# Full-Cycle Simulations of The Fermilab Booster ††thanks: This manuscript has
been authored by Fermi Research Alliance, LLC under Contract No. DE-
AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office
of High Energy Physics.

J.-F. Ostiguy<EMAIL_ADDRESS>C. M. Bhat
Fermilab  Batavia  IL  USA

###### Abstract

The Proton Improvement Plan phase II (PIP-II) project currently under
construction at FNAL will replace the existing 400 MeV normal conducting linac
with a new 800 MeV superconducting linac. The beam power in the downstream
rapid-cycling Booster synchrotron will be doubled by raising the machine cycle
frequency from 15 to 20 Hz and by increasing the injected beam intensity by a
factor 1.5. This has to be accomplished without raising uncontrolled losses
beyond the administrative limit of 500 W. In addition, slip-stacking
efficiency in the Recycler, the next machine in the accelerator chain, sets an
upper limit on the longitudinal emittance of the beam delivered by the
Booster. As part of an effort to better understand potential losses and
emittance blow-up in the Booster, we have been conducting full cycle 6D
simulations using the code PyORBIT. The simulations include space charge, wall
impedance effects and transition crossing. In this paper, we discuss our
experience with the code and present representative results for possible
operational scenarios.

##  1 INTRODUCTION

The Proton Improvement Plan phase II (PIP-II) will double the beam power
available for neutrino production in support of the laboratory flagship DUNE
experiment. In addition to a new superconducting linac, modifications to the
existing complex will be needed. For safe operation and on-hands maintenance,
a rule of thumb is that uncontrolled particle losses in circular machines in
the accelerator chain should not exceed 1 W/m. Since the Booster already
operates near this limit, an increase in beam power during the PIP-II era
implies a reduction in particle losses while preserving beam stability and
limiting emittance blowup. Therefore, we have been conducting full 6D particle
tracking simulations in combination with extensive experimental
investigations. In this paper we focus primarily on excessive longitudinal
emittance blowup at transition, an issue of concern for the Booster. During
injection, both the transverse and longitudinal distributions will be painted
in phase space to enhance the charge spatial density uniformity, thereby
reducing space charge tune shift and peak longitudinal line density. At the
start of the acceleration cycle, the painted emittance is 0.06 eV-s (95%). To
prevent excessive losses during slip-stacking at injection in the Recycler
ring, the longitudinal emittance of the beam transferred out from the Booster
should not exceed 0.1 eV-s (95%).

##  2 BOOSTER

The PIP-II era Booster will accelerate protons from 0.8 to 8 GeV. Its lattice
comprises 24 cells with basic structure OFDOODF, where F and D respectively
represent focusing and defocusing gradient bending magnets. The acceleration
voltage is provided by a total of 22 cavities. To prevent multipactoring, they
are operated at full voltage in two paraphased groups; the net ring voltage is
controlled by altering the phase between the two groups. The maximum total
available ring voltage is 1.16 MV.

The current injection scheme is based on charge-exchange multi-turn injection
followed by rf adiabatic capture. In contrast, PIP-II will transfer linac
bunches directly into already formed stationary rf buckets. Because the linac
and ring frequencies are not commensurable, bunches that would not land inside
ring rf buckets will be removed in the MEBT using a fast bunch-by-bunch
kicker. Injection into a stationary bucket allows for transverse and
longitudinal phase space painting to reduce the incoherent tune shift and the
peak longitudinal density by making the charge spatial density more uniform.

A distintive feature of the Fermilab Booster is the absence of a metallic
vacuum chamber; instead, the entire magnet aperture region is under vacuum.
This approach side-steps issues with magnetic shielding and magnetic field
perturbations caused by eddy currents induced in the chamber wall. A downside
is that the beam is directly exposed to the magnet laminations resulting in an
unusually lossy wall impedance. This impedance causes significant distortion
of the phase space at transition.

The Booster crosses transition at $\gamma_{t}\simeq 5.45$. The non-adiabatic
dynamics at transition results in an increase in longitudinal emittance; due
to space charge and impedance, a portion of this increase is intensity
dependent. Keeping losses at an acceptable level during slip-stacking in the
Recycler ring imposes a limit of 0.1 eV-s on the final 95% longitudinal
emittance. Currently, there is no dedicated $\gamma_{t}$ jump system; however
emittance blowup can be mitigated using an active quadrupole mode damper. The
Booster is equipped with 48 quadrupole corrector magnets and investigations
are ongoing to determine to what extent these could be used to emulate a
$\gamma_{t}$ jump system.

##  3 SIMULATIONS

Although a number tracking codes capable of handling space charge have been
developed, few of them are adequate to simulate a rapid-cycling synchrotron.
In addition, many codes were developed decades ago and do not support parallel
execution. Other important considerations are the existence of an active user
community and the availability of the source code under non-restrictive terms.
All these factors led us to select PyORBIT, a code developed at ORNL [1]
originally to model the the SNS accumulation ring. While PyORBIT supports a
variety of space charge solvers, the bunch longitudinal to transverse size
ratio in the Booster is high $(\gg 10)$ making it appropriate to employ a so-
called 2.5D solver where the transverse and longitudinal space charge forces
are computed separately. Longitudinally, the force is obtained from the
derivative of the longitudinal line density while transversely, it is obtained
by solving Poisson’s equation in the beam frame using FFTs and weighting the
resultant field pattern by the local longitudinal line density. The 2.5D
approximation dramatically reduces computational load. To fix ideas, on a
modern small computer cluster, simulating a typical full 25 ms booster
acceleration cycle (15000 turns) involving 50 to 100k particles) on 20 CPU
cores typically takes 6 hours.

Figure 1: Longitudinal phase space distributions of a bunch at injection and
extraction.

We encountered a number of issues when attempting to simulate a full cycle
through transition. In contrast with a longitudinal only code, the precise
value of $\gamma_{t}$, the dependence of the the momentum compaction factor on
$\Delta p/p$ are not direct input parameters; rather, they are implicitly
determined by the lattice. Since the precise value of $\gamma_{t}$ associated
with a lattice model can be affected by the element slicing scheme, we found
that the best approach is to have PyORBIT itself compute $\gamma_{t}$. While
this is possible, the linear optics functionality of the code has limitations
that often need to be circumvented. Just like in real machine, unless the
lattice transverse tunes are carefully set a priori, the change in $\Delta
p/p$ at transition can easily excite transverse resonances causing rapid beam
loss. While physically obvious, this type of issue can be difficult to
diagnose numerically.

As already mentioned, the Booster wall impedance is unsually lossy. Near
transition, the bunch becomes narrow, the voltage induced by the wall reaches
a maximum and the resistive part of the impedance causes energy loss. Like
many tracking codes, PyORBIT particle coordinates are expressed with respect
to an a-priori defined reference, the so-called synchronous particle. Given an
rf voltage amplitude curve $V(t)$, the rf phase $\phi_{s}(t)$ of this
reference particle is set by condition
$V(t)\sin\phi_{s}(t)=V_{\text{sync}}(t)$ where $V_{\text{sync}}$ is the
voltage required to preserve synchronism.

Due to the lossy impedance, to remain in synchronism the bunch centroid will
move adiabatically to a phase $\phi_{s,\text{bunch}}$. In the Booster, near
transition, $\phi_{s,\text{bunch}}-\phi_{s}$ is on the order of $10$ deg. To
preserve the dynamics across transition, the bunch phase must jump from
$\phi_{s,\text{bunch}}$ to $\pi-\phi_{s,\text{bunch}}$. It is important to
appreciate that in the presence of a lossy impedance, the bunch centroid phase
and the synchronous phase $\phi_{s}$ will not be the same. As a result, simply
jumping the rf phase from $\phi_{s}$ to $\pi-\phi_{s}$ would cause spurious
mismatch. In the real machine, there is no notion of the reference phase
$\phi_{s}$; the beam centroid (with respect to the rf phase) is simply phase
locked to the rf frequency. At transition both the rf phase signal and the
beam phase signal are inverted resulting in a net change
$\pi-2\phi_{s,\text{bunch}}$. There are different ways to address this issue
in simulation. Our approach mimics the phase lock mechanism. At each turn, the
bunch centroid phase and the reference phase are compared; both the rf phase
and the longitudinal particle coordinates are then shifted so that the
coordinate of the particles are always expressed with respect to the bunch
centroid. Investigating longitudinal emittance blowup required the inclusion
of a quadrupole damper in the simulations. While PyORBIT does not readily
provide direct support for dampers or feedback systems, it is a relatively
straightforward matter to implement such functionality at the python level.
One can show [2] that in the linear approximation, the second moment of the
bunch longitudinal coordinate obeys the equation

$\frac{d^{2}}{dt^{2}}\Delta\sigma^{2}_{\phi}+4\omega_{S}^{2}\Delta\sigma^{2}_{\phi}=-2\omega_{S}\sigma^{2}_{0}\epsilon(t)$
(1)

where $\omega_{S}$ is the synchrotron frequency, $\sigma_{0}^{2}$ is the
matched (equilibrium) bunch length,
$\Delta\sigma^{2}_{\phi}=\sigma^{2}_{\phi}-\sigma^{2}_{0}$ and
$\epsilon(t)=\frac{\Delta V}{V}$ is an rf voltage modulation. If one uses as
modulation a signal proportional to the derivative of the second moment

$\epsilon(t)=\frac{2\xi}{\sigma_{0}\omega_{S}}\frac{d}{dt}\Delta\sigma^{2}_{\phi}$
(2)

where $0<\xi<1$, Eq. (2) becomes

$\frac{d^{2}}{dt^{2}}\Delta\sigma^{2}_{\phi}+4\omega_{S}\frac{d}{dt}\Delta\sigma^{2}_{\phi}+4\omega_{S}^{2}\Delta\sigma^{2}_{\phi}=0$
(3)

This is a standard second order ODE; its solution is a damped oscillation with
damping time $\tau=\frac{1}{2\xi\omega_{S}}$. In the code the damper is
implemented by taking a properly scaled and filtered version of the derivative
of the envelope of the second moment signal to modulate the rf cavity voltage.

##  4 RESULTS

Figure 1 (a) shows the initial longitudinal phase space at the start of the
acceleration cycle. The hollowed out region is created during injection
painting to reduce the peak line density. The 95% emittance is 0.052 eV-s. For
comparison purposes, we first present Fig. 1(b) which shows the phase space
near the end of the acceleration cycle assuming only a simple phase jump at
transition. The final 95% emittance is 0.197 eV-s, about a factor 2 larger
than the requirement.

Fig. 2 shows the rms bunch length oscillations triggered by crossing
transition ( transition occurs at turn 7100). The green trace correspond to
the case where the damper is turned off. The magenta trace correspond to the
case where the damper is turned on, showing that a damper is clearly effective
in reducing the amplitude of the bunch shape oscillations and preventing the
phase space filamentation responsible for the emittance blowup. The phase
space at the end of a cycle where the damper is turned on is shown in Fig.
1(c); the 95% emittance has been reduced to $0.137$ eV-s. An issue with the
damper is that the amplitude of the modulation is limited by the available
ring voltage; this is amounts to a feedback gain saturation effect. The
availability of a modest voltage overhead can significantly affect the damper
effectiveness. Fig. 3 illustrates this effect. In the simulation, the maximum
available voltage (initially set to 1.1 MV) is increased to 1.16 (cyan) and
1.25 (green) MV;the final bunch length is (therefore the final emittance) can
clearly be further reduced if additional voltage is available.

