# DFL-TORO: A One-Shot Demonstration Framework for Learning Time-Optimal
Robotic Manufacturing Tasks

Alireza Barekatain1, Hamed Habibi1, and Holger Voos1 1Authors are with the
Automation and Robotics Research Group, Interdisciplinary Centre for Security,
Reliability, and Trust (SnT), University of Luxembourg, Luxembourg. Holger
Voos is also associated with the Faculty of Science, Technology, and Medicine,
University of Luxembourg, Luxembourg. {alireza.barekatain, hamed.habibi,
<EMAIL_ADDRESS>work is supported by FNR ”Fonds national de la
Recherche” (Luxembourg) through Industrial Fellowship Ph.D. grant (ref.
15882013).Special thanks to Dr. Paul Kremer for his guidance and support
throughout the experiments of this project.

###### Abstract

This paper presents DFL-TORO, a novel Demonstration Framework for Learning
Time-Optimal Robotic tasks via One-shot kinesthetic demonstration. It aims at
optimizing the process of Learning from Demonstration (LfD), applied in the
manufacturing sector. As the effectiveness of LfD is challenged by the quality
and efficiency of human demonstrations, our approach offers a streamlined
method to intuitively capture task requirements from human teachers, by
reducing the need for multiple demonstrations. Furthermore, we propose an
optimization-based smoothing algorithm that ensures time-optimal and jerk-
regulated demonstration trajectories, while also adhering to the robot’s
kinematic constraints. The result is a significant reduction in noise, thereby
boosting the robot’s operation efficiency. Evaluations using a Franka Emika
Research 3 (FR3) robot for a reaching task further substantiate the efficacy
of our framework, highlighting its potential to transform kinesthetic
demonstrations in contemporary manufacturing environments.

##  I Introduction

In the manufacturing industry, the transition from mass production to mass
customization [1] necessitates flexible robot programming adaptable to
frequently changing tasks[2]. Traditional programming, with its reliance on
robot experts, increases costs and downtime [3]. As a solution, Learning from
Demonstration (LfD) [4] allows robots to learn tasks flexibly from non-experts
[5]. However, deploying LfD in practical manufacturing settings poses
performance challenges [6, 4], which motivates the current work.

In robotic manufacturing, the performance can be evaluated based on
“Implementation Efficiency” and “Operation Efficiency” metrics [4].
Implementation efficiency pertains to the resources required for teaching a
task, while operation efficiency considers production speed against ongoing
costs such as maintenance and robot wear and tear. Often, the quality and the
procedure of acquiring demonstrations are sub-optimal with respect to the
mentioned metrics [7]. Our focus in this paper is to address the challenges
faced when attempting to acquire information-rich demonstrations in an
efficient manner, in order to enable LfD algorithms to optimally learn and
generalize tasks.

Figure 1: Advantages of DFL-TORO

Human demonstrations inherently dictate the requirements of a “correct”
execution of a task. While LfD’s aim is not to mirror the demonstration but to
grasp the core recipe, it is vital to assign significance to each segment of
the demonstration. This highlights how much variation the robot can tolerate
while still fulfilling the task reliably. For instance, in a Reaching task,
high precision is required while approaching a target, but the initial motion
can vary without undermining reliability. This variation can be encoded as a
set of tolerance values, capturing the task’s pivotal points and providing
flexibility. However, providing such detailed demonstrations is challenging.
Existing methods rely on multiple demonstration statistics to deduce these
tolerances, which reduces implementation efficiency.

Humans demonstrate tasks slowly due to the cognitive complexity of
illustrating various aspects simultaneously [8]. As minimizing the execution
time is a crucial factor, it is essential to obtain the most optimal timing
law. Most existing LfD methods address this by letting humans speed up initial
demonstrations [9]. However, the robot’s kinematic limits are often
overlooked, with no guarantee of achieving the optimal time in practice.
Complicating this, human demonstrations can be noisy from hand trembling or
sensor errors, resulting in jerky trajectories. Speeding up such
demonstrations can amplify the irregularities, causing erratic motions,
raising maintenance costs, and wasting energy. Moreover, LfD algorithms are
likely to overfit and mimic this noise. These issues collectively diminish
operation efficiency, underscoring the importance of smooth and time-optimal
demonstrations.

Given the outlined challenges, this paper introduces DFL-TORO, a novel
Demonstration Framework for Learning Time-Optimal Robotic tasks via One-shot
kinesthetic demonstration. DFL-TORO intuitively captures human demonstrations
and obtains task tolerances, yielding smooth, jerk-regulated, timely, and
noise-free trajectories, as illustrated in Fig. 1. Our main contributions are
as follows:

  * •

An optimization-based smoothing algorithm, considering the robot’s kinematic
limits and task tolerances, which delivers time-and-jerk optimal trajectories
and filters out the noise, enhancing operation efficiency.

  * •

A method for intuitive refinement of velocities and acquisition of tolerances,
reducing the need for repetitive demonstrations and boosting operational
efficiency.

  * •

Evaluation of DFL-TORO for a Reaching task via Franka Emika Research 3 (FR3)
robot, highlighting its superiority over conventional kinesthetic
demonstrations, using Dynamic Movement Primitives (DMPs) [10].

##  II Background and Related Work

Recent literature has shifted from focusing on robustness against sub-optimal
demonstrations [11] towards assessing the information gain and quality of
demonstrations [7, 12]. To do so, several studies have employed incremental
learning to simplify the demonstration process, enabling humans to teach task
components in phases. Authors in [13] let humans demonstrate path and timing
separately, by allowing them to adjust the speed of their initial
demonstration. The proposed framework in [9] enables path and speed refinement
through kinesthetic guidance, while the one in [8] uses teleoperated feedback
for efficient task execution. Building on these, our approach incorporates the
robot’s kinematic limits into an optimization problem, solving for the best
feasible timing law for the demonstrated path. Then, instead of merely
speeding up, we permit users to “slow down” demonstrations until reliable
execution, and hence ensure the optimal timing.

Task variability tolerances are primarily used to adjust a robot’s impedance,
e.g., low impedance when there is high variability [14], but they are also
pivotal for noise removal. Given that demonstrations do not have a set ground
truth, distinguishing signals from noise is challenging. Using tolerances as
trajectory guidelines ensures the trajectory is with the desired accuracy,
preserving the demonstration’s authenticity [15]. The study in [16] offers a
detailed comparison of various smoothing algorithms, and the noise-removing
effect of RBF-based methods like DMPs. However, without careful hyperparameter
tuning, they risk overfitting and noise replication. The study concludes that
optimization-based smoothing is more effective in preserving demonstration
features and improving information gain. Within this context, some works have
omitted tolerances [17] or used fixed user-defined tolerances [18]. However,
our approach extracts these tolerances directly from human demonstrations and
incorporates them into the optimization.

To deduce tolerances from human demonstrations, it is common to consider the
variance of multiple demonstrations, as an indicator of variability tolerance
[19, 20, 15]. To enhance implementation efficiency, we propose a novel
approach to discern tolerances from a single demonstration. Grounded in the
psychological observation that humans naturally slow down when aiming for
precision [21], our method captures tolerances during periods where the
demonstrator slows down. Although demonstrations are presented in joint space,
we extract tolerances in the Cartesian space, emphasizing the end-effector
pose as the task’s crucial point.

It has remained a challenge to simultaneously optimize for time and jerk, as
might create an infeasible problem. This stems from the fact that applying
path constraints requires knowing the relative time when each part of the path
is reached. On the other hand, such timings cannot be considered as decision
variables, since the underlying trajectory representation is a piece-wise
polynomial [22]. Also, the original timing of the demonstration is not
optimal. We address these challenges by solving the problem in two steps,
where first the timing is optimized and then the jerk profile is regulated by
removing the noise.

##  III DFL-TORO Methodology

Figure 2: DFL-TORO workflow. Red arrows indicate interactive procedures with
the human-robot in the loop.

