to the semi-magnetic topological insulator phase and then to the axion
insulator phase.

The above calculations take into account the contributions from all the
occupied bands and we show that the half quantization can be extracted in the
three phases. However, we find that the half quantization cannot be extracted
if only one single band is considered. This is shown in Figs. 5(c-d) and Figs.
5(e-f), which are the same as Figs. 5(a-b), expect that the numerically
calculated layer-resolved Hall conductance only considers the contributions
from the first valence band and the second valence band, respectively. The
layer-resolved Hall conductances of one single band have divergent values for
each layer. This implies that the half-quantized surface Hall effect is a
consequence of all the occupied bands, rather than the individual gapped
surface bands.

##  VIII Different thickness and parameters

Figure 6: (a) Wave function distribution as a function of layer index $z$ for
the highest valence band of the axion insulator phase for different $k_{x}.$
(b) and (c) are the same as (a), except that they depict the second and third
valence bands. The wave function distributions of the semi-magnetic
topological insulator phase and the Chern insulator phase are similar to the
axion insulator phase. (d) The logarithm of the energy gap between the first
and second valence bands as a function of $\Delta_{t}$ for different
thicknesses $n_{z}$. Each point is obtained by find the minimum energy
difference between the first and second valence bands by scanning in the two-
dimensional Brillouin zone. The $k$ points used in the calculation is
$N_{k}\times N_{k},$with $N_{k}=200$. The parameters are taken as
$\Delta_{b}=-0.2$. In (a-c), we take $\Delta_{t}=0.05$ and $n_{z}=50$. Figure
7: (a) Energy spectra of the axion insulator phase. Here the color scheme of
the bands indicates the wave function distribution. (d) Numerically calculated
Hall conductance $\sigma_{1}^{c,v}$ (blue), $\sigma_{2}^{c,v}$ (red),
$\sigma_{3}^{c,v}$ (green), and $\sigma_{t}$ (orange) as a function of
$E_{F}$. Here, $\sigma_{t}$ depicts the conductance contributed from all the
bands, $\sigma_{1}^{c,v}$ depict the conductance contributed from the lowest
conduction and highest valence bands, $\sigma_{2}^{c,v}$ depict the
conductance contributed from the second lowest conduction and second highest
valence bands, and so on. The solid black line corresponds to
$\sigma=e^{2}/2h.$ In (a) and (d), the yellow and light blue regions
correspond to the magnetic gap on the top and bottom surfaces, respectively.
(b, e) and (c, f) are the same as (a, d), except that they depict the semi-
magnetic topological insulator phase and the Chern insulator phase,
respectively. (g) Probability distribution as a function of layer index $z$
for the highest valence band of the axion insulator phase for different
$k_{x}.$ (h) and (i) are the same as (a), except that they depict the second
and third valence bands. The wave function distributions of the semi-magnetic
topological insulator phase and the Chern insulator phase are similar to the
axion insulator phase. We take $\Delta_{t}=0.05$ for the axion insulator
phase, $\Delta_{t}=0$ for the semi-magnetic topological insulator phase, and
$\Delta_{t}=-0.05$ for the Chern insulator phase. The Zeeman splitting term
for the bottom surface is $\Delta_{b}=-0.1$ for all three phases. The film
thickness is taken as $n_{z}=50$.

Above, we consider a thinfilm case with $n_{z}=10$. Now, we show that the
conclusions are irrelevant to the film thickness. Figures 6(a-c) show the
probability distributions of the first, second, and third highest valence
bands for different $k_{z}$ with $n_{z}=50$, respectively. The conclusions are
similar to the thinfilm case shown in Figs. 2(a-c). Figure 6(d) shows the
logarithm of the energy gap between the first and second valence bands as a
function of $\Delta_{t}$ for different thicknesses $n_{z}$. Though the energy
difference decreases with the increasing film thickness, the first and second
valence bands are still well separated in energy scale, as long as
$\Delta_{t}\neq\pm\Delta_{b}$.

Moreover, we consider the case for $\Delta_{b}=-0.1$. The first row in Fig. 7
shows the spectra of the three topological phases, which are similar to those
shown in the second row in Fig. 1, except that the high energies for the first
and second highest valence and lowest conduction bands merge into the bulk.
This can be observed more clearly in the third row in Fig. 7, which shows the
probability distributions of the first, second, and third highest valence
bands, respectively. Thus, the probability distribution of the highest valence
band at high energy with $k_{x}=\pi$ can be localized not only at the surface
[the red lines in Figs. 2(a) and 6(a)], but also at the bulk [the red line in
Fig. 7(g)]. Moreover, by checking the Hall conductance (the second row in Fig.
7), the probability distributions of the state at high energy will not change
the fact that the Hall conductance is dominated by the lowest conduction and
highest valence bands, when the Fermi energy resides inside the magnetic gap.

##  IX Conclusion

In this work, we have investigated the energy spectra, wave function
distribution, and the corresponding Berry curvature in the three distinct
topological phases in magnetic topological insulator films, including the
axion insulator phase, the semi-magnetic topological insulator phase, and the
Chern insulator phase. In the axion and Chern insulator phases, a nearly half-
quantized Hall conductance is observed at low energy near $\Gamma$ point for
the gapped Dirac cone (which is in accordance with the previous consensus),
however, another nearly half-quantized Hall conductance must be compensated at
high energy away from $\Gamma$ point to ensure that the total Hall conductance
is an integer-quantized number in units of $e^{2}/h$. In the semi-magnetic
topological insulator phase, the gapless Dirac cone hosts a vanishing Hall
conductance at low energy near $\Gamma$ point as expected, however, another
half-quantized Hall conductance emerges at high energy away from $\Gamma$
point. This explains how the contradiction between the previous consensus
based on the low-energy effective model and the scenarios based on a lattice
model is reconciled. Moreover, due to the coupling between the low-energy
state and the high-energy state, the exact half quantization is only revealed
in the gapless surface state in the semi-magnetic topological insulator phase,
but not found in the gapped surface state in the axion and Chern insulator
phases from the band calculation.

On the other hand, we adopt the layer-resolved Hall conductance to
characterize the three phases. The half-quantized surface Hall effect is
revealed in the three topological phases and it is contributed by all the
occupied bands. This is distinct to the previous consensus that the half
quantization originates from the individual gapped band.

###### Acknowledgements.

We thank helpful discussions with Bo Fu and Huan-Wen Wang. This work was
supported by the Research Grants Council, University Grants Committee, Hong
Kong under Grant Nos. C7012-21G and 17301220 and the National Key R&D Program
of China under Grant No. 2019YFA0308603.