An option under consideration is to emulate the old system by pulsing the
existing quadrupole correctors (there is a focusing and a defocusing corrector
in each of the lattice 24 cells). By exciting a subset of the correctors in a
spatial pattern that excites a closed orbit resonance, it appears feasible to
create enough of a perturbation to change $\gamma_{t}$ somewhere between 0.3
and 0.6 units. How much is achievable in practice depends on a number of
factors that are still under study. Another issue is the corrector power
supply slew rate limit. As an example, Fig. 1(d) shows the final phase space
distribution obtained using such an emulated $\gamma_{t}$ jump while the
quadrupole damper is turned off. In this particular example
$\Delta\gamma_{t}\sim 0.6$ and the jump duration is $\sim 100\mu$s (somewhat
shorter than the Booster non-adiabatic time – about 210 $\mu{\rm s}$ at 20
Hz). In this case, it is seen that the final 95 % emittance ($0.103$ eV-s)
would meet the PIP-II requirement.

Figure 2: Damping quadrupole oscillations. Figure 3: Voltage overhead and
damper performance.

##  5 CONCLUSION

We demonstrated full cycle tracking simulations using the code PyORBIT. Albeit
based on a simplified model, the simulation predicts that using the existing
quadrupole damper, the Booster longitudinal emittance would come slightly
above the PIP-II requirement. To further reduce emittance blowup, a possible
scenario would be to combine the quadrupole damper with a weak $\gamma_{t}$
jump based on pulsing existing correctors. Both experimental and simulation
work are on-going to determine the viability of this approach.

## References

  * [1] A. Shislo et al., The particle accelerator simulation code PyORBIT, Procedia Comput. Sci., vol 51, 2015, pp. 1272-1281
doi:10.1016/j.procs.2015.05.312

  * [2] H. Klingbeil et al., Modeling longitudinal oscillations of bunched beams in synchrotrons
doi:10.48550/arXiv.101.3957


# A construction method of the quasi-monolithic compact interferometer based
on UV-adhesives bonding

Xiang Lin  Hao Yan∗<EMAIL_ADDRESS>Yiqiu Ma  Zebing Zhou

###### Abstract

Quasi-monolithic interferometers play a crucial role in high-precision
measurement experiments, including gravitational wave detection, inertial
sensing, vibrometry, and seismology. Achieving high stability and accuracy in
such interferometers requires a method for bonding optical components to a
baseplate. While optical contact bonding and silicate bonding are common
methods, UV adhesives offer advantages such as controlled curing and low
geometrical requirements for optical components and baseplates. This paper
presents a detailed construction method for a quasi-monolithic compact
interferometer based on UV-adhesive bonding. We built two types of
interferometers using this method: a $100\,{\rm mm}\times 100\,{\rm mm}\times
20\,{\rm mm}$ Mach-Zender homodyne interferometer with unequal arm lengths of
about $100\,{\rm mm}$ for laser frequency noise monitoring, and a heterodyne
interferometer as a displacement sensing head sizing $20\,{\rm mm}\times
30\,{\rm mm}\times 20\,{\rm mm}$. Our Mach-Zender interferometer achieved a
phase noise level of $2\,\mu{\rm rad}\sqrt{{\rm Hz}}$ at $1\,{\rm Hz}$ and a
equivalent laser frequency noise monitoring sensitivity of about $1\,{\rm
kHz}/\sqrt{{\rm Hz}}$ at $1\,{\rm Hz}$. The compact heterodyne interferometer
sensing head showed a sensitivity level of $1\,{\rm pm}/\sqrt{{\rm Hz}}$ in
translation and $0.2\,{\rm nrad}/\sqrt{{\rm Hz}}$ in two tilts above
$0.4\,{\rm Hz}$. Our tests demonstrate that quasi-monolithic compact
interferometers based on UV-adhesive bonding can achieve high sensitivity
levels at the pico-meter and nano-radian scales.

††preprint: AIP/123-QED

##  I INTRODUCTION

High-precision compact interferometers find applications in various fields
such as gravitational wave detectors,Armano  _et al._ (2021) inertial
sensors,Zhang and Guzman (2022); Isleif  _et al._ (2019); Hines  _et al._
(2023); Carter  _et al._ (2020) and vibrometers.Yan  _et al._ (2022); Wan  _et
al._ (2009); Mahadevan  _et al._ (2008) For these applications, sensitivity
requirements are often at the sub-nanometer or even picometer level. Opto-
mechanical stability is crucial for such interferometers. Not only should the
material of the baseplate and the optical components be ultra-stable, but the
contact between them should also be ultra-stable.Dehne (2012)

The opto-mechanical stability of the interferometer bench mainly depends on
the coefficient of thermal expansion and the mechanical stiffness. An optical
bench with a low coefficient of thermal expansion and high mechanical
stiffness is the key to achieving a high-precision interferometer.Wan, Ge, and
Chen (2011) Conventional interferometers typically use commercial metal mirror
mounts made of aluminum, stainless steel, or invar alloys, which have a
relatively large coefficient of thermal expansion of about
$10^{-5}-10^{-6}\,/{\rm K}$. Furthermore, the mirror mounts that use screws
and springs to facilitate positional adjustment and reuse of optical
components are further reduced in mechanical stiffness and thermal stability.
In pursuit of higher accuracy and stability, interferometers tend to use
quasi-monolithic designs, i.e., purely optical materials and contact
methods.Dehne (2012)

One such technique is the optical contact bonding that joins two highly
polished glass surfaces purely by van der Waals forces.Greco, Marchesini, and
Molesini (2001) However, the main limitation is the low and unstable breaking
strength, making it unsuitable for industrial metrology and space science. The
second optical bonding technique is known as silicate bonding,Van Veggel  _et
al._ (2008); van Veggel and Killow (2014) which was successfully used on the
optical bench of the LISA Pathfinder mission with fm-level accuracy.Armano
_et al._ (2021); Schwarze  _et al._ (2019) The basic principle is to use
chemical reactions to break and reconnect the Si-O bond between the optical
components and the bench, resulting in excellent mechanical stiffness and
thermal stability. However, the main disadvantage is the short chemical
reaction time (about tens of seconds) left for position and attitude
adjustments of the optical components, resulting in complex manufacturing
processes and high cost. In addition, this method requires two highly polished
glass surfaces to be bonded.

Another optical bonding method is adhesive bonding, i.e., UV adhesives or
epoxy resin adhesives, etc.Yi  _et al._ (2017); Preston, Thorpe, and Miner
(2012) The robustness of the two-component epoxy resin adhesive bonds against
mechanical and thermal inputs has been tested, Ressel  _et al._ (2010) making
this new bonding technique a potential option for interferometric applications
in future space missions. Epoxy adhesives have a relatively high bond
strength, but still require a long curing time, waiting approximately two days
for a single adhesive to cure. The bonding process with UV adhesives can be
controlled by UV light and takes only a few tens of seconds with a UV lamp.
Compared to optical contact and silicate bonding, UV-adhesive bonding has a
longer time scale for alignment, as well as shorter curing times, which is
convenient for the manufacture of quasi-monolithic interferometers. The main
disadvantages of UV adhesive are the relatively poor bond strength and thermal
stability.

The quasi-monolithic interferometer construction method presented in the paper
uses UV adhesive bonding and low thermal expansion materials like fused silica
to achieve stable and robust bonds between the optical components and the
baseplate. The construction process includes the construction of a quasi-
monolithic fiber collimator, positioning and pointing adjustment scheme of
optical components, and the UV-adhesive bonding process.

Unequal arm length control is critical for interferometers to achieve common-
mode rejection of ambient noise or laser frequency noise monitoring. A wide-
range swept laser is used in the setup to monitor unequal arm length with sub-
millimetre accuracy during the position tuning and assembly process of the
optical components.

The Mach-Zender homodyne interferometer built using this construction method
has a large unequal-arm length and can be used for noise monitoring, including
laser frequency noise and temperature fluctuations. The heterodyne
interferometer built with this method can achieve picometre or nano-radian
levels of displacement measurement.

##  II EXPERIMENTAL KEY TECHNOLOGIES

There are three key technologies for constructing a quasi-monolithic glass
interferometer based on UV-adhensive bonding: (1) the construction of the
quasi-monolithic fiber collimator, (2) the positioning and pointing adjustment
of the optical components (fiber collimator, beam splitter, reflecting mirror)
on the baseplate, and (3) the UV bonding process. The details are presented as
follows.

###  II.1  Quasi-monolithic glass fiber collimator

A fiber optic collimator plays a crucial role in an interferometer by
converting light from a fiber into collimated spatial light. It typically
consists of a fiber ceramic ferrule, one or more lenses, and an external
holder. However, commercial collimators often use metal mounts to adjust the
focus and attitudes of the lenses, making them unsuitable for constructing
all-glass interferometers. Alternatively, some glass collimators bond the
fiber ceramic ferrule and the lens inside a glass ferrule using a C-lens or
G-lens.

To create a fiber collimator suitable for quasi-monolithic glass
interferometers, we used a small custom-made glass cube with a cylindrical
hole as a bonding mount for the G-lens. The G-lens and the cubic glass mount
were bonded using UV adhesives to form our quasi-monolithic glass fiber
collimator, as shown in Fig. 1. The outgoing laser beam from this collimator
was found to have a beam waist of approximately $0.2\,{\rm mm}$, corresponding
to a Rayleigh distance of about $0.12\,{\rm m}$. The optical power
distribution of the laser emitted from the quasi-monolithic collimator
followed a Gaussian distribution with an ellipticity of $0.99$, as shown in
Fig. 1(d).

Figure 1:  The quasi-monolithic glass fiber collimator. (a) is the G-lens with
a size of $4\,{\rm mm}$ in diameter and $8\,{\rm mm}$ in length, (b) is the
$10\,{\rm mm}$ cubic glass mount with a cylindrical hole in the center, (c)
shows the quasi-monolithic glass fiber collimators, and (d) shows the optical
power distribution of the laser emitted from the quasi-monolithic glass fiber
collimator.

###  II.2 Optical alignment

In constructing a quasi-monolithic glass interferometer, precise positioning
and angular alignment of optical components are essential. There are several
reasons for this, including achieving high contrast interference, reducing
cosine errors, and controlling unequal arm lengths. The latter is especially
important because smaller unequal arm lengths lead to lower interferometer
noise.

To facilitate precise alignment, a six-degree-of-freedom mirror mount is used
in the setup, as shown in Fig. 2. This mirror mount has three degrees of
freedom of translation and three degrees of freedom of rotation, allowing for
sub-millimeter and sub-milliradian adjustments. The main steps of optical
component alignment are as follows: (1) fix the target component (collimator,
beamsplitter, reflector, etc.) on the six-degree-of-freedom mirror mount as in
Fig. 2(b), (2) place the target component in an approximately suitable
position in the optical system according to the optical path design, (3)
adjust the height of the target component from the substrate to a few
millimeters, (4) adjust the remaining five degrees of freedom of the target
component until the positions and attitudes are as designed as in Fig. 2(c),
(5) lower the height of the target component to the substrate.

Among the various optical components, the positioning and pointing of the
final component (typically a beam splitter) responsible for the
interferometric beam combination are crucial. It directly impacts the
interference contrast and requires high-precision adjustment. In contrast, the
positioning and pointing accuracy requirements of other optical components are
relatively lower. During the optical bench construction, non-critical optical
elements are adjusted with the aid of a small aperture diaphragm to ensure
that all laser beams are in the same plane. The final beam combination is
adjusted by observing the interference contrast as feedback, as shown in Fig.
2(d).

