11institutetext: University of Texas at Arlington, Arlington TX, USA
11email<EMAIL_ADDRESS><EMAIL_ADDRESS>

#  Thermal Image Super-Resolution Using Second-Order Channel Attention with
Varying Receptive Fields

Nolan B. Gutierrez  William J. Beksi  The authors acknowledge the Texas
Advanced Computing Center (TACC) at the University of Texas at Austin for
providing software, computational, and storage resources that have contributed
to the research results reported within this paper.

###### Abstract

Thermal images model the long-infrared range of the electromagnetic spectrum
and provide meaningful information even when there is no visible illumination.
Yet, unlike imagery that represents radiation from the visible continuum,
infrared images are inherently low-resolution due to hardware constraints. The
restoration of thermal images is critical for applications that involve
safety, search and rescue, and military operations. In this paper, we
introduce a system to efficiently reconstruct thermal images. Specifically, we
explore how to effectively attend to contrasting receptive fields (RFs) where
increasing the RFs of a network can be computationally expensive. For this
purpose, we introduce a deep attention to varying receptive fields network
(AVRFN). We supply a gated convolutional layer with higher-order information
extracted from disparate RFs, whereby an RF is parameterized by a dilation
rate. In this way, the dilation rate can be tuned to use fewer parameters thus
increasing the efficacy of AVRFN. Our experimental results show an improvement
over the state of the art when compared against competing thermal image super-
resolution (SR) methods.

###### Keywords:

T

hermal Imaging, Super-Resolution, Compression via Dilations

##  1 Introduction

The purpose of single image super-resolution (SISR) restoration is to
determine the mapping between a possibly degraded low-resolution (LR) image
and its high-resolution (HR) counterpart. Finding this arrangement is
difficult due to the intractable nature of the problem. Techniques for
discovering the mapping can be divided into two areas: interpolated and
learning-based. Contemporary SISR has been dominated by deep learning which
has demonstrated superiority over hand-crafted methods such as bicubic and
bilinear interpolation.

Convolutional neural networks (CNNs) have been shown to be successful at
attending to visual images on tasks such as SISR. This includes squeeze-and-
excitation methods for global excitation of feature maps [14, 23], and the use
of weight excitations [32, 25]. Furthermore, ablation studies have
conclusively shown that excitation networks for feature maps bring performance
gains [40]. On another note, a variety of methods exist for modifying the RF
of a CNN either through concatenation [22], deformation [10], or dilation
[37]. In this work, we study how dilated convolutions offer _compression
through dilations_. Concretely, we show how they modify the effective
receptive fields (ERFs) of a CNN and how they interact with an existing
enhancement known as second-order channel attention (SOCA) [11].

The development of infrared thermographic cameras has spurred researchers to
carry out innovative research in the thermal image domain. Representing
traditional SR, Mandanici et al. [28] combined geometric registration with
projection and interpolation to produce HR thermal images. Naturally,
researchers have recently investigated color-guided thermal image SR [8, 2].
For example, a pyramidal network provided by Gupta et al. [12] attains
accurate results by extracting edge-maps from RGB images at various levels of
the network. Chudasama et al. [9] present an efficient SR network for thermal
images by eliciting high-frequency details with a limited number of feature
extraction modules. Other works have introduced popular loss functions to the
field of thermal image SR [1, 17].

In this work we not only explore the use of efficient thermal SR, but we also
provide complete benchmarks on three thermal imagery datasets. Furthermore, we
compare the results of four architectural variants to assess performance gains
or losses due to the compression of parameters in our SR network. In contrast
to previous work on SISR, our proposed model applies SOCA to a concatenation
of features produced from convolutions with changing RFs. An intermediate
convolutional layer quantifies the importance of the values from each RF
before passing this information to SOCA. The key contributions of our work are
the following: (i) we show the effectiveness of SOCA for thermal image SR;
(ii) we present a novel approach to sample from a foveated RF; (iii) we
demonstrate an efficient network for multiple upscaling factors; (iv) we
establish new benchmarks on public thermal image datasets. Our source code is
available at [6].

The rest of this paper is organized as follows. We provide a summary of
related thermal image SR work in Sec. 2, and the basics of SR in Sec. 3. In
Sec. 4, we propose an architecture for transforming LR input images to super-
resolved output images. In Sec. 5, we demonstrate our dilation-rate driven
deep attention to varying receptive fields network through experimental
results. Finally, we conclude in Sec. 6.

##  2 Related Work

Most recent developments in the image restoration domain focus on the visual
image space [31, 39, 4]. The known importance of deeper CNNs in improving
representational power has spurred the development of architectures that
improve stability and provide better representations [18, 24]. This is done
not only through more residual connections [19, 3, 26], but also through
structural preservation [27, 16], constrained hypotheses spaces [7], fast
Fourier transform [33] and generative techniques [35], and student-teacher
networks [21]. Additionally, SR works that focus on improving contextual
information have employed different attention mechanisms [29] and enhanced
inception modules [30].

##  3 Preliminaries

The task of super-resolving an LR image to its HR counterpart may be
summarized by its image space and degradation model. Formally, the
relationship can be defined as

$\bm{I}_{x}=D(\bm{I}_{y};\bm{\theta}),$ (1)

where $D$ (known as the degradation function) maps an HR image
$\textbf{I}_{y}$ to an LR image $\textbf{I}_{x}$ with degradation parameters
$\bm{\theta}$. Hence, SR can be reduced to finding the parameters of $D$,
however it is an intractable process. Learning an SR model, $F$, can be
formalized as

$\bm{I}_{y}=F(\bm{I}_{x};\bm{\theta}).$ (2)

Furthermore, any optimization algorithm can be applied to find these
parameters by minimizing an objective function,

$\hat{\bm{\theta}}=\underset{\bm{\theta}}{\operatorname{arg}\,\operatorname{min}}\;L(\hat{\bm{\theta}}_{y},\bm{I}_{y})+\lambda\psi(\bm{\theta}),$
(3)

where $\lambda$ is a small value that represents the importance of the
regularization term $\psi$. This term may aid a model’s ability to generalize
to data never before seen. A common objective function to minimize is the mean
squared error (MSE) otherwise known as the 2-norm,

$\text{MSE}(\bm{I}_{x},\bm{I}_{y})=\frac{1}{N}\sum\limits_{i=1}^{N}(\bm{I}_{x}(i)-\bm{I}_{y}(i))^{2},$
(4)

where $N$ is the number of samples in a batch.

##  4 Second-Order Channel Attention with Varying Receptive Fields

Figure 1: An overview of our thermal imaging SR model.

We establish a novel deep learning architecture for thermal image SR as
follows. First (Sec. 4.1), we provide a detailed description of SOCA. Second
(Sec. 4.2), we expound upon how dilated convolutions are computed. Third (Sec.
4.3), we describe how CNNs can be compressed with respect to an ERF via
compression through dilations. Fourth (Sec. 4.4), we incorporate residual in
residual to aid in the stabilization of our network. Finally (Sec. 4.5), we
apply SOCA to separate RFs (by dilation rates) within residual in residual to
assist in the SR of LR images obtained with a bicubic degradation model.

###  4.1 Second-Order Channel Attention

We utilize an alternative SOCA network to enhance convolutional blocks by
supplying a covariance matrix that allows for more discriminative
representations. To produce these second-order statistics, the covariance
normalization is obtained through Newton-Schulz iteration [13]. Additionally,
this serves to speed up the computation. First, a feature map of dimension
$H\times W\times C$ is reshaped into a feature map $\bm{X}$ of shape $HW\times
C$. Second, the covariance matrix is calculated,

$\bm{\Sigma}=\bm{X}\bm{I_{f}}\bm{X}^{\top},$ (5)

where $\bm{I_{f}}=\frac{1}{s}(\bm{I}-\frac{1}{s}\bm{1})$ and $s=HW$. $\bm{I}$
and $\bm{1}$ are the $m\times m$ identity matrix and the matrix of all ones,
respectively. Then, the covariance matrix is pre-normalized,

$\bm{\hat{\Sigma}}=\frac{1}{\text{tr}(\bm{\Sigma})}\bm{\Sigma},$ (6)

where tr($\cdot$) denotes the matrix trace. Let $\bm{Y}_{0}=\bm{\hat{\Sigma}}$
and $\bm{Z}_{0}=\bm{I},$ then $\bm{Y_{n}}$ and $\bm{Z}_{n}$ are obtained by

$\displaystyle\bm{Y}_{n}$
$\displaystyle=\frac{1}{2}\bm{Y}_{n-1}(3\bm{I}-\bm{Z}_{n-1}\bm{Y}_{n-1}),$ (7)
$\displaystyle\bm{Z}_{n}$
$\displaystyle=\frac{1}{2}(3\bm{I}-\bm{Z}_{n-1}\bm{Y}_{n-1})\bm{Z}_{n-1},$ (8)

with $\bm{Y_{n}}$ and $\bm{Z}_{n}$ quadratically converging to $\bm{Y}$ and
$\bm{Y}^{-1}$, respectively. The final normalized matrix after five iterations
of Newton-Schulz is found by compensating the pre-normalization step with

$\bm{\hat{Y}}=\sqrt{tr(\bm{\Sigma})}\bm{Y}_{N}.$ (9)

Afterwards, global covariance pooling is applied to obtain a scalar-valued
statistic $z_{i}$ for each channel $i$,

$\bm{z}_{i}=\frac{1}{C}\sum\limits_{j}^{C}\bm{\hat{Y}}_{ij}.$ (10)

This permits the channel attention to capture correlations higher than the
first order. In the next step, the sigmoid activation function serves as a
gating mechanism that entrusts the network to selectively choose what to add
to the incoming input features. To create this gating mechanism, we use two
convolutional layers, $\bm{W}_{0}$ and $\bm{W}_{1}$, with rectified linear
unit (ReLU) and sigmoid activation functions. Concretely,

$G(\bm{z})=\bm{W}_{1}\ast(\bm{W}_{0}\ast\bm{z}),$ (11)

where $\ast$ is the convolution operation and $G(\bm{z})$ is an attention map.

###  4.2 Dilated Convolutions

A dilated convolution multiplies a rate $l$ by $\Delta$ during the convolution
operation, i.e.,

$(F\ast_{l}k)(\textbf{p})=\sum\limits_{\Delta\in\Omega_{r}}F(\textbf{p}-l\cdot\Delta)k(\Delta),$
(12)

where $\Omega_{r}=[-r,r]^{2}\cap\mathbb{Z}^{2}$, $k:\Omega_{r}\to\mathbb{R}$,
$\bm{p}\in\mathbb{Z}^{2}$ is a location on $\bm{X}$,
$F:\mathbb{Z}^{2}\to\mathbb{R}$, and $*_{l}$ is an $l$-dilated convolution. In
(12), $k$ is known as the kernel function which slides over $\bm{X}$. This
allows a convolutional network to sample pixel values from a larger RF over
the input features.

In the case of SR, it is advantageous to sample from different sized RFs
depending on a number of factors including depth, resolution, and the scaling
factor. We use dilated convolutions to extract features within each residual
block. Specifically, the feature sets of three convolutional layers with
varying dilation rates are concatenated and passed to an intermediate layer
which pools the information from contrasting RFs. This intermediate layer
effectively pools information at each feature map’s location from a foveated
RF where more parameters are concentrated towards the center of the field.

###  4.3 Compression through Dilations

We utilize dilated convolutions to artificially increase the ERFs of our CNN.
An ERF is defined as the area containing any input pixel with a non-negligible
impact on a particular output unit within a feature map [27]. In addition, we
introduce the concept of compression through dilations as the case in which a
CNN uses fewer parameters to increase an ERF with dilated convolutions
compared to without dilated convolutions. For example, assuming we are using
bias, two single-layer CNNs defined by
$\text{Conv2D}(\text{Input\\_Shape}=(32,32,3),64,(5,5))$ and
$\text{Conv2D}(\text{Input\\_Shape}=(32,32,3),64,(3,3),\text{Dilation\\_Rate}=2)$
have an ERF area of 25. However, our CNN has 160,064 and 57,664 parameters,
respectively, giving a compression ratio of 2.776.