## References

  * Qi  _et al._ (2008) X.-L. Qi, T. L. Hughes, and S.-C. Zhang, “Topological field theory of time-reversal invariant insulators”, Phys. Rev. B 78, 195424 (2008).
  * Nomura and Nagaosa (2011) K. Nomura and N. Nagaosa, “Surface-quantized anomalous hall current and the magnetoelectric effect in magnetically disordered topological insulators”, Phys. Rev. Lett. 106, 166802 (2011).
  * Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors”, Rev. Mod. Phys. 83, 1057 (2011).
  * Haldane (1988) F. D. M. Haldane, “Model for a quantum hall effect without landau levels: Condensed-matter realization of the "parity anomaly"”, Phys. Rev. Lett. 61, 2015 (1988).
  * Fu  _et al._ (2007) L. Fu, C. L. Kane, and E. J. Mele, “Topological insulators in three dimensions”, Phys. Rev. Lett. 98, 106803 (2007).
  * Niemi and Semenoff (1983) A. J. Niemi and G. W. Semenoff, “Axial-anomaly-induced fermion fractionization and effective gauge-theory actions in odd-dimensional space-times”, Phys. Rev. Lett. 51, 2077 (1983).
  * Jackiw (1984) R. Jackiw, “Fractional charge and zero modes for planar systems in a magnetic field”, Phys. Rev. D 29, 2375 (1984).
  * Semenoff (1984) G. W. Semenoff, “Condensed-matter simulation of a three-dimensional anomaly”, Phys. Rev. Lett. 53, 2449 (1984).
  * Mong  _et al._ (2010) R. S. K. Mong, A. M. Essin, and J. E. Moore, “Antiferromagnetic topological insulators”, Phys. Rev. B 81, 245209 (2010).
  * Wilczek (1987) F. Wilczek, “Two applications of axion electrodynamics”, Phys. Rev. Lett. 58, 1799 (1987).
  * Essin  _et al._ (2009) A. M. Essin, J. E. Moore, and D. Vanderbilt, “Magnetoelectric polarizability and axion electrodynamics in crystalline insulators”, Phys. Rev. Lett. 102, 146805 (2009).
  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators”, Rev. Mod. Phys. 82, 3045 (2010).
  * Wang  _et al._ (2015) J. Wang, B. Lian, X.-L. Qi, and S.-C. Zhang, “Quantized topological magnetoelectric effect of the zero-plateau quantum anomalous Hall state”, Phys. Rev. B 92, 081107(R) (2015).
  * Varnava and Vanderbilt (2018) N. Varnava and D. Vanderbilt, “Surfaces of axion insulators”, Phys. Rev. B 98, 245117 (2018).
  * Fu  _et al._ (2021) B. Fu, Z.-A. Hu, and S.-Q. Shen, “Bulk-hinge correspondence and three-dimensional quantum anomalous Hall effect in second-order topological insulators”, Phys. Rev. Research 3, 033177 (2021).
  * Chu  _et al._ (2011) R.-L. Chu, J. Shi, and S.-Q. Shen, “Surface edge state and half-quantized Hall conductance in topological insulators”, Phys. Rev. B 84, 085312 (2011).
  * Liu  _et al._ (2021) Z. Liu, D. Qian, Y. Jiang, and J. Wang, “Dissipative edge transport in disordered axion insulator films”, arXiv:2109.06178 (2021).
  * Sun  _et al._ (2020) H.-P. Sun, C. M. Wang, S.-B. Zhang, R. Chen, Y. Zhao, C. Liu, Q. Liu, C. Chen, H.-Z. Lu, and X. C. Xie, “Analytical solution for the surface states of the antiferromagnetic topological insulator MnBi2Te4”, Phys. Rev. B 102, 241406(R) (2020).
  * Shen (2017) S.-Q. Shen,  _Topological Insulators_ (Springer Singapore, 2017).
  * Shan  _et al._ (2010) W.-Y. Shan, H.-Z. Lu, and S.-Q. Shen, “Effective continuous model for surface states and thin films of three-dimensional topological insulators”, New J. Phys. 12, 043048 (2010).
  * Lu  _et al._ (2013) H.-Z. Lu, A. Zhao, and S.-Q. Shen, “Quantum transport in magnetic topological insulator thin films”, Phys. Rev. Lett. 111, 146802 (2013).
  * Mogi  _et al._ (2022) M. Mogi, Y. Okamura, M. Kawamura, R. Yoshimi, K. Yasuda, A. Tsukazaki, _et al._ , “Experimental signature of the parity anomaly in a semi-magnetic topological insulator”, Nat. Phys. 18, 390 (2022).
  * Gao  _et al._ (2021) A. Gao, Y.-F. Liu, C. Hu, J.-X. Qiu, C. Tzschaschel, B. Ghosh, _et al._ , “Layer Hall effect in a 2D topological axion antiferromagnet”, Nature 595, 521 (2021).
  * Mogi  _et al._ (2017a) M. Mogi, M. Kawamura, A. Tsukazaki, R. Yoshimi, K. S. Takahashi, M. Kawasaki, and Y. Tokura, “Tailoring tricolor structure of magnetic topological insulator for robust axion insulator”, Sci. Adv. 3, eaao1669 (2017a).
  * Hao  _et al._ (2019) Y.-J. Hao, P. Liu, Y. Feng, X.-M. Ma, E. F. Schwier, M. Arita, _et al._ , “Gapless surface Dirac cone in antiferromagnetic topological insulator MnBi2Te4”, Phys. Rev. X 9, 041038 (2019).
  * Zou  _et al._ (2022) J.-Y. Zou, B. Fu, H.-W. Wang, Z.-A. Hu, and S.-Q. Shen, “Half-quantized hall effect and power law decay of edge-current distribution”, Phys. Rev. B 105, L201106 (2022).
  * Zou  _et al._ (2023) J.-Y. Zou, R. Chen, B. Fu, H.-W. Wang, Z.-A. Hu, and S.-Q. Shen, “Half-quantized hall effect at the parity-invariant fermi surface”, Phys. Rev. B 107, 125153 (2023).
  * Fu  _et al._ (2022) B. Fu, J.-Y. Zou, Z.-A. Hu, H.-W. Wang, and S.-Q. Shen, “Quantum Anomalous Semimetals”, npj Quantum Materials 7, 94 (2022).
  * Yu  _et al._ (2010) R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and Z. Fang, “Quantized anomalous hall effect in magnetic topological insulators”, Science 329, 61 (2010).
  * Chang  _et al._ (2013) C. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, _et al._ , “Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator.”, Science 340, 167 (2013).
  * Chang  _et al._ (2015) C.-Z. Chang, W. Zhao, D. Y. Kim, H. Zhang, B. A. Assaf, D. Heiman, S.-C. Zhang, C. Liu, M. H. W. Chan, and J. S. Moodera, “High-precision realization of robust quantum anomalous hall state in a hard ferromagnetic topological insulator”, Nat. Mater. 14, 473 (2015).
  * Mogi  _et al._ (2015) M. Mogi, R. Yoshimi, A. Tsukazaki, K. Yasuda, Y. Kozuka, K. S. Takahashi, M. Kawasaki, and Y. Tokura, “Magnetic modulation doping in topological insulators toward higher-temperature quantum anomalous Hall effect”, Appl. Phys. Lett. 107, 182401 (2015).
  * Mogi  _et al._ (2017b) M. Mogi, M. Kawamura, R. Yoshimi, A. Tsukazaki, Y. Kozuka, N. Shirakawa, K. S. Takahashi, M. Kawasaki, and Y. Tokura, “A magnetic heterostructure of topological insulators as a candidate for an axion insulator”, Nat. Mater. 16, 516 (2017b).
  * Xiao  _et al._ (2018) D. Xiao, J. Jiang, J.-H. Shin, W. Wang, F. Wang, Y.-F. Zhao, _et al._ , “Realization of the axion insulator state in quantum anomalous Hall sandwich heterostructures”, Phys. Rev. Lett. 120, 056801 (2018).
  * Tokura  _et al._ (2019) Y. Tokura, K. Yasuda, and A. Tsukazaki, “Magnetic topological insulators”, Nature Rev. Phys. 1, 126 (2019).
  * Zhao  _et al._ (2013) Y. Zhao, C.-Z. Chang, Y. Jiang, A. DaSilva, Y. Sun, H. Wang, _et al._ , “Demonstration of surface transport in a hybrid Bi2Se3/Bi2Te3 heterostructure”, Sci. Rep. 3, 3060 (2013).
  * Fijalkowski  _et al._ (2021) K. M. Fijalkowski, N. Liu, M. Hartl, M. Winnerlein, P. Mandal, A. Coschizza, _et al._ , “Any axion insulator must be a bulk three-dimensional topological insulator”, Phys. Rev. B 103, 235111 (2021).