Figure 2:  The positioning and pointing adjustment scheme. (a) shows the six
degree-of-freedom mirror mount, (b) shows the quasi-monolithic glass
collimator on the mirror mount, (c) shows the position and attitude adjustment
of the optical components, (d) shows the allignment of two laser beams by
maximising their overlap to achieve good interferometric contrast.

###  II.3  UV-adhesives bonding

The UV adhesive bonding process is a critical step in the construction of
quasi-monolithic glass interferometers. In comparison to other bonding methods
such as optical contacting bonding and silicate bonding, UV adhesive bonding
is more convenient and suitable for situations where high mechanical rigidity
is not required.

Fig. 3 illustrates the UV adhesive bonding process. Initially, the target
component is positioned and adjusted with a large gap to the baseplate using
the 5-Dofs adjustment mount. Then, UV adhesive is dispensed using a special
stick and applied evenly while ensuring the correct amount is used. The target
component is then lowered to fit the upper surface of the optical baseplate
while maintaining its position and attitudes. Finally, the optics are
illuminated with a UV lamp until the adhesive has cured. This process is
repeated until all the optical components are adhesive-bonded in their
designed positions and attitudes.

Figure 3:  UV-adhesives bonding process. (a) shows the UV-adhesives dispensing
with a special stick in the gap between the baseplate and the optical
component, (b) shows the process of curing the adhesives with a UV lamp.

##  III INTERFEROMETER CONSTRUCTION

Using the techniques described above, we have successfully constructed two
classic quasi-monolithic glass interferometers. The first one is a homodyne
Mach-Zender interferometer with a large unequal arm length, which is suitable
for noise monitoring, including laser frequency noise and temperature
fluctuations. The second one is a compact heterodyne interferometer with an
optical path configuration consisting of two symmetrical Michelson structures.
When combined with differential wavefront sensing (DWS), it enables high-
precision measurements in three degrees of freedom, including one translation
and two tilts.

###  III.1  Mach-Zender interferometer

Figure 4:  Schematic design of the optical path of the homodyne Mach-Zender
interferometer with a unequal arm length of about $100\,{\rm mm}$. FC: fiber
collimator, BS: beam splitter, PD: photo detector. Figure 5:  UV-adhesives
bonding process of the quasi-monolithic glass homodyne Mach-Zender
interferometer. (a) is the fused-silica bench with a size of $100\,{\rm
mm}\times 100\,{\rm mm}\times 20\,{\rm mm}$, (b) and (c) show the adjustment
of the quasi-monolithic glass collimator, (d)-(g) show the UV bonding process
of the remaining optical components (two beam splitters and two reflectors),
(h) shows the finished quasi-monolithic glass Mach-Zender interferometer.
Figure 6:  The test of the completed quasi-monolithic glass Mach-Zender
interferometer.

Conventional Mach-Zender interferometer is of equal arm length. In order to
achieve unequal-arm interference, a slight modification was made to the
optical path design of a conventional Mach-Zender interferometer, as shown in
Fig. 4. The laser beam is injected from the left and divided into two paths.
The upper short path transmits directly, while the lower long path is
reflected twice by two mirrors (${\rm M_{1}}$ and ${\rm M_{2}}$) before
interfering with the upper path. Finally, the interference signal is balanced
by a pair of photodetectors (${\rm PD_{1}}$ and ${\rm PD_{2}}$). The unequal
arm length is approximately $100\,{\rm mm}$ according to the drawing design.

Figure 5 shows the UV-adhesive bonding process of the quasi-monolithic glass
homodyne Mach-Zender interferometer. The optical baseplate and components are
made of fused silica with a thermal expansion coefficient of about $0.5\times
10^{-6}\,{\rm/K}$. The completed quasi-monolithic glass Mach-Zender
interferometer, with dimensions of $100\,{\rm mm}\times 100\,{\rm mm}\times
30\,{\rm mm}$, was tested and shown in Fig. 6. The adjustment method shown in
Fig. 2 is effective. After careful adjustment of the position and attitude of
the optical components, the laser frequency sweep test showed an interference
contrast of over $0.8$, indicating that the laser beams were well coincided
and aligned.

The laser frequency noise can be measured by an unequal-arm interferometer.
Gerberding  _et al._ (2017) The measurement equation is given by:

$\delta f=\frac{c}{2\pi\Delta L}\delta\varphi,.$ (1)

where $\delta f$ is the laser frequency noise, $c$ is the speed of light, and
$\Delta L$ is the unequal armlength of the Mach-Zender interferometer. The
frequency-phase coefficient $c/{2\pi\Delta L}$ is about $500\,{\rm Hz}/\mu{\rm
rad}$ with a unequal armlength of about $100\,{\rm mm}$ in our setup.

###  III.2  Heterodyne interferometer

In addition to the homodyne Mach-Zender interferometer, we also developed a
compact heterodyne interferometer that is ideal for large dynamic measurements
due to its linear response to the translational displacement of the target
reflector. Moreover, the differential wavefront sensing techniques utilized in
this interferometer allow for two additional tilt measurements.

To enhance the sensitivity of the interferometer, we designed two symmetrical
Michelson-type optical path structures that suppress ambient noise
coupling.Yan  _et al._ (2020, 2015); Gillmer  _et al._ (2014); Yu, Gillmer,
and Ellis (2015) The interferometer, as shown in Fig. 7, uses two laser beams
($f_{1}$ and $f_{2}$) with a fixed frequency difference, which are injected by
collimators (${\rm FC_{1}}$ and ${\rm FC_{2}}$) from the left and split into
two beams by two beam splitters (${\rm BS_{1}}$ and ${\rm BS_{1}}$),
respectively. The transmitted beams hit the reference mirror (${\rm M_{ref}}$)
and the target measurement mirror (${\rm M_{mea}}$), respectively, and are
reflected back into the photodetector (PD and QPD) by ${\rm BS_{1}}$ and ${\rm
BS_{2}}$. The reference beams directly enter the photodetector (PD and QPD).
Finally, two interference beat signals are obtained: the upper one is the
measurement signal (measured by QPD), and the lower one is the reference
signal (measured by PD). The three degrees of freedom motion of the target
mirror can be obtained by equations:

$\displaystyle
x=\frac{\lambda}{4\pi}\left(\frac{\varphi_{1}+\varphi_{2}+\varphi_{3}+\varphi_{4}}{4}-\varphi_{5}\right),$
(2) $\displaystyle\theta_{\rm
pitch}=k\left(\varphi_{1}+\varphi_{2}-\varphi_{3}-\varphi_{4}\right),$ (3)
$\displaystyle\theta_{\rm
yaw}=k\left(\varphi_{1}+\varphi_{4}-\varphi_{2}-\varphi_{3}\right),$ (4)

where $\\{x,\theta_{\rm pitch},\theta_{\rm yaw}\\}$ denotes the displacement
of translation and two tilts (pitch and yaw), respectively, and $\lambda$
represents the wavelength of the laser. The four quadrant phase signal of the
QPD is represented by $\\{\varphi_{1},\ \varphi_{2},\ \varphi_{3},\
\varphi_{4}\\}$, while $\varphi_{5}$ denotes the phase signal of the PD. The
coefficient of the phase to the angle, denoted by $k$, requires experimental
calibration and depends on the waist diameter, wavefront, PD size, etc. The
compact heterodyne interferometer has a size of $20\,{\rm mm}\times 30\,{\rm
mm}\times 20\,{\rm mm}$, as shown in Fig. 8.

Figure 7:  Schematic design of the optical path of the compact heterodyne
interferometer. QPD: quadrant photo detector, ${\rm M_{ref}}$: reference
mirror, ${\rm M_{mea}}$: measurement mirror. Figure 8:  Pictures of the
compact heterodyne interferometer with a size of $20\,{\rm mm}\times 30\,{\rm
mm}\times 20\,{\rm mm}$. (a) shows the adjustment of the optical components,
(b) shows the finished compact bench.

##  IV  EXPERIMENTAL RESULTS

###  IV.1  Mach-Zender interferometer

To test the phase noise floor of the quasi-monolithic glass Mach-Zender
interferometer, we placed the optical bench inside a temperature-insulated
tank to isolate temperature fluctuations and allow temperature modulation
tests. We used a commercial laser with low power and frequency noise to inject
light into the Mach-Zender interferometer. The sensitivity curve of the Mach-
Zender interferometer is shown in Fig. 9, which achieved a phase noise floor
of $2\,\mu{\rm rad}/\sqrt{{\rm Hz}}$ above $1\,{\rm Hz}$. According to Eqs.
(1), the laser frequency noise floor of $1\,{\rm kHz}/\sqrt{{\rm Hz}}$ above
$1\,{\rm Hz}$ can be obtained with an unequal armlength of $100\,{\rm mm}$, as
shown in the right coordinate.

Figure 9:  Phase noise of the qutasi-monolithic glass MZI (Mach-Zender
interferometer). The solid red curve shows the phase noise of the MZI inside a
insulated tank. The dashed black curve shows the electronic noise floor,
obtained by replacing the interferometric signal with an single laser beam for
photoelectric detection. The coordinates on the right shows the corresponding
equivalent laser frequency noise measurement sensitivity.

###  IV.2  Heterodyne interferometer

The quasi-monolithic glass heterodyne interferometer has the capability to
measure motion in three degrees of freedom using differential wavefront
sensing (DWS). Unlike the homodyne Mach-Zender interferometer, the heterodyne
interferometer generates an AC beat signal which requires a phasemeter to
measure. To evaluate the noise floor, a target reflector is attached to an
optical stage and the heterodyne interferometer is used to measure its motion
in three degrees of freedom. The translation and two tilts (pitch and yaw) can
be obtained from the signals of the quadrant photodetector (QPD) and
photodetector (PD) using Eq. (4).

The noise floor for the two tilts is depicted in Fig. 10. A tilt sensitivity
of $0.2\,{\rm nrad}/\sqrt{{\rm Hz}}$ is achieved above $0.4\,{\rm Hz}$, with
the noise curve increasing rapidly and reaching $30\,{\rm nrad}/\sqrt{{\rm
Hz}}$ at $0.01\,{\rm Hz}$. The situation is similar for translation
measurement, as shown in Fig. 11. A translation sensitivity of $1\,{\rm
pm}/\sqrt{{\rm Hz}}$ is achieved above $0.4\,{\rm Hz}$, with the noise curve
increasing rapidly and reaching $200\,{\rm pm}/\sqrt{{\rm Hz}}$ at $0.01\,{\rm
Hz}$.

Figure 10:  Tilts noise of the quasi-monolithic compact heterodyne
interferometer. (a) shows the time-domain noise curve without displacemnet
input, and (b) shows the noise amplitude spectral density curve. Figure 11:
Translation noise of the quasi-monolithic compact heterodyne interferometer.
(a) is the time-domain curve, and (b) is the noise amplitude spectral density
curve.