###  4.4 Residual in Residual

We stabilize our deep channel attention network by the addition of residual in
residual (RIR) connections [40]. RIR entails two levels of connections with
groups on the outer level and blocks on the inner level. More precisely,

$\hat{{\bf Y}}={\bf X}+{\bf W}\ast R_{g}(R_{g-1}(\ldots R_{1}({\bf
X})\ldots)).$ (13)

$R_{g}$ is the $g$-th residual group and it is formulated as

$R_{g}({\bf X})={\bf X}+{\bf W}\ast B_{t}(B_{t-1}(\ldots B_{1}({\bf
X})\ldots)),$ (14)

where $B_{t}$ is the $t$-th residual channel attention block. We apply SOCA to
the features extracted from the convolutional layers with unique RFs. To
obtain the individual RFs, we make the dilation rate of each convolutional
layer exclusive and not equal to one for two of the layers. For example, if
$\bf W_{1}$, $\bf W_{2}$, and $\bf W_{3}$ are the weight sets associated with
three convolutional layers, then the residual block is derived as

$B_{t}({\bf X})={\bf X}+\text{SOCA}([{\bf W_{1}}\ast{\bf X},{\bf
W_{2}}\ast_{2}{\bf X},{\bf W_{3}}\ast_{3}{\bf X}])$ (15)

where $[{\bf W_{1}},{\bf W_{2}},{\bf W_{3}}]$ constitutes the concatenation of
$\bf W_{1}$, $\bf W_{2}$, and $\bf W_{3}$ along the channel axis.

###  4.5 Model Overview

Our overall model is shown in Fig. 1. To upscale an input image, features are
extracted from a series of residual groups and blocks within the RIR
architecture similar to RCAN [40]. Pixel shuffle [36] is used to rearrange
$\bm{X}$ of shape $(H,W,C\cdot r^{2})$ to $(H\cdot r,W\cdot r,C)$ by
periodically building a new feature map $PS(\bm{X})$ with pixel values from
dissimilar channels according to the equation

$PS(\bm{X})_{x,y,c}=\bm{X}_{\lfloor x/r\rfloor,\lfloor y/r\rfloor,C\cdot
r\cdot mod(y,r)+C\cdot mod(x,r)+c}.$ (16)

Finally, a single-channel convolutional layer reduces the number of channels
to the same number as in the LR image.

##  5 Experimental Results

Our experiments model the downscaling of HR thermal images using a bicubic
degradation model with statistical noise.

###  5.1 Datasets

We use the Thermal Image Super-Resolution (TISR) 2020 Challenge dataset, the
FLIR Thermal Dataset for Algorithm Training (TDAT) [38], and the KAIST
multispectral pedestrian detection benchmark dataset [15]. The TISR dataset
consists of three sets of 1,021 images from three distinct cameras. These
cameras include a Domo, Axis, and FLIR with a resolution of ($160\times 120$),
($320\times 240$), and ($640\times 480$), respectively. Of these images, 60
were kept private, leaving 951 in the training set and 50 in the test set for
each camera. For TDAT, we evaluate on only the first 100 images captured by a
FLIR FC-6320 camera. Lastly, for the KAIST dataset, we collect every 200-th
image from the day and night scenes and then evaluate on the set of 226
images. The images from the KAIST dataset were captured by a FLIR-A35 thermal
camera with a resolution of $640\times 480$.

The ground-truth dataset was created by first forming batches of 16 single-
channel image patches where each patch is of size $scale\times 48$. The LR
images were then obtained by bicubicly interpolating these patches to a size
of $48\times 48$. For both training and testing, the images were preprocessed
by adding Gaussian noise with a mean of 0 and a variance of 10 for the bicubic
with noise degradation model. Finally, all elements of each LR patch are
normalized and clipped to [0,1].

###  5.2 Implementation

Our final architecture uses three residual groups with six residual blocks per
group. Each convolutional layer has 64 filters resulting in a highly-efficient
network. During training, the Adam optimizer [20], parameterized by a learning
rate of $10^{-4},\beta_{1}=0.9,\beta_{2}=0.999$, and $\epsilon=10^{-7}$, is
applied to minimize the MSE of each batch for 300 epochs. Training with four
NVIDIA GeForce GTX 1080 Ti GPUs took less than three hours per model.

###  5.3 Evaluation

For the experiments, we tested four variants of our architecture to evaluate
the performance gains of the network. The variants are as follows.

  * •

Dilated residual in residual (DDRR): SOCA in the residual block of Fig. 1 is
replaced by a convolutional layer with a $3\times 3$ kernel size and no
activation function.

  * •

Residual in residual with SOCA (RRSOCA): Our different dilation rate module in
the residual block of Fig. 1 is replaced with a series of two convolutional
layers each with a kernel size of $3\times 3$ and a ReLU activation function.

  * •

Compressed RCAN (CRCAN) via dilated convolutions: This architecture is similar
to RRSOCA, but the first and second convolutional layers have a dilation rate
of 1 and 2, respectively.

  * •

Attention to varying receptive fields network (AVRFN): This is our proposed
model as shown in Fig. 1.

Test Set | Scale | Parameters | PSNR | SSIM
---|---|---|---|---
AXIS Domo P1290 | 2 | 2491009 | $30.097$ | $0.846$
AXIS Q2901 | 2491009 | $34.416$ | $0.928$
FLIR FC-6320 | 2491009 | $37.116$ | $0.942$
TDAT | 2491009 | $31.184$ | $0.733$
KAIST | 2491009 | $40.612$ | $0.962$
AXIS Domo P1290 | 3 | 2511809 | $26.831$ | $0.747$
AXIS Q2901 | 2511809 | $30.007$ | $0.850$
FLIR FC-6320 | 251180 | $33.668$ | $0.898$
TDAT | 2511809 | $30.055$ | $0.730$
KAIST | 2511809 | $39.065$ | $0.958$
AXIS Domo P1290 | 4 | 2507649 | $25.146$ | $0.671$
AXIS Q2901 | 2507649 | $27.724$ | $0.784$
FLIR FC-6320 | 2507649 | $31.574$ | $0.857$
TDAT | 2507649 | $28.110$ | $0.626$
KAIST | 2507649 | $37.793$ | $0.951$
Table 1: The results of our AVRFN model on images captured by the TISR [34],
TDAT [38] and KAIST [15] datasets.

(a)

(b) 24.43/0.64

(c) GT

(d)

(e) 26.64/0.83

(f) GT

(g)

(h) 28.54/0.78

(i) GT

Figure 2: Examples of (left column) downsampled images from (top row) low-
resolution, (middle row) medium-resolution, and (bottom row) high-resolution
thermal cameras, their $\times 4$ upscaled counterparts (middle column), and
the ground truth (GT) (right column). Additionally, (b), (e), and (h) show the
PSNR and SSIM scores, respectively, when evaluated against the ground truth.

Fig. 2 highlights example inputs, SR predictions, and the ground-truth images
from the datasets. The added Gaussian noise produces heavily pixelated input
images which presents very difficult conditions to evaluate our methods on. In
all of our experiments, we use the peak signal-to-noise ratio (PSNR) and
structural similarity index (SSIM) to evaluate each architectural variant.
Table 1 shows the performance of our proposed AVRFN when evaluated on each of
the datasets. Additionally, Table 2 provides an ablation study which compares
the performance of the various types of compression with different channel
attention networks. Note that we were unable to make a fair comparison with
related work found in [9] since the performance of our baseline method did not
match. However, each of our variants performed better than the baseline RCAN
architecture.

An interesting finding is that adding compression through dilations in the
residual block of RCAN leads to improved performance. After each residual
connection, the ERF resets to the kernel size of the succeeding CNN due to the
easy pass-through of low-level information found in residual networks [5].
Contemporary work has found a correlation between larger ERFs and performance
gains [5]. We hypothesize that by introducing compression through dilations in
each residual block, we increase the ERF at a faster rate thus allowing for
performance gains. An unexpected result is that DDRR, the only variant without
any form of channel-attention, performs significantly worse. This confirms
previous ablation studies [40], but it also shows that most of the performance
gains arise from channel attention and not compression through dilations. In
addition, our varying dilation rate module (AVRFN) improves performance over
the baseline which shows that attending to different RFs can improve
performance. Nonetheless, our highest performance gains are obtained when we
simply add compression through dilations to the RCAN baseline.

##  6 Conclusion

In this work, we showed the advantage of attending to varying resolutions for
the reconstruction of thermal images by efficiently parametrizing a
convolutional layer with a dilation rate. Together with SOCA, our model
achieves state-of-the-art results on the task of thermal image SR and yields
up-to-date benchmarks for the research community. In the future, we intend to
look at ways in which training may be further stabilized and how attention to
uncertainty maps can improve the computational efficiency of thermal image SR.

Test Set | Model | Scale | Parameters | PSNR | SSIM
---|---|---|---|---|---
AXIS Domo P1290 | RRSOCA | 4 | 1661377 | $25.487$ | $0.691$
DDRR | 4 | 2839873 | $25.458$ | $0.691$
CRCAN | 4 | 2839873 | $\bf 25.491$ | $\bf 0.692$
RCAN | 4 | 1661377 | $25.239$ | $0.682$
AVRFN | 4 | 1917313 | $25.368$ | $0.685$
AXIS Q2901 | RRSOCA | 4 | 1661377 | $28.167$ | $0.802$
DDRR | 4 | 2839873 | $28.159$ | $0.801$
CRCAN | 4 | 2839873 | $28.189$ | $0.802$
RCAN | 4 | 1661377 | $27.923$ | $0.795$
AVRFN | 4 | 1917313 | $27.990$ | $0.797$
FLIR FC-6320 | RRSOCA | 4 | 1661377 | $31.978$ | $0.867$
DDRR | 4 | 2839873 | $31.985$ | $0.867$
CRCAN | 4 | 2839873 | $32.002$ | $0.867$
RCAN | 4 | 1661377 | $31.756$ | $0.861$
AVRFN | 4 | 1917313 | $31.824$ | $0.864$
TDAT | RRSOCA | 4 | 1661377 | $28.388$ | $0.641$
DDRR | 4 | 2839873 | $28.427$ | $0.645$
CRCAN | 4 | 2839873 | $28.426$ | $0.645$
RCAN | 4 | 1661377 | $28.271$ | $0.636$
AVRFN | 4 | 1917313 | $28.298$ | $0.637$
KAIST | RRSOCA | 4 | 1661377 | $37.977$ | $0.949$
DDRR | 4 | 2839873 | $37.456$ | $0.918$
CRCAN | 4 | 2839873 | $37.573$ | $0.922$
RCAN | 4 | 1661377 | $37.089$ | $0.938$
AVRFN | 4 | 1917313 | $37.827$ | $0.943$
Table 2: The results of our $\times 4$ model variants on images captured by
the TISR [34], TDAT [38] and KAIST [15] datasets.

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# Superalgebras with homogeneous structures of Lie type

Sami Mabrouk and Othmen Ncib  University of Gafsa, Faculty of Sciences Gafsa,
2112 Gafsa, Tunisia<EMAIL_ADDRESS><EMAIL_ADDRESS>University of Gafsa, Faculty of Sciences Gafsa, 2112 Gafsa, Tunisia
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

In this paper, we extend the concept of Lie superalgebras to a more
generalized framework called Super-Lie superalgebras. In addition, they seem
to be exploring various super-generalizations of other algebraic structures,
such as Super-associative, left (right) Super-Leibniz, and Super-
left(right)-symmetric superalgebras, then we give some examples and related
fundamental results. The notion of Rota-Baxter operators with any parity on
the Super-Lie superalgebras is given. Moreover, we study a representations of
Super-Lie superalgebras and its associate dual representations. The notion of
derivations of Super-Lie superalgebras is introduced thus we show that the
converse of a bijective derivation defines a Rota-Baxter operator. Finally, we
give a generalization of the Super-Lie superalgebras and some other structures
in the ternary case which we supported this with some examples and interesting
results.

Keywords: Lie superalgebra, Super-Lie superalgebra, (Super)-left-symmetric
superalgebra, Rota-Baxter operator, representation, derivation, $3$-super-Lie
superalgebra.

Mathematics Subject Classification (2020): 17B05, 17B10, 17B70, 17B38, 17A40.