  * Li  _et al._ (2021a) H. Li, H. Jiang, C.-Z. Chen, and X. C. Xie, “Critical behavior and universal signature of an axion insulator state”, Phys. Rev. Lett. 126, 156601 (2021a).
  * Liu and Wang (2020) Z. Liu and J. Wang, “Anisotropic topological magnetoelectric effect in axion insulators”, Phys. Rev. B 101, 205130 (2020).
  * Zhou  _et al._ (2022) H. Zhou, H. Li, D.-H. Xu, C.-Z. Chen, Q.-F. Sun, and X. C. Xie, “Transport theory of half-quantized hall conductance in a semimagnetic topological insulator”, Phys. Rev. Lett. 129, 096601 (2022).
  * Liu and Hesjedal (2021) J. Liu and T. Hesjedal, “Magnetic topological insulator heterostructures: A review”, Advanced Materials , 2102427 (2021).
  * Deng  _et al._ (2020) Y. Deng, Y. Yu, M. Z. Shi, Z. Guo, Z. Xu, J. Wang, X. H. Chen, and Y. Zhang, “Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi2Te4”, Science 367, 895 (2020).
  * Zhang  _et al._ (2020) R.-X. Zhang, F. Wu, and S. Das Sarma, “Möbius insulator and higher-order topology in MnBi2nTe3n+1”, Phys. Rev. Lett. 124, 136407 (2020).
  * Shi  _et al._ (2019) M. Z. Shi, B. Lei, C. S. Zhu, D. H. Ma, J. H. Cui, Z. L. Sun, J. J. Ying, and X. H. Chen, “Magnetic and transport properties in the magnetic topological insulators MnBi2Te4(Bi2Te3)n$(n=1,2)$”, Phys. Rev. B 100, 155144 (2019).
  * Rienks  _et al._ (2019) E. D. L. Rienks, S. Wimmer, J. Sánchez-Barriga, O. Caha, P. S. Mandal, J. Ruzicka, _et al._ , “Large magnetic gap at the dirac point in Bi2Te3/MnBi2Te4 heterostructures”, Nature 576, 423 (2019).
  * Otrokov  _et al._ (2019a) M. M. Otrokov, I. P. Rusinov, M. Blanco-Rey, M. Hoffmann, A. Y. Vyazovskaya, S. V. Eremeev, A. Ernst, P. M. Echenique, A. Arnau, and E. V. Chulkov, “Unique thickness-dependent properties of the van der Waals interlayer antiferromagnet MnBi2Te4 films”, Phys. Rev. Lett. 122, 107202 (2019a).
  * Lin  _et al._ (2022) W. Lin, Y. Feng, Y. Wang, Z. Lian, H. Li, Y. Wu, _et al._ , “Influence of the dissipative topological edge state on quantized transport in ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$”, Phys. Rev. B 105, 165411 (2022).
  * Li  _et al._ (2021b) Y. Li, C. Liu, Y. Wang, Z. Lian, H. Li, Y. Wu, J. Zhang, and Y. Wang, “Nonlocal Transport and One-dimensional Conduction in the Axion Insulator State of MnBi2Te4”, arXiv:2105.10390 (2021b).
  * Li  _et al._ (2019a) J. Li, Y. Li, S. Du, Z. Wang, B.-L. Gu, S.-C. Zhang, K. He, W. Duan, and Y. Xu, “Intrinsic magnetic topological insulators in van der Waals layered MnBi2Te4-family materials”, Sci. Adv. 5, eaaw5685 (2019a).
  * Li  _et al._ (2019b) H. Li, S.-Y. Gao, S.-F. Duan, Y.-F. Xu, K.-J. Zhu, S.-J. Tian, _et al._ , “Dirac surface states in intrinsic magnetic topological insulators EuSn2As2 and MnBi2nTe3n+1”, Phys. Rev. X 9, 041039 (2019b).
  * Li  _et al._ (2019c) J. Li, C. Wang, Z. Zhang, B.-L. Gu, W. Duan, and Y. Xu, “Magnetically controllable topological quantum phase transitions in the antiferromagnetic topological insulator MnBi2Te4”, Phys. Rev. B 100, 121103 (2019c).
  * Li  _et al._ (2019d) Y. Li, Z. Jiang, J. Li, S. Xu, and W. Duan, “Magnetic anisotropy of the two-dimensional ferromagnetic insulator MnBi2Te4”, Phys. Rev. B 100, 134438 (2019d).
  * Lee  _et al._ (2019) S. H. Lee, Y. Zhu, Y. Wang, L. Miao, T. Pillsbury, H. Yi, _et al._ , “Spin scattering and noncollinear spin structure-induced intrinsic anomalous Hall effect in antiferromagnetic topological insulator MnBi2Te4”, Phys. Rev. Research 1, 012011 (2019).
  * Chen  _et al._ (2019a) B. Chen, F. Fei, D. Zhang, B. Zhang, W. Liu, S. Zhang, _et al._ , “Intrinsic magnetic topological insulator phases in the Sb doped MnBi2Te4 bulks and thin flakes”, Nature Commun. 10, 4469 (2019a).
  * Chen  _et al._ (2019b) Y. J. Chen, L. X. Xu, J. H. Li, Y. W. Li, H. Y. Wang, C. F. Zhang, _et al._ , “Topological electronic structure and its temperature evolution in antiferromagnetic topological insulator MnBi2Te4”, Phys. Rev. X 9, 041040 (2019b).
  * Ding  _et al._ (2020) Y.-R. Ding, D.-H. Xu, C.-Z. Chen, and X. C. Xie, “Hinged quantum spin Hall effect in antiferromagnetic topological insulators”, Phys. Rev. B 101, 041404(R) (2020).
  * Otrokov  _et al._ (2019b) M. M. Otrokov, I. I. Klimovskikh, H. Bentmann, D. Estyunin, A. Zeugner, Z. S. Aliev, _et al._ , “Prediction and observation of an antiferromagnetic topological insulator”, Nature 576, 416 (2019b).
  * Chen  _et al._ (2021) R. Chen, S. Li, H.-P. Sun, Q. Liu, Y. Zhao, H.-Z. Lu, and X. C. Xie, “Using nonlocal surface transport to identify the axion insulator”, Phys. Rev. B 103, L241409 (2021).
  * Gu  _et al._ (2021) M. Gu, J. Li, H. Sun, Y. Zhao, C. Liu, J. Liu, H. Lu, and Q. Liu, “Spectral signatures of the surface anomalous hall effect in magnetic axion insulators”, Nat. Commun. 12, 3524 (2021).
  * Liu  _et al._ (2020) C. Liu, Y. Wang, H. Li, Y. Wu, Y. Li, J. Li, K. He, Y. Xu, J. Zhang, and Y. Wang, “Robust axion insulator and Chern insulator phases in a two-dimensional antiferromagnetic topological insulator”, Nat. Mater. 19, 522 (2020).
  * Chen  _et al._ (2022) R. Chen, H.-P. Sun, M. Gu, C.-B. Hua, Q. Liu, H.-Z. Lu, and X. C. Xie, “Layer Hall effect induced by hidden Berry curvature in antiferromagnetic insulators”, Natl. Sci. Rev. , nwac140 (2022).
  * Lei and MacDonald (2023) C. Lei and A. H. MacDonald, “Kerr, Faraday, and Magnetoelectric Effects in MnBi2Te4 Thin Films”, arXiv:2303.14635 (2023).
  * Mei  _et al._ (2023) R. Mei, Y.-F. Zhao, C. Wang, Y. Ren, D. Xiao, C.-Z. Chang, and C.-X. Liu, “Electrically Controlled Anomalous Hall Effect and Orbital Magnetization in Topological Magnet MnBi2Te4”, arXiv:2303.06204 (2023).
  * Morimoto  _et al._ (2015) T. Morimoto, A. Furusaki, and N. Nagaosa, “Topological magnetoelectric effects in thin films of topological insulators”, Phys. Rev. B 92, 085113 (2015).
  * Qi  _et al._ (2009) X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, “Inducing a magnetic monopole with topological surface states”, Science 323, 1184 (2009).
  * Maciejko  _et al._ (2010) J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, “Topological quantization in units of the fine structure constant”, Phys. Rev. Lett. 105, 166803 (2010).
  * Tse and MacDonald (2010) W.-K. Tse and A. H. MacDonald, “Giant magneto-optical Kerr effect and universal Faraday effect in thin-film topological insulators”, Phys. Rev. Lett. 105, 057401 (2010).
  * Yu  _et al._ (2019) J. Yu, J. Zang, and C.-X. Liu, “Magnetic resonance induced pseudoelectric field and giant current response in axion insulators”, Phys. Rev. B 100, 075303 (2019).
  * Thouless  _et al._ (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized hall conductance in a two-dimensional periodic potential”, Phys. Rev. Lett. 49, 405 (1982).
  * Lu  _et al._ (2010) H.-Z. Lu, W.-Y. Shan, W. Yao, Q. Niu, and S.-Q. Shen, “Massive Dirac fermions and spin physics in an ultrathin film of topological insulator”, Phys. Rev. B 81, 115407 (2010).