DFL-TORO workflow (Fig. 2) is detailed as follows:

  * A)

The human teacher provides an inherently noisy demonstration $q_{o}(t)$ to a
$n$ DoF manipulator via kinesthetic guidance, where the underlying path is
extracted and represented as $m$ waypoints, denoted by $w_{i}$ for $\forall
i=1,\cdots,m$. The waypoints encode the joint configuration
$Q_{w_{i}}\in\mathbb{R}^{n}$, as well as the end-effector’s position
$p_{w_{i}}=[x_{w_{i}},y_{w_{i}},z_{w_{i}}]^{T}\in\mathbb{R}^{3}$ and
orientation $\theta_{w_{i}}\in\mathbb{S}^{3}$.

  * B)

“Time Optimization” module computes the ideal timing $\tau_{i}\in[0,1]$ for
$\forall i=1,\cdots,m$ for each waypoint.

  * C)

“Trajectory Generation” module solves a comprehensive optimization problem to
minimize the jerk. Given that specific task tolerances are not extracted yet,
default tolerance values $\epsilon^{d}_{p}\in\mathbb{R}^{3}$ and
$\epsilon^{d}_{\theta}\in\mathbb{R}$ are incorporated to generate an initial
trajectory. $\epsilon^{d}_{p}$ represents the tolerance in Cartesian position,
while $\epsilon^{d}_{\theta}$ signifies the angular difference tolerance.
Consequently, the outcome of this hierarchical optimization procedure is a
minimum-time, smooth trajectory $q_{f}(t)$.

  * D)

“Refinement Phase” allows the teacher to interactively slow down and correct
the timing law. The robot replays the trajectory, allowing the human teacher
to visually assess the execution speed. The teacher can pinpoint areas where
the speed is either unreliable or unsafe and determine which segments require
slowing down. Since the current trajectory $q_{f}(t)$ executes very rapidly,
it leaves no time for humans to observe the motion and provide feedback.
Therefore, the Refinement Phase operates at a reduced speed
$V_{0}^{r}\in\mathbb{R}_{+}$, giving humans enough reaction time to make
changes.

  * E)

Refinement Phase provides revised timings for the trajectory, denoted as
$\tau_{i}^{r}\in[0,1]$, as well as the task tolerances
$\epsilon_{p}^{i}\in\mathbb{R}^{3}$ and $\epsilon_{\theta}^{i}\in\mathbb{R}$
extracted from human feedback. These updated values are subsequently fed to
Trajectory Generation module which leads to a fine-tuned trajectory
$q_{f}^{r}(t)$ in accordance with the new timings and tolerances.

The steps of the proposed demonstration framework is provided in Algorithm 1,
and elaborated in the following.

###  III-A Optimization-based Smoothing

We employ B-Splines to represent trajectories in our approach. B-Splines are
ideal for robot manipulator trajectory optimization due to their smoothness,
local control, and efficient computation [23]. Their piecewise polynomial
nature ensures smooth motion paths, making them highly suitable for complex
motion planning. A B-spline curve is defined as a linear combination of
control points $P_{i^{*}},{i^{*}}=1,\cdots,\bar{n}$ and B-spline basis
functions $N_{{i^{*}},k}(s)$. Control points $P_{i^{*}}$ determine the shape
of the curve. A B-Spline $\xi(s)$ of $k^{th}$ order with $\bar{n}$ control
points $P_{1},P_{2},...,P_{\bar{n}}$ is expressed as:

$\displaystyle\xi(s)=\sum_{{i^{*}}=1}^{\bar{n}}N_{{i^{*}},k}(s)P_{i^{*}},$ (1)

where $s\in[0,1]$ is the normalized time. Let $u_{1}\leq u_{2}\leq\ldots\leq
u_{\bar{m}}$ be the knots and $\bar{m}=\bar{n}+k$, with

$\displaystyle N_{{i^{*}},1}(s)$ $\displaystyle=\begin{cases}1&\text{if
}u_{i^{*}}\leq s<u_{{i^{*}}+1}\\\ 0&\text{otherwise}\end{cases},$
$\displaystyle N_{{i^{*}},k}(s)$
$\displaystyle=\frac{s-u_{i^{*}}}{u_{{i^{*}}+k-1}-u_{i^{*}}}N_{{i^{*}},k-1}(s)$
$\displaystyle+\frac{u_{{i^{*}+k}}-s}{u_{{i^{*}+k}}-u_{{i^{*}+1}}}N_{{i^{*}+1},k-1}(s).$

$\xi(s)$ is associated with a duration parameter $T$, which maps the
normalized time $s$ into actual time $t$ [23]. $P_{i^{*}}$ and $T$ are the
optimization variables. In the following, we describe two modules of
optimization-based smoothing.

####  III-A1 Time Optimization Module

The objective of this optimizer is to find the minimum-time trajectory
$q_{t}(t)$ that strictly passes through all the waypoints $Q_{w_{i}}$. At this
stage, we ignore constraints related to acceleration and jerk since the noise
is still present in the path. Adding such constraints would prevent the
optimizer from finding the ideal timings. B-Spline sub-trajectories $\xi_{j}$
for $\forall j=1,\cdots,m-1$ are used to represent the trajectory between
every two adjacent waypoints $w_{j}$ and $w_{j+1}$, with control points
$P_{{i^{*}},j}$ and durations $T_{j}$. Then, the time optimization problem is
formulated as:

$\displaystyle(P^{*}_{{i^{*}},j},T^{*}_{j})$
$\displaystyle=\underset{P_{{i^{*}},j},T_{j}}{\mathrm{arg\,min}}\sum_{j=1}^{m-1}T_{j},$
(2a) $\displaystyle q_{j}(t_{j})$
$\displaystyle=\xi_{j}(\frac{t_{j}}{T_{j}}),$ (2b) $\displaystyle
q_{\text{min}}$ $\displaystyle\leq q_{j}(t_{j})\leq q_{\text{max}},$ (2c)
$\displaystyle v_{\text{min}}$ $\displaystyle\leq\dot{q}_{j}(t_{j})\leq
v_{\text{max}},$ (2d) $\displaystyle\xi_{j}(0)$ $\displaystyle=Q_{w_{j}},$
(2e) $\displaystyle\xi_{j}(1)$ $\displaystyle=Q_{w_{j+1}},$ (2f)
$\displaystyle\dot{\xi}_{1}(0)$ $\displaystyle=\dot{\xi}_{m-1}(1)=0,$ (2g)

for $\forall t_{j}\in[0,T_{j}]$. (2b) relates the normalized and actual
trajectories. We set bounds $q_{\text{min}}$, $q_{\text{max}}$,
$v_{\text{min}}$ and $v_{\text{max}}\in\mathbb{R}^{n}$ for joint position and
velocity as (2c) and (2d), respectively. Note that “$\leq$” represents
element-wise inequality of vectors. Continuity constraints (2e) and (2f) are
applied at the intersection of these sub-trajectories. (2g) enforces the
trajectory to start and rest with zero velocity. The optimization yields an
unrealistic trajectory due to the existing noise, with high acceleration and
jerk values. (2a) determines the ideal timings $T^{*}_{j}$ to move through all
waypoints with highest velocity, which provides normalized time $\tau_{i}$
defined as

$\displaystyle\tau_{i}$
$\displaystyle=\begin{cases}\frac{\sum_{j=1}^{i-1}T^{*}_{j}}{T^{*}_{m-1}}&i\geq
2\\\ 0&i=1\end{cases},$ (3)

which sets the groundwork for Trajectory Generation module, that addresses the
full optimization problem.