###### Contents

  1. 1 Introduction
    1. 1.1 Layout of the paper
    2. 1.2 Conventions and notations
  2. 2 Super-Lie superalgebras and Admissibilty
    1. 2.1 Definitions and Examples
    2. 2.2 Super-Lie admissible superalgebras
  3. 3 Rota-Baxter of Super-Lie superalgebras
  4. 4 Represenations and derivations of Super-Lie superlagebras
    1. 4.1 Represenations of Super-Lie superalgebras
    2. 4.2 Derivations of Super-Lie superalgebras
  5. 5 $3$-super-Lie superalgebras
  6. 6 Further discussion

000 Corresponding author (Sami Mabrouk<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1\. Introduction

Graded Lie algebras are a topic of interest in physics in the context of
”supersymmetries”, relating particles of differing statistics. In mathematics,
graded Lie algebras have been known fore same time in the context of
deformation theory. A superalgebra is a $\mathbb{Z}_{2}$-graded vector space
$\mathfrak{g}=\mathfrak{g}_{\overline{0}}\oplus\mathfrak{g}_{\overline{1}}$
equipped with an even bilinear map (i.e. if
$x\in\mathfrak{g}_{i},\;y\in\mathfrak{g}_{j};\;i,j\in\mathbb{Z}_{2}=\\{\overline{0},\overline{1}\\}$,
then $x\cdot y\in\mathfrak{g}_{i+j}$). A Lie superalgebra is a superalgebra
$(\mathfrak{g},[\cdot,\cdot])$ where

$\displaystyle[x,y]$
$\displaystyle=-(-1)^{ij}[y,x],\;\;\text{for}\;x\in\mathfrak{g}_{i},\;y\in\mathfrak{g}_{j},$
$\displaystyle[x,[y,z]]$
$\displaystyle=[[x,y],z]+(-1)^{ij}[y,[x,z]],\;\;\text{for}\;x\in\mathfrak{g}_{i},\;y\in\mathfrak{g}_{j}.$

In [13], the authors studies the theory of Lie superalgebras as Lie algebras
of certain generalized groups, nowadays called Lie supergroups, whose function
algebras are algebras with commuting and anticommuting variables. Recently, a
satisfactory theory, similar to Lie’s theory, has been developed on the
connection between Lie supergroups and Lie superalgebras [14].

Another class of non-associative superalgebras called Lie antialgebras is
introduced by V. Ovsienko in [32]. The axioms were established after
encountering two “unusual” algebraic structures in a context of symplectic
geometry: the algebras, named $\text{asl}_{2}$ and the conformal antialgebra
$\mathcal{AK}(1)$. Ovsienko found that both of them are related to an odd
bivector fields on $\mathbb{R}^{2|1}$ invariant under the action of the
orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$. The algebra
$\mathcal{AK}(1)$ is also related to the famous NeveuSchwartz Lie superalgebra
(see [28, 29] for more details).

The study of Rota- Baxter algebra appeared for the first time in the work of
Baxter [11] in $1960$ and were then intensively studied by Atkinson [6],
Miller [30], Rota [34], Cartier [17], and more recently, they reappeared in
the work of Guo [25] and Ebrahimi-Fard [23]. A few years later, the Rota-
Baxter operators were studied in the super case of some algebraic structures.
We are interested in the Rota-Baxter operators of weight
$\lambda\in\mathbb{K}$ on Lie superalgebras which is introduced in [1] and
defined as an even linear map $\mathfrak{R}:\mathfrak{g}\to\mathfrak{g}$
satisfying for any homogeneous elements $x,y\in\mathfrak{g}$, the identity

(1.1)
$[\mathfrak{R}(x),\mathfrak{R}(y)]=\mathfrak{R}\Big{(}[\mathfrak{R}(x),y]+[x,\mathfrak{R}(y)]+\lambda[x,y]\Big{)}.$

Pre-Lie algebras (called also left-symmetric algebras, Vinberg algebras,
quasi-associative algebras) are a class of a natural algebraic systems
appearing in many fields in mathematics and mathematical physics. They were
first mentioned by Cayley in $1890$ [18]. They play an important role in the
study of symplectic and complex structures on Lie groups and Lie algebras [5,
19, 20, 21, 27], phases spaces of Lie algebras [9, 26], classical and quantum
Yang–Baxter equations [22] etc. Recently, the $\mathbb{Z}_{2}$-graded version
of pre-Lie algebras called pre-Lie superalgebras are defined as a linear
superspace $\mathfrak{g}$ equipped with an even bilinear map $"\circ"$
satisfying for any homogeneous elements $x,y\in\mathfrak{g}$, the identity

(1.2) $\mathfrak{ass}(x,y,z)=(-1)^{|x||y|}\mathfrak{ass}(y,x,z),$

where $\mathfrak{ass}(x,y,z)=x\circ(y\circ z)-(x\circ y)\circ z$, also
appeared in many others fields (see [15, 24, 35] for more details). They were
first introduced by Gerstenhaber in $1963$ to study the cohomology structure
of associative algebras [24].

When we look for the relationships between Lie superalgebras and pre-Lie
superalgebras structures we find that we can construct pre-Lie superalgebras
from Lie superalgebras uses Rota–Baxter operators. Let $\mathfrak{R}$ be a
Rota-Baxter operator on a Lie superalgebra $(\mathfrak{g},[\cdot,\cdot])$,
then, the product

(1.3) $x\circ y=[\mathfrak{R}(x),y],\;\forall
x,y\in\mathcal{H}(\mathfrak{g}),$

defines a pre-Lie superalgebra. The reader notices that the Rota-Baxter
operator is always considered as an even linear map and he has the right to
ask the following question: why is the operator $\mathfrak{R}$ not considered
odd? In [10], the authors answer this question and define the homogeneous
Rota-Baxter operators with parity as the data in the following:

A Rota-Baxter operator on Lie-superalgebra $(\mathfrak{g},[\cdot,\cdot])$ is a
linear map $\mathfrak{R}:\mathfrak{g}\to\mathfrak{g}$ defined by

(1.4)
$[\mathfrak{R}(x),\mathfrak{R}(y)]=\mathfrak{R}\Big{(}(-1)^{|\mathfrak{R}|(|x|+|\mathfrak{R}|)}[\mathfrak{R}(x),y]+[x,\mathfrak{R}(y)]\Big{)},\;\forall
x,y\in\mathcal{H}(\mathfrak{g}).$

Motiveted by this works, we introduce the notion of superalgebra with
homogenous structures of Lie type. We study this phenomenon for other
algebraic structures (associative superalgebras, pre-Lie superalgebras etc)
and their applications and associated operators such as representations,
derivations and Rota-Baxter operators. The ternary extending superalgebra with
homogenous structures of Lie type is studied.

###  1.1. Layout of the paper

Here is an outline of the paper. In Section 2, we introduce a Super-
generalization of some superalgebraic structures that we are interested in,
and to give some examples and related results which gives value and importance
to this new structures. In Section 3, we study a generalization of Rota-Baxter
operators on Super-Lie superalgebras, which requires us to provide to
introduce the right and left Rota-Baxter operators of Super-Lie superalgebras
which allows us to obtain a relation between Super-Lie superalgebras
structures and its associated Super-left(right)-symmetric superalgebras
structures via left (right) Rota-Baxter operators. We also give some
interesting examples. Representation theoy of Super-Lie superalgebras and its
associated dual representation are introduced in Section 4, then we
characterize it by semi-direct product. In what follows allowing us to
introduce the notions of right and left derivations of Super-Lie superalgebras
and we show that, the converse of an invertible left (resp. right) derivation
define a right (resp. left) Rota-Baxter operator. We finished this section by
a generalization of the Super-Lie superalgebras and some other structures in
the ternary case which we supported this with some examples and interesting
results. Section 5 is devoted to give a generalization of the Super-Lie
superalgebras and some other structures in the ternary case which we supported
this with some examples and interesting results. The aim of Section 6 is to
open the horizons by asking and discussing some important questions about
certain notions related to these new developments, the most important of which
are concerning deformations and cohomology, Maurer-Cartan characterization,
$\mathcal{O}$-operators etc.

###  1.2. Conventions and notations

  1. (1)

Throughout this paper, $\mathbb{K}$ denotes an algebraically closed field of
characteristic zero, and all vector spaces are over $\mathbb{K}$ and finite-
dimensional.

  2. (2)

A vector space $V$ is said to be $\mathbb{Z}_{2}$-graded if we are given a
family $(V_{i})_{i\in\mathbb{Z}_{2}}$ of vector subspaces of $V$ such that
$V=V_{\overline{0}}\oplus V_{\overline{1}}.$

  3. (3)

The symbol $\overline{x}$ always implies that $x$ is a
$\mathbb{Z}_{2}$-homogeneous element and it is their $\mathbb{Z}_{2}$-degree.
We denote by $\mathcal{H}(V)$ the set of all homogeneous elements of $V$ and
$\mathcal{H}(V^{n})$ refers to the set of tuples with homogeneous elements.

  4. (4)

We refer by $\overline{F}$ to the parity of a multilinear map
$F:V^{1}\times\dots\times V^{n}\rightarrow W$ by

$F(V_{i_{1}}^{1},\dots,V_{i_{n}}^{1})\subset
W_{i_{1}+\cdots+i_{n}+\overline{F}}.$

  5. (5)

Let $End(V)$ be the $\mathbb{Z}_{2}$-graded vector space of endomorphisms of a
$\mathbb{Z}_{2}$-graded vector space $V.$ The graded binary commutator
$[f,g]=f\circ g-(-1)^{\overline{f}\overline{g}}g\circ f$ induces a structure
of Lie superalgebra in $End(V)$.

##  2\. Super-Lie superalgebras and Admissibilty

In this section, we introduce the Super-generalization of some algebraic
structures such as Super-associative superalgebras, left(right) Super-Leibniz
superalgebras, Super-Lie superalgebras, Super-left(right)-symmetric
superalgebras and we give some examples and related results of these
structures. All structures studied throughout this work are considered
homogeneous.

###  2.1. Definitions and Examples

######  Definition 2.1.

A Super-associative superalgebra is a couple $(\mathfrak{g},\mu)$ consisting
of a $\mathbb{Z}_{2}$-graded vector space $\mathfrak{g}$ and a bilinear map
$\mu:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$ satisfying

(2.1) $\mathfrak{ass}_{\mu}(x,y,z)=0,\;\forall
x,y,z\in\mathfrak{g},\;(\mathbb{Z}_{2}\text{-graded associativity condition})$

where $\mathfrak{ass}_{\mu}(x,y,z)=\mu(\mu(x,y),z)-\mu(x,\mu(y,z))$. If, in
addition
$\mu(x,y)=(-1)^{(\overline{x}+\overline{\mu})(\overline{y}+\overline{\mu})}\mu(y,x),\;\forall
x,y\in\mathcal{H}(\mathfrak{g})$, the super-associative superalgebra is called
supercommutative.

######  Remark 2.1.

If $\overline{\mu}=0$, we recover the classical associative superalgebras
structures also called even associative superalgebras. If $\overline{\mu}=1$,
we call it odd associative superalgebras.

######  Example 2.1.

Let
$\mathfrak{g}=\mathfrak{g}_{\overline{0}}\oplus\mathfrak{g}_{\overline{1}}$
where $\mathfrak{g}_{\overline{0}}=<e_{1}>$ and
$\mathfrak{g}_{\overline{1}}=<e_{2}>$ be a two dimentional
$\mathbb{Z}_{2}$-graded vector space. Let use define the bilinear map
$\mu:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$ on the elements of the
basis of $\mathfrak{g}$ by

$\mu(e_{1},e_{2})=-\mu(e_{2},e_{1})=e_{1},\;\;\mu(e_{2},e_{2})=e_{2}.$

Then $(\mathfrak{g},\mu)$ is a super-noncommutative odd associative
superalgebra.

######  Example 2.2.

Let $\mathfrak{g}=<e_{1},e_{2}\;|\;e_{3}>$ be a three dimentional
$\mathbb{Z}_{2}$-graded vector space. Then, the odd bilinear map
$\mu:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$ defined by

$\mu(e_{1},e_{2})=-\mu(e_{2},e_{1})=e_{3},$

defines on $\mathfrak{g}$ a supercommutative odd associative superalgebra
structure.

######  Definition 2.2.

A left Super-Leibniz superalgebra is a couple $(\mathfrak{g},[\cdot,\cdot])$
where, $\mathfrak{g}$ is a $\mathbb{Z}_{2}$-graded vector space and
$[\cdot,\cdot]$ be a bilinear map satisfying the following identity

(2.2)
$[x,[y,z]]=[[x,y],z]+(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}[y,[x,z]],$

for any homogeneous elements $x,y,z\in\mathfrak{g}$. The identity (2.2) is
called left Super-Leibniz identity.