# Jacobson program for Lie algebras

A. Goswami  [1] Department of Mathematics and Applied Mathematics, University
of Johannesburg, P.O. Box 524, Auckland Park 2006, South Africa. [2] National
Institute for Theoretical and Computational Sciences (NITheCS), South Africa.
<EMAIL_ADDRESS>
###### Abstract.

Following the structure theory approach for rings, the aim of this paper is to
study some distinguished classes of Lie algebras. We introduce the notion of a
Lie-module and discuss some relations of it with various classes of ideals of
a Lie algebra. We give several Lie-module-related characterizations of
primitive Lie algebras, and prove a representation theorem of a Lie algebra in
terms of primitive Lie algebras. We prove a density theorem. Finally, we
highlight on the importance of nuclear ideals of prime and semiprime Lie
algebras in determining them.

###### Key words and phrases:

Lie algebra; annihilator; density theorem; nuclear ideal

###### 1991 Mathematics Subject Classification:

17B05, 17B99,17A01, 17A60.

##  1\. Introduction

Rowen [9] gave a Jacobson’s structure theoretic scheme for rings in an
increasing order of generality. One may consider the same strategy for proving
theorems for Lie algebras. Due to the lack of associativity, we see the first
indication of different behaviour already in the definition of a module over a
Lie algebra. To avoid this problem, one may consider the Poincaré-Birkhoff-
Witt embedding of a Lie algebra into an associative algebra. However, if a Lie
algebra is primitive then the corresponding enveloping algebra may not be
primitive. Furthermore, Kucherov, Pikhtilkova, and Pikhtilkov [5] gave
extensive list of examples (see Example 2.9) of primitive Lie algebras which
naturally creates an expectation of a structure-theoretic study of this class
of Lie algebras. On the other hand, Kawamoto [3] defined prime and semiprime
ideals of a Lie algebra which further has been considered by Pikhtilkov [8],
once again indicates a possibility of looking into prime and semiprime Lie
algebras.

Similar to Jacobson program for rings, the purpose of this paper is to
initiate a study of the following distinguished classes of Lie algebras and
some correlations between them:

  1. (i)

Simple.

  2. (ii)

Primitive.

  3. (iii)

Semiprimitive.

  4. (iv)

Semiprime.

  5. (v)

Prime.

  6. (vi)

General case.

In Section 2 we shall introduce a module over a Lie algebra and briefly study
some properties of it. We correlate it with maximal, minimal, and primitive
ideals of a Lie algebra. We characterize primitive Lie algebras several ways,
and represent a Lie algebra in terms of primitive Lie algebras (Theorem 2.16).
We prove a density theorem (Theorem 2.17). Section 3 is about studying prime
and semiprime Lie algebras. Here the notion of a commutator and an associator
play important role in building the notion of nuclear ideal which determines
existence of prime and semiprime Lie algebras (Theorem 3.3 and Theorem 3.6).

##  2\. Primitive and semiprimitive Lie algebras

To define a primitive Lie algebra as a faithful irreducible representation, we
require the notion of a module over a Lie algebra. A (_left_) _Lie-module_ is
an abelian group $M$ endowed with a map $\mathfrak{g}\times M\to M$ (denoted
by $(x,m)\mapsto xm$) satisfying the identities:

  1. (i)

$[x,y]m=x(ym)-y(xm);$

  2. (ii)

$(\alpha x+\beta y)m=\alpha(xm)+\beta(ym);$

  3. (iii)

$x(m+m^{\prime})=xm+xm^{\prime}$

for all $x,y\in\mathfrak{g},$ $\alpha,\beta\in\mathds{k}$, and
$m,m^{\prime}\in M$.

Since a Lie algebra $\mathfrak{g}$ is a module over itself, the expression for
$[x,y]m$ in Axiom (i) then must satisfy the Jacobi identity:
$[[x,y],z]=[x,[y,z]]-[y,[x,z]].$ When we say a Lie-module $M$, we always mean
a module $M$ over the Lie algebra $\mathfrak{g}$. What we have called a Lie-
module, strictly speaking, should be called a $\mathfrak{g}$-module, but the
terminology “$\mathfrak{g}$-module” already exists in the literature for an
action of a Lie algebra over a vector space.

Before we embark for primitive Lie algebras, let us first briefly discuss on
some of the basic concepts and elementary results of Lie-modules. This study
of Lie-modules will show us their relationships with various types of ideals
of a Lie algebra. Furthermore, we shall want to ‘realize’ a Lie algebra as Lie
algebra of endomorphisms of a Lie-module. To this end, consider the set
$\mathrm{End}_{\mathfrak{g}}(M)$ of group endomorphisms of a Lie-module $M$.
We wish to impose a Lie algebra structure on $\mathrm{End}_{\mathfrak{g}}(M)$
and that is done in the following

######  Lemma 2.1.

For all $m\in M,$ for all $f,g\in\mathrm{End}_{\mathfrak{g}}(M),$ and for all
$\alpha\in\mathds{k}$, the set $\mathrm{End}_{\mathfrak{g}}(M)$ endowed with
the operations:

  1. (i)

$(f+g)(m)=f(m)+g(m);$

  2. (ii)

$(\alpha f)(m)=\alpha f(m);$

  3. (iii)

$[f,g]=f\circ g-g\circ f$

forms a Lie algebra over $\mathds{k}$.

Now to obtain a Lie algebra homomorphism from $\mathfrak{g}$ into
$\mathrm{End}_{\mathfrak{g}}(M)$, a natural choice of our map (guided by
rings) is nothing but a ‘left multiplication map’. So, we have

######  Lemma 2.2.

The map $f\colon\mathfrak{g}\to\mathrm{End}_{\mathfrak{g}}(M)$ defined by
$x\mapsto(\phi_{x}\colon\phi_{x}(m)=xm)$ is a Lie algebra homomorphism.

###### Proof.

Suppose $x,y\in\mathfrak{g}$ and $\alpha\in\mathds{k}.$ Then for all $m\in M,$

$f(x+y)(m)=\phi_{x+y}(m)=(x+y)m=xm+ym=(\phi_{x}+\phi_{y})(m)=(f(x)+f(y)(m),\;\text{and}$
$\alpha f(x)(m)=\alpha\phi_{x}(m)=\alpha(xm)=(\alpha x)m=\phi_{\alpha
x}(m)=f(\alpha x)(m).$

Finally,

$f([x,y])(m)=\phi_{[x,y]}(m)=[x,y]m=x(ym)-y(xm)=\phi_{x}\phi_{y}(m)-\phi_{y}\phi_{x}(m)=[f(x),f(y)](m).\qed$

A _Lie-realization_ of $\mathfrak{g}$ is a Lie algebra homomorphism
$\mathcal{R}_{\mathfrak{g}}\colon\mathfrak{g}\to\mathrm{End}_{\mathfrak{g}}(M)$
of $\mathfrak{g}$ into the Lie algebra $\mathrm{End}_{\mathfrak{g}}(M)$ of
endomorphisms of some Lie-module $M$. A Lie-realization
$\mathcal{R}_{\mathfrak{g}}$ is called _faithful_ if
$\mathrm{ker}\,\mathcal{R}_{\mathfrak{g}}$ is trivial. Note that in our
definition of Lie-realization, if we consider the codomain of
$\mathcal{R}_{\mathfrak{g}}$ as the $\mathfrak{g}$-module $\mathfrak{gl}(V)$,
then $\mathcal{R}_{\mathfrak{g}}$ is known as a _representation of_
$\mathfrak{g}.$

Once we have Lie-modules, the next thing we shall immediately like to consider
is homomorphisms between them. If $M$, $M^{\prime}$ are Lie-modules, then a
_Lie-module homomorphism_ from $M$ into $M^{\prime}$ is a group homomorphism
$f\colon M\to M^{\prime}$ such that

$f(xm)=xf(m)-mf(x)$ (1)

for all $x\in\mathfrak{g}$ and for all $m\in M.$ Since every Lie algebra is a
Lie-module over itself, our definition (1) of the scalar multiplication now
agrees with Axiom (i). A _Lie-submodule_ $S$ of a Lie-module $M$ is a subgroup
of $M$ such that $xs\in S,$ for all $x\in\mathfrak{g}$ and for all $s\in S.$
If $S$ is a Lie-submodule of $M$, then the _quotient_ Lie-module $M/S$ is the
quotient abelian group $M/S$ regarding as a module over $\mathfrak{g}$ by
defining $x(m+S)=xm+S$ for all $x\in\mathfrak{g}$ and for all $m\in M.$ Like
for modules over rings, we also have the fundamental theorem of homomorphisms
for Lie-modules.

######  Proposition 2.3.

If $f\colon M\to M^{\prime}$ is a Lie-module homomorphism, then
$M/\mathrm{ker}\,f$ is isomorphic to $\mathrm{im}\,f.$

Although by a Lie-module $M$, we always mean $M$ as a module over the Lie
algebra $\mathfrak{g},$ but it is sometimes convenient to regard $M$ as a Lie-
module over a different Lie-algebra, and to this end we have the following
result similar to rings (cf. Jacobson [2]).