####  III-A2 Trajectory Generation Module

Given $\tau_{i}$, we fit one B-Spline across all the waypoints, denoted as
$\xi_{f}$ with control points $P_{i,f}$ and duration variable $T_{f}$. We
match the number of control points with the waypoints, i.e., $\bar{n}=m$,
giving flexibility to the optimizer to locally adjust the trajectory around
each waypoint. The cost function is formulated as:

$\displaystyle J_{f}=\alpha
T_{f}+\beta\int_{0}^{1}||\dddot{\xi}_{f}||^{2}ds+\gamma\sum_{i=1}^{m}||P_{i,f}-Q_{w_{i}}||^{2},$
(4)

where $\alpha,\beta$ and $\gamma$ are positive weights. The term
$\int_{0}^{1}||\dddot{\xi}_{f}||^{2}ds$ attempts to minimize the normalized
jerk profile, exploiting the tolerances of waypoints. This strategy is crucial
for eliminating noise while respecting the tolerances. The term
$\sum_{i=1}^{m}||P_{i,f}-Q_{w_{i}}||^{2}$ ensures that control points align
closely with the original joint configurations at the waypoints. Given the
robot manipulator’s kinematic redundancy, an end-effector pose can correspond
to multiple configurations. This term limits the robot’s configuration null
space, prompting the optimizer to remain near the initial configuration. The
trajectory generation problem is formulated as follows:

$\displaystyle(P^{*}_{i,f},T^{*}_{f})$
$\displaystyle=\underset{P_{i,f},T_{f}}{\mathrm{arg\,min}}\,J_{f},$ (5a)
$\displaystyle q_{f}(t)$ $\displaystyle=\xi_{f}(\frac{t}{T_{f}}),$ (5b)
$\displaystyle q_{\text{min}}$ $\displaystyle\leq q_{f}(t)\leq
q_{\text{max}},$ (5c) $\displaystyle v_{\text{min}}$
$\displaystyle\leq\dot{q}_{f}(t)\leq v_{\text{max}},$ (5d) $\displaystyle
a_{\text{min}}$ $\displaystyle\leq\ddot{q}_{f}(t)\leq a_{\text{max}},$ (5e)
$\displaystyle J_{\text{min}}$ $\displaystyle\leq\dddot{q}_{f}(t)\leq
J_{\text{max}},$ (5f) $\displaystyle\xi_{f}(0)$ $\displaystyle=Q_{w_{1}},$
(5g) $\displaystyle\xi_{f}(1)$ $\displaystyle=Q_{w_{m}},$ (5h)
$\displaystyle\dot{\xi}_{f}(0)$ $\displaystyle=\dot{\xi}_{f}(1)=0,$ (5i)
$\displaystyle(p_{f}(t),\theta_{f}(t))$ $\displaystyle=F_{K}(q_{f}(t)),$ (5j)
$\displaystyle-\epsilon_{p}^{i}\leq p_{f}($
$\displaystyle\tau_{i}T_{f})-p_{w_{i}}\leq\epsilon_{p}^{i},$ (5k)
$\displaystyle\mathsf{QuatDiff}$
$\displaystyle(\theta_{f}(\tau_{i}T_{f}),\theta_{w_{i}})\in[0,\epsilon_{\theta}^{i}],$
(5l)

for $\forall t\in[0,T_{f}]$. We have introduced acceleration and jerk bounds
$a_{\text{min}}$, $a_{\text{max}}$, $J_{\text{min}}$ and
$J_{\text{max}}\in\mathbb{R}^{n}$ in (5e) and (5f), respectively, to ensure
the trajectory’s feasibility. Constraints (5g)-(5i) enforce the trajectory to
start at $Q_{w_{1}}$ and rest at $Q_{w_{m}}$ with zero velocity. (5j)
represents the forward kinematic function $F_{K}$, which is used to assign the
tolerance in the end-effector’s Cartesian space with
$p_{f}(t)=[x_{f}(t),y_{f}(t),z_{f}(t)]\in\mathbb{R}^{3}$ and
$\theta_{f}(t)\in\mathbb{S}^{3}$. The goal is to limit the position and angle
deviation of the end-effector at each waypoint $w_{i}$ to $\epsilon_{p}^{i}$
and $\epsilon_{\theta}^{i}$, via (5k) and (5l), respectively, at normalized
time $\tau_{i}$. In (5k), “$\leq$” represents element-wise inequality of
vectors. $\mathsf{QuatDiff}(.,.)\in[0,\pi]$ is a conventional function to
compute the absolute angle difference of two quaternions [24]. Solving (5a)
yields $P^{*}_{i,f}$, with which $q_{f}(t)$ is computed using (1) and (5b).

(A): Kinesthetic Guidance

Input: Noisy human demonstration $q_{o}(t)$

Output: Path waypoints $W_{i}=\left(Q_{w_{i}},p_{w_{i}},\theta_{w_{i}}\right)$

(B): Time Optimization

Input: $W_{i}$

Find ideal timings using (2a)-(2g) and (3)

Output: Initial timings $\tau_{i}$

(C): Trajectory Generation

Input: $W_{i}$, $\tau_{i}$, default tolerances
$\epsilon_{p}^{d},\epsilon_{\theta}^{d}$

Solve for smooth trajectory using (5a)-(5l)

Output: Trajectory $q_{f}(t)$

(D): Refinement Phase

Input: $q_{f}(t)$, human command $C(t),\eta$

Initialize $v(0)=V_{0}^{r},s_{r}(0)=V_{0}^{r}\cdot t$

while  _Replaying Trajectory_ do

Compute $R(t)$ using (6)

Compute $s_{r}(t)$ using (7)

Send $\xi(s_{r}(t))$ to robot for interactive feedback

end while

Compute new timings $\tau_{i}^{r}$ using (8)

for  _each $w_{i}$_ do

$\epsilon_{p}^{i}\leftarrow\Gamma_{p}(\tau_{i})$ using (9)

$\epsilon_{\theta}^{i}\leftarrow\Gamma_{\theta}(\tau_{i})$ using (9)

end for

Output: New timings $\tau_{i}^{r}$, new tolerances
$\epsilon_{p}^{i},\epsilon_{\theta}^{i}$

(E): Trajectory Fine-Tuning

Input: $W_{i},\tau_{i}^{r},\epsilon_{p}^{i},\epsilon_{\theta}^{i}$

Re-optimize $q_{f}(t)$ via updated timings and tolerances using (5a)-(5l)

Output: Optimal Demonstration $q_{f}^{r}(t)$

Algorithm 1 DFL-TORO Architecture

(a) Original Demonstration $q_{o}(t)$

(b) Time Optimization Output $q_{t}(t)$

(c) Trajectory Generation Output $q_{f}(t)$

Figure 3: Evolution of RT1 from original to smooth time-optimal joint
trajectory. Each color represents a distinct DoF.

###  III-B Refinement Phase

During Refinement Phase, the teacher can practically slow down the time
progression and hence the trajectory velocity. To do so, the trajectory
$q_{f}(t)$ with duration $T_{f}$ is replayed with a reduced speed
$V_{0}^{r}=\frac{1}{\eta T_{f}}$ for $\eta>1$, letting the human teacher
observe and provide interactive feedback in the loop. This is achieved through
a teleoperated command, $C(t)\in[-1,0]$, which functions like a brake pedal.
It allows the teacher to safely and intuitively interact with the robot in
real-time, controlling its velocity during execution. To attenuate noise or
abrupt changes in $C(t)$, we pass it as the attractor point of a virtual
spring-damper system:

$\displaystyle\tau\ddot{R}(t)=K(C(t)-R(t))-D\dot{R}(t).$ (6)

By carefully setting the values of $\tau$, $K$, and $D$, we can configure (6)
to be critically damped, as well as adjust its settling time which, in turn,
influences the delay in response to human command. The output $R(t)\in[-1,0]$
is used as a deceleration term to adjust the trajectory velocity, as follows.

####  III-B1 Velocity Adjustment

Since $q_{f}(t)=\xi_{f}(s(t))$ where $s(t)=V_{0}^{r}\cdot t$, by adjusting
$s(t)$ we can directly modify the timing law of the trajectory $q_{f}(t)$.
Slowing down specific portions of $\xi_{f}(s(t))$ locally can be done by
incorporating $R(t)$ as a notion of time deceleration, which leads to a
modified mapping $s_{r}(t)$. We compute this new mapping throughout the
refinement phase via a straightforward kinematic model:

$\displaystyle\left\\{\begin{matrix}\dot{v}(t)=R(t)\\\
\dot{s}_{r}(t)=v(t)\end{matrix}\right.,\quad\begin{matrix}v(t)\in[V_{min}^{r},V^{r}_{0}],\\\
s_{r}(t)\in[0,1]\end{matrix}$ (7)

To prevent the robot from completely stopping, we bound $v(t)$ to a minimum
execution speed $V_{min}^{r}\in\mathbb{R}_{+}$. $s_{r}(t)$ and $\tau_{i}$ are
used to determine the updated timing $\tau_{i}^{r}$ as

$\displaystyle\tau_{i}^{r}=\frac{s_{r}^{-1}(\tau_{i})}{s_{r}^{-1}(\tau_{m})},$
(8)

where $s_{r}^{-1}$ is the inverse of $s_{r}$ obtained via interpolation. It is
worth mentioning that refinement at the reduced speed $V_{0}^{r}$ applies the
same proportional adjustment on the real speed of execution, as $\tau_{i}^{r}$
are in normalized time.