Let us fixed an element $x\in\mathcal{H}(\mathfrak{g})$. Define the linear map
$ad^{L}_{x}:\mathfrak{g}\to\mathfrak{g}$ by

(2.3) $ad^{L}_{x}(y)=[x,y],\;\forall y\in\mathcal{H}(\mathfrak{g}).$

Then, $\overline{ad^{L}_{x}}=\overline{x}+\overline{[\cdot,\cdot]}$ and the
left Super-Leibniz identity (2.2) can be written as follow

(2.4)
$ad^{L}_{x}[y,z]=[ad^{L}_{x}(y),z]+(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}[y,ad^{L}_{x}(z)].$

######  Definition 2.3.

A right Super-Leibniz superalgebra is a $\mathbb{Z}_{2}$-graded vector space
equipped with a bilinear map $[\cdot,\cdot]$ such that, for any homogeneous
elements $x,y,z\in\mathfrak{g}$, the following identity hold

(2.5)
$[[x,y],z]=[x,[y,z]]+(-1)^{(\overline{y}+\overline{[\cdot,\cdot]})(\overline{z}+\overline{[\cdot,\cdot]})}[[x,z],y].$

The identity (2.5) is called right Super-Leibniz identity.

For any fixed homogeneous element $x$ of $\mathfrak{g}$, we define the linear
map $ad^{R}_{x}:\mathfrak{g}\to\mathfrak{g}$ by

(2.6) $ad^{R}_{x}(y)=[y,x],\;\forall y\in\mathcal{H}(\mathfrak{g}).$

Then, $\overline{ad^{R}_{x}}=\overline{x}+\overline{[\cdot,\cdot]}$ and the
right Super-Leibniz identity (2.5) can be written as follow

(2.7)
$ad^{R}_{z}[x,y]=[x,ad^{R}_{z}(y)]+(-1)^{(\overline{y}+\overline{[\cdot,\cdot]})(\overline{z}+\overline{[\cdot,\cdot]})}[ad^{R}_{z}(x),y].$

######  Definition 2.4.

A Super-Lie superalgebra is a $\mathbb{Z}_{2}$-graded vector space
$\mathfrak{g}$ equipped with a bilinear product
$[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$ satisfying

(2.8)
$\displaystyle[x,y]=-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(|y|+\overline{[\cdot,\cdot]})}[y,x]\;\;\;\;\;\;\;\;(\mathbb{Z}_{2}\text{-graded
super-skewsymmetry}),$ (2.9)
$\displaystyle\displaystyle\circlearrowleft_{x,y,z}(-1)^{(\overline{x}+\overline{[,\cdot,\cdot]})(\overline{z}+\overline{[,\cdot,\cdot]})}[x,[y,z]]=0\;\;(\mathbb{Z}_{2}\text{-graded
super-Jacobi identity})$

for any $x,y,z\in\mathcal{H}(\mathfrak{g})$. Such Super-Lie superalgebra is
denoted by $(\mathfrak{g},[\cdot,\cdot])$.

######  Remark 2.2.

If $\overline{[\cdot,\cdot]}=0$, we recover the (even) Lie superalgebra
structure and if $\overline{[\cdot,\cdot]}=1$, we recall it an odd Lie
superalgebra. A Super-Lie superalgebra is just a left Super-Leibniz (resp.
right Super-Leibniz) superalgebra $(\mathfrak{g},[\cdot,\cdot])$ in which, the
product $[\cdot,\cdot]$ is $\mathbb{Z}_{2}$-graded super-skewsymmetric.

######  Example 2.3.

Let $\mathfrak{g}=<e_{1},e_{2},e_{3}>$ be a three dimentional
$\mathbb{Z}_{2}$-graded vector space where
$\mathfrak{g}_{\overline{0}}=<e_{1}>$ and
$\mathfrak{g}_{\overline{1}}=<e_{2},e_{3}>$. Define the bilinear product
$[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$ by

$[e_{2},e_{3}]=e_{2},$

and the other products are zeros. Then $(\mathfrak{g},[\cdot,\cdot])$ is an
odd Lie superalgebra.

###  2.2. Super-Lie admissible superalgebras

######  Definition 2.5.

A Super-Lie admissible superalgebra is a $\mathbb{Z}_{2}$-graded vector space
$\mathfrak{g}$ equipped with a bilinear map
$\mu:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$ such that, the bracket
defined for all homogeneous elements $x,y\in\mathfrak{g}$ by

(2.10)
$[x,y]=\mu(x,y)-(-1)^{(\overline{x}+\overline{\mu})(\overline{y}+\overline{\mu})}\mu(y,x),$

satisfies the $\mathbb{Z}_{2}$-graded super-Jacobi identity (2.9).

######  Definition 2.6.

Let $\mathfrak{g}$ be a $\mathbb{Z}_{2}$-graded vector space. Then,
$\mathfrak{g}$ is called:

  1. (1)

Super-left-symmetric superalgebra if it is equipped with a product
$\vartriangleright$ satisfying

(2.11)
$\mathfrak{ass}_{\vartriangleright}(x,y,z)=(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{\vartriangleright})}\mathfrak{ass}_{\vartriangleright}(y,x,z),\;\forall
x,y,z\in\mathcal{H}(\mathfrak{g}).$

  2. (2)

Super-right-symmetric superalgebra if it is equipped with a product
$\vartriangleleft$ satisfying

(2.12)
$\mathfrak{ass}_{\vartriangleleft}(x,y,z)=(-1)^{(\overline{y}+\overline{\vartriangleleft})(\overline{z}+\overline{\vartriangleleft})}\mathfrak{ass}_{\vartriangleleft}(x,z,y),\;\forall
x,y,z\in\mathcal{H}(\mathfrak{g}).$

######  Remark 2.3.

Any Super-associative superalgebra is a Super-left-symmetric (resp. Super-
right-symmetric) superalgebra.

######  Theorem 2.1.

Any Super-left-symmetric (resp. Super-right-symmetric) superalgebra is a
Super-Lie admissible superalgebra.

###### Proof.

Let $(\mathfrak{g},\vartriangleright)$ be a Super-left-symmetric superalgebra
Then, it is easy to see that, $[\cdot,\cdot]$ is $\mathbb{Z}_{2}$-graded
super-skewsymmetric. It remains to show that $\mathbb{Z}_{2}$-graded super-
Jacobi identity is satisfied. Let $x,y,z\in\mathcal{H}(\mathfrak{g})$, we have

$\displaystyle[x,[y,z]]$
$\displaystyle=x\vartriangleright[y,z])-(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{[y,z]}+\overline{\vartriangleright})}[y,z]\vartriangleright
x)$ $\displaystyle=x\vartriangleright(y\vartriangleright
z)-(-1)^{(\overline{y}+\overline{\vartriangleright})(\overline{z}+\overline{\vartriangleright})}x\vartriangleright(z\vartriangleright
y)-(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{z}+2\overline{\vartriangleright})}(y\vartriangleright
z)\vartriangleright x$
$\displaystyle+(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{z}+2\overline{\vartriangleright})}(-1)^{(\overline{y}+\overline{\vartriangleright})(\overline{z}+\overline{\vartriangleright})}(z\vartriangleright
y)\vartriangleright x$ $\displaystyle=x\vartriangleright(y\vartriangleright
z)-(-1)^{(\overline{y}+\overline{\vartriangleright})(\overline{z}+\overline{\vartriangleright})}x\vartriangleright(z\vartriangleright
y)-(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{z})}(y\vartriangleright
z)\vartriangleright x$
$\displaystyle+(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{z})}(-1)^{(\overline{y}+\overline{\vartriangleright})(\overline{z}+\overline{\vartriangleright})}(z\vartriangleright
y)\vartriangleright x.$

Similarly, we have:

$\displaystyle[[x,y],z]$ $\displaystyle=(x\vartriangleright
y)\vartriangleright
z-(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{\vartriangleright})}(y\vartriangleright
x)\vartriangleright
z-(-1)^{(\overline{z}+\overline{\vartriangleright})(\overline{x}+\overline{z})}z\vartriangleright(x\vartriangleright
y)$
$\displaystyle+(-1)^{(\overline{z}+\overline{\vartriangleright})(\overline{x}+\overline{y})}(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{\vartriangleright})}z\vartriangleright(y\vartriangleright
x)$

and

$\displaystyle[y,[x,z]_{\mu}]_{\mu}$
$\displaystyle=y\vartriangleright(x\vartriangleright
z)-(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{z}+\overline{\vartriangleright})}y\vartriangleright(z\vartriangleright
x)-(-1)^{(\overline{y}+\overline{\vartriangleright})(\overline{x}+\overline{z})}(x\vartriangleright
z)\vartriangleright y$
$\displaystyle+(-1)^{(\overline{y}+\overline{\vartriangleright})(\overline{x}+\overline{z})}(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{z}+\overline{\vartriangleright})}(z\vartriangleright
x)\vartriangleright y.$

By, the fact that $(\mathfrak{g},\vartriangleright)$ is
$\mathbb{Z}_{2}$-graded left-symmetric superalgebra and a direct computation,
we get

$\displaystyle R$
$\displaystyle=[x,[y,z]]-[[x,y],z]-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}[y,[x,z]]$
$\displaystyle=\Big{(}x\vartriangleright(y\vartriangleright
z)-(x\vartriangleright y)\vartriangleright
z\Big{)}+(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{\vartriangleright})}\Big{(}(y\vartriangleright
x)\vartriangleright z-y\vartriangleright(x\vartriangleright z)\Big{)}$
$\displaystyle+(-1)^{(\overline{y}+\overline{\vartriangleright})(\overline{z}+\overline{\vartriangleright})}\Big{(}(x\vartriangleright
z)\vartriangleright y-x\vartriangleright(z\vartriangleright
y)\Big{)}-(-1)^{zv+\overline{\vartriangleright})(\overline{x}+\overline{y})}\Big{(}(z\vartriangleright
x)\vartriangleright y-z\vartriangleright(x\vartriangleright y)\Big{)}$
$\displaystyle-(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{z})}\Big{(}(y\vartriangleright
z)\vartriangleright x-y\vartriangleright(z\vartriangleright
x)\Big{)}+(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{z})}(-1)^{(\overline{y}+\overline{\vartriangleright})(\overline{z}+\overline{\vartriangleright})}\Big{(}(z\vartriangleright
y)\vartriangleright x-z\vartriangleright(y\vartriangleright x)\Big{)}$
$\displaystyle=-\mathfrak{ass}_{\vartriangleright}(x,y,z)+(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{\vartriangleright})}\mathfrak{ass}_{\vartriangleright}(y,x,z)+(-1)^{(\overline{y}+\overline{\vartriangleright})(\overline{z}+\overline{\vartriangleright})}\Big{(}\mathfrak{ass}_{\vartriangleright}(x,z,y)-(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{z}+\overline{\vartriangleright})}\mathfrak{ass}_{\vartriangleright}(z,x,y)\Big{)}$
$\displaystyle-(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{z})}\Big{(}\mathfrak{ass}_{\vartriangleright}(y,z,x)-(-1)^{(\overline{y}+\overline{\vartriangleright})(\overline{z}+\overline{\vartriangleright})}\mathfrak{ass}_{\vartriangleright}(z,y,x)\Big{)}$
$\displaystyle=0.$

Then, $(\mathfrak{g},\vartriangleright)$ is a Super-Lie admissible
superalgebra. Similarly, we can show the same result for the Super-right-
symmetric superalgebra. ∎

By Theorem 2.1 and Remark 2.3, we can get the following result.

######  Corollary 2.1.

Any Super-associative superalgebra is a Super-Lie admissible superalgebra.

######  Example 2.4.

Let us consider the Super-associative superalgebra $(\mathfrak{g},\mu)$ given
in Example 2.1. Then, by Corollary 2.1, the odd bilinear map
$[\cdot,\cdot]_{\mu}$ defined on the basis $\\{e_{1},e_{2}\\}$ by

$[e_{1},e_{2}]\mu=\mu(e_{1},e_{2})-(-1)^{\overline{\mu}(1+\overline{\mu})}\mu(e_{2},e_{1})=2e_{1},$

defines on $\mathfrak{g}$ an odd Lie superalgebra structure.