##  V  CONCLUSION

We have successfully demonstrated a method for constructing quasi-monolithic
compact interferometers using UV-adhesives, resulting in the creation of two
optical benches. One of these benches is a Mach-Zender homodyne interferometer
with unequal arm lengths of $100\,{\rm mm}$, which is capable of noise
monitoring. The phase noise of this interferometer reaches $2\,\mu{\rm
rad}/\sqrt{{\rm Hz}}$ at $1\,{\rm Hz}$, and its equivalent frequency
monitoring sensitivity is about $1\,{\rm kHz}/\sqrt{{\rm Hz}}$ at $1\,{\rm
Hz}$. The other bench is a 3-Dofs compact heterodyne interferometer with a
sensitivity of $1\,{\rm pm}/\sqrt{{\rm Hz}}$ in translation and $0.2\,{\rm
nrad}/\sqrt{{\rm Hz}}$ in two tilts (pitch and yaw) above $0.4\,{\rm Hz}$.

In comparison to other bonding methods such as optical contact bonding,
silicate bonding, and epoxy resin adhesives bonding, UV-adhesives bonding
offers advantages such as controlled curing time and low geometrical machining
requirements for the optical components and baseplates, making it a suitable
option for constructing quasi-monolithic interferometers at the pico-meter and
nano-radian levels.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grant Nos. 12105375).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## Author Contributions

Xiang Lin: Experiment; Investigation (equal); Hao Yan: Conceptualization
(equal); Investigation (equal); Writing – original draft. Yiqiu Ma:
Conceptualization (equal); Writing – review & editing (equal). Zebing Zhou:
Investigation (equal); Writing – review.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# Local derivations and automorphisms of nilpotent Lie algebras

Abror Khudoyberdiyev1,2 1V.I.Romanovskiy Institute of Mathematics Uzbekistan
Academy of Sciences. 9, University street, 100174, Tashkent, Uzbekistan
2National University of Uzbekistan named after Mirzo Ulugbek. 4, University
street, 100174, Tashkent, Uzbekistan  and  Doston Jumaniyozov1 E-mail
addresses<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The paper is devoted to the study of local derivations and automorphisms of
nilpotent Lie algebras. Namely, we proved that nilpotent Lie algebras with
indices of nilpotency $3$ and $4$ admit local derivation (local automorphisms)
which is not a derivation (automorphisms). Further, it is presented a
sufficient condition under which a nilpotent Lie algebra admits a local
derivation which is not a derivation. With the same condition, it is proved
the existence of pure local automorphism on a nilpotent Lie algebra. Finally,
we present an $n$-dimensional non-associative algebra for which the space of
local derivations coincides with the space of derivations.

###### Key words and phrases:

Lie algebra, nilpotent algebra, derivation, local derivation, automorphism,
local automorphism

##  1\. Introduction

Local mappings on non-associative algebras have been one of the main topic of
many research works in recent years. The notions that are close to derivations
and automorphisms (so-called local derivations and local automorphisms) were
introduced and investigated independently by R.V. Kadison [14] and D.R. Larson
and A.R. Sourour [17]. The papers [14, 17] gave rise to a series of works
devoted to the description of local automorphisms and local derivations of
$C^{\ast}$-algebras and operator algebras. Namely, for the algebra
$\mathcal{A}$ of all bounded linear operators on a Banach space $X$ it is
proved that every invertible local automorphism of $\mathcal{A}$ is an
automorphism [17].

The coincidence of the space of local derivations ${\rm LocDer}(-)$ with the
space of derivations ${\rm Der}(-)$ on semisimple Lie algebra over an
algebraically closed field of characteristic zero is established, while for
some finite-dimensional nilpotent Lie algebras these spaces are different (see
in [1]). Similar result for standard Borel subalgebras of a finite-dimensional
simple Lie algebra over an algebraically closed field of characteristic zero
was proved in [19]. Moreover, it is proved that every local derivation is a
derivation on a complex solvable Lie algebra with nilradical of maximal rank
(see [15]). Further, it is proved that every local derivation is a derivation
on infinite-dimensional Lie algebra $W(2,2)$ (see [18]). In general, the
behavior of local derivations is not definite on solvable Lie algebras because
of the existence of solvable Lie algebras which admit local derivations which
are not ordinary derivation and also algebras for which every local derivation
is a derivation [5]. The first example of a simple (ternary) algebra with non-
trivial local derivations is constructed in [12]. After that, the first
example of a simple (binary) algebra with non-trivial local derivations was
constructed in [4].

As for local automorphism, there are several results which devoted to the
description of properties of local automorphisms on some classes of algebras.
Namely, it was proved that every local automorphism on the special linear Lie
algebra $\mathfrak{sl}_{n}$ is either automorphism or anti-automorphism [2].
Further Constantini [9] extended the above result for an arbitrary complex
simple Lie algebra $\mathfrak{g}.$ Namely, his main result can be formulated
as follows:

${\rm LocAut}(\mathfrak{g})={\rm Aut}(\mathfrak{g})\rtimes\\{{\rm
Id}_{\mathfrak{g}},-{\rm Id}_{\mathfrak{g}}\\},$

where ${\rm Id}_{\mathfrak{g}}$ is the identical automorphism of complex
simple Lie algebra $\mathfrak{g}.$

In fact, the algebra of all square matrices over the field of complex numbers,
finite-dimensional simple Leibniz algebras over an algebraically closed field
of the characteristic zero, infinite-dimensional simple Witt algebra over an
algebraically closed field of zero characteristic do not admit pure local
automorphisms (see [3], [6], [8]). On the other hand, some types of finite-
dimensional nilpotent Lie and Leibniz algebras, as well as the algebra of
lower triangular $n\times n$-matrices admit local automorphisms which are not
automorphisms (see [2], [3], [11]). Moreover, in [16], it is proved that any
local automorphism is an automorphism on complex solvable Lie algebra of
maximal rank.

It should be noted that the space
$\operatorname{\mathrm{LocDer}}(\mathcal{A})$ coincides with the space
$\operatorname{\mathrm{Der}}(\mathcal{A})$ mainly when $\mathcal{A}$ has no
outer derivations (see [1, 7]). But, it is known that every nilpotent Lie
algebra has an outer derivation. Moreover, the fact that every linear
transformation on a Lie algebra of nilindex $2$ is a derivation yields that
the spaces $\operatorname{\mathrm{LocDer}}(\mathcal{A})$ and
$\operatorname{\mathrm{Der}}(\mathcal{A})$ coincide. So, it is natural to
conjecture that all nilpotent Lie algebras with nilindex greater than $3$
admit pure local derivations. It is one of the open problems regarding local
mappings on non-associative algebras. In this paper, authors attack this
problem on nilpotent Lie algebras with small indices of nilpotency. Namely, it
is proved that nilpotent Lie algebras with nilindex $3$ and $4$ admit pure
local derivations. Moreover, we presented a sufficient condition under which
every nilpotent Lie algebra admits local derivation which is not derivation.
In addition, we address the existence of pure local automorphism on nilpotent
Lie algebras. Interestingly, the situations of local derivations and
automorphisms are similar on nilpotent Lie algebras, that is under the same
conditions a nilpotent Lie algebra admits pure local automorphism and
derivation.

##  2\. General definitions

######  Definition 2.1.

A Lie algebra over the field $\mathbb{F}$ is a vector space $\mathfrak{g}$
endowed with bilinear mapping
$[-,-]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g},$ satisfying the
following identities:

$[x,x]=0,$ $[[x,y],z]+[[y,z],x]+[[z,x],y]=0.$

The center of a Lie algebra $\mathfrak{g}$ is defined as

$Z(\mathfrak{g})=\\{x\in\mathfrak{g}\ |\ [x,y]=0,\mbox{ for all
}y\in\mathfrak{g}\\}.$

For a Lie algebra $\mathfrak{g}$, consider the following sequence:

$\mathfrak{g}^{1}=\mathfrak{g},\quad\mathfrak{g}^{k+1}=[\mathfrak{g},\mathfrak{g}^{k}],\quad
k\geq 1.$

If $\mathfrak{g}^{p}=0$ for some $p\in\mathbb{N},$ then $\mathfrak{g}$ is said
to be nilpotent. The minimal number with this property is called an index of
nilpotency (shortly nilindex) of $\mathfrak{n}.$ If the index of nilpotency of
$\mathfrak{g}$ is $3,$ then $\mathfrak{n}$ is called $2$-step nilpotent Lie
algebra.

Note that the center of a nilpotent Lie algebra $\mathfrak{g}$ is always
nonempty as $[x,y]=0$ for all $x\in\mathfrak{g}^{p-1},\ y\in\mathfrak{g},$
where $p$ is the index of nilpotency of $\mathfrak{g}.$

Let $\mathfrak{g}$ be a nilpotent Lie algebra and
$\left\\{e_{1},\ldots,e_{k}\right\\}$ be generators of $\mathfrak{g}.$ Then,
due to a result of [13], $\mathfrak{g}$ admits a basis $\mathcal{B}$ of the
following form:

$\left\\{e_{1},\ldots,e_{k},\left[e_{i_{1}},[e_{i_{2}},\cdots,[e_{i_{m-1}},e_{i_{m}}],\cdots]\right]:1\leq
i_{1},\ldots,i_{m}\leq k,1\leq m\leq k\right\\}.$

A linear transformation $d$ of a Lie algebra $\mathfrak{g}$ is a derivation if
it satisfies Leibniz rule, that is, equality

$d([x,y])=[d(x),y]+[x,d(y)]$

holds for any $x,y\in\mathfrak{g}.$ The set of all derivations of
$\mathfrak{g}$ is denoted by ${\rm Der}(\mathfrak{g}).$ Note that ${\rm
Der}(\mathfrak{g})$ forms a Lie algebra under commutator. It is known that any
linear transformation of $\mathfrak{g}$ sending every element to the element
of center of $\mathfrak{g}$ is a derivation. Such derivations are called
central derivations. Moreover, a linear transformation $d_{0}$ sending every
element $x$ to zero element is called trivial derivation. Throughout this
paper, when we require or claim the existence of a derivation, we always mean
a non-trivial derivation.

Let $x\in\mathfrak{g}.$ A linear mapping defined as

${\rm ad}x(y)=[x,y]$

is a derivation of $\mathfrak{g}.$ Such derivations are called inner
derivations and the set of all inner derivations is denoted by ${\rm
Inn}(\mathfrak{g}).$

An automorphism $\varphi$ of a Lie algebra $\mathfrak{g}$ is defined as a
linear bijection $\varphi:\mathfrak{g}\to\mathfrak{g}$ such that

$\varphi([x,y])=[\varphi(x),\varphi(y)]$

for any $x,y\in\mathfrak{g}.$ The set of all automorphisms of $\mathfrak{g}$
forms a group, which is denoted by ${\rm Aut}(\mathfrak{g}).$ Note that, the
identity map ${\rm Id}$ on $\mathfrak{g}$ is an automorphism. It is called a
trivial automorphism. An automorphism $\varphi$ is called unipotent, if
$\varphi-{\rm Id}$ is nilpotent.

Let $D$ be a derivation of a Lie algebra $\mathfrak{g}$ over $\mathbb{F}.$ If
$char\mathbb{F}=0,$ then the exponential map defined as

${\rm exp}(D)=\sum_{k=0}^{\infty}\frac{D^{k}}{k!}$

is an automorphism of $\mathfrak{g}.$ If $D$ is an inner derivation, then
${\rm exp}(D)$ is called inner automorphism.