####  III-B2 Tolerance Extraction

To extract the tolerances, we posit a direct correlation between the value of
$R(t)$ and the desired level of accuracy at a given point on the trajectory.
This means, the more intensively the brake pedal is pressed, the greater the
criticality of that trajectory portion, necessitating increased precision.
This is captured by introducing the functions $\Gamma_{p}(t)$ and
$\Gamma_{\theta}(t)$, defined as:

$\displaystyle\Gamma_{\kappa}(t)=(\epsilon_{\kappa}^{max}-\epsilon_{\kappa}^{min})(1-(-R(t))^{n+\zeta})+\epsilon_{\kappa}^{min},$
(9)

where $\kappa\in\\{p,\theta\\}$. In (9), $R(t)=0$ is associated with maximum
tolerances $\epsilon_{p}^{max}$ and $\epsilon_{\theta}^{max}$, whereas a
$R(t)=-1$ corresponds to the minimum tolerances $\epsilon_{p}^{min}$ and
$\epsilon_{\theta}^{min}$. The parameters $\zeta$ and $n$ determine the shape
and curve of $\Gamma_{\kappa}$. By computing $\Gamma_{\kappa}(\tau_{i})$ for
each waypoint, we extract the requisite tolerances and pass it along with
$\tau_{i}^{r}$ for re-optimization.

Figure 4: $s(t)$ and $s_{r}(t)$ during the Refinement Phase of RT1. $t^{*}$
marks the moment the teacher starts giving command $C(t)$.

##  IV Validation Experiments

Figure 5: RT1 fine tuning. Upper: $x_{f}^{r}(t)$, with the extracted tolerance
ranges via $R(t)$. Lower: waypoints timing law before and after refinement.
Figure 6: $x_{f}^{r}(t)$ with low and high tolerance ranges, for RT2.

(a) 3D path

(b) Kinematic violations

(c) Jerk profiles

Figure 7: Performance comparison of ${\mathrm{DMP}}_{o}$ and
${\mathrm{DMP}}_{f}$ with respect to kinematic feasibility and jerk profiles.

The effectiveness of DFL-TORO is assessed on a 7 DoF FR3 robot in two reaching
task scenarios. The performance of Optimization-based Smoothing is analyzed in
Sec. IV-B, covering stages (B) and (C) of Algorithm 1. In Sec. IV-C, we
examine Refinement Phase (stages (D) and (E)). Using DMP as the learning
method, a baseline comparison to the state-of-the-art kinesthetic
demonstration is conducted in Sec. IV-D. The choice of DMP is due to the fact
that it requires only one demonstration to train. Nevertheless, our proposed
method is generic and can be applied to any LfD approach. Note that in order
to meaningfully evaluate trajectories with different durations, Maximum
Absolute Normalized Jerk (MANJ) in normalized time $s\in[0,1]$ is used.
Furthermore, we only illustrate one axis of position trajectory for brevity.

###  IV-A Experimental Setup

Demonstrations are recorded through kinesthetic guidance using a ROS interface
at a frequency of 10 Hz. For teleoperation, we utilize a PS5 wireless
controller, receiving continuous brake commands. Waypoints are automatically
selected based on an end-effector movement threshold, requiring a shift of at
least 1 cm or 0.1 radians for Reaching Task #1 (RT1) and Reaching Task #2
(RT2). The kinematic limits of FR3 are obtained from the robot’s datasheet
[25]. B-Splines of order $k=4$ are selected for smoothness up to the velocity
level. $u_{1},\cdots,u_{\bar{m}}$ are distributed in a clamped uniform manner
[23] to create basis functions in (1). Time Optimization and Trajectory
Generation modules are implemented using the Drake toolbox [26]. The weights
in (4) are set as $\alpha=1$, $\beta=0.04$, and $\gamma=1$. Finally, default
tolerance values are chosen as $\epsilon^{d}_{p}=[2,2,2]$ cm and
$\epsilon^{d}_{\theta}=0.1$ radians. In Refinement Phase, we select $\eta=5$,
$V_{min}^{r}=0.2V_{0}^{r}$, $\tau=0.1$, $D=20$, and $K=100$. Also, we choose
$\epsilon_{p}^{max}=[5,5,5]$ cm, $\epsilon_{p}^{min}=[1,1,1]$ cm,
$\epsilon_{\theta}^{max}=0.3$ radians, and $\epsilon_{\theta}^{min}=0.1$
radians. For illustration,
$p_{f}^{r}(t)=[x_{f}^{r}(t),y_{f}^{r}(t),z_{f}^{r}(t)]$, representing the
Cartesian position trajectory associated with $q_{f}^{r}(t)$, is considered.
Besides lowering tolerances in the precise portions, we assign maximum
tolerance for $C(t)=0$, providing more freedom for Trajectory Generation to
adapt the motion. This is best shown via RT2, where higher tolerances allow
the optimizer to find better solutions. We implement DMP in the joint space
via [27], to study the effectiveness of DFL-TORO on RT1 in terms of kinematic
feasibility and jerk regulation.

###  IV-B Performance of Optimization-based Smoothing

The original demonstration’s joint trajectory $q_{o}(t)$ is depicted in Fig.
3(a). The original demonstration’s velocity, acceleration, and jerk are
differentiated in a non-casual manner [27]. Notably, the demonstration
trajectory is slow, while noise reflects itself in the jerk profile. Fig. 3(b)
shows the output of the Time Optimization module, stage (B), yielding
$q_{t}(t)$ with $\tau_{i}$ extracted via (3). Using $\tau_{i}$ along with
$\epsilon^{d}_{p}$ and $\epsilon^{d}_{\theta}$ in stage (C) leads to
$q_{f}(t)$, a smooth, noise-free, and time-optimal trajectory shown in Fig.
3(c). $q_{o}(t)$ and $q_{f}(t)$ are compared in Table I considering the
metrics of execution time and MANJ. Evidently, DFL-TORO has significantly
improved the demonstration quality.

###  IV-C Demonstration Refinement

Given $C(t)$, the progression of $s(t)$, $s_{r}(t)$ and $R(t)$ (using (6)) are
depicted in Fig. 4 (stage (D)). Initially, without any command, $s(t)$ and
$s_{r}(t)$ overlap, leaving the timing law unchanged. Upon receiving the
command at $t^{*}$, $s_{r}(t)$ progression changes to slow down the motion and
thus extend execution time. This revised mapping leads to the derivation of
$\tau_{i}^{r}$ using (8). Also, we extract $\epsilon_{p}^{i}$ and
$\epsilon_{\theta}^{i}$ via (9). Fig. 5 shows $x_{f}^{r}(t)\in p_{f}^{r}(t)$,
the fine-tuned trajectory (stage (E)). The tolerance range is adjusted to be
more precise towards the end of reaching, where slow speed was commanded,
shown in Fig. 4. Moreover, to showcase the velocity adjustment effect on the
trajectory’s timing law, we present the distribution of $y_{w_{i}}$ over the
trajectory before and after refinement. Evidently, the waypoints are stretched
towards the end of the task. It is also noticeable that while the timing law
of the beginning of the trajectory is left unchanged during refinement, the
waypoint distribution does not overlap. This is due to the fact that since the
tolerance values are increased from $\epsilon_{p}^{d}$ to
$\epsilon_{p}^{max}$, as in (9), the optimizer has managed to find a faster
trajectory to traverse these waypoints. Fig. 6 shows the effect of increasing
tolerance values in RT2, which leads to a 15% reduction in cycle time and a
50% improvement in reducing the maximum jerk.