######  Proposition 2.4.

Suppose $\mathfrak{g}$ is a Lie algebra and $\mathfrak{a}$ is an ideal of
$\mathfrak{g}.$ If $M$ is a Lie-module over $\mathfrak{g}/\mathfrak{a}$ then
$M$ can be considered as a Lie-module over $\mathfrak{g}$. Conversely, if $M$
is a Lie-module over $\mathfrak{g}$ and $\mathfrak{a}$ is contained in the
kernel of the Lie-realization then $M$ can be regarded as a Lie-module over
$\mathfrak{g}/\mathfrak{a}$. In either case the Lie-submodules of $M$ as Lie-
module over $\mathfrak{g}$ are the same as those of $M$ as Lie-module over
$\mathfrak{g}/\mathfrak{a}$ and if $\mathfrak{d}$ is the kernel of the Lie-
realization for $\mathfrak{g}$ then $\mathfrak{d}/\mathfrak{a}$ is that for
$\mathfrak{g}/\mathfrak{a}$.

###### Proof.

Consider the following diagrams:

$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}_{\mathfrak{g}}}$$\scriptstyle{\pi_{\mathfrak{a}}}$$\textstyle{\mathfrak{g}/\mathfrak{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{R}_{\mathfrak{g}/\mathfrak{a}}}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x+\mathfrak{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{End}_{\mathfrak{g}}(M)}$$\textstyle{\mathrm{End}_{\mathfrak{g}/\mathfrak{a}}(M)}$$\textstyle{\phi_{x}}$$\textstyle{\phi_{x+\mathfrak{a}}}$

Suppose $\mathfrak{a}$ is an ideal of $\mathfrak{g}$ and $M$ is a Lie-module
over $\mathfrak{g}/\mathfrak{a}$. The above diagrams indicates that if we
define $xm:=(x+\mathfrak{a})m$ for $m\in M$, $x\in\mathfrak{g},$ then we
obtain $M$ as a Lie-module over $\mathfrak{g}.$ Conversely, if $M$ is a Lie-
module over $\mathfrak{g}$ and if
$\mathfrak{a}\subseteq\mathrm{ker}\,\mathcal{R}_{\mathfrak{g}},$ then
$\mathcal{R}_{\mathfrak{g}}(\mathfrak{a})(m)=\phi_{\mathfrak{a}}(m)=\mathfrak{a}m=0.$
Hence we can define $(x+\mathfrak{a})m:=xm$ and this makes $M$ a Lie-module
over $\mathfrak{g}/\mathfrak{a}$. ∎

A Lie-module $M$ is called _cyclic_ if $M=\\{xm\mid x\in\mathfrak{g}\\}$ for
some $m\in M$. Here is one way to construct cyclic Lie-submodules of $M$. We
also need this construction to characterize simple Lie-modules (see
Proposition 2.6).

######  Lemma 2.5.

If $\mathfrak{a}$ is an ideal of $\mathfrak{g}$ and $m\in M$, then
$\mathfrak{a}m=\\{am\mid a\in\mathfrak{a},m\in M\\}$ is a Lie-submodule.

###### Proof.

Note that $am-a^{\prime}m=(a-a^{\prime})m\in\mathfrak{a}m$ and
$x(am)=[x,a]m+a(xm)\in\mathfrak{a}m$ for all $a,a^{\prime}\in\mathfrak{a}$ and
$x\in\mathfrak{g},$ ∎

A Lie-module $M$ is called _simple_ or _irreducible_ if $\mathfrak{g}M\neq 0$
and $M$ has no submodules other than $0$ and $M.$ A characterization of simple
Lie-modules is given in the following

######  Proposition 2.6.

A nonzero Lie-module $M$ is simple if and only if $M=\mathfrak{g}m$ for every
nonzero $m\in M$.

###### Proof.

Suppose $M$ is a nonzero simple Lie-module. Given a nonzero $m\in M,$ since by
Lemma 2.5, $\mathfrak{g}m$ is a Lie-submodule of $M$ for every $m\in M$,
either $\mathfrak{g}m=0$ or $\mathfrak{g}m=M.$ Since $\mathfrak{g}M\neq 0,$ we
must have $\mathfrak{g}m=M.$ Conversely, let $M=\mathfrak{g}m$ for every
nonzero $m$ in $M$ and $S$ be a nonzero Lie-submodule of $M$. Then
$M=\mathfrak{g}s\subseteq S$ for every nonzero $s\in S$, and that implies
$M=S.$ Hence $M$ is simple. ∎

A nonzero ideal $\mathfrak{a}$ of a Lie algebra $\mathfrak{g}$ is called
_minimal_ if $\mathfrak{a}$ does not properly contain any other nonzero ideals
of $\mathfrak{g},$ whereas a proper ideal of $\mathfrak{g}$ is called
_maximal_ if it is not properly contained in another ideal of $\mathfrak{g}.$
The next two propositions respectively show the relationships of minimal and
maximal ideals (of a Lie algebra) with Lie-modules.

######  Proposition 2.7.

If $\mathfrak{g}$ is a simple Lie algebra with a minimal ideal $\mathfrak{a},$
then every simple Lie-module is isomorphic to $\mathfrak{a}.$

###### Proof.

Suppose $M$ is a simple Lie-module. Since $\mathfrak{g}$ is simple,
$\mathfrak{a}\mathfrak{g}=\mathfrak{g}.$ Moreover, we have

$\mathfrak{a}M\supseteq\mathfrak{a}(\mathfrak{g}M)=(\mathfrak{a}\mathfrak{g})M+\mathfrak{g}(\mathfrak{a}M)=\mathfrak{g}M+\mathfrak{g}(\mathfrak{a}M)=\mathfrak{g}M\neq
0.$

This implies there exists $m$ in $M$ such that $\mathfrak{a}m\neq 0.$ But $M$
is simple and so we must have $\mathfrak{a}m=M.$ Then evidently the map
$f\colon\mathfrak{a}\to M$ defined by $f(a)=am$ is a surjective Lie-module
homomorphism. Since $\mathrm{ker}(f)\subsetneq\mathfrak{a}$ and $\mathfrak{g}$
is simple, we must have $\mathrm{ker}(f)=0.$ This proves $f$ is an isomorphism
as required.∎

######  Proposition 2.8.

Every simple Lie-module is isomorphic to $\mathfrak{g}/\mathfrak{m}$ for some
maximal ideal $\mathfrak{m}$ of $\mathfrak{g}$. Conversely, if $\mathfrak{m}$
is a maximal ideal of $\mathfrak{g}$ such that
$\mathfrak{g}^{2}\nsubseteq\mathfrak{m}$, then $\mathfrak{g}/\mathfrak{m}$ is
a simple Lie-module.

###### Proof.

Suppose $M$ is a simple Lie-module and $m$ is a nonzero element in $M$. This
implies $\mathfrak{g}m=M$ and hence the map $f\colon\mathfrak{g}\to M$ defined
by $f(x)=xm$ is a surjective Lie-module homomorphism. If $\mathfrak{m}$ is the
kernel of $f$, then by Proposition 2.3, $\mathfrak{g}/\mathfrak{m}$ is
isomorphic to $M$. We need to show that $\mathfrak{m}$ is a maximal ideal of
$\mathfrak{g}.$ Let $\mathfrak{m}\subseteq\mathfrak{a}\subseteq\mathfrak{g}$
for some ideal $\mathfrak{a}$ of $\mathfrak{g}.$ Then
$\mathfrak{a}/\mathfrak{m}$ is a Lie-submodule of $\mathfrak{g}/\mathfrak{m}$
and hence $\mathfrak{a}/\mathfrak{m}$ is isomorphic to $M$. Since $M$ is
simple, $\mathfrak{a}/\mathfrak{m}$ must be equal to either $0$ or
$\mathfrak{g}/\mathfrak{m}$, which respectively implies either
$\mathfrak{a}=\mathfrak{m}$ or $\mathfrak{a}=\mathfrak{g}.$ Therefore,
$\mathfrak{m}$ is maximal. For the converse, let $\mathfrak{m}$ is a maximal
ideal of $\mathfrak{g}$. Let $S$ be a Lie-submodule of
$\mathfrak{g}/\mathfrak{m}.$ Since $\mathfrak{a}=\\{x\in\mathfrak{g}\mid
x+\mathfrak{m}\in S\\}$ is an ideal of $\mathfrak{g}$ containing
$\mathfrak{m},$ we must have either $\mathfrak{a}=\mathfrak{m}$ or
$\mathfrak{a}=\mathfrak{g},$ and that respectively implies either $S=0$ or
$S=\mathfrak{g}/\mathfrak{m}.$ Hence, if
$\mathfrak{g}^{2}\nsubseteq\mathfrak{m}$, then $\mathfrak{g}/\mathfrak{m}$ is
simple. ∎

The annihilator of a Lie-module will now lead us towards the definition of a
primitive Lie algebra and the (_left_) _annihilator_ of a Lie-module $M$ is
defined by $\mathrm{Ann}_{\mathfrak{g}}(M)=\\{x\in\mathfrak{g}\mid
xm=0\;\;\text{for all}\;\;m\in M\\}.$ When $M=\\{m\\},$ we write
$\mathrm{Ann}_{\mathfrak{g}}(m)$ for $\mathrm{Ann}_{\mathfrak{g}}(\\{m\\})$.
If $\mathrm{Ann}_{\mathfrak{g}}(M)=0$ then $M$ is said to be a _faithful_ Lie-
module. A Lie algebra $\mathfrak{g}$ is called _primitive_ if $\mathfrak{g}$
has a faithful simple Lie-module. There is another notion of primitive Lie
algebras introduced by López in [7], which we are not going to discuss here.