A linear mapping $\Delta$ is said to be a local derivation, if for every
$x\in\mathfrak{g}$ there exists a derivation $d_{x}$ on $\mathfrak{g}$
(depending on $x$) such that $\Delta(x)=d_{x}(x)$. Similarly, a linear mapping
$\nabla$ is called a local automorphism, if for every $x\in\mathfrak{g}$ there
exists automorphism $\varphi_{x}$ such that $\nabla(x)=\varphi_{x}(x)$. The
set of local derivations (local automorphisms) of $\mathfrak{g}$ is denoted by
${\rm LocDer}(\mathfrak{g})$ (respectively ${\rm LocAut}(\mathfrak{g})$) Note
that the set of all local automorphisms of an algebra form a multiplicative
group, which contains the group of automorphisms as a subgroup (see [10, Lemma
4]).

Let $f$ be a linear transformation on a Lie algebra $\mathfrak{g}$ and $A$ be
a subspace of $\mathfrak{g}.$ $\mathcal{B}=\\{e_{1},e_{2},\dots,e_{n}\\}$ is a
basis of $\mathfrak{g}$ such that
$\mathcal{B_{k}}=\\{e_{1},e_{2},\dots,e_{k}\\}$ basis of $A$. By the
expression $f_{|A},$ we mean the linear mapping which is defined as follows:

$\displaystyle f_{|A}(e_{i})=\left\\{\begin{array}[]{cccll}f(e_{i}),&\mbox{ if
}&e_{i}\in A,\\\ 0,&\mbox{ if }&e_{i}\in\mathfrak{g}\setminus
A.\end{array}\right.$

##  3\. Local derivations of nilpotent Lie algebras

Let $\mathfrak{n}$ be a finite-dimensional nilpotent Lie algebra over a field
$\mathbb{F}$ with index of nilpotency $p,$ that is, $\mathfrak{n}^{p}=0.$ Let
$\mathcal{B}=\\{e_{1},e_{2},\dots,e_{n}\\}$ be a basis of $\mathfrak{n}.$ At
first, let us show that, if $\mathfrak{n}$ is $2$-step nilpotent Lie algebra,
that is $p=3$, then it has a diagonal derivation. Indeed, consider a linear
mapping ${\rm D}$ which is defined as

$\displaystyle{}\begin{array}[]{cccccccll}{\rm D}_{\lambda}(e_{i})=\lambda
e_{i},&&\mbox{if }&e_{i}\in\mathfrak{n}\setminus\mathfrak{n}^{2},\\\ {\rm
D}_{\lambda}(e_{i})=2\lambda e_{i},&&\mbox{if
}&e_{i}\in\mathfrak{n}^{2}.\end{array}$ (3.3)

We show that ${\rm D}_{\lambda}$ is a derivation. As expected, let
$x,y\in\mathfrak{n}$ and $x=\sum\limits_{i}\alpha_{i}e_{i},\
y=\sum\limits_{j}\beta_{j}e_{j}.$ Then, we have

$\displaystyle{\rm D}([x,y])$ $\displaystyle={\rm
D}_{\lambda}\Big{(}\sum_{i,j}\alpha_{i}\beta_{j}[e_{i},e_{j}]\Big{)}=\sum_{i,j}\alpha_{i}\beta_{j}{\rm
D}_{\lambda}\left([e_{i},e_{j}]\right)$
$\displaystyle=2\lambda\sum_{i,j}\alpha_{i}\beta_{j}[e_{i},e_{j}]=\sum_{i,j}\alpha_{i}\beta_{j}[\lambda
e_{i},e_{j}]+\sum_{i,j}\alpha_{i}\beta_{j}[e_{i},\lambda e_{j}]$
$\displaystyle=\sum_{i,j}\alpha_{i}\beta_{j}[{\rm
D}_{\lambda}(e_{i}),e_{j}]+\sum_{i,j}\alpha_{i}\beta_{j}[e_{i},{\rm
D}_{\lambda}(e_{j})]=[{\rm D}_{\lambda}(x),y]+[x,{\rm D}_{\lambda}(y)].$

This implies that ${\rm D}_{\lambda}$ is a derivation.

######  Theorem 3.1.

Let $\mathfrak{n}$ be a $2$-step nilpotent Lie algebra over a field
$\mathbb{F}$ $(char\mathbb{F}\neq 2).$ Then, $\mathfrak{n}$ admits a local
derivation which is not derivation.

###### Proof.

Let ${\rm D}$ be a derivation of $\mathfrak{n}$ which is defined as (3.3) and
$\mathcal{B}=\\{e_{1},e_{2},\dots,e_{m},\dots,e_{n}\\}$ be a basis of
$\mathfrak{n},$ where the number $m$ means the number of generators of
$\mathfrak{n}.$ Consider a linear mapping $\Delta$ defined as

$\displaystyle\Delta(e_{i})=\left\\{\begin{array}[]{cccll}0,&\mbox{ if
}&e_{i}\in\mathfrak{n}\setminus\mathfrak{n}^{2},\\\ 2e_{i},&\mbox{ if
}&e_{i}\in\mathfrak{n}^{2}.\end{array}\right.$

We shall show that $\Delta$ is a local derivation which is not derivation.
First, let us show that it is not derivation. Let $[e_{r},e_{t}]=e_{q}$ for
some $e_{r},e_{t}\in\mathfrak{n}\setminus\mathfrak{n}^{2},\
e_{q}\in\mathfrak{n}^{2}.$ Then, we have

$[\Delta(e_{r}),e_{t}]+[e_{r},\Delta(e_{t})]=0\neq
2e_{q}=2[e_{r},e_{t}]=\Delta([e_{r},e_{t}]).$

Thus, $\Delta$ is not derivation.

Now, we show that it is a local derivation.

Let $x=\sum\limits_{i=1}^{n}\lambda_{i}e_{i},$ then
$\Delta(x)=2\sum\limits_{i=m+1}^{n}\lambda_{i}e_{i}.$ Consider the following
possible cases:

  * 1.

Let $\lambda_{j}\neq 0$ for some $j$ $(1\leq j\leq m).$ Then, consider a
derivation defined as

$\displaystyle D_{x}(e_{j})$
$\displaystyle=\frac{2}{\lambda_{j}}\sum\limits_{i=m+1}^{n}\lambda_{i}e_{i},$
$\displaystyle D_{x}(e_{i})$ $\displaystyle=0,\quad i=1,\dots,n,\quad i\neq
j.$

Hence, we derive

$D_{x}(x)=\sum\limits_{i=1}^{n}\lambda_{i}D_{x}(e_{i})=\lambda_{j}D_{x}(e_{j})=\lambda_{j}\frac{2}{\lambda_{j}}\sum\limits_{i=m+1}^{n}\lambda_{i}e_{i}=2\sum\limits_{i=m+1}^{n}\lambda_{i}e_{i}=\Delta(x).$

  * 2.

Let $\alpha_{j}=0$ for all $j$ $(1\leq j\leq m).$ Then, we have
$\Delta(x)=2x.$ Define a derivation $D_{x}$ as $D_{x}={\rm D}_{\bf 1},$ i.e.,

$\begin{array}[]{cccccccll}{\rm D}_{x}(e_{i})=e_{i},&&\mbox{if
}&e_{i}\in\mathfrak{n}\setminus\mathfrak{n}^{2},\\\ {\rm
D}_{x}(e_{i})=2e_{i},&&\mbox{if }&e_{i}\in\mathfrak{n}^{2}.\end{array}$

Thus, we have $D_{x}(x)={\rm D}_{1}(x)=2x=\Delta(x).$

The proof is complete. ∎

To illustrate Theorem 3.1, consider the following example of $2$-step
nilpotent Lie algebra:

######  Example 3.2.

Let $\mathfrak{h}_{n}$ be complex $(2n+1)$-dimensional Heisenberg algebra with
a basis $\mathcal{B}=\\{e_{-n},\dots,e_{-1},e_{0},e_{1},\dots,e_{n}\\}.$ The
multiplication table is given as follows:

$\displaystyle{}[e_{-i},e_{i}]=-[e_{i},e_{-i}]=e_{0},$ (3.4)

where $i=1,\dots,n.$ The omitted products are supposed to be zero.

A linear mapping $\Delta$ defined as

$\displaystyle\Delta(e_{0})=e_{0},\qquad\Delta(e_{i})=0,\quad i\neq 0,$

is a local derivation by Theorem 3.1. Since

$[\Delta(e_{-1}),e_{1}]+[e_{-1},\Delta(e_{1})]=0\neq
e_{0}=\Delta(e_{0})=\Delta([e_{-1},e_{1}]),$

we conclude that $\Delta$ is not derivation.

######  Remark 3.3.

It should be noted that, in Theorem 3.1 the condition $char\mathbb{F}\neq 2$
is essential. General behaviour of local derivations on nilpotent Lie algebras
over the field of characteristic $2$ is different. There is a nilpotent Lie
algebra over the field of characteristic $2$ for which the spaces of local
derivations and derivations coincide.

The following example shows that Theorem 3.1 is not true over the field of
characteristic $2:$

######  Example 3.4.

Let $\mathfrak{s}$ be a $12$-dimensional nilpotent Lie algebra over the field
$\mathbb{Z}_{2}$ with a basis $\mathcal{B}=\\{e_{1},e_{2},\dots,e_{12}\\}$ and
the following table of multiplication:

$\displaystyle[e_{2},e_{1}]$ $\displaystyle=e_{8},$
$\displaystyle[e_{1},e_{6}]$ $\displaystyle=e_{8},$
$\displaystyle[e_{1},e_{3}]$ $\displaystyle=e_{9},$
$\displaystyle[e_{1},e_{5}]$ $\displaystyle=e_{9},$
$\displaystyle[e_{5},e_{6}]$ $\displaystyle=e_{9},$
$\displaystyle[e_{1},e_{4}]$ $\displaystyle=e_{10},$
$\displaystyle[e_{2},e_{3}]$ $\displaystyle=e_{10},$
$\displaystyle[e_{1},e_{7}]$ $\displaystyle=e_{10},$
$\displaystyle[e_{5},e_{7}]$ $\displaystyle=e_{10},$
$\displaystyle[e_{4},e_{3}]$ $\displaystyle=e_{11},$
$\displaystyle[e_{7},e_{6}]$ $\displaystyle=e_{11},$
$\displaystyle[e_{2},e_{7}]$ $\displaystyle=e_{12},$
$\displaystyle[e_{3},e_{6}]$ $\displaystyle=e_{12},$

where the omitted products are supposed to be zero.

Direct computations show that every derivation ${\rm D}$ of $\mathfrak{s}$
acts on basis elements as follows:

$\displaystyle{}{\rm D}(e_{i})=\left\\{\begin{array}[]{cccll}\alpha e_{i}+{\rm
T}(e_{i}),&\mbox{ if }&1\leq i\leq 7,\\\ 0,&\mbox{ if }&8\leq i\leq
12,\end{array}\right.$ (3.7)

where $\alpha\in\mathbb{Z}_{2}$ and ${\rm T}$ is a central derivation.

Let $\Delta$ be a local derivation of $\mathfrak{s}.$ Then, $\Delta$ is
defined as follows:

$\displaystyle\Delta(e_{i})=\left\\{\begin{array}[]{cccll}\beta_{i}e_{i}+{\rm
T_{e_{i}}}(e_{i}),&\mbox{ if }&1\leq i\leq 7,\\\ 0,&\mbox{ if }&8\leq i\leq
12,\end{array}\right.$

where $\beta_{i}\in\mathbb{Z}_{2}$ and ${\rm T}_{e_{i}}$ are central
derivations.