######  Proposition 2.1.

Let $(\mathfrak{g},\vartriangleright)$ be a Super-left-symmetric superalgebra.
Then $(\mathfrak{g},\vartriangleleft)$ is a Super-right-symmetric
superalgebra, where

(2.13) $x\vartriangleleft
y=(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{\vartriangleright})}y\vartriangleright
x,\;\forall x,y\in\mathcal{H}(\mathfrak{g}).$

######  Definition 2.7.

A $\mathbb{Z}_{2}$-graded superalgebra $(\mathfrak{g},\circ)$ is called Super-
flexible superalgebra, if it satisfies:

(2.14)
$\mathfrak{ass}_{\circ}(x,y,z)=(-1)^{(\overline{x}+\overline{\circ})(\overline{y}+\overline{z})+(\overline{y}+\overline{\circ})(\overline{z}+\overline{\circ})}\mathfrak{ass}_{\circ}(z,y,x),\;\forall
x,y,z\in\mathcal{H}(\mathfrak{g}).$

######  Proposition 2.2.

If $\mathfrak{g}$ is both super-left and super-right-symmetric superalgebra,
then it is a Super-flexible superalgebra. Moreover, any Super-flexible
superalgebra is an admissible Super-Lie superalgebra.

######  Remark 2.4.

Any Super-associative superalgebra is a Super-flexible superalgebra.

##  3\. Rota-Baxter of Super-Lie superalgebras

In this section, we give an extension of the notion of Rota-Baxter operators
on the Super-Lie Superalgebras, for this we are obliged to introduce the
notions of the left Rota-Baxter operators, and the right Rota-Baxter operators
on Super-Lie superalgebras wich is a generalization of what given in [10].
Thus, we give the relation between Super-Lie superalgebras and Super-
left(right)-symmetric superalgebras from the left (right) Rota-Baxter
operators on Super-Lie superalgebras. We also enriched it with some examples.

######  Definition 3.1.

Let $(\mathfrak{g},[\cdot,\cdot])$ be a Super-Lie superalgebra. A left Rota-
Baxter operator on $\mathfrak{g}$ is an homogeneous linear map
$\mathfrak{R}:\mathfrak{g}\to\mathfrak{g}$ satisfying

(3.1)
$[\mathfrak{R}(x),\mathfrak{R}(y)]=\mathfrak{R}\Big{(}[\mathfrak{R}(x),y]+(-1)^{\overline{\mathfrak{R}}(\overline{y}+\overline{\mathfrak{R}}+\overline{[\cdot,\cdot]})}[x,\mathfrak{R}(y)]\Big{)},\;\forall
x,y\in\mathcal{H}(\mathfrak{g}).$

######  Theorem 3.1.

Let $\mathfrak{R}$ be left Rota-Baxter operator on a Super-Lie superalgebra
$(\mathfrak{g},[\cdot,\cdot])$. Then, the binary operation defined by

(3.2) $x\vartriangleright y=[\mathfrak{R}(x),y],\;\forall
x,y\in\mathcal{H}(\mathfrak{g}),$

defines on $\mathfrak{g}$ a Super-left-symmetric superalgebra structure.

###### Proof.

It is easy to see that Eq. (3.2) gives,
$\overline{\vartriangleright}=\overline{\mathfrak{R}}+\overline{[\cdot,\cdot]}$.
Let $x,y,z\in\mathcal{H}(\mathfrak{g})$. Then, by using Eqs. (2.2) and (3.1),
we have

$\displaystyle\mathfrak{ass}_{\vartriangleright}(x,y,z)-(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{\vartriangleright})}\mathfrak{ass}_{\vartriangleright}(y,x,z)$
$\displaystyle=(x\vartriangleright y)\vartriangleright
z-x\vartriangleright(y\vartriangleright
z)-(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{\vartriangleright})}(y\vartriangleright
x)\vartriangleright z$
$\displaystyle+(-1)^{(\overline{x}+\overline{\vartriangleright})(\overline{y}+\overline{\vartriangleright})}y\vartriangleright(x\vartriangleright
z)$
$\displaystyle=[\mathfrak{R}[\mathfrak{R}(x),y],z]-(-1)^{(\overline{x}+\overline{\mathfrak{R}}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{\mathfrak{R}}+\overline{[\cdot,\cdot]})}[\mathfrak{R}[\mathfrak{R}(y),x],z]$
$\displaystyle-[\mathfrak{R}(x),[\mathfrak{R}(y),z]]+(-1)^{(\overline{x}+\overline{\mathfrak{R}}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{\mathfrak{R}}+\overline{[\cdot,\cdot]})}[\mathfrak{R}(y),[\mathfrak{R}(x),z]]$
$\displaystyle=\Big{[}\Big{(}\mathfrak{R}[\mathfrak{R}(x),y]+(-1)^{\overline{\mathfrak{R}}(\overline{y}+\overline{\mathfrak{R}}+\overline{[\cdot,\cdot]})}\mathfrak{R}[x,\mathfrak{R}(y)]\Big{)},z\Big{]}$
$\displaystyle-[[\mathfrak{R}(x),\mathfrak{R}(y)],z]$
$\displaystyle=[[\mathfrak{R}(x),\mathfrak{R}(y)],z]-[[\mathfrak{R}(x),\mathfrak{R}(y)],z]$
$\displaystyle=0.$

Then, $(\mathfrak{g},\vartriangleright)$ is a Super-left-symmetric
superalgebra. ∎

######  Example 3.1.

Let us consider the orthosymplectic superalgebra
$\mathfrak{osp}(1|2)=\mathfrak{osp}(1|2)_{\overline{0}}\oplus\mathfrak{osp}(1|2)_{\overline{1}}$
by the respective basis $\\{Y,F,G\\}$ of $\mathfrak{osp}(1|2)_{\overline{0}}$,
$\\{E_{+},E_{-}\\}$ of $\mathfrak{osp}(1|2)_{\overline{1}}$ and the even
bilinear product define on the basis of $\mathfrak{osp}(1|2)$ by

$[Y,F]=F,\;\;[Y,G]=-G,\;\;[F,G]=2Y,$
$[Y,E_{\pm}]=\pm\frac{1}{2}E_{\pm},\;\;[F,E_{+}]=[G,E_{-}]=0,$
$[F,E_{-}]=-E_{+},\;\;\;[G,E_{+}]=-E_{-},$
$[E_{+},E_{+}]=F,\;\;[E_{-},E_{-}]=-G,\;\;[E_{+},E_{-}]=Y.$

Then $(\mathfrak{osp}(1|2),[\cdot,\cdot])$ is an even Lie superalgebra (See
[33] for more details).

Now, let us define the odd linear map
$\mathfrak{R}_{L}:\mathfrak{osp}(1|2)\to\mathfrak{osp}(1|2)$ on the elements
of basis of $\mathfrak{osp}(1|2)$ by

$\mathfrak{R}_{L}(F)=E_{-},\;\;\;\;\mathfrak{R}_{L}(E_{+})=-\frac{1}{2}G,$

and the other terms are zeros. Then $\mathfrak{R}$ is a left Rota-Baxter
operator on $\mathfrak{osp}(1|2)$. By using Theorem (3.1), the bilinear map
$\vartriangleright:\mathfrak{osp}(1|2)\times\mathfrak{osp}(1|2)\to\mathfrak{osp}(1|2)$
defined by

$F\vartriangleright
Y=[\mathfrak{R}_{L}(F),Y]=\frac{1}{2}E_{-},\;\;F\vartriangleright
F=[\mathfrak{R}_{L}(F),F]=E_{+},$ $F\vartriangleright
E_{+}=[\mathfrak{R}_{L}(F),E_{+}]=Y,\;\;F\vartriangleright
E_{-}=[\mathfrak{R}_{L}(F),E_{-}]=-G,$ $E_{+}\vartriangleright
Y=[\mathfrak{R}_{L}(E_{+}),Y]=\frac{1}{2}G,\;\;E_{+}\vartriangleright
F=[\mathfrak{R}_{L}(E_{+}),F]=-Y,$ $E_{+}\vartriangleright
E_{+}=[\mathfrak{R}_{L}(E_{+}),E_{+}]=\frac{1}{2}E_{-},$

defines an odd Super-left-symmetric superalgebra structure on
$\mathfrak{osp}(1|2)$.

######  Definition 3.2.

Let $(\mathfrak{g},[\cdot,\cdot])$ be a Super-Lie superalgebra. A right Rota-
Baxter operator on $\mathfrak{g}$ is an homogeneous linear map
$\mathfrak{R}:\mathfrak{g}\to\mathfrak{g}$ satifying

(3.3)
$[\mathfrak{R}(x),\mathfrak{R}(y)]=\mathfrak{R}\Big{(}(-1)^{\overline{\mathfrak{R}}(\overline{x}+\overline{\mathfrak{R}}+\overline{[\cdot,\cdot]})}[\mathfrak{R}(x),y]+[x,\mathfrak{R}(y)]\Big{)},\;\forall
x,y\in\mathcal{H}(\mathfrak{g}).$

######  Theorem 3.2.

Let $\mathfrak{R}$ be a right Rota-Baxter operator on a Super-Lie superalgebra
$(\mathfrak{g},[\cdot,\cdot])$. Then $(\mathfrak{g},\vartriangleleft)$ is a
Super-right-symmetric superalgebra, where the bilinear map
$"\vartriangleleft"$ is defined by

(3.4) $x\vartriangleleft y=[x,\mathfrak{R}(y)],\;\forall
x,y\in\mathcal{H}(\mathfrak{g}).$

###### Proof.

Let $x,y,z\in\mathcal{H}(\mathfrak{g})$. By using Eqs. (2.5) and (3.3), a
direct computation gives that

$\displaystyle\mathfrak{ass}_{\vartriangleleft}(x,y,z)-(-1)^{(\overline{y}+\overline{\vartriangleleft})(\overline{z}+\overline{\vartriangleleft})}\mathfrak{ass}_{\vartriangleleft}(x,z,y)$
$\displaystyle=(x\vartriangleleft y)\vartriangleleft
z-x\vartriangleleft(y\vartriangleleft
z)-(-1)^{(\overline{y}+\overline{\vartriangleleft})(\overline{z}+\overline{\vartriangleleft})}(x\vartriangleleft
z)\vartriangleleft y$
$\displaystyle+(-1)^{(\overline{y}+\overline{\vartriangleleft})(\overline{z}+\overline{\vartriangleleft})}x\vartriangleleft(z\vartriangleleft
y)$
$\displaystyle=[[x,\mathfrak{R}(y)],\mathfrak{R}(z)]-[x,\mathfrak{R}[y,\mathfrak{R}(z)]]$
$\displaystyle-(-1)^{(\overline{y}+\overline{[\cdot,\cdot]}+\overline{\mathfrak{R}})(\overline{z}+\overline{[\cdot,\cdot]}+\overline{\mathfrak{R}})}[[x,\mathfrak{R}(z)],\mathfrak{R}(y)]$
$\displaystyle+(-1)^{(\overline{y}+\overline{[\cdot,\cdot]}+\overline{R})(\overline{z}+\overline{[\cdot,\cdot]}+\overline{\mathfrak{R}})}[x,\mathfrak{R}[z,\mathfrak{R}(y)]]$
$\displaystyle=-\Big{(}[x,\mathfrak{R}[y,\mathfrak{R}(z)]]+(-1)^{\overline{\mathfrak{R}}(\overline{y}+\overline{[\cdot,\cdot]}+\overline{\mathfrak{R}})}[x,\mathfrak{R}[\mathfrak{R}(y),z]]\Big{)}$
$\displaystyle+\Big{(}[[x,\mathfrak{R}(y)],\mathfrak{R}(z)]-(-1)^{(\overline{y}+\overline{[\cdot,\cdot]}+\overline{\mathfrak{R}})(\overline{z}+\overline{[\cdot,\cdot]}+\overline{\mathfrak{R}})}\Big{)}[[x,\mathfrak{R}(z)],\mathfrak{R}(y)]$
$\displaystyle=-[x,[\mathfrak{R}(y),\mathfrak{R}(z)]]+[[x,\mathfrak{R}(y)],\mathfrak{R}(z)]$
$\displaystyle+(-1)^{(\overline{y}+\overline{[\cdot,\cdot]}+\overline{\mathfrak{R}})(\overline{x}+\overline{[\cdot,\cdot]})}[\mathfrak{R}(y),[x,\mathfrak{R}(z)]]$
$\displaystyle=-[x,[\mathfrak{R}(y),\mathfrak{R}(z)]]+[x,[\mathfrak{R}(y),\mathfrak{R}(z)]]$
$\displaystyle=0.$

Then, $(\mathfrak{g},\vartriangleleft)$ is a Super-right-symmetric
superalgebra. ∎

######  Example 3.2.

Let us consider the even Lie Superalgebra $\mathfrak{osp}(1|2)$ given in the
Example 3.1. Define the odd linear map
$\mathfrak{R}_{R}:\mathfrak{osp}(1|2)\to\mathfrak{osp}(1|2)$ by

$\mathfrak{R}_{R}(F)=E_{-},\quad\mathfrak{R}_{R}(E_{+})=\frac{1}{2}G,$

and the other terms are zeros. Then $\mathfrak{R}_{R}$ is a right Rota-Baxter
operator on $\mathfrak{osp}(1|2)$. So, by using Theorem 3.2, the bilinear map
$\vartriangleleft:\mathfrak{osp}(1|2)\times\mathfrak{osp}(1|2)\to\mathfrak{osp}(1|2)$
defined by

$Y\vartriangleleft
F=[Y,\mathfrak{R}_{R}(F)]=-\frac{1}{2}E_{-},\;\;F\vartriangleleft
F=[F,\mathfrak{R}_{R}(F)]=-E_{+},$ $F\vartriangleleft
E_{+}=[F,\mathfrak{R}_{R}(E_{+})]=Y,\;\;E_{-}\vartriangleleft
F=[E_{-},\mathfrak{R}_{R}(F)]=-G,$ $Y\vartriangleleft
E_{+}=[Y,\mathfrak{R}_{R}(E_{+})]=-\frac{1}{2}G,\;\;E_{+}\vartriangleleft
F=[E_{+},\mathfrak{R}_{R}(F)]=Y,$ $E_{+}\vartriangleleft
E_{+}=[E_{+},\mathfrak{R}_{R}(E_{+})]=\frac{1}{2}E_{-},$

defines an odd Super-right-symmetric superalgebra structure on
$\mathfrak{osp}(1|2)$.