######  Example 2.9.

_Here are some examples of primitive Lie algebras in our sense. For a more
complete discussion see[5]._

  1. (i)

_A nilpotent finite-dimensional Lie algebra of degree 2 can be primitive._

  2. (ii)

_Every radical Lie algebra is primitive._

  3. (iii)

_If an Artinian Lie algebra over a field has a unique minimal ideal, then it
is primitive._

  4. (iv)

_The semisimple finite-dimensional Lie algebras over an algebraically closed
field of characteristic zero are primitive._

  5. (v)

_If a semisimple finite-dimensional Lie algebra over a field of characteristic
zero is the direct sum of central simple Lie algebras, then it is primitive._

  6. (vi)

_Every free Lie algebra is primitive._

  7. (vii)

_All commutative Lie algebras over the fields $\mathds{Z}_{p}$, where $p$ is
prime, and over the field $\mathds{Q}$ are also primitive._

  8. (viii)

_Infinite-dimensional commutative Lie algebras are primitive over any field._

  9. (ix)

_If the enveloping algebra $U(\mathfrak{g})$ is primitive, then the Lie
algebra $\mathfrak{g}$ is also primitive. However, the converse is not true in
general._

  10. (x)

_There is a non-Artinian noncommutative primitive Lie algebra._

An ideal $\mathfrak{p}$ of a Lie algebra $\mathfrak{g}$ is called _primitive_
if $\mathfrak{g}/\mathfrak{p}$ is a primitive Lie algebra. In general, the two
notions: left primitive and right primitive are not equivalent (see Bergman
[1] for an example). Since every ideal in a Lie algebra is two-sided, we do
not have such complexity.

######  Lemma 2.10.

An annihilator $\mathrm{Ann}_{\mathfrak{g}}(M)$ is an ideal of $\mathfrak{g}$.

###### Proof.

For all $x,y\in\mathrm{Ann}_{\mathfrak{g}}(M)$ and for all
$\alpha,\beta\in\mathds{k},$ $(\alpha x-\beta y)m=\alpha(xm)-\beta(ym)=0$ for
all $m\in M$. Also, for all $z\in\mathfrak{g}$,
$x\in\mathrm{Ann}_{\mathfrak{g}}(M),$ $[z,x]m=z(xm)-x(zm)=0$ for all $m\in M.$
∎

Since now we know that an annihilator of a Lie-module $M$ is an ideal of
$\mathfrak{g},$ we obtain an equivalent (but an easy formulation) of primitive
ideals of Lie algebras in the following

######  Proposition 2.11.

An ideal $\mathfrak{p}$ of a Lie algebra $\mathfrak{g}$ is primitive if and
only if $\mathfrak{p}$ is the annihilator of a simple Lie-module $M$.

###### Proof.

Suppose $\mathfrak{p}$ is a primitive ideal of a Lie algebra $\mathfrak{g}.$
Then $\mathfrak{g}/\mathfrak{p}$ is a primitive ring and hence has a faithful
simple Lie-module $M$. Then by Proposition 2.4, $\mathfrak{p}$ is an
annihilator of $\mathfrak{g}/\mathfrak{p}.$ Conversely, if $\mathfrak{a}$ is
an annihilator of a simple Lie-module $M$, then $M$ is obviously a faithful
simple Lie-module over $\mathfrak{g}/\mathfrak{p},$ that is, $\mathfrak{p}$ is
a primitive ideal. ∎

In Proposition 2.6 we have seen a characterization of simple Lie-module in
terms of cyclic Lie-submodules. Now we shall (partially) characterize cyclic
Lie-modules in terms of right regular ideals of Lie algebras, originally
introduced by Segal [11] for rings. An ideal $\mathfrak{a}$ of a Lie algebra
$\mathfrak{g}$ is called _right regular_ if there exists an element
$e\in\mathfrak{g}$ such that $x-xe\in\mathfrak{a}$ for all $x\in\mathfrak{g}.$

######  Proposition 2.12.

If $M$ is a cyclic Lie-module then $M$ is isomorphic to
$\mathfrak{g}/\mathfrak{a}$ for some primitive ideal $\mathfrak{a}$ of
$\mathfrak{g}.$ Conversely, if a Lie-module $M$ is isomorphic to
$\mathfrak{g}/\mathfrak{a}$ for some right regular ideal $\mathfrak{a}$ of
$\mathfrak{g},$ then $M$ is cyclic and $\mathfrak{a}$ is a primitive ideal of
$\mathfrak{g}.$

###### Proof.

Suppose $M=\\{xm\mid x\in\mathfrak{g}\\}$ is a Lie-module with generator $m.$
Consider the map $f\colon\mathfrak{g}\to M$ defined by $f(x)=xm$. From (1), it
follows that $f$ is a surjective Lie-module homomorphism with kernel
$\mathrm{Ann}_{\mathfrak{g}}(m),$ and hence
$\mathfrak{g}/\mathrm{Ann}_{\mathfrak{g}}(m)$ is isomorphic to $M$ by
Proposition 2.3. Conversely, let $M=\mathfrak{g}/\mathfrak{a}$ for some right
regular ideal $\mathfrak{a}$ of $\mathfrak{g}.$ Since
$x+\mathfrak{a}=x(e+\mathfrak{a}),$ $M$ is definitely a cyclic module with
generator $e+\mathfrak{a}$. Furthermore,
$\mathrm{Ann}_{\mathfrak{g}}(e+\mathfrak{a})=\mathfrak{a}.$ ∎

For an associative algebra $A$, we have above the complete characterization,
that is, if $M$ is a cyclic module over $A$, then $M$ is isomorphic to
$A/\mathfrak{a}$ for some right regular ideal $\mathfrak{a}$ of $A.$ In
Proposition 2.12, we can only have $\mathfrak{a}$ as a primitive ideal,
whereas the converse statement is independent of associativity of the
algebraic structure.

######  Proposition 2.13.

Every maximal ideal of a Lie algebra is a primitive ideal.

###### Proof.

Let $\mathfrak{m}$ be a maximal ideal of $\mathfrak{g}.$ Then
$\mathfrak{g}/\mathfrak{m}$ is a simple module over $\mathfrak{g}$ and

$\mathfrak{m}=\mathrm{ker}(\mathfrak{g}\to\mathfrak{g}/\mathfrak{m})=\mathrm{Ann}_{\mathfrak{g}}(\mathfrak{g}/\mathfrak{m}).\hfill\qed$

Another relation between primitive and maximal ideals of a Lie algebra is
given in the following proposition, and the proof essentially follows from
Lemma 2.8.

######  Proposition 2.14.

If $\mathfrak{p}$ is a primitive ideal of a Lie algebra $\mathfrak{g}$, then
there exists a maximal ideal $\mathfrak{m}$ of $\mathfrak{g}$ such that
$\mathfrak{p}=\\{x\in\mathfrak{g}\mid x\mathfrak{g}\subseteq\mathfrak{m}\\}$.
Conversely, if $\mathfrak{m}$ is a maximal ideal of $\mathfrak{g}$ and
$\mathfrak{g}^{\scriptscriptstyle 2}\nsubseteq\mathfrak{m}$, then
$\mathfrak{p}=\\{x\in\mathfrak{g}\mid x\mathfrak{g}\subseteq\mathfrak{m}\\}$
is a primitive ideal of $\mathfrak{g}.$

We say an ideal to be _semiprimitive_ if it is an intersection of primitive
ideals. The intersection of all primitive ideals of a Lie algebra
$\mathfrak{g}$ is called the _Jacobson radical_ of $\mathfrak{g}$ and we
denote it by $\mathrm{J}(\mathfrak{g})$. A Lie algebra $\mathfrak{g}$ is
called _semiprimitive_ if $\mathrm{J}(\mathfrak{g})$ is the zero ideal.
Kucherov, Pikhtilkova, and Pikhtilkov [5, Lemma 1] proved that
$\mathrm{J}(\mathfrak{g})$ is trivial for every Lie algebra $\mathfrak{g},$
and hence we have the following striking property.