Now, we show that every local derivation $\Delta$ on $\mathfrak{s}$ is a
derivation. Indeed, consider an element $x=\sum\limits_{i=1}^{7}e_{i}.$
Applying $\Delta(x)={\rm D}_{x}(x)$ we have

$\displaystyle\Delta(x)=\sum_{i=1}^{7}\beta_{i}e_{i}+y=\alpha\sum_{i=1}^{7}e_{i}+z={\rm
D}_{x}(x),$

where $y,z\in Z(\mathfrak{s}).$ By comparing the corresponding coefficients at
basis elements we obtain $\beta_{1}=\beta_{2}=\dots=\beta_{7}.$ Hence,
$\Delta$ is of the form (3.7), that is, $\Delta$ is a derivation.

Now, we consider more general case. We define a necessary condition under
which a nilpotent Lie algebra admits a pure local derivation.

######  Theorem 3.5.

Let $\mathfrak{n}$ be a nilpotent Lie algebra over a field $\mathbb{F}$ with
nilindex $p$ $(p\geq 4).$ If there exists a derivation ${\rm D}\in{\rm
Der}({\mathfrak{n}}),$ such that ${\rm D}(\mathfrak{n}^{2})\subseteq
Z(\mathfrak{n}),$ then $\mathfrak{n}$ admits a local derivation which is not
derivation.

###### Proof.

Let $\mathcal{B}=\\{e_{1},e_{2},\dots,e_{m},\dots,e_{n}\\}$ be a basis of
$\mathfrak{n},$ where the number $m$ means the number of generator elements of
$\mathfrak{n}.$ Suppose that, $[e_{r},e_{t}]=e_{q}$ for some
$e_{r},e_{t}\in\mathfrak{n}\setminus\mathfrak{n}^{2},\
e_{q}\in\mathfrak{n}^{2}\setminus\mathfrak{n}^{3}.$

If ${\rm D}(e_{i})=0$ for all
$e_{i}\in\mathfrak{n}^{2}\setminus\mathfrak{n}^{3},$ then ${\rm D}$ would be a
trivial derivation which contradicts the existence of derivation with ${\rm
D}(\mathfrak{n}^{2})\subseteq Z(\mathfrak{n}).$ Therefore, there exists
$e_{i_{0}}\in\mathfrak{n}^{2}$ such that ${\rm D}(e_{i_{0}})\neq 0.$

Without loss of generality we may assume that ${\rm D}(e_{q})\neq 0.$ Then,
define a linear mapping $\Delta:\mathfrak{n}\to\mathfrak{n}$ as follows:

$\Delta={\rm D}_{|\mathfrak{n}^{2}}.$

We shall show that it is a local derivation which is not derivation. Since

$[\Delta(e_{r}),e_{t}]+[e_{r},\Delta(e_{t})]=0\neq{\rm
D}(e_{q})=\Delta([e_{r},e_{t}]),$

we conclude that $\Delta$ is not derivation.

However, $\Delta$ is a local derivation. Indeed, let
$x=\sum\limits_{i=1}^{n}\lambda_{i}e_{i}.$ Then, $\Delta(x)=y\in
Z(\mathfrak{n}).$

Consider the following possible cases:

  * •

Let $\lambda_{j}\neq 0$ for some $j$ $(1\leq j\leq m).$ Then, define a
derivation as follows:

$\displaystyle D_{x}(e_{j})$ $\displaystyle=\frac{1}{\lambda_{j}}y,$
$\displaystyle D_{x}(e_{i})$ $\displaystyle=0,\quad i=1,\dots,n,\quad i\neq
j.$

Hence, we obtain that

$D_{x}(x)=\lambda_{j}D_{x}(e_{j})=\lambda_{j}\frac{1}{\lambda_{j}}y=y=\Delta(x).$

  * •

Let $\lambda_{j}=0$ for all $j$ $(1\leq j\leq m).$ Then, consider a derivation
$D_{x}$ defined as

$D_{x}={\rm D}.$

Hence, we derive

$D_{x}(x)={\rm D}(x)={\rm
D}\left(\sum\limits_{i=m+1}^{n}\lambda_{i}e_{i}\right)=y=\Delta(x).$

The proof is complete. ∎

Now, we state some implications of Theorem 3.5.

######  Proposition 3.6.

Let $\mathfrak{n}$ be a nilpotent Lie algebra with nilindex $4.$ Then,
$\mathfrak{n}$ admits a local derivation which is not derivation.

###### Proof.

According to the result above, it is sufficient to show that there exists a
derivation $D$ which sends $\mathfrak{n}$ to the center. Since $\mathfrak{n}$
is an algebra of nilindex $4,$ we have $\mathfrak{n}^{4}=0$ and
$0\neq\mathfrak{n}^{3}\subseteq Z(\mathfrak{n}).$ Thus, there exist
$x\in\mathfrak{n},\ y\in\mathfrak{n}^{2}$ such that $[x,y]\neq 0.$ Consider an
inner derivation $ad_{x}.$ Having
$[x,y]\in[\mathfrak{n},\mathfrak{n}^{2}]\subseteq\mathfrak{n}^{3}\subseteq
Z(\mathfrak{n})$ in mind, we have a derivation $D=ad_{x}$ which satisfies with
the condition of Theorem 3.5. ∎

######  Corollary 3.7.

Let $\mathfrak{n}$ be a nilpotent Lie algebra with $\mathfrak{n}^{p}=0$
$(p\geq 5)$. If $[\mathfrak{n}^{p-3},\mathfrak{n}^{2}]\neq 0,$ then
$\mathfrak{n}$ admits a local derivation which is not derivation.

###### Proof.

Let $\mathfrak{n}$ be a nilpotent Lie algebra of nilindex $p$ $(p\geq 5).$ The
condition $[\mathfrak{n}^{p-3},\mathfrak{n}^{2}]\neq 0$ guarantees the
existence of $x\in\mathfrak{n}^{p-3},\ y\in\mathfrak{n}^{2}$ such that
$[x,y]\neq 0.$ Consider an inner derivation $ad_{x}.$ Since
$[x,y]\in[\mathfrak{n}^{p-3},\mathfrak{n}^{2}]\subseteq\mathfrak{n}^{p-1}\subseteq
Z(\mathfrak{n}),$ we have a derivation $d=ad_{x}$ which satisfies with the
condition of Theorem 3.5. ∎

Here, for this kind of algebras there is an inner derivation providing the
existence of pure local derivation. However, there are many nilpotent Lie
algebras for which $[\mathfrak{n}^{p-3},\mathfrak{n}^{2}]=0,$ so we can not
construct a local derivation by using inner derivations. Interestingly, for
these algebras there exists outer derivations fulfilling the condition of
Theorem 3.5.

For instance, consider the following example:

######  Example 3.8.

Let $n\geq 3$ and $\mathfrak{w}$ be a complex $n$-dimensional Witt algebra
with basis $\mathcal{B}=\\{e_{1},e_{2},\dots,e_{n}\\}$ and the following basis
products:

$[e_{i},e_{j}]=(j-i)e_{i+j},\quad 1\leq i,j\leq n,\ i+j\leq n.$

Note that $\mathfrak{w}^{k}=span\\{e_{k+1},\dots,e_{n}\\}$ and
$Z(\mathfrak{w})=\\{e_{n}\\}.$ Thus, $\mathfrak{w}^{n}=0$ which implies
$[\mathfrak{w}^{n-3},\mathfrak{w}^{2}]=0.$ It means that we can not construct
an inner derivation satisfying the condition of Theorem 3.5. However, a linear
mapping $d$ defined as

$\displaystyle d(e_{2})=e_{n-1},$ $\displaystyle d(e_{3})=(n-2)e_{n},$
$\displaystyle d(e_{i})=0,\quad i\neq 2,3,$

is a derivation of $\mathfrak{w}.$ Moreover, this derivation sends $e_{3}$ to
the center of $\mathfrak{w}.$ Thus, by Theorem 3.5, a linear transformation
$\Delta$ defined as

$\displaystyle\Delta(e_{3})=(n-2)e_{n},$
$\displaystyle\quad\Delta(e_{i})=0,\quad i\neq 3,$

is local derivation which is not derivation.

Now, we generalize Example 3.8.

######  Proposition 3.9.

Let $\mathfrak{n}$ be a two-generated nilpotent Lie algebra. Then,
$\mathfrak{n}$ admits a local derivation which is not derivation.

###### Proof.

At first, we construct a derivation which sends $\mathfrak{n}^{2}$ to the
center of the algebra $Z(\mathfrak{n})$. Let
$\mathcal{B}=\\{e_{1},e_{2},\dots,e_{n}\\}$ be a basis and $e_{1},e_{2}$ be
generator elements of $\mathfrak{n}.$ Suppose that, $\mathfrak{n}^{p}=0$ for
some $p.$ Without loss of generalization, we assume that
$[e_{1},e_{2}]=e_{3}.$ Since $\mathfrak{n}^{p-1}\neq 0,$ there exists
$x\in\mathfrak{n}^{p-2}$ such that either $[e_{1},x]\neq 0$ or $[e_{2},x]\neq
0.$ Let $[e_{1},x]=y\in\mathfrak{n}^{p-1}.$ Now, we show that a linear mapping
$\delta$ defined as

$\displaystyle\delta(e_{2})=x,\quad\delta(e_{3})=y,\quad\quad\delta(e_{i})=0,\quad
i\neq 2,3$

is a derivation of $\mathfrak{n}.$ Indeed, we have

$\displaystyle\delta([e_{1},e_{2}])$
$\displaystyle=\delta(e_{3})=y=[e_{1},x]=[\delta(e_{1}),e_{2}]+[e_{1},\delta(e_{2})],$
$\displaystyle\delta([e_{1},e_{i}])$ $\displaystyle=\delta\big{(}\sum_{j\geq
4}(\ast)e_{j}\big{)}=0=[\delta(e_{1}),e_{i}]+[e_{1},\delta(e_{i})],\quad i\geq
3,$ $\displaystyle\delta([e_{2},e_{i}])$
$\displaystyle=\delta\big{(}\sum_{j\geq
4}(\ast)e_{j}\big{)}=0=[\delta(e_{2}),e_{i}]+[e_{2},\delta(e_{i})],\quad i\geq
3,$ $\displaystyle\delta([e_{i},e_{j}])$
$\displaystyle=0=[\delta(e_{i}),e_{j}]+[e_{i},\delta(e_{j})],$ $\displaystyle
i,j\neq 2,3.$

Hence, $\delta$ is a derivation. Moreover, $\delta$ fulfills the condition
$\delta(\mathfrak{n}^{2})\subseteq Z(\mathfrak{n}).$ Thus, by Theorem 3.5,
$\mathfrak{n}$ admits a local derivation which is not derivation. The case
$[e_{1},x]=0,\ [e_{2},x]\neq 0$ can be proved similarly. ∎

In fact, a Lie algebra can not be generated by one element. Generally, we do
not know the situation in other non-associative algebras. There exists
nilpotent algebras for which every local derivation is derivation. For
instance, consider the following example:

######  Example 3.10.

Let $\mathfrak{a}$ be a complex $n$-dimensional algebra with a basis
$\mathcal{B}=\\{e_{1},e_{2},\dots,e_{n}\\}$ $(n\geq 3)$ and multiplication
table

$\displaystyle{}e_{i}e_{i}=e_{i+1},\quad 1\leq i\leq n-1.$ (3.8)

Direct computations show that every derivation $D$ of $\mathfrak{a}$ acts on
basis elements as follows:

$\displaystyle D(e_{1})=\alpha e_{1}+\beta e_{n},\qquad D(e_{i})=2^{i-1}\alpha
e_{i},$

where $\alpha,\beta\in\mathbb{C},$ $2\leq i\leq n.$

We shall show that every local derivation on $\mathfrak{a}$ is a derivation.
From the definition of local derivation, we derive that any local derivation
$\Delta$ acts on basis elements as

$\displaystyle\Delta(e_{1})=\delta e_{1}+\zeta
e_{n},\qquad\Delta(e_{i})=\xi_{i}e_{i},$

where $\delta,\zeta,\xi_{i}\in\mathbb{C},$ $2\leq i\leq n.$

To prove that ${\rm LocDer}(\mathfrak{a})={\rm Der}(\mathfrak{a}),$ we need to
show that $\xi_{i}=2^{i-1}\delta$ for any $i=1,\dots n.$

Take an element $x=e_{1}+e_{i},\ (2\leq i\leq n-1).$ Then, there exists a
derivation $D_{x}$ such that $D_{x}(x)=\Delta_{x}(x).$ Hence, we derive

$\displaystyle D_{x}(x)$
$\displaystyle=\alpha_{x}e_{1}+2^{i-1}\alpha_{x}e_{i}+\beta_{x}e_{n}=\delta
e_{1}+\xi_{i}e_{i}+\zeta e_{n}=\Delta(x).$

Comparing the corresponding coefficients we have $\xi_{i}=2^{i-1}\delta.$
Thus, it only remains to show that $\xi_{n}=2^{n-1}\delta.$ By taking an
element $x=e_{n-1}+e_{n}$ and applying $D_{x}(x)=\Delta_{x}(x)$ we have

$D_{x}(x)=2^{n-2}\alpha_{x}e_{n-1}+2^{n-1}\alpha_{x}e_{n}=2^{n-2}\delta
e_{n-1}+\xi_{n}e_{n}=\Delta(x).$

Comparing respective coefficients, we obtain the required relation. So,
$\Delta$ is a derivation.

The example above is one generated algebra. In general, there exists a two-
generated algebra for which every local derivation is a derivation.

######  Example 3.11.

Let $\mathfrak{c}$ be a complex $6$-dimensional two-generated commutative
algebra with a basis $\mathcal{B}=\\{e_{1},e_{2},\dots,e_{6}\\}$ $(n\geq 3)$
and multiplication table

$\displaystyle e_{1}e_{2}$ $\displaystyle=e_{3},$ $\displaystyle e_{1}e_{3}$
$\displaystyle=e_{4},$ $\displaystyle e_{1}e_{4}$ $\displaystyle=e_{5},$
$\displaystyle e_{2}e_{3}$ $\displaystyle=e_{5},$ $\displaystyle e_{2}e_{4}$
$\displaystyle=e_{6},$

Direct computations show that every derivation $D$ of $\mathfrak{c}$ has the
following form:

$\begin{pmatrix}\alpha&0&0&0&0&0\\\ 0&2\alpha&0&0&0&0\\\ 0&0&3\alpha&0&0&0\\\
0&0&0&4\alpha&0&0\\\ a_{51}&a_{52}&0&0&5\alpha&0\\\
a_{61}&a_{62}&0&0&0&6\alpha\\\ \end{pmatrix}.$

Let $\Delta$ be a local derivation of $\mathfrak{c}.$ Then, it has the
following matrix form:

$\begin{pmatrix}\mu_{1}&0&0&0&0&0\\\ 0&\mu_{2}&0&0&0&0\\\ 0&0&\mu_{3}&0&0&0\\\
0&0&0&\mu_{4}&0&0\\\ b_{51}&b_{52}&0&0&\mu_{5}&0\\\
b_{61}&b_{62}&0&0&0&\mu_{6}\\\ \end{pmatrix}.$

Consider the elements $e_{1}+e_{2}+e_{3}+e_{4}\in\mathfrak{c}.$ By definition,
we have $\Delta(e_{1}+e_{2}+e_{3}+e_{4})=D(e_{1}+e_{2}+e_{3}+e_{4}).$ Hence,
we have

$\displaystyle\Delta(e_{1}+e_{2}+e_{3}+e_{4})$ $\displaystyle=\alpha
e_{1}+2\alpha e_{2}+3\alpha e_{3}+4\alpha
e_{4}+(a_{51}+a_{52})e_{5}+(a_{61}+a_{62})e_{6}$
$\displaystyle=\mu_{1}e_{1}+\mu_{2}e_{2}+\mu_{3}e_{3}+\mu_{4}e_{4}+(b_{51}+b_{52})e_{5}+(b_{61}+b_{62})e_{6}$
$\displaystyle=D(e_{1}+e_{2}+e_{3}+e_{4}).$

Comparing the corresponding coefficients, we have $\mu_{2}=2\mu_{1},\
\mu_{3}=3\mu_{1},\ \mu_{4}=4\mu_{1}.$ Applying the same argument for the
element $e_{4}+e_{5}+e_{6},$ we obtain $\mu_{5}=5\mu_{1},\ \mu_{6}=6\mu_{1}.$
Thus, $\Delta$ is a derivation.

##  4\. Local automorphisms of nilpotent Lie algebras

In this part, we will show that any $2$-step nilpotent Lie algebra admits a
local automorphism which is not derivation. Moreover, a nilpotent Lie algebra
admits pure local automorphism under the condition of Theorem 3.5. At first,
let us show that there exists a non unipotent automorphism on a $2$-step
nilpotent Lie algebra over the field of characteristic zero.

Let $\mathfrak{n}$ be a $2$-step nilpotent Lie algebra with a basis
$\mathcal{B}=\\{e_{1},e_{2},\dots,e_{n}\\}.$ Consider a linear mapping $\psi$
defined as

$\displaystyle\psi(e_{i})$ $\displaystyle=\varepsilon e_{i},\quad\mbox{if
}\quad e_{i}\in\mathfrak{n}\setminus\mathfrak{n}^{2},$
$\displaystyle\psi(e_{i})$ $\displaystyle=\varepsilon^{2}e_{i},\quad\mbox{if
}\quad e_{i}\in\mathfrak{n}^{2},$

where $\varepsilon\in\mathbb{F}\setminus\\{0\\}.$ Then, $\psi$ is an
automorphism of $\mathfrak{n}.$ Indeed, let $x,y\in\mathfrak{n}$ and
$x=\sum\limits_{i}\alpha_{i}e_{i},\ y=\sum\limits_{j}\beta_{j}e_{j}.$ Then, we
have

$\displaystyle\psi([x,y])$
$\displaystyle=\psi\Big{(}\sum_{i,j}\alpha_{i}\beta_{j}[e_{i},e_{j}]\Big{)}=\sum_{i,j}\alpha_{i}\beta_{j}\psi\left([e_{i},e_{j}]\right)$
$\displaystyle=\varepsilon^{2}\sum_{i,j}\alpha_{i}\beta_{j}[e_{i},e_{j}]=\sum_{i,j}\alpha_{i}\beta_{j}[\varepsilon
e_{i},\varepsilon e_{j}]$
$\displaystyle=\sum_{i,j}\alpha_{i}\beta_{j}[\psi(e_{i}),\psi(e_{j})]=[\psi(x),\psi(y)].$

This implies that $\psi$ is an automorphism.

Consider the derivation ${\rm D}_{\lambda}$ which is defined as (3.3). Then,
${\rm exp}({\rm D}_{\lambda})$ is an automorphism that acts on basis elements
as follows:

$\displaystyle{}\begin{array}[]{cccccccll}{\rm exp}({\rm
D}_{\lambda})(e_{i})={\bf e}^{\lambda}e_{i},&&\mbox{if
}&e_{i}\in\mathfrak{n}\setminus\mathfrak{n}^{2},\\\ {\rm exp}({\rm
D}_{\lambda})(e_{i})={\bf e}^{2\lambda}e_{i},&&\mbox{if
}&e_{i}\in\mathfrak{n}^{2},\end{array}$ (4.3)

where ${\bf e}$ is the exponential constant.

######  Theorem 4.1.

Let $\mathfrak{n}$ be a $2$-step nilpotent Lie algebra over a field
$\mathbb{F}$ $(char\mathbb{F}\neq 2).$ Then, $\mathfrak{n}$ admits a local
automorphism which is not automorphism.

###### Proof.

Let ${\rm D}_{\bf 1}$ be a derivation defined as (3.3) and $\Phi={\rm
exp}({\rm D}_{\bf 1})$ be corresponding automorphism as (4.3). Consider a
linear mapping $\nabla$ defined as follows:

$\displaystyle\nabla(x)$ $\displaystyle={\rm Id}(x),\ \mbox{ if
}x\in\mathfrak{n}\setminus\mathfrak{n}^{2},$ $\displaystyle\nabla(x)$
$\displaystyle=\Phi(x),\ \mbox{ if }x\in\mathfrak{n}^{2},$

where ${\rm Id}$ is the identity map.

We shall show that $\nabla$ is a local automorphism which is not an
automorphism. First, let us prove that it is not an automorphism. Since
$\mathfrak{n}$ is nonabelian, there exist
$e_{r},e_{t}\in\mathfrak{n}\setminus\mathfrak{n}^{2},\
e_{k}\in\mathfrak{n}^{2}$ such that $[e_{r},e_{t}]=e_{k}.$ Hence, we derive

$[\nabla(e_{r}),\nabla(e_{t})]=[e_{r},e_{t}]=e_{k}\neq{\bf
e}^{2}e_{k}=\nabla(e_{k})=\nabla([e_{r},e_{t}]),$

where ${\bf e}$ is the exponential constant. Thus, $\nabla$ is not
automorphism.

Now, we show that $\nabla$ is a local automorphism. Let
$\mathcal{B}=\\{e_{1},e_{2},\dots,e_{m},\dots,e_{n}\\}$ be a basis of
$\mathfrak{n},$ where the number $m$ means the number of generators of
$\mathfrak{n}.$ For an element $x=\sum\limits_{i=1}^{n}\lambda_{i}e_{i},$ we
have

$\nabla(x)=\sum_{i=1}^{m}\lambda_{i}e_{i}+{\bf
e}^{2}\sum_{i=m+1}^{n}\lambda_{i}e_{i}.$

Consider the following possible cases:

  * 1.

If $\lambda_{j}\neq 0$ for some $j,\ (1\leq j\leq m),$ then we choose a
derivation $D_{x}$ that acts on basis elements as follows:

$\displaystyle D_{x}(e_{j})$ $\displaystyle=\frac{{\bf
e}^{2}-1}{\lambda_{j}}\sum_{i=m+1}^{n}\lambda_{i}e_{i},$ $\displaystyle
D_{x}(e_{r})$ $\displaystyle=0,$

for all $r\in\\{1,2,\dots,m\\},\ r\neq j.$ Using this derivation, we construct
an automorphism $\Phi_{x}={\rm exp}(D_{x}).$ Having in mind $D_{x}^{2}=0,$ we
have

${\rm exp}(D_{x})={\rm Id}+D_{x}.$

Hence, we derive

$\displaystyle{\rm exp}(D_{x})(x)$ $\displaystyle={\rm
Id}(x)+D_{x}(x)=x+\sum_{i=1}^{n}\lambda_{i}D_{x}(e_{i})$
$\displaystyle=x+\lambda_{j}D_{x}(e_{j})=\sum\limits_{i=1}^{n}\lambda_{i}e_{i}+\lambda_{j}\frac{{\bf
e}^{2}-1}{\lambda_{j}}\sum_{i=m+1}^{n}\lambda_{i}e_{i}$
$\displaystyle=\sum_{i=1}^{m}\lambda_{i}e_{i}+{\bf
e}^{2}\sum_{i=m+1}^{n}\lambda_{i}e_{i}=\nabla(x).$

  * 2.