##  4\. Represenations and derivations of Super-Lie superlagebras

In this section, we define the notion of representation with parity of Super-
Lie superalgebras and to construct its associated dual representation. A
characterisation of representations on a Super-Lie superalgebra structure via
semi-direct product has been studied. In addition, we are required to
introduce the notions of right and left derivations of Super-Lie superalgebras
in which, we show that, the converse of an invertible left (right) derivation
define a right (left) Rota-Baxter operator. Still we have not forgotten to
give some examples and associated fundamental results.

###  4.1. Represenations of Super-Lie superalgebras

Let $(\mathfrak{g},[\cdot,\cdot])$ be a Super-Lie superalgebra and $V$ be a
$\mathbb{Z}_{2}$-graded vector space.

######  Definition 4.1.

A linear map $\rho:\mathfrak{g}\to End(V)$ is called representation of
$\mathfrak{g}$ if $\overline{\rho}=\overline{[\cdot,\cdot]}$ and it satisfies

(4.1)
$\rho([x,y])=\rho(x)\rho(y)-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}\rho(y)\rho(x),\;\forall
x,y\in\mathcal{H}(\mathfrak{g}).$

We also say that $V$ is a $\mathfrak{g}$-module. This representation is
denoted by $(V,\rho)$.

######  Example 4.1.

For any $x\in\mathcal{H}(\mathfrak{g})$, the linear map $ad:\mathfrak{g}\to
End(\mathfrak{g});\;x\mapsto ad(x)$ (or $ad_{x}$) defined by

$ad_{x}(y)=[x,y],\;\forall y\in\mathcal{H}(\mathfrak{g}),$

defines a representation on $\mathfrak{g}$ called adjoint representation.
Indeed, for any $x,y,z\in\mathcal{H}(\mathfrak{g})$, by using the identity
(2.2), we have

$\displaystyle ad_{[x,y]}(z)$
$\displaystyle=[[x,y],z]=[x,[y,z]]-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}[y,[x,z]]$
$\displaystyle=ad_{x}\circ
ad_{y}(z)-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}ad_{y}\circ
ad_{x}(z).$

Then, $ad[x,y]=ad_{x}\circ
ad_{y}-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}ad_{y}\circ
ad_{x},\;\forall x,y\in\mathcal{H}(\mathfrak{g})$, which gives that, the map
”$ad$” is a representation of $\mathfrak{g}$.

######  Proposition 4.1.

Let $V$ be a $\mathbb{Z}_{2}$-graded vector space and $\rho:\mathfrak{g}\to
End_{\mathbb{K}}(V)$ be a linear map. Then $(V,\rho)$ is a representation of
$\mathfrak{g}$ if and only if,
$\mathfrak{g}\ltimes_{\rho}V:=(\mathfrak{g}\oplus
V,[\cdot,\cdot]_{\mathfrak{g}\oplus V})$ is a Super-Lie superalgebra, where
the bracket $[\cdot,\cdot]_{\mathfrak{g}\oplus V}$ is defined by

(4.2) $[x+u,y+v]_{\mathfrak{g}\oplus
V}=[x,y]+\rho(x)(v)-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}\rho(y)(u),$

for all $x,y\in\mathcal{H}(\mathfrak{g})$ and $u,v\in\mathcal{H}(V)$. We call
$\mathfrak{g}\ltimes_{\rho}V$ the semi-direct product of the Super-Lie
superalgebra $(\mathfrak{g},[\cdot,\cdot])$ and $V$. Note that
$\overline{x+u}=\overline{x}=\overline{u}$.

###### Proof.

Let $x,y\in\mathcal{H}(\mathfrak{g})$ and $u,v\in\mathcal{H}(V)$, we have

$\displaystyle[x+u,y+v]_{\mathfrak{g}\oplus V}$
$\displaystyle=[x,y]+\rho(x)(v)-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}\rho(y)(u)$
$\displaystyle=-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}[y,x]-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}\rho(y)(u)+\rho(x)(v)$
$\displaystyle=-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}([y,x]+\rho(y)(u)-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}\rho(x)(v))$
$\displaystyle=(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}[y+v,x+u]_{\mathfrak{g}\oplus
V}.$

Then, the product $[\cdot,\cdot]_{\mathfrak{g}\oplus V}$ is
$\mathbb{Z}_{2}$-graded super-antisymmetric. It remains to show that, the
identity (2.2) is satisfied. Let $x,y,z\in\mathcal{H}(\mathfrak{g})$ and
$u,v,w\in\mathcal{H}(V)$, then by a direct computation, we have:

$\displaystyle[x+u,[y+v,z+w]_{\mathfrak{g}\oplus V}]_{\mathfrak{g}\oplus V}$
$\displaystyle=[x,[y,z]]+\rho(x)\Big{(}\rho(y)w-(-1)^{(\overline{y}+\overline{[\cdot,\cdot]})(\overline{z}+\overline{[\cdot,\cdot]})}\rho(z)v\Big{)}$
$\displaystyle-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{z})}\rho([y,z])u,$

$\displaystyle[[x+u,y+v]_{\mathfrak{g}\oplus V},z+w]_{\mathfrak{g}\oplus V}$
$\displaystyle=[[x,y],z]+\rho([x,y])w$
$\displaystyle-(-1)^{(\overline{x}+\overline{y})(\overline{z}+\overline{[\cdot,\cdot]})}\rho(z)\Big{(}\rho(x)v-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}\rho(y)u\Big{)},$

and

$\displaystyle[y+v,[x+u,z+w]_{\mathfrak{g}\oplus V}]_{\mathfrak{g}\oplus V}$
$\displaystyle=[y,[x,z]]+\rho(y)\Big{(}\rho(x)w-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{z}+\overline{[\cdot,\cdot]})}\rho(z)u\Big{)}$
$\displaystyle-(-1)^{(\overline{y}+\overline{[\cdot,\cdot]})(\overline{x}+\overline{z})}\rho([x,z])v.$

Then, $\mathfrak{g}\ltimes_{\rho}V:=(\mathfrak{g}\oplus
V,[\cdot,\cdot]_{\mathfrak{g}\oplus V})$ is a Super-Lie superalgebra if and
only if

$\displaystyle[x+u,[y+v,z+w]_{\mathfrak{g}\oplus V}]_{\mathfrak{g}\oplus V}$
$\displaystyle=[[x+u,y+v]_{\mathfrak{g}\oplus V},z+w]_{\mathfrak{g}\oplus V}$
$\displaystyle+(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}[y+v,[x+u,z+w]_{\mathfrak{g}\oplus
V}]_{\mathfrak{g}\oplus V},$

for any, $x,y,z\in\mathcal{H}(\mathfrak{g})$ and $u,v,w\in\mathcal{H}(V)$.
Then, by applying the fact that $(\mathfrak{g},[\cdot,\cdot])$ is a Super-Lie
superalgebra and we take $u=v=0$, we get

$\rho[x,y]w=\rho(x)\rho(y)w-(-1)^{(\overline{x}+\overline{[\cdot,\cdot]})(\overline{y}+\overline{[\cdot,\cdot]})}\rho(y)\rho(x)w$

for any homogeneous element $w\in V$, which gives the result. ∎

Let $(V,\rho)$ be a representation of $\mathfrak{g}$. Define
$\rho^{*}:\mathfrak{g}\to End(V^{*})$ by

(4.3)
$\rho^{*}(x)(\xi)=-(-1)^{\overline{\xi}(\overline{x}+\overline{\rho})}\xi\circ\rho(x),\quad\forall\quad
x\in\mathcal{H}(\mathfrak{g})\quad\text{and }\quad\xi\in\mathcal{H}(V^{*}).$

######  Proposition 4.2.

With the above notations, $\rho^{*}$ is a represenation of $\mathfrak{g}$ on
the dual space $V^{*}$, which is called the dual representation of $\rho$.

###### Proof.

Let $x,y\in\mathcal{H}(\mathfrak{g})$ and $\xi\in\mathcal{H}(V^{*})$. Then, by
using Eq. (4.1), we have

$\displaystyle\rho^{*}([x,y])(\xi)$
$\displaystyle=-(-1)^{\overline{\xi}(\overline{x}+\overline{y}+\overline{\rho}+\overline{[\cdot,\cdot]})}\xi\rho[x,y]$
$\displaystyle=-(-1)^{\overline{\xi}(\overline{x}+\overline{y})}\xi\big{(}\rho(x)\rho(y)-(-1)^{(\overline{x}+\overline{\rho})(\overline{y}+\overline{\rho})}\rho(y)\rho(x)\big{)}$
$\displaystyle=-(-1)^{(\overline{x}+\overline{\rho})(\overline{y}+\overline{\rho})}\rho^{*}(y)\rho^{*}(x)\xi+\rho^{*}(x)\rho^{*}(y)\xi.$

Then,
$\rho^{*}([x,y])=\rho^{*}(x)\rho^{*}(y)-(-1)^{(\overline{x}+\overline{\rho})(\overline{y}+\overline{\rho})}\rho^{*}(y)\rho^{*}(x),\;\forall
x,y\in\mathcal{H}(\mathfrak{g})$. Thus, $\rho^{*}$ is a representation of
$\mathfrak{g}$ on $V^{*}$. The proof is finished. ∎

###  4.2. Derivations of Super-Lie superalgebras

######  Definition 4.2.

A left derivation of a Super-Lie superalgebra $(\mathfrak{g},[\cdot,\cdot])$
is an homogeneous linear map $\mathfrak{D}^{L}:\mathfrak{g}\to\mathfrak{g}$
satisfying the following relation

(4.4)
$\mathfrak{D}^{L}[x,y]=[\mathfrak{D}^{L}(x),y]+(-1)^{\overline{\mathfrak{D}^{L}}(\overline{x}+\overline{[\cdot,\cdot]})}[x,\mathfrak{D}^{L}(y)]$

for all $x,y\in\mathcal{H}(\mathfrak{g})$. We denoted by
$\mathfrak{D}er^{L}(\mathfrak{g})$, the set of all left derivations of
$\mathfrak{g}$.

######  Lemma 4.1.

The $\mathbb{Z}_{2}$-graded vector space $\mathfrak{D}er^{L}(\mathfrak{g})$
with commutator define a Lie superalgebra.