######  Proposition 2.15.

Every Lie algebra is semiprimitive.

It is well known that a nonzero ring $R$ is semiprimitive if and only if $R$
is isomorphic to a subdirect product of primitive rings. Thanks to Proposition
2.15, the following stronger result holds for a Lie algebra.

######  Theorem 2.16.

Every nonzero Lie algebra is isomorphic to a subdirect product of primitive
Lie algebras.

###### Proof.

Suppose $\\{\mathfrak{p}_{\lambda}\mid\lambda\in\Lambda\\}$ is the collection
of all primitive ideals of a Lie algebra $\mathfrak{g}.$ By Proposition 2.15,
$\bigcap_{\lambda\in\Lambda}\mathfrak{p}_{\lambda}=0$ and by Lemma 2.11, each
$\mathfrak{g}/\mathfrak{p}_{\lambda}$ is a primitive Lie algebra. Therefore,
the Lie algebra homomorphism
$\phi\colon\mathfrak{g}\to\prod_{\lambda\in\Lambda}\mathfrak{g}/\mathfrak{p}_{\lambda}$
is injective. Also, the image of $\phi$ is a subdirect product of the
collection $\\{\mathfrak{p}_{\lambda}\mid\lambda\in\Lambda\\}$. ∎

As part of Jacobson program, _Jacobson Density Theorem_ is one of the corner
stone of ring theory. We shall now obtain a generalized version of it for Lie
algebras. Our formulation is an adaptation of the corresponding ring-theoretic
counterpart proved in Lang [6].

######  Theorem 2.17 (Density Theorem).

Let $M$ be a semisimple Lie-module and suppose
$[\mathfrak{g},\mathfrak{g}]\subseteq\mathrm{Ann}_{\mathfrak{g}}(M).$ Let
$f\in\mathrm{End}_{\mathrm{End}_{\mathfrak{g}}(M)}(M).$ If
$x_{1},\ldots,x_{n}\in M$ then there exists an element $\alpha$ in
$\mathfrak{g}$ such that $\alpha x_{i}=f(x_{i})$ for $1\leqslant i\leqslant
n.$

###### Proof.

Let $\mathfrak{g}^{\prime}:=\mathrm{End}_{\mathfrak{g}}(M).$ From Lemma 2.1 it
is clear that $\mathfrak{g}^{\prime}$ is a Lie-algebra. Next we claim that $M$
is also a Lie-module over $\mathfrak{g}^{\prime}$. Indeed, we define the
action $\mathfrak{g}^{\prime}\times M\to M$ by $(\phi,x)\mapsto\phi(x)$ for
$\phi\in\mathfrak{g}^{\prime}$ and $x\in M.$ Now each $\alpha\in\mathfrak{g}$
induces a $\mathfrak{g}^{\prime}$-homomorphism $f_{\alpha}\colon M\to M$ by
the map $f_{\alpha}(x):=\alpha x.$ Indeed, for any $\beta\in\mathfrak{g},$

$f_{\alpha}(\beta x)=\alpha(\beta x)=\beta(\alpha x)=\beta f_{\alpha}(x),$

where the second equality follows from our hypothesis
$[\mathfrak{g},\mathfrak{g}]\subseteq\mathrm{Ann}_{\mathfrak{g}}(M).$ Suppose
$\mathfrak{g}^{\prime\prime}:=\mathrm{End}_{\mathfrak{g}^{\prime}}(M).$ Then
the map $\alpha\mapsto f_{\alpha}$ induces a Lie-algebra homomorphism
$\mathfrak{g}\to\mathfrak{g}^{\prime\prime}.$

To obtain our desired claim, first we prove it for $n=1.$ Since $M$ is
semisimple, $M$ can be represented as $M=\mathfrak{g}x_{1}\oplus N$, where $N$
is a Lie-submodule of $M$, and from Lemma 2.5 it follows that $\mathfrak{g}x$
is also a Lie-submodule of $M$. Note that the projection map $\pi\colon
M\to\mathfrak{g}x_{1}$ belongs to $\mathfrak{g}^{\prime}$ and hence

$f(x_{1})=f(\pi x_{1})=\pi f(x_{1}),$

where the second equality follows from our hypothesis. This implies
$f(x_{1})\in\mathfrak{g}x_{1}$.

To obtain the result for an arbitrary $n,$ consider the map

$\bar{f}:=(f,\ldots,f)\colon M^{n}\to M^{n}$

and let $\mathfrak{g}^{\prime}_{n}:=\mathrm{End}_{\mathfrak{g}}(M^{n}).$ Then
$\bar{f}\in\mathrm{End}_{\mathfrak{g}_{n}^{\prime}}(M^{n}).$ By the above,
there exists $\alpha\in\mathfrak{g}$ such that

$\alpha(x_{1},\ldots,x_{n})=\bar{f}(x_{1},\ldots,x_{n})=(f(x_{1}),\ldots,f(x_{n})),$

and this completes the proof. ∎

##  3\. Prime and semiprime Lie algebras

As per our plan, in Section 2, we have talked about (i) – (iii), and (vi)
types of Lie algebras from our list (see Introduction 1). Now we shall discuss
about (iv) and (v). We shall start with the definitions of these types of Lie
algebras. Following Kawamoto [3], a Lie algebra $\mathfrak{g}$ is called
_prime_ if $[\mathfrak{a},\mathfrak{b}]=0$ implies either $\mathfrak{a}=0$ or
$\mathfrak{b}=0$ for all ideals $\mathfrak{a},\mathfrak{b}$ of $\mathfrak{g}$.
A Lie algebra $\mathfrak{g}$ is called _semiprime_ if
$[\mathfrak{a},\mathfrak{a}]=0$ implies $\mathfrak{a}=0$ for all ideals
$\mathfrak{a}$ of $\mathfrak{g}$.

The notion of a commutator and an associator are going to play crucial role in
determining properties of these types of Lie algebras. If $x,y$ are elements
of a Lie algebra $\mathfrak{g}$, the _commutator_ of $x$ and $y$ is defined by
$\langle x,y\rangle=[x,y]-[y,x].$ As commutator is a measurement of lack of
commutativity, the notion of the _associator_ $(x,y,z)=[[x,y],z]-[x,[y,z]]$
for any three elements $x,y,z\in\mathfrak{g}$ measures lack of associativity
in $\mathfrak{g}$. Notice that $(x,y,z)=[y,[z,x]]$ for all
$x,y,z\in\mathfrak{g}.$ Since every Lie algebra $\mathfrak{g}$ satisfies the
identities:

  1. (i)

$(x,x,x)=0;$

  2. (ii)

$\alpha_{1}(x,y,z)+\alpha_{2}(y,z,x)+\alpha_{3}(z,x,y)+\alpha_{4}(x,z,y)+\alpha_{5}(z,y,x)+\alpha_{6}(y,x,z)=0$

for fixed $\alpha_{i}\in\mathds{k}$ and for all $x,y,z\in\mathfrak{g},$ in the
sense of Kleinfeld [4], every Lie algebra $\mathfrak{g}$ is an _associator
dependent algebra_. The proposition below gives some additional properties of
commutators and associators of Lie algebras.

######  Proposition 3.1.

Suppose $\mathfrak{g}$ is a Lie algebra and $a,x,y,z\in\mathfrak{g}$. Then

  1. (i)

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

  2. (ii)

$[a,(x,y,z)]+[(a,x,y),z]=([a,x],y,z)-(a,[x,y],z)+(a,x,[y,z]).$

  3. (iii)

If the characteristic of $\mathfrak{g}$ is $2$, then $\langle
x,(y,x,y)\rangle=0$.