If $\lambda_{j}\neq 0$ for all $j$ $(1\leq j\leq m),$ then we choose a
derivation ${\rm D}_{\bf 1}$ and construct an automorphism $\Phi={\rm
exp}({\rm D}_{\bf 1}).$ Choosing $\Phi_{x}=\Phi$ and taking ${\rm D}_{\bf
1}^{2}=0$ into account, we derive

$\Phi_{x}(x)={\rm exp}({\rm D}_{\bf 1})(x)=x+{\rm D}_{\bf
1}(x)=\Phi(x)=\nabla(x).$

Thus, $\nabla$ is a local automorphism. The proof is complete. ∎

To illustrate Theorem 4.1, consider the following example:

######  Example 4.2.

Let $\mathfrak{h}_{n}$ be complex $(2n+1)$-dimensional Heisenberg algebra with
a basis $\mathcal{B}=\\{e_{-n},\dots,e_{-1},e_{0},e_{1},\dots,e_{n}\\}$ and
multiplication table as (3.4). Consider a linear mapping $\nabla$ defined as

$\displaystyle\nabla(e_{i})=e_{i},\quad i\neq 0,\qquad\nabla(e_{0})=2e_{0}.$

By Theorem 4.1, $\nabla$ is a local automorphism. Since

$[\nabla(e_{1}),\nabla(e_{-1})]=[e_{1},e_{-1}]=e_{0}\neq
2e_{0}=\nabla(e_{0})=\nabla([e_{1},e_{-1}]),$

we conclude that $\nabla$ is not an automorphism.

Now, let us give a sufficient condition under which a nilpotent Lie algebra
admits pure local automorphism. Interestingly, the conditions under which a
nilpotent Lie algebra admitting pure local derivation coincide with the
conditions of admitting pure local automorphism.

######  Theorem 4.3.

Let $\mathfrak{n}$ be a nilpotent Lie algebra with nilindex $p$ $(p\geq 4).$
If there exists a derivation ${\rm D}\in{\rm Der}({\mathfrak{n}})$ such that,
${\rm D}(\mathfrak{n}^{2})\subseteq Z(\mathfrak{n}),$ then $\mathfrak{n}$
admits a local automorphism which is not automorphism.

###### Proof.

Let ${\rm D}$ be derivation such that ${\rm D}(\mathfrak{n}^{2})\subseteq
Z(\mathfrak{n})$ and $\Phi={\rm exp}({\rm D})$ be an automorphism of
$\mathfrak{n}.$ Let $\mathcal{B}=\\{e_{1},e_{2},\dots,e_{m},\dots,e_{n}\\}$ be
a basis of $\mathfrak{n},$ where the number $m$ means the number of generators
of $\mathfrak{n}.$ Consider a linear mapping $\nabla$ defined as follows:

$\displaystyle\nabla(x)$ $\displaystyle={\rm Id}(x),\ \mbox{ if
}x\in\mathfrak{n}\setminus\mathfrak{n}^{2},$ $\displaystyle\nabla(x)$
$\displaystyle=\Phi(x),\ \mbox{ if }x\in\mathfrak{n}^{2},$

where ${\rm Id}$ is the identity map.

We shall prove that $\nabla$ is a local automorphism which is not an
automorphism. First, we show that it is not an automorphism. Let

$[e_{r},e_{t}]=e_{k},$

where $e_{r},e_{t}\in\mathfrak{n}\setminus\mathfrak{n}^{2},\
e_{k}\in\mathfrak{n}^{2}\setminus\mathfrak{n}^{3}.$ Without loss of
generality, we may assume that ${\rm D}(e_{k})\neq 0.$ Hence, having in mind
${\rm D}^{2}=0,$ we derive

$[\nabla(e_{r}),\nabla(e_{t})]=[e_{r},e_{t}]=e_{k}\neq e_{k}+{\rm
D}(e_{k})={\rm exp}({\rm D})(e_{k})=\nabla(e_{k})=\nabla([e_{r},e_{t}]).$

Thus, $\nabla$ is not automorphism.

Now, we show that it is a local automorphism. For an element
$x=\sum\limits_{i=1}^{n}\lambda_{i}e_{i},$ we have

$\nabla(x)=\sum_{i=1}^{n}\lambda_{i}e_{i}+\sum_{i=m+1}^{n}\lambda_{i}{\rm
D}(e_{i}).$

Consider the following possible cases:

  * 1.

If $\lambda_{j}\neq 0$ for some $j$ $(1\leq j\leq m),$ then we choose a
derivation $D_{x}$ that acts on basis elements as follows:

$\displaystyle D_{x}(e_{j})$
$\displaystyle=\frac{1}{\lambda_{j}}\sum_{i=m+1}^{n}\lambda_{i}{\rm
D}(e_{i}),$ $\displaystyle D_{x}(e_{r})$ $\displaystyle=0,$

for all $r\in\\{1,2,\dots,m\\},$ $r\neq j.$ Using this derivation, we
construct an automorphism $\Phi_{x}={\rm exp}(D_{x}).$ Taking $D_{x}^{2}=0$
into account, we have

${\rm exp}(D_{x})={\rm Id}+D_{x}.$

Hence, we derive

$\displaystyle\Phi_{x}(x)$ $\displaystyle={\rm
Id}(x)+D_{x}(x)=x+\sum_{i=1}^{n}\lambda_{i}D_{x}(e_{i})$
$\displaystyle=x+\lambda_{j}D_{x}(e_{j})=\sum\limits_{i=1}^{n}\lambda_{i}e_{i}+\lambda_{j}\frac{1}{\lambda_{j}}\sum_{i=m+1}^{n}\lambda_{i}{\rm
D}(e_{i})$
$\displaystyle=\sum_{i=1}^{n}\lambda_{i}e_{i}+\sum_{i=m+1}^{n}\lambda_{i}{\rm
D}(e_{i})=\nabla(x).$

  * 2.

If $\lambda_{j}\neq 0$ for all $j$ $(1\leq j\leq m),$ then we choose a
derivation ${\rm D}$ and construct an automorphism $\Phi={\rm exp}({\rm D}).$
Choosing $\Phi_{x}=\Phi$ and taking ${\rm D}^{2}=0$ and $x\in\mathfrak{n}^{2}$
into account, we derive

$\Phi_{x}(x)={\rm exp}({\rm D})(x)=x+{\rm D}(x)=\Phi(x)=\nabla(x).$

Thus, $\nabla$ is a local automorphism. The proof is complete. ∎

Now, we state some results which come from Theorem 4.3.

######  Corollary 4.4.

Let $\mathfrak{n}$ be a nilpotent Lie algebra with $\mathfrak{n}^{p}=0$. If
$[\mathfrak{n}^{p-3},\mathfrak{n}^{2}]\neq 0,$ then $\mathfrak{n}$ admits a
local automorphism which is not automorphism.

###### Proof.

The condition $[\mathfrak{n}^{p-3},\mathfrak{n}^{2}]\neq 0$ yields the
existence of elements $x\in\mathfrak{n}^{p-3},\ y\in\mathfrak{n}^{2}$ such
that $[x,y]\neq 0.$ Consider an inner derivation $ad_{x}.$ Since
$[x,y]\in[\mathfrak{n}^{p-3},\mathfrak{n}^{2}]\subseteq\mathfrak{n}^{p-1}\subseteq
Z(\mathfrak{n}),$ we have a derivation $D=ad_{x}$ which satisfies with the
condition of Theorem 4.3. ∎

######  Corollary 4.5.

Let $\mathfrak{n}$ be a nilpotent Lie algebra with nilindex $4$. Then,
$\mathfrak{n}$ admits a local automorphism which is not automorphism.

###### Proof.

Since the nilindex of $\mathfrak{n}$ is $4,$ we have
$[\mathfrak{n},\mathfrak{n}^{2}]=\mathfrak{n}^{3}\neq 0.$ So, by Corollary
4.4, $\mathfrak{n}$ admits a pure local derivation. ∎

######  Corollary 4.6.

Let $\mathfrak{n}$ be a two-generated nilpotent Lie algebra. Then,
$\mathfrak{n}$ admits a local automorphism which is not automorphism.

###### Proof.

In the proof of Proposition 3.9, it is shown that there exists a derivation
satisfying the condition of Theorem 4.3. ∎

As was in local derivations, the general situation of local automorphisms
might be different in one generated algebras. There exists nilpotent algebras
for which every local automorphism is automorphism. For instance, consider the
following example:

######  Example 4.7.

Let $\mathfrak{a}$ be a complex $n$-dimensional algebra with a basis
$\mathcal{B}=\\{e_{1},e_{2},\dots,e_{n}\\}$ $(n\geq 3)$ and multiplication
table is defined as (3.8). Then, direct computations show that every
automorphism $\Phi$ of $\mathfrak{a}$ acts on basis elements as follows:

$\displaystyle\Phi(e_{1})$ $\displaystyle=\alpha e_{1}+\beta e_{n},$
$\displaystyle\Phi(e_{i})$ $\displaystyle=\alpha^{2^{i-1}}e_{i},$

where $\alpha,\beta\in\mathbb{C},\ 2\leq i\leq n.$
We shall show that ${\rm LocAut(\mathfrak{a})}={\rm Aut}(\mathfrak{a}).$ From
the definition of local automorphism, we derive that any local automorphism
$\nabla$ acts on basis elements as

$\displaystyle\nabla(e_{1})$ $\displaystyle=\delta e_{1}+\zeta e_{n},$
$\displaystyle\nabla(e_{i})$ $\displaystyle=\xi_{i}e_{i},$

where $\delta,\zeta,\xi_{i}\in\mathbb{C},\ 2\leq i\leq n.$

It is enough to show that $\xi_{i}=\delta^{2^{i-1}}.$ Take an element
$x=e_{1}+e_{i},\ (2\leq i\leq n-1).$ Then, there exists a automorphism
$\Phi_{x}$ such that $\Phi_{x}(x)=\nabla_{x}(x).$ Hence, we derive

$\displaystyle\Phi_{x}(x)$
$\displaystyle=\alpha_{x}e_{1}+\alpha_{x}^{2^{i-1}}e_{i}+\beta_{x}e_{n}=\delta
e_{1}+\xi_{i}e_{i}+\zeta e_{n}=\nabla(x).$

Comparing the corresponding coefficients we have $\xi_{i}=\delta^{2^{i-1}}.$
Thus, it only remains to show that $\xi_{n}=\delta^{2^{n-1}}.$ By taking an
element $x=e_{n-1}+e_{n}$ and applying $\Phi_{x}(x)=\nabla_{x}(x)$ we have

$\displaystyle\Phi_{x}(x)$
$\displaystyle=\alpha_{x}^{2^{n-2}}e_{n-1}+\alpha_{x}^{2^{n-1}}e_{n}=\delta^{2^{n-2}}e_{n-1}+\xi_{n}e_{n}=\nabla(x).$

Comparing respective coefficients, we obtain the required relation. So,
$\nabla$ is an automorphism.

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