###### Proof.

Let
$\mathfrak{D}^{L}_{1},\;\mathfrak{D}^{L}_{2}\in\mathfrak{D}er^{L}(\mathfrak{g})$
and $\mathfrak{D}=[\mathfrak{D}^{L}_{1},\mathfrak{D}^{L}_{2}]$. Then, for any
$x,y\in\mathcal{H}(\mathfrak{g})$, we have

$\displaystyle\mathfrak{D}[x,y]$
$\displaystyle=[\mathfrak{D}^{L}_{1},\mathfrak{D}^{L}_{2}][x,y]$
$\displaystyle=\mathfrak{D}^{L}_{1}(\mathfrak{D}^{L}_{2}[x,y])-(-1)^{\overline{\mathfrak{D}^{L}_{1}}\overline{\mathfrak{D}^{L}_{2}}}\mathfrak{D}^{L}_{2}(\mathfrak{D}^{L}_{1}[x,y])$
$\displaystyle=\mathfrak{D}^{L}_{1}([\mathfrak{D}_{2}^{L}(x),y]+(-1)^{\overline{\mathfrak{D}_{2}^{L}}(\overline{x}+\overline{[\cdot,\cdot]})}[x,\mathfrak{D}_{2}^{L}(y)])$
$\displaystyle-(-1)^{\overline{\mathfrak{D}^{L}_{1}}\overline{\mathfrak{D}^{L}_{2}}}\mathfrak{D}^{L}_{2}([\mathfrak{D}_{1}^{L}(x),y]+(-1)^{\overline{\mathfrak{D}_{1}^{L}}(\overline{x}+\overline{[\cdot,\cdot]})}[x,\mathfrak{D}_{1}^{L}(y)])$
$\displaystyle=[\mathfrak{D}_{1}^{L}\mathfrak{D}^{L}_{2}(x),y]+(-1)^{\overline{\mathfrak{D}_{1}^{L}}(\overline{x}+\overline{\mathfrak{D}_{2}^{L}}+\overline{[\cdot,\cdot]})}[\mathfrak{D}_{2}^{L}(x),\mathfrak{D}_{1}^{L}(y)]+(-1)^{\overline{\mathfrak{D}_{2}^{L}}(\overline{x}+\overline{[\cdot,\cdot]})}[\mathfrak{D}_{1}^{L}(x),\mathfrak{D}_{2}^{L}(y)]$
$\displaystyle+(-1)^{(\overline{\mathfrak{D}_{1}^{L}}+\overline{\mathfrak{D}_{2}^{L}})(\overline{x}+\overline{[\cdot,\cdot]})}[x,\mathfrak{D}_{1}^{L}\mathfrak{D}_{2}^{L}(y)]-(-1)^{\overline{\mathfrak{D}^{L}_{1}}\overline{\mathfrak{D}^{L}_{2}}}[\mathfrak{D}^{L}_{2}\mathfrak{D}_{1}^{L}(x),y]-(-1)^{\overline{\mathfrak{D}^{L}_{2}}(\overline{x}+\overline{[\cdot,\cdot]})}[\mathfrak{D}_{1}^{L}(x),\mathfrak{D}^{L}_{2}(y)]$
$\displaystyle-(-1)^{\overline{\mathfrak{D}_{1}^{L}}(\overline{x}+\overline{\mathfrak{D}_{2}^{L}}+\overline{[\cdot,\cdot]})}[\mathfrak{D}_{2}^{L}(x),\mathfrak{D}_{1}^{L}(y)]-(-1)(-1)^{(\overline{\mathfrak{D}_{1}^{L}}+\overline{\mathfrak{D}_{2}^{L}})(\overline{x}+\overline{[\cdot,\cdot]})}[x,\mathfrak{D}_{2}^{L}\mathfrak{D}_{1}^{L}(y)]$
$\displaystyle=[(\mathfrak{D}_{1}^{L}\mathfrak{D}_{2}^{L}-(-1)^{\overline{\mathfrak{D}_{1}^{L}}\overline{\mathfrak{D}_{2}^{L}}}\mathfrak{D}_{2}^{L}\mathfrak{D}_{1}^{L})(x),y]+(-1)^{(\overline{\mathfrak{D}_{1}^{L}}+\overline{\mathfrak{D}_{2}^{L}})(\overline{x}+\overline{[\cdot,\cdot]})}[x,(\mathfrak{D}_{1}^{L}\mathfrak{D}_{2}^{L}-(-1)^{\overline{\mathfrak{D}_{1}^{L}}\overline{\mathfrak{D}_{2}^{L}}}\mathfrak{D}_{2}^{L}\mathfrak{D}_{1}^{L})(y)]$
$\displaystyle=[\mathfrak{D}(x),y]+(-1)^{\overline{\mathfrak{D}}(\overline{x}+\overline{[\cdot,\cdot]})}[x,\mathfrak{D}(y)].$

Then,
$\mathfrak{D}=[\mathfrak{D}_{1}^{L},\mathfrak{D}_{2}^{L}]\in\mathfrak{D}er^{L}(\mathfrak{g})$,
which gives that, $\mathfrak{D}er^{L}(\mathfrak{g})$ is a Lie sub-superalgebra
of $End(\mathfrak{g})$. Therefore, $\mathfrak{D}er^{L}(\mathfrak{g})$ is a Lie
superalgebra. ∎

######  Example 4.2.

For any fixed homogeneous element $x$ of a Super-Lie superalgebra
$(\mathfrak{g},[\cdot,\cdot])$,
the map $ad^{L}_{x}$ defined by Eq. (5.5) define a left derivation of
$\mathfrak{g}$, with the consideration that
$\overline{ad_{x}^{L}}=\overline{x}+\overline{[\cdot,\cdot]}$ as an immediate
relation by the definition of $ad^{L}_{x}$.

######  Definition 4.3.

A right derivation of a Super-Lie superalgebra $(\mathfrak{g},[\cdot,\cdot])$
is an homogeneous linear map $\mathfrak{D}^{R}:\mathfrak{g}\to\mathfrak{g}$
such that, the following condition hold

(4.5)
$\mathfrak{D}^{R}[x,y]=[x,\mathfrak{D}^{R}(y)]+(-1)^{\overline{\mathfrak{D}^{R}}(\overline{y}+\overline{[\cdot,\cdot]})}[\mathfrak{D}^{R}(x),y]$

for all $x,y\in\mathcal{H}(\mathfrak{g})$. The set of all right derivations of
$g$ is generally denoted by $\mathfrak{D}er^{R}(\mathfrak{g})$.

######  Example 4.3.

For any fixed homogeneous element $x$ of a Super-Lie superalgebra
$(\mathfrak{g},[\cdot,\cdot])$, the map $ad^{R}_{x}$ defined by Eq. (2.6)
define a right derivation of $\mathfrak{g}$.

######  Lemma 4.2.

Let $(\mathfrak{g},[\cdot,\cdot])$ be a Super-Lie superalgebra, then
$(\mathfrak{D}er^{R}(\mathfrak{g}),[\cdot,\cdot])$ is a Lie superalgebra.

######  Theorem 4.1.

Let $\mathfrak{D}$ be an homogeneous invertible linear map of a Super-Lie
superalgebra $(\mathfrak{g},[\cdot,\cdot])$. Then:

  1. (1)

$\mathfrak{D}$ is a left drivation of $\mathfrak{g}$ if and only if
$\mathfrak{D}^{-1}$ is a right Rota-Baxter operator on $\mathfrak{g}.$

  2. (2)

$\mathfrak{D}$ is a right drivation of $\mathfrak{g}$ if and only if
$\mathfrak{D}^{-1}$ is a left Rota-Baxter operator on $\mathfrak{g}.$

###### Proof.

  1. (1)

Let $\mathfrak{D}$ be a left derivation of a Super-Lie superalgebra
$(\mathfrak{g},[\cdot,\cdot])$, then, for any two homogeneous elements
$x,y\in\mathfrak{g}$, we have

$\mathfrak{D}[x,y]=[\mathfrak{D}(x),y]+(-1)^{\overline{\mathfrak{D}}(\overline{x}+\overline{[\cdot,\cdot]})}[x,\mathfrak{D}(y)].$

Then $\mathfrak{D}$ is invertible if and only if

$[x,y]=\mathfrak{D}^{-1}[\mathfrak{D}(x),y]+(-1)^{\overline{\mathfrak{D}}(\overline{x}+\overline{[\cdot,\cdot]})}\mathfrak{D}^{-1}[x,\mathfrak{D}(y)].$

Suppose that $X=\mathfrak{D}(x)$ and $Y=\mathfrak{D}(y)$, we obtain

$[\mathfrak{D}^{-1}(X),\mathfrak{D}^{-1}(Y)]=\mathfrak{D}^{-1}([X,\mathfrak{D}^{-1}(Y)]+(-1)^{\overline{\mathfrak{D}^{-1}}(\overline{\mathfrak{D}^{-1}}+\overline{X}+\overline{[\cdot,\cdot]})}[\mathfrak{D}^{-1}(X),Y]),$

since the fact that $\overline{\mathfrak{D}}=\overline{\mathfrak{D}^{-1}}$.
Then $\mathfrak{D}^{-1}$ is a right Rota-Baxter operator on $\mathfrak{g}$.

  2. (2)

This is a direct computation by a similar way given in $(1)$.

∎

##  5\. $3$-super-Lie superalgebras

In this section we introduce the case of $\mathbb{Z}_{2}$-graded $3$-Lie
superalgebras which called $3$-Super-Lie superalgebras in which we give some
properties, examples and related interesting results.

######  Definition 5.1.

A left ternary Super-Leibniz superalgebra is a couple
$(\mathfrak{g},[\cdot,\cdot,\cdot])$ where, $\mathfrak{g}$ is a
$\mathbb{Z}_{2}$-graded vector space and $[\cdot,\cdot,\cdot]$ be an
homogeneous trilinear map with degree $\mathfrak{q}\in\mathbb{Z}_{2}$
satisfying for any homogeneous elements $x,y,z\in\mathfrak{g}$ the following
identity

(5.1) $\displaystyle[x_{1},x_{2},[y_{1},y_{2},y_{3}]]$
$\displaystyle=[[x_{1},x_{2},y_{1}],y_{2},y_{3}]+(-1)^{(\overline{x_{1}}+\overline{x_{2}}+\mathfrak{q})(\overline{y_{1}}+\mathfrak{q})}[y_{1},[x_{1},x_{2},y_{2}],y_{3}]$
$\displaystyle+(-1)^{(\overline{x_{1}}+\overline{x_{2}}+\mathfrak{q})(\overline{y_{1}}+\overline{y_{2}})}[y_{1},y_{2},[x_{1},x_{2},y_{3}]].$

The identity (5.1) is called left ternary Super-Leibniz identity.

Let $x_{1},x_{2}$ be two fixed homogeneous elements of $\mathfrak{g}$. Define
the linear map $ad^{L}_{x_{1},x_{2}}:\mathfrak{g}\to\mathfrak{g}$ by

(5.2) $ad^{L}_{x_{1},x_{2}}(y)=[x_{1},x_{2},y],\;\forall
y\in\mathcal{H}(\mathfrak{g}).$

Then,
$\overline{ad^{L}_{x_{1},x_{2}}}=\overline{x_{1}}+\overline{x_{2}}+\mathfrak{q}$
and the left ternary Super-Leibniz identity (5.1) can be written as follow

(5.3) $\displaystyle ad^{L}_{x_{1},x_{2}}[y_{1},y_{2},y_{3}]$
$\displaystyle=[ad^{L}_{x_{1},x_{2}}(y_{1}),y_{2},y_{3}]+(-1)^{(\overline{x_{1}}+\overline{x_{2}}+\mathfrak{q})(\overline{y_{1}}+\mathfrak{q})}[y_{1},ad^{L}_{x_{1},x_{2}}(y_{2}),y_{3}]$
$\displaystyle+(-1)^{(\overline{x_{1}}+\overline{x_{2}}+\mathfrak{q})(\overline{y_{1}}+\overline{y_{2}})}[y_{1},y_{2},ad^{L}_{x_{1},x_{2}}(y_{3})].$

######  Definition 5.2.

A right ternary Super-Leibniz superalgebra is a couple
$(\mathfrak{g},[\cdot,\cdot,\cdot])$ consisting of a $\mathbb{Z}_{2}$-graded
vector space $\mathfrak{g}$ and an homogeneous trilinear map
$[\cdot,\cdot,\cdot]$ on $\mathfrak{g}$ with degree
$\mathfrak{q}\in\mathbb{Z}_{2}$ satisfying, for any homogeneous elements
$x,y,z\in\mathfrak{g}$, the following identity

(5.4) $\displaystyle[[x_{1},x_{2},y_{1}],y_{2},y_{3}]$
$\displaystyle=[x_{1},x_{2},[y_{1},y_{2},y_{3}]]+(-1)^{(\overline{y_{2}}+\overline{y_{3}}+\mathfrak{q})(\overline{y_{1}}+\mathfrak{q})}[x_{1},[x_{2},y_{2},y_{3}],y_{1}]$
$\displaystyle+(-1)^{(\overline{x_{2}}+\overline{y_{1}})(\overline{y_{2}}+\overline{y_{3}}+\mathfrak{q})}[[x_{1},y_{2},y_{3}],x_{2},y_{1}].$

The identity (5.4) is called right ternary Super-Leibniz identity.