###  IV-D Baseline Comparison

For the same start and goal configuration, we train ${\mathrm{DMP}}_{o}$ via
$q_{o}(t)$, and ${\mathrm{DMP}}_{f}$ via $q_{f}(t)$, reproducing
$q_{o,dmp}(t)$ and $q_{f,dmp}(t)$, respectively. Since $q_{o}(t)$ is slower
than $q_{f}(t)$, we leverage the inherent scaling factor in
${\mathrm{DMP}}_{o}$ to produce $q_{s,dmp}(t)$ [10], to have the same duration
as $q_{f,dmp}(t)$ for meaningful comparison. Notation-wise, $p_{f,dmp}(t)$ ,
$p_{o,dmp}(t)$ and $p_{s,dmp}(t)$ represent the associated Cartesian position
trajectories. Fig. 7(a) shows the path of $p_{o,dmp}(t)$ and $p_{f,dmp}(t)$.
As shown, $p_{o,dmp}(t)$ has produced unnecessary curves at the beginning of
the motion, replicating the existing noise. The noise causes $q_{s,dmp}(t)$ to
become an infeasible trajectory. The kinematic violations are highlighted in
Fig. 7(b), illustrating the infeasibility of $q_{s,dmp}(t)$ while
$q_{f,dmp}(t)$ remains feasible for the same execution time. Furthermore, a
comparison of the jerk values for $q_{s,dmp}(t)$ and $q_{f,dmp}(t)$ is shown
in Fig. 7(c), showcasing that $q_{f,dmp}(t)$ creates significantly lower jerk
values. Table I summarizes the time and jerk values for different trajectories
associated with RT1. Obviously, the execution time and MANJ metrics are
significantly reduced by DFL-TORO. This is further verified by comparing
${\mathrm{DMP}}_{o}$ and ${\mathrm{DMP}}_{f}$. Even though DMP naturally
smoothens the original demonstration, $q_{f,dmp}(t)$ still outperforms
$q_{o,dmp}(t)$ by a significant margin. Furthermore, although $q_{s,dmp}(t)$
is scaled to the optimal time, it yields an infeasible trajectory with high
jerk values, comparing $q_{s,dmp}(t)$ and $q_{f,dmp}(t)$.

TABLE I: Comparison of time and Maximum Absolute Normalized Jerk (MANJ) for
RT1.

| Trajectory | Time [$s$] | MANJ [$rad/s^{3}$]
---|---|---|---
Demonstration | $q_{o}(t)$ | $8.88$ | $15\,358$
$q_{f}(t)$ | 1.2 | 78.3
DMP | $q_{o,dmp}(t)$ | $8.88$ | $2305.9$
$q_{s,dmp}(t)$ | $1.2$ | $1837$
$q_{f,dmp}(t)$ | 1.2 | 101.5

##  V Conclusion

In this paper, we presented DFL-TORO, a novel demonstration framework within
the LfD process, addressing the poor quality and efficiency of human
demonstrations. The framework captures task requirements intuitively with no
need for multiple demonstrations, as well as locally adjusting the velocity
profile. The obtained trajectory was ensured to be time-optimal, noise-free,
and jerk-regulated while satisfying the robot’s kinematic constraints. The
effectiveness of DFL-TORO was experimentally evaluated using a FR3 in a
reaching task scenario. Future research can explore the richness of
information in tolerance values and integrate the task’s semantic nuances.
Additionally, by tailoring the learning algorithm, we can improve DMP
performance, ensuring optimal output trajectories and more effective motion
generalization. Reinforcement Learning is a suitable option for improving DMP
generalization in manufacturing contexts.

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On the Schur multipliers of Lie superalgebras of maximal class

###### Abstract.

Let $L$ be a non-abelian nilpotent Lie superalgebra of dimensiom $(m|n)$.
Nayak shows there is a non-negative $s(L)$ such that
$s(L)=\frac{1}{2}(m+n-2)(m+n-1)+n+1-\dim{\mathcal{M}(L)}$. Here we intend that
classify all non-abelian nilpotent Lie superalgebras, when $1\leq s(L)\leq
10$. Moreover, we classify the structure of all Lie superalgebras of dimension
at most $5$ such that $\dim{L^{2}}=\dim{\mathcal{M}(L)}$.

###### Key words and phrases:

Schur multiplier, Nilpotent Lie superalgebra, Maximal class.

Mathematics Subject Classification 2010. 17B01, 17B05, 17B30, 19C09.

##  1\. Introduction and preliminaries

The theory of Lie superalgebra which physicists know as
$\mathbb{Z}_{2}$-graded Lie algebra was first introduced by Kac in [8]. This
theory is important and useful, because of the connection between Lie
superalgebras and Lie groups. Recently, many results in representation and
classification are obtained by using Lie superalgebra theory. For more
information see [1, 5, 8, 10, 11, 12, 16] and references given in them. We
know for non-zero integer $t(L)$, the dimension of the Schur multiplier is
$\frac{1}{2}n(n-1)-t(L)$. Looking at the dimension of the Schur multiplier is
one of the ways to determine the structure of all nilpotent Lie algebras. This
method was studied by several authors for $0\leq t(L)\leq 8$ in [2, 6, 7].
Also, the Lie algebras of maximal class have been classified for $0\leq
t(L)\leq 16$. On the other hand for a non-abelian nilpotent Lie algebra $L$,
let $s(L)=\frac{1}{2}(n-1)(n-2)+1-\dim\mathcal{M}(L)$. We know $s(L)>0$ and
the structure of all Lie algebras are given where $0\leq s(L)\leq 3$ (for more
information see [13, 14, 18]). Also in [19], the structure of all Lie algebras
of maximal class for $0\leq s(L)\leq 15$ is given. Recently, Nayak in [11]
introduced a bound for the dimension of Schur multipliers of Lie superalgebras
which play similar role in characterization of finite dimensional nilpotent
Lie superalgebras as that of Lie algebras. In this paper, we will characterize
the structure of all Lie superalgebras of maximal class $L$, when $1\leq
s(L)\leq 10$, and similar to the result of [19], we also give the structure of
all nilpotent Lie superalgebra of maximal class $L$ when
$\dim\mathcal{M}(L)=\dim L^{2}$. Note that throughout this paper, all modules
and algebras are defined over an unital commutative ring $\mathbb{K}$. Here,
we give some notations on Lie superalgebras that are given in [5]. Let
$\mathbb{Z}_{2}=\\{0,1\\}$ be a field and we put $(-1)^{\bar{0}}=1$ and
$(-1)^{\bar{1}}=-1$. A $\mathbb{Z}_{2}$-graded algebra (or superalgebra) $M$
is a direct sum of algebras $M_{\bar{0}}$ and $M_{\bar{1}}$
($M=M_{\bar{0}}\oplus M_{\bar{1}}$), whose elements are called even and odd,
respectively. Non-zero elements of $M_{\bar{0}}\cup M_{\bar{1}}$ are said to
be homogeneous. For a homogeneous element $m\in M_{\bar{\alpha}}$ with
$\alpha\in\mathbb{Z}_{2}$, $|m|=\bar{\alpha}$ is the degree of $m$. So
whenever we have the notation $|m|$, $m$ will be a homogeneous element. A
subalgebra $N$ of $M$ is called $\mathbb{Z}_{2}$-graded subalgebra (or sub
superalgebra), if $N=N_{\bar{0}}\oplus N_{\bar{1}}$ where $N_{\bar{0}}=N\cap
M_{\bar{0}}$ and $N_{\bar{1}}=N\cap M_{\bar{1}}$.

######  Definition 1.1.

[5] A Lie superalgebra is a superalgebra $M=M_{\bar{0}}\oplus M_{\bar{1}}$
with a multiplication denoted by $[.,.]$, called super bracket operation,
satisfying the following identities

  1. (i)

$[x,y]=-(-1)^{|x||y|}[y,x],$

  2. (ii)

$[x,[y,z]]=[[x,y],z]+(-1)^{|x||y|}[y,[x,z]],$

  3. (iii)

$[m_{\bar{0}},m_{\bar{0}}]=0$

for all homogeneous elements $x,y,z\in M$ and $m_{\bar{0}}\in M_{\bar{0}}$.