###### Proof.

(i). We notice that

$\displaystyle[x,\langle y,z\rangle]+[\langle x,z\rangle,y]$
$\displaystyle=[x,[y,z]-[z,y]]+[[x,z]-[z,x],y]$
$\displaystyle=[x,[y,z]]-[x,[z,y]]+[[x,z],y]-[[z,x],y]$
$\displaystyle=[x,[y,z]]+[y,[z,x]]+[x,[y,z]]+[y,[z,x]]$
$\displaystyle=-[z,[x,y]]+[[x,y],z]$ $\displaystyle=\langle[x,y],z\rangle.$

(ii). The left hand side is:

$\displaystyle[a,(x,y,z)]+[(a,x,y),z]$
$\displaystyle=[a,[[x,y],z]]-[a,[x,[y,z]]]$
$\displaystyle\qquad\quad+[[[a,x],y],z]-[[a,[x,y]],z],$

whereas, the right hand side is:

$\displaystyle([a,x],y,z)-(a,[x,y],z)+(a,x,[y,z])$
$\displaystyle=[[[a,x],y],z]-[[a,x],[y,z]]$
$\displaystyle\quad-[[a,[x,y]],z]+[a,[[[x,y],z]]$
$\displaystyle\quad\quad+[[a,x],[y,z]]-[a,[x,[y,z]]].$

(iii). We observe that

$\displaystyle\langle x,(y,x,y)\rangle$
$\displaystyle=[x,(y,x,y)]-[(y,x,y),x]$
$\displaystyle=[x,[[y,x],y]-[y,[x,y]]]-[[[y,x],y]-[y,[x,y]],x]$
$\displaystyle=[x,[[y,x],y]]-[x,[y,[x,y]]]-[[[y,x],y],x]-[[y,[x,y]],x]$
$\displaystyle=[x,[[y,x],y]]-[x,[y,[x,y]]]+[x,[[y,x],y]]+[x,[y,[x,y]]]$
$\displaystyle=2[x,[[y,x],y]]$ $\displaystyle=0.\qed$

Note that in proving (i) we have used the Jacobi identity. A general
formulation of (i) for an arbitrary nonassociative structure can be found in
Slater [12]. The identity (ii) is known as Teichmüller identity and it holds
for any non-associative algebras (see Schafer [10]).

The _nucleus_ $\mathcal{N}(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ is
the set of elements of $\mathfrak{g}$ which associate with every pair of
elements in $\mathfrak{g}$, that is,
$\mathcal{N}(\mathfrak{g})=\\{g\in\mathfrak{g}\mid(g,x,y)=(x,g,y)=(x,y,g)=0\;\;\text{for
all}\;\;x,y\in\mathfrak{g}\\}.$

######  Lemma 3.2.

If $x,y,z\in\mathfrak{g}$ and $n\in\mathcal{N}(\mathfrak{g}),$ then

  1. (i)

$(x,y,[z,n])=[(x,y,z),n];$

  2. (ii)

$([n,x],y,z)=[n,(x,y,z)];$

  3. (iii)

$(x,\mathfrak{g},x)\subseteq\mathcal{N}(\mathfrak{g}).$

###### Proof.

For (i), we see

$\displaystyle(x,y,[z,n])$ $\displaystyle=[[x,y],[z,n]]-[x,[y,[z,n]]]$
$\displaystyle=[[[x,y],z],n]-[x,[[y,z],n]]$
$\displaystyle=[[[x,y],z],n]-[[x,[y,z]],n]$ $\displaystyle=[(x,y,z),n].$

(ii). We have

$\displaystyle([n,x],y,z)$ $\displaystyle=[[[n,x],y],z]-[[n,x],[y,z]]$
$\displaystyle=[n,[[x,y],z]]-[n,[x,[y,z]]]$ $\displaystyle=[n,(x,y,z)].$

(iii). We notice that $(x,y,x)=[y,[x,x]]=[y,0]=0\in\mathcal{N}(\mathfrak{g})$
for all $x,y\in\mathfrak{g}.$ ∎

The _associator ideal_ $\mathfrak{a}_{\mathfrak{g}}$ of a Lie algebra
$\mathfrak{g}$ is the ideal generated by all the associators. A _nuclear
ideal_ $\mathfrak{n}_{\mathfrak{g}}$ of a Lie algebra $\mathfrak{g}$ is a
nonzero ideal of $\mathfrak{g}$ such that
$\mathfrak{n}_{\mathfrak{g}}\subseteq\mathcal{N}(\mathfrak{g}).$ The _maximum
nuclear ideal_ $\mathfrak{m}_{\mathfrak{g}}$ of $\mathfrak{g}$ is the sum of
all nuclear ideals of $\mathfrak{g}$.

######  Theorem 3.3.

If $\mathfrak{g}$ is a prime Lie algebra, then either $\mathfrak{g}$ is
associative or $\mathfrak{m}_{\mathfrak{g}}$ is trivial.

###### Proof.

We observe that $\mathfrak{a}_{\mathfrak{g}}\mathfrak{m}_{\mathfrak{g}}$ and
$\mathfrak{m}_{\mathfrak{g}}\mathfrak{a}_{\mathfrak{g}}$ are ideals, and by
(i) and (ii) of Lemma 3.2, we have
$\mathfrak{a}_{\mathfrak{g}}\mathfrak{m}_{\mathfrak{g}}=0=\mathfrak{m}_{\mathfrak{g}}\mathfrak{a}_{\mathfrak{g}}.$
Since $\mathfrak{g}$ is prime, we must have either
$\mathfrak{a}_{\mathfrak{g}}=0$ (i.e., $\mathfrak{g}$ is associative) or
$\mathfrak{m}_{\mathfrak{g}}=0.$ ∎

######  Corollary 3.4.

A nonassociative prime Lie algebra does not have any nuclear ideals.

A Lie algebra is called _purely nonassociative_ if
$\mathfrak{m}_{\mathfrak{g}}=0.$ The _centre_ $\mathcal{C}(\mathfrak{g})$ of a
Lie algebra $\mathfrak{g}$ is the set
$\mathcal{C}(\mathfrak{g})=\\{c\in\mathcal{N}(\mathfrak{g})\mid\langle
g,c\rangle=0\;\;\text{for all}\;\;g\in\mathfrak{g}\\}.$ It is easy to see that
$\mathcal{C}(\mathfrak{g})$ is a commutative associative Lie subalgebra of
$\mathfrak{g}.$

######  Proposition 3.5.

If $\mathfrak{g}$ is a prime Lie algebra, then the set of annihilators of a
nonzero element in the centre of $\mathfrak{g}$ is zero.

###### Proof.

Suppose $c$ is a nonzero element in $\mathcal{C}(\mathfrak{g})$. Then
$\mathrm{Ann}_{\mathfrak{g}}(c)$ annihilates the ideal generated by $c$. Since
$\mathfrak{g}$ is prime and $c$ is nonzero, $\mathrm{Ann}_{\mathfrak{g}}(c)$
must be zero. ∎

The intersection of all prime ideals of a Lie algebra is called the _prime
radical_ or (_Baer radical_) of the algebra. The following result
characterises semiprime Lie algebras in terms of prime radicals. Pikhtilkov
[8, Theorem 2.3.2] gave the following characterization of semiprime Lie
algebras.

######  Theorem 3.6.

A Lie algebra is semiprime if and only if its prime radical be zero.

## Acknowledgement

I am indebted to Louis Rowen for a number of valuable suggestions that led to
this problem of investigation.

## References

  * [1] G. M. Bergman, A ring primitive on the right but not on the left, Proc. Amer. Math. Soc., 15 (1964), 473–475.
  * [2] N. Jacobson, Structure of rings, Amer. Math. Soc. Colloquium Publications, vol. 37, Providence, 1956.
  * [3] N. Kawamoto, On prime ideals of Lie algebras, Hiroshima Math. J., 4 (1974) 679–684.
  * [4] E. Kleinfeld, Associator dependent rings, Arch. Math. 13 (1962), 203–212.
  * [5] A. A. Kucherov, O. A. Pikhtilkova, and S. A. Pikhtilkov, On primitive Lie algebras, J. Math. Sci., 186(4) (2012) 651–654.
  * [6] S. Lang, Algebra, 3rd ed., Springer, 2002.
  * [7] A. F. López, Jordan structures in Lie algebras, AMS, 2019.
  * [8] S. A. Pikhtilkov, The structural theory of special Lie algebras, Izd. Tul’skogo Gos. Ped. Univ., Tula, 2005.
  * [9] L. H. Rowen, Ring theory (volume I), Academic Press, Inc., 1988.
  * [10] R. D. Schafer, An introduction to nonassociative algebras, Academic Press Inc., New York, 1966.
  * [11] I. E. Segal, The group algebra of a locally compact group, Trans. Amer. Math. Soc., 61(1) (1947), 69–105.
  * [12] M. Slater, Nucleus and center in alternative rings, J. Alg., 7 (1967) 372–388.