Let $x_{1},x_{2}$ be two fixed homogeneous elements of $\mathfrak{g}$. Define
the linear map $ad^{R}_{x_{1},x_{2}}:\mathfrak{g}\to\mathfrak{g}$ by

(5.5) $ad^{R}_{x_{1},x_{2}}(y)=[y,x_{1},x_{2}],\;\forall
y\in\mathcal{H}(\mathfrak{g}).$

Then,
$\overline{ad^{R}_{x_{1},x_{2}}}=\overline{x_{1}}+\overline{x_{2}}+\mathfrak{q}$
and the left ternary Super-Leibniz identity (5.1) can be written as follow

(5.6) $\displaystyle ad^{R}_{y_{2},y_{3}}[x_{1},x_{2},y_{1}]$
$\displaystyle=[x_{1},x_{2},ad^{R}_{y_{2},y_{3}}(y_{1})]+(-1)^{(\overline{y_{2}}+\overline{y_{3}}+\mathfrak{q})(\overline{y_{1}}+\mathfrak{q})}[x_{1},ad^{R}_{y_{2},y_{3}}(x_{2}),y_{1}]$
$\displaystyle+(-1)^{(\overline{x_{2}}+\overline{y_{1}})(\overline{y_{2}}+\overline{y_{3}}+\mathfrak{q})}[ad^{R}_{y_{2},y_{3}}(x_{1}),x_{2},y_{1}].$

######  Definition 5.3.

Let
$\mathfrak{g}=\mathfrak{g}_{\overline{0}}\oplus\mathfrak{g}_{\overline{1}}$ be
a $\mathbb{Z}_{2}$-graded vector space. A trilinear mapping
$(x,y,z)\in\mathfrak{g}\otimes\mathfrak{g}\otimes\mathfrak{g}\mapsto[x,y,z]\in\mathfrak{g}$
is said to be an homogeneous ternary Super-Lie bracket with degree
$\mathfrak{q}\in\mathbb{Z}_{2}$ if
$\overline{[x,y,z]}=\overline{x}+\overline{y}+\overline{z}+\mathfrak{q}$, it
is $\mathbb{Z}_{2}$-graded super-skewsymmetric, i.e.,

(5.7)
$[x,y,z]=-(-1)^{(\overline{x}+\mathfrak{q})(\overline{y}+\mathfrak{q})}[y,x,z]=-(-1)^{(\overline{y}+\mathfrak{q})(\overline{z}+\mathfrak{q})}[x,z,y],$

and satisfies the $\mathbb{Z}_{2}$-graded Filippov-Jacobi identity

(5.8) $\displaystyle[x,y,[z,t,u]]$
$\displaystyle=[[x,y,z],t,u]+(-1)^{(\overline{x}+\overline{y}+\mathfrak{q})(\overline{z}+\mathfrak{q})}[z,[x,y,t],u]$
$\displaystyle+(-1)^{(\overline{x}+\overline{y}+\mathfrak{q})(\overline{z}+\overline{t})}[z,t,[x,y,u]]$

A vector superspace $\mathfrak{g}$, together with an homogeneous ternary
Super-Lie bracket defined on it, is called a $3$-Super-Lie superalgebra.

######  Remark 5.1.

If $\mathfrak{q}=0$, we recover the (even) $3$-Lie superalgebra structure (see
[16] for more details) and if $\mathfrak{q}=1$, which is called odd $3$-Lie
superalgebra.

######  Example 5.1.

Let $\mathfrak{g}=<e_{0},e_{1}>$ be a two dimentional $\mathbb{Z}_{2}$-graded
vector space where $\mathfrak{g}_{\overline{0}}=<e_{0}>$ and
$\mathfrak{g}_{\overline{1}}=<e_{1}>$. Define the trilinear product
$[\cdot,\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$ by

$[e_{1},e_{1},e_{1}]=e_{0},$

and the other products are zeros. Then $(\mathfrak{g},[\cdot,\cdot,\cdot])$
defines an odd $3$-Lie superalgebra.

######  Definition 5.4.

Let $(\mathfrak{g},[\cdot,\cdot,\cdot])$ be a $3$-Super-Lie superalgebra. An
homogeneous linear map $\mathfrak{R}:\mathfrak{g}\to\mathfrak{g}$ is called
Rota-Baxter operator (called also left Rota-Baxter operator) with degree
$\mathfrak{r}\in\mathbb{Z}_{3}$ on $\mathfrak{g}$, if it satisfies the
following rule

(5.9) $\displaystyle[\mathfrak{R}x,\mathfrak{R}y,\mathfrak{R}z]$
$\displaystyle=\mathfrak{R}\Big{(}[\mathfrak{R}x,\mathfrak{R}y,z]+(-1)^{\mathfrak{r}(\mathfrak{r}+\overline{y}+\overline{[\cdot,\cdot,\cdot]})}[\mathfrak{R}x,y,\mathfrak{R}z]+(-1)^{\mathfrak{r}(\overline{x}+\overline{y})}[x,\mathfrak{R}y,\mathfrak{R}z]\Big{)}$

for any homogeneous elements $x,y,z$ of $\mathfrak{g}$.

By a straightforward computation, we obtain the following results

######  Proposition 5.1.

Let $\mathfrak{R}$ be a Rota-Baxter operator on a $3$-Super-Lie superalgebra
$(\mathfrak{g},[\cdot,\cdot,\cdot])$. Then
$(\mathfrak{g},[\cdot,\cdot,\cdot]_{\mathfrak{R}})$ is a $3$-Super-Lie
superalgebra, where

$\displaystyle[x,y,z]_{\mathfrak{R}}$
$\displaystyle=[\mathfrak{R}x,\mathfrak{R}y,z]+(-1)^{\mathfrak{r}(\mathfrak{r}+\overline{y}+\overline{[\cdot,\cdot,\cdot]})}[\mathfrak{R}x,y,\mathfrak{R}z]+(-1)^{\mathfrak{r}(\overline{x}+\overline{y})}[x,\mathfrak{R}y,\mathfrak{R}z]$

for any homogeneous elements $x,y,z$ of $\mathfrak{g}$. Moreover,
$\overline{[\cdot,\cdot,\cdot]_{\mathfrak{R}}}=\overline{[\cdot,\cdot,\cdot]}$
and ${\mathfrak{R}}$ is $3$-super-Lie superalgeras morphism from
$(\mathfrak{g},[\cdot,\cdot,\cdot]_{\mathfrak{R}})$ to
$(\mathfrak{g},[\cdot,\cdot,\cdot])$.

######  Theorem 5.1.

Let $(\mathfrak{g},[\cdot,\cdot,\cdot])$ be a $3$-Super-Lie superalgebra and
$a\in\mathcal{H}(\mathfrak{g})$ such that
$\overline{a}=\overline{[\cdot,\cdot,\cdot]}=\mathfrak{q}\in\mathbb{Z}_{2}$.
Then $\mathfrak{g}_{a}=(\mathfrak{g},[\cdot,\cdot]_{a}=[a,\cdot,\cdot])$ is a
Lie superalgebra. Morover
$a\in\mathcal{Z}({\mathfrak{g_{a}}})=\\{x\in\mathfrak{g};\;[x,y]_{a}=0,\;\forall
y\in\mathfrak{g}\\}$.

###### Proof.

It’s obvious to show that $[\cdot,\cdot]_{a}$ is $\mathbb{Z}_{2}$-graded
super-skewsymmetric and $[a,x,a]=[a,a,x]=0,\;\forall x\in\mathfrak{g}$. Let
$x,y,z\in\mathcal{H}(\mathfrak{g})$, then, by using (5.8), we have

$\displaystyle[x,[y,z]_{a}]_{a}=$ $\displaystyle[a,x,[a,y,z]]$
$\displaystyle=$
$\displaystyle[[a,x,a],y,z]+(-1)^{(\overline{a}+\overline{[\cdot,\cdot,\cdot]})(\overline{a}+\overline{x}+\overline{[\cdot,\cdot,\cdot]})}[a,[a,x,y],z]$
$\displaystyle+(-1)^{(\overline{a}+\overline{x}+\overline{[\cdot,\cdot,\cdot]})(\overline{a}+\overline{y}+\overline{[\cdot,\cdot,\cdot]})}[a,y,[a,x,z]]$
$\displaystyle=$
$\displaystyle[[x,y]_{a},z]_{a}+(-1)^{\overline{x}\overline{y}}[y,[x,z]].$

This completed the proof. ∎

######  Example 5.2.

Let us consider the odd $3$-Lie superalgebra $\mathfrak{g}=<e_{0},e_{1}>$
given in the Example 5.1. Then, by Theorem 5.1, the bilinear map
$[\cdot,\cdot]_{e_{1}}$ given by

$[e_{1},e_{1}]_{e_{1}}=e_{0},$

defines on $\mathfrak{g}$ a Lie superalgebra structure.

##  6\. Further discussion

In this paper, we introduce a generalization of Lie superalgebras, when the
bracket is not necessary even. We study some related algebraic structure
called Super-Lie admissible superalgebras constructed by Rota-Baxter
operators. Their representations are studied and characterized by semi-direct
product on the sum direct of superalgebra and super-module. Here we collect
some further questions regarding this new algebraic structure:

Deformations and Cohomolgy: In Section 4, we study representations of the new
algebraic structures called Super-Lie superalgebras as a generalization of Lie
superalgebras. In [4] the authors study deformations and cohomlogy of Lie
superalgebra with a given representation. Motivated by this work, will this
allow us to generalize the studies of cohomology and deformations to the
Super-Lie superalgebras?

Maurer-Cartan characterization: In [4], the authors give Maurer-Cartan
characterizations as well as a cohomology theory of Lie superalgebras.
Explicitly, they introduce the notion of a bidifferential graded Lie algebra
and thus give Maurer-Cartan characterizations of Lie superalgebras. One may
now ask the following question: Can the graded Lie algebra introduced by the
authors characterize the cohomolgy of Super-Lie superalgebra when the bracket
is odd?

$\mathcal{O}$-operators: An $\mathcal{O}$-operator is a relative
generalization of Rota-Baxter operators on a Lie superalgebra associated to a
given representation (see [10]). A homogeneous linear map
$T:V\longrightarrow{\mathfrak{g}}$ is called an $\mathcal{O}$-operator of a
Lie superalgebra $({\mathfrak{g}},[\cdot,\cdot])$ associated to the
representation $(V,\rho)$ if it satisfies

(6.1)
$[T(v),T(w)]=T\Big{(}(-1)^{(|T|+|v|)|T|}\rho(T(v))w-(-1)^{|v|(|T|+|w|)}\rho(T(w))v\Big{)},\;\;\forall
v,w\in\mathcal{H}(V).$

In Section 3, we introduce a generalization of Rota-Baxter operator on a Lie
superalgebra to Super-Lie superalgebra case. It is interesting to generalize
this notion for the relative case and study their cohomolgies and
deformations.

Classification of odd Rota-Baxter operators: In [1, 3], the authors give a
classification of the even homogeneous Rota-Baxter operators on a Lie
superalgebras and construct pre-Lie superalgebras from the homogeneous Rota-
Baxter operators and their subadjacent Lie superalgebras. We can also classify
the odd homogeneous Rota-Baxter operators on Lie superalgebras like Witt
superalgebras and Witt superalgebras of Block type. We construct their
associated subadjacent odd Lie superalgebras.

Induced $3$-super-Lie superalgebras: In [7], the structure of $3$-Lie algebras
induced by Lie algebras have been investigated. The super case of this
construction using an even trace function is studied by V. Abramov in [2] (see
more [12, 31]). By this works, can we induce an odd $3$-Lie superalgebra using
an odd trace function on a Lie superalgebra?

## Acknowledgment

The authors would like to thank Prof. Abdenacer Makhlouf and Prof. Taoufik
Chtioui for helpful discussions and suggestions.

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