Note that the last equation is easily derived from the first equation, in this
case, $2$ is invertible in $\mathbb{K}$. The second equation is equivalent to
the following graded Jacobi identity
$(-1)^{|x||z|}[x,[y,z]]+(-1)^{|y||x|}[y,[z,x]]+(-1)^{|z||y|}[z,[x,y]]=0$. By
using above identities, it can be seen that for a Lie superalgebra
$M=M_{\bar{0}}\oplus M_{\bar{1}}$, the even part $M_{\bar{0}}$ is a Lie
algebra and the odd part $M_{\bar{1}}$ is a $M_{\bar{0}}$-module. Hence if
$M_{\bar{1}}=0$, then $M$ is a Lie algebra and if $M_{\bar{0}}=0$, then $M$ is
an abelian Lie superalgebra (i.e. for all $x,y\in M$, $[x,y]=0$). But in
general a Lie superalgebra is not a Lie algebra. The sub superalgebra of $L$
is a $\mathbb{Z}_{2}$-graded vector subspace which is closed under bracket
operation. Take $[L,L]$, it is an graded subalgebra of $L$ and is denoted as
$L^{2}$. A $\mathbb{Z}_{2}$-graded subspace $I$ is a graded ideal of $L$ if
$[I,L]\subseteq I$ and for all $x\in L$ the ideal $Z(L)=\\{z\in L;[z,x]=0\\}$
is a graded ideal and it is called the center of $L$. If $I$ is an ideal of
$L$, the quotient Lie superalgebra $L/I$ inherits a canonical Lie superalgebra
structure such that the natural projection map becomes a homomorphism. The
notions of epimorphisms, isomorphisms and auotomorphisms have the obvious
meaning. According to the super dimension structure of Lie superalgebras over
a field, we say that $L=L_{\bar{0}}\oplus L_{\bar{1}}$ is an $(m,n)$ Lie
superalgebra, if $\dim L_{\bar{0}}=m$ and $\dim L_{\bar{1}}=n$. Also
throughout $A(m|n)$ denotes an abelian Lie superalgebra with $\dim A=(m|n)$.
The descending central sequence of a Lie superalgebra $L$ is defined by
$L^{1}=L$ and $L^{c+1}=[L^{c},L]$, for all $c\geq 1$. If for some positive
integer $c$, $L^{c+1}=0$ and $L^{c}\neq 0$, then $L$ is called nilpotent with
nilpotency class $c$. Also we have $|[m,n]|=|m|+|n|$. By a homomorphism
between superspaces $f:V\to W$ of degree $|f|\in\mathbb{Z}_{2}$, we mean a
linear map satisfying $f(V_{\bar{\alpha}})\subseteq W_{\bar{\alpha}+|f|}$ for
$\alpha\in\mathbb{Z}_{2}$. In particular, if $|f|=\bar{0}$, then the
homomorphism $f$ is called homogeneous linear map of even degree. A Lie
superalgebra homomorphism $f:M\to N$ is a homogeneous linear map of even
degree such that $f([x,y])=[f(x),f(y)]$ holds for all $x,y\in L$. Now we
present some results related to the Schur multiplier of a Lie superalgebra L.

######  Theorem 1.2 (11, Theorem 3.3).

Let $L$ be a Lie superalgebra with $\dim{L}=(m|n)$. Then
$\dim{\mathcal{M}(L)}\leq\frac{1}{2}[(n+m)^{2}+(n-m)]$.

As a consequence, there is a non-negative integer $t(L)$, such that
$\dim{\mathcal{M}(L)}=\frac{1}{2}[(n+m)^{2}+(n-m)]-t(L)$.

######  Theorem 1.3 (11, Theorem 3.4).

Let $L$ be a Lie superalgebra with $\dim{L}=(m|n)$. Then
$\dim{\mathcal{M}(L)}=\frac{1}{2}[(n+m)^{2}+(n-m)]$ if and only if $L$ is
abelian.

As a result, we can say that $t(L)=0$ if and only if $L$ is abelian.

######  Theorem 1.4 (11, Theorem 5.1).

Let $L=L_{\bar{0}}\oplus L_{\bar{1}}$ be a nilpotent Lie superalgebra of
$\dim{L}=(m|n)$ and $\dim{L^{2}}=r+s$ with $r+s\geq 1$. Then

$\dim{\mathcal{M}(L)}\leq\frac{1}{2}[(m+n+r+s-2)(m+n-r-s-1)]+n+1.$

Moreover if $r+s=1$, then equality holds if and only if $L\cong H(1,0)\oplus
A(m-3|n)$, where $A(m-3|n)$ is an abelian Lie superalgebra with dimension
$(m-3|n)$ and $H(1,0)$ is a special Heisenberg Lie superalgebra of dimension
$(3|0)$. Define the function $s(L)$ as
$s(L)=\frac{1}{2}(m+n-2)(m+n-1)+n+1-\dim{\mathcal{M}(L)}$. So,
$\dim{\mathcal{M}(L)}=\frac{1}{2}(m+n-2)(m+n-1)+n+1-s(L)$. Where $s(L)\geq 0$.
Nevertheles, we have $t(L)=m+n-2+s(L)$. It seems that the classification of
nilpotent Lie superalgebras $L$ by $s(L)$ leads to the classification of $L$
in terms of $t(L)$.

######  Theorem 1.5.

If $L$ is a finite dimensional nilpotent Lie superalgebra of dimension greater
than $1$ and class $(p,q)$. Then $\mathcal{M}(L)\neq 0$.

###### Proof.

The result obtained [3, Theorem $3.2$].

∎

######  Definition 1.6 (15, Definition 4.1).

A Lie algebra $L$ is called capable if $L\cong H/Z(H)$ for a Lie algebra $H$.

We denote $Z^{*}(L)$ to be the smallest graded ideal in $L$ such that
$L/Z^{*}(L)$ is capable, (see [15]).

######  Lemma 1.7 (15, Lemma 4.3).

A Lie superalgebra $L$ is capable if and only if $Z^{*}(L)=\\{0\\}$.

######  Theorem 1.8 (15, Theorem 4.9).

Let $N$ be a central ideal in a Lie superalgebra $L$. Then the following
conditions are equivalent

  1. (i)

$\frac{\mathcal{M}(L/N)}{\mathcal{M}(L)}\cong N\cap L^{2}$

  2. (ii)

$N\subseteq Z^{*}(L)$

  3. (iii)

$\mathcal{M}(L)\to\mathcal{M}(L/N)$ is monomorphism.

##  2\. Main Results

This section is devoted to obtain the Main Result.

######  Theorem 2.1.

Let $L$ be a nilpotent Lie superalgebra with dimensional greater than $2$ and
$n\geq 1$ such that $\dim{L^{2}}=m+n-2$, then

$(n+m)(n+m-3)+4\leq 2t(L)<(n+m)^{2}+n-m.$

###### Proof.

Using Theorem $5.4$ in [11], we have $\dim{\mathcal{M}(L)}\leq m+2n-2$. Since
$\dim{\mathcal{M}(L)}=\frac{1}{2}[(n+m)^{2}+n-m]-t(L)$, we can see that

$\frac{1}{2}[(n+m)^{2}+n-m]-t(L)\leq m+2n-2.$

Thus, $(n+m)(n+m-3)+4\leq 2t(L)$. Now, Theorem 1.5 implies that
$\dim{\mathcal{M}(L)}>0$, so $\frac{1}{2}[(n+m)^{2}+n-m]-t(L)>0$. Hence
$2t(L)<(n+m)^{2}+n-m$ and the result follows. ∎

######  Theorem 2.2.

Let $L$ be a nilpotent Lie superalgebra with dimensional greater than $2$ with
$n\geq 1$ such that $\dim{L^{2}}=m+n-2$, then

$(n+m)(n+m-5)+8\leq 2s(L)<(n+m)(n+m-1)-2m+4.$

###### Proof.

Since $t(L)=(n+m-2)+s(L)$, by using Theorem 2.1, we have

$(n+m)(n+m-3)+4\leq 2(n+m-2)+2s(L)<(n+m)^{2}+(n-m).$

Thus,

$\displaystyle(n+m)(n+m-3)+4-2(n+m-2)$ $\displaystyle\leq 2(n+m-2)+2s(L)$
$\displaystyle<(n+m)^{2}+(n-m)-2(n+m-2)$

So we have $(n+m)(n+m-5)+8\leq 2s(L)$ and $2s(L)<(n+m)(n+m-1)-2m+4$ and the
result is obtained. ∎

Now we need to the classification of nilpotent Lie superalgebras of dimension
at most $5$ which are not Lie algebras. N. Backhouse and N. L. Matiadou in [1]
and [9], respectively, classified these Lie superalgebras into two types
trivial and non trivial. Note that the Lie superalgebra $L$ is trivial, if
$[L_{\bar{1}},L_{\bar{1}}]=0$, otherwise $L$ is non trivial. According to the
all notations in [1], we also denote the elements of $L_{\bar{0}}$ $($resp
$L_{\bar{1}})$ by Latin letters $($resp Greek letters$)$ taken from the
begining of the alphabet. Using this classification, we have following Lie
superalgebras with maximal class $(\dim L^{2}=m+n-2)$. Later we will compute
the Schur multiplier of them in the Theorem 2.3.

Table $1$, $(1,2)$-Lie superalgebra |  |
---|---|---
Name | Relations | $\dim{\mathcal{M}(L)}$
Trivial LS |  |
$L_{1,2}^{(3)}$ | $[\alpha,\beta]=\alpha$ | $2$
Non-Trivial LS |  |
$L_{1,2}^{(1)}$ | $[\alpha,\alpha]=a,[\beta,\beta]=a$ | $2$
$L_{1,2}^{(2)}$ | $[\alpha,\alpha]=a,[\beta,\beta]=-a$ | $2$

Table $2$, $(1,3)$-Lie superalgebra |  |
---|---|---
Name | Relations | $\dim{\mathcal{M}(L)}$
Trivial LS |  |
$L_{1,3}^{(5)}$ | $[a,\beta]=\alpha,[a,\gamma]=\beta$ | $3$

Table $3$, $(2,2)$-Lie superalgebra |  |
---|---|---
Name | Relations | $\dim{\mathcal{M}(L)}$
Non-Trivial LS |  |
$L_{2,2}^{(9)}$ | $[\alpha,\alpha]=a,[\beta,\beta]=b$ | $2$
$L_{2,2}^{(10)}$ | $[\alpha,\alpha]=a,[\beta,\beta]=b,[\alpha,\beta]=a$ | $1$
$L_{2,2}^{(11)}$ | $[\alpha,\alpha]=a,[\beta,\beta]=b,$ | $1$
| $[\alpha,\beta]=p(a+b)\ ;\ p>0$ |
$L_{2,2}^{(12)}$ | $[\alpha,\alpha]=a,[\beta,\beta]=b,$ | $1$
| $[\alpha,\beta]=p(a-b)\ ;\ p>0$ |

Table $4$, $(1,4)$-Lie superalgebra |  |
---|---|---
Name | Relations | $\dim{\mathcal{M}(L)}$
Trivial LS |  |
$E^{22}$ | $[a,\alpha]=\beta,[a,\beta]=\gamma,[a,\gamma]=\delta,$ | $6$

Table $5$, $(3,2)$-Lie superalgebra |  |
---|---|---
Name | Relations | $\dim{\mathcal{M}(L)}$
Non-Trivial LS |  |
$3A_{1,1}+2A$ | $[\alpha,\alpha]=a,[\beta,\beta]=b,[\alpha,\beta]=c$ | $3$

Table $6$, $(2,3)$-Lie superalgebras |  |
---|---|---
Name | Relations | $\dim{\mathcal{M}(L)}$
Non-Trivial LS |  |
$(D^{15}+A_{1,1})^{1}$ | $[a,\beta]=\alpha,[a,\gamma]=\beta,[\gamma,\gamma]=b$ | $1$
$(D^{15}+A_{1,1})^{2}$ | $[a,\beta]=\alpha,[a,\gamma]=\beta,[\beta,\beta]=b$ | $2$
| $[a,\gamma]=-b$ |
$(D^{15}+A_{1,1})^{3}$ | $[a,\beta]=\alpha,[a,\gamma]=\beta,[\beta,\beta]=b$ | $3$
| $[\gamma,\gamma]=b,[\alpha,\gamma]=-b$ |
$(D^{15}+A_{1,1})^{4}$ | $[a,\beta]=\alpha,[a,\gamma]=\beta,[\beta,\beta]=b$ | $2$
| $[\gamma,\gamma]=-b,[\alpha,\gamma]=-b$ |

According to the free presentations in the previous tables, we want to compute
the Schur multiplier of them.

######  Theorem 2.3.

The Schur multiplier of Lie superalgebras is given in the Table $7$.  Table
$7$, Schur Multiplier LS $\mathcal{M}(L)$ LS $\mathcal{M}(L)$ $L_{1,2}^{(1)}$
$A(2|0)$ $(D^{15}+A_{1,1}))^{3}$ $A(2|1)$ $L_{1,2}^{(2)}$ $A(2|0)$
$L_{2,2}^{(10)}$ $A(1|0)$ $L_{1,2}^{(3)}$ $A(1|1)$ $L_{2,2}^{(11)}$ $A(1|0)$
$L_{2,2}^{(9)}$ $A(1|1)$ $L_{2,2}^{(12)}$ $A(1|0)$ $(D^{15}+A_{1,1})^{2}$
$A(1|1)$ $E^{22}$ $A(5|1)$ $(D^{15}+A_{1,1})^{4}$ $A(1|1)$ $3A_{1,1}+2A$
$A(1|2)$ $L_{1,3}^{(5)}$ $A(2|1)$ $(D^{15}+A_{1,1})^{1}$ $A(0|1)$

###### Proof.

We state the proof in details to $L_{1,2}^{(3)}$ and $(D^{15}+A_{1,1})^{4}$,
the remaining case be proved in a similar way. Let $L\cong L_{1,2}^{(3)}$.
According to the Definition $1.3$ in [17],
$\alpha\wedge\alpha=\alpha\wedge\beta=0$. Thus $L\wedge
L=<a\wedge\alpha,a\wedge\beta,\beta\wedge\beta>$. Hence for all
$w\in\mathcal{M}(L)$, there exist
$\alpha_{1},\alpha_{2},\alpha_{3}\in\mathbb{R}$, such that
$w={\alpha_{1}}(a\wedge\alpha)+{\alpha_{2}}(a\wedge\beta)+{\alpha_{3}}(\beta\wedge\beta)$.
Now let $\tilde{\kappa}:L\wedge L\to[L,L]$ be given by $(x\wedge y\to[x,y])$.
Since $\tilde{\kappa}(w)=0$, we have
${\alpha_{1}}[a,\alpha]+{\alpha_{2}}[a,\beta]+{\alpha_{3}}[\beta,\beta]$, so
${\alpha_{2}}{\alpha}=0$ and $\alpha_{2}=0$. Thus
$w={\alpha_{1}}(a\wedge\alpha)+{\alpha_{3}}(\beta\wedge\beta)$. Therefore
$\mathcal{M}(L)=<a\wedge\alpha,\alpha\wedge\alpha>$ and
$\dim\mathcal{M}(L)=2$. On the other hand by using Definition $1.3$ in [17],
$\tilde{\kappa}$ is a homogeneous linear map of even degree. Also, we know
$|[a,\alpha]|=|a|+|\alpha|=0+1=1$ and $|[\beta,\beta]|=|\beta|+|\beta|=1+1=0$.
Hence $a\wedge\alpha$ and $\beta\wedge\beta$ are odd and even elements of
$L\wedge L$, respectively. Finally, since the Schur multiplier of a Lie
superalgebra is abelian, we have $\mathcal{M}(L)\cong A(1|1)$.
Let $L\cong(D^{15}+A_{1,1})^{4}$, by using to the Definition $1.3$ in [17], we
have $a\wedge
b=b\wedge\alpha=b\wedge\beta=b\wedge\gamma=\alpha\wedge\alpha=\alpha\wedge\beta=\beta\wedge\gamma=0,$
and $\beta\wedge\beta=-\gamma\wedge\alpha$. Thus $L\wedge
L=<a\wedge\alpha,a\wedge\beta,a\wedge\gamma,\alpha\wedge\alpha,\gamma\wedge\gamma>$.
Hence for all $w\in\mathcal{M}(L)$, there exist
$\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5}\in\mathbb{R}$, such
that
$w={\alpha_{1}}(a\wedge\alpha)+{\alpha_{2}}(a\wedge\beta)+{\alpha_{3}}(a\wedge\gamma)+{\alpha_{4}}(\alpha\wedge\alpha)+{\alpha_{5}}(\gamma\wedge\gamma)$.
Now let $\tilde{\kappa}:L\wedge L\to[L,L]$ be given by $(x\wedge y\to[x,y])$.
Since $\tilde{\kappa}(w)=0$, we have
${\alpha_{1}}[a,\alpha]+{\alpha_{2}}[a,\beta]+{\alpha_{3}}[a,\gamma]+{\alpha_{4}}[\alpha,\alpha]+{\alpha_{5}}[\gamma,\gamma]=0$,
so, ${-\alpha_{1}}{\beta}+{\alpha_{2}}{\alpha}+{(\alpha_{4}+\alpha_{5})}{b}=0$
and $\alpha_{1},\alpha_{2}=0,\alpha_{5}=-\alpha_{4}$. Thus
$w={\alpha_{3}}(a\wedge\gamma)+{\alpha_{4}}(\alpha\wedge\alpha-\gamma\wedge\gamma)$.
Therefore
$\mathcal{M}(L)=<a\wedge\gamma,\alpha\wedge\alpha-\gamma\wedge\gamma>$ and
$\dim\mathcal{M}(L)=2$. On the other hand, by using Definition $1.3$ in [17],
$\tilde{\kappa}$ is a homogeneous linear map of even degree. Also, we know
$|[a,\gamma]|=1,|[\alpha,\alpha]|=0$ and $|[\gamma,\gamma]|=0$. Hence
$a\wedge\gamma$, $\alpha\wedge\alpha-\gamma\wedge\gamma$ are odd and even
elements of $L\wedge L$, respectively. Finally, since the Schur multiplier of
a Lie superalgebra is abelian, we have $\mathcal{M}(L)\cong A(1|1)$. ∎

Now we want to classify non-abelian nilpotent Lie superalgebras of maximal
class for $1\leq s(L)\leq 10$.

######  Theorem 2.4.

Let $L$ be a non-abelian $(m|n)$-dimensional nilpotent Lie superalgebra of
maximal class and $m+n\geq 3$. Then $1\leq s(L)\leq 10$ if and only if $L$ is
isomorphic to the one of the Lie superalgebras listed in the table $8$.

Table $8$, Summary of Results |
---|---
$s(L)$ | Name
$1$ | There is no such Lie superalgebra
$2$ | $L_{1,2}^{(1)},L_{1,2}^{(2)},L_{1,2}^{(3)}$
$3$ | There is no LS
$4$ | $L_{1,3}^{(5)},L_{2,2}^{(9)}$
$5$ | $L_{2,2}^{(10)},L_{2,2}^{(11)},L_{2,2}^{(12)},E^{22},$
$6$ | $3A_{1,1}+2A$
$7$ | $(D^{15}+A_{1,1})^{3}$
$8$ | $(D^{15}+A_{1,1})^{2},(D^{15}+A_{1,1})^{4}$
$9$ | $(D^{15}+A_{1,1})^{1}$
$10$ | There is no such Lie superalgebra

###### Proof.

Using Theorem 2.2, if $s(L)=1$ then $m+n=3$ and $\dim{\mathcal{M}(L)}=3$, so
by looking at the Table $1$, there is no such Lie superalgebra with these
properties.
Let $s(L)=2$, by looking of Tables $1,2,3$, if $m+n=3$, then we should have
$L\cong L_{1,2}^{(1)}$, $L_{1,2}^{(2)}$ and $L\cong L_{1,2}^{(3)}$. Also when
$m+n=4$, there is no such a Lie superalgebra with our condition.
Let $s(L)=3$, by looking at Tables $1,2,3$, if $m+n=3,4$, there is no such Lie
superalgebra for the cases. Now, let $s(L)=4$, then $m+n=4$ and by using
Tables $2,3$ we have $L\cong L_{1,3}^{(5)}$ or $L_{2,2}^{(9)}$. Similarly for
$5\leq s(L)\leq 10$, we can have a classification for non-abelian nilpotent
Lie superalgebra of maximal class and dimension at most $5$. We put the
summary of the results in the Table $8$. ∎

Now we want to determine the structure of all $(m|n)$-dimensional nilpotent
Lie superalgebras $L$ when $\dim L^{2}=\dim\mathcal{M}(L)=m+n-2$ for all
$m+n\leq 5$.

######  Proposition 2.5.

Let $L$ be an $(m|n)$-dimensional nilpotent Lie superalgebra such that $\dim
L^{2}=\dim\mathcal{M}(L)=m+n-2$ and $m+n\leq 5$. Then $L$ is isomorphic to one
of the nilpotent Lie superalgebras $L_{2,2}^{(9)}$, $3A_{1,1}+2A$ or
$(D^{15}+A_{1,1})^{3}$.

###### Proof.

By looking at the classification of all nilpotent Lie superalgebras of
dimension at most $5$ in [1, 9], we can see that only Lie superalgebras whose
derived superalgebra is of dimension $m+n-2$ are those listed in Tables $1$ to
$6$. By using a similar way as in the proof of theorem 2.3, we have
$\dim{\mathcal{M}(L_{2,2}^{(9)})}=2$, $\dim{\mathcal{M}(3A_{1,1}+2A)}=3$ and
$\dim{\mathcal{M}((D^{15}+A_{1,1})^{3})}=3$. Hence $L$ should be isomorphic to
one of the nilpotent Lie superalges $L_{2,2}^{(9)}$, $3A_{1,1}+2A$ or
$(D^{15}+A_{1,1})^{3}$. ∎

######  Proposition 2.6.

Let $L$ be an $(m|n)$-dimensional nilpotent Lie algebra of maximal class. If
$\dim\mathcal{M}(L)=\dim L^{2}$, then $L$ is capable.

###### Proof.

According to the Proposition 2.5, $L$ is isomorphic to the one of the Lie
superalges $L_{2,2}^{(9)}$, $3A_{1,1}+2A$ and $(D^{15}+A_{1,1})^{3}$. Now by
contray let $L\cong 3A_{1,1}+2A$ be non capable. By using Theorem , there is a
non-zero element $x$ in $Z^{*}(L)$. Two cases can be considered. First, if $x$
be an even element of $Z^{*}(L)$, then according to the Table $2$, $L$ is
isomorphic to the one of the $(2|2)$-dimensional Lie superalgebras
$L_{2,2}^{(9)},L_{2,2}^{(10)},L_{2,2}^{(11)},L_{2,2}^{(12)}$ which the
dimension their Schur multipliers are $1$ or $2$. So in this case, the map
$\mathcal{M}(L)\to\mathcal{M}(\frac{L}{<x>})$ is not injective that is in
contradiction with part $(iii)$ of the Theorem 2. Hence $L$ is capable.
Second, let $x$ be an odd element of $Z^{*}(L)$, then by using Black house’s
classification in [1], $L\cong L_{3,1}^{(1)}=<a,b,c,\alpha\ |\
[b,c]=a,[b,\alpha]=\alpha>$ and similar to the Theorem 2.3, it can be shown
that $\dim\mathcal{M}(L_{1,3}^{(1)})=2$. Thus, the map
$\mathcal{M}(L)\to\mathcal{M}(\frac{L}{<x>})$ is not injective and similarly,
we have a contradiction. Therefore, in this case $L$ is capable. It can also
be easily and similarly seen that $L_{2,2}^{(9)}$ and $(D^{15}+A_{1,1})^{3}$
are capable. ∎

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