# WiCV@CVPR2023: The Eleventh Women In Computer Vision Workshop at the Annual
CVPR Conference

Doris Antensteiner1, Marah Halawa2, Asra Aslam3, Ivaxi Sheth4,
Sachini Herath5, Ziqi Huang6, Sunnie S. Y. Kim7, Aparna Akula8, Xin Wang9

1Austrian Institute of Technology, 2Technical University of Berlin,
3University of Leeds,
4CISPA – Helmholtz Center for Information Security, 5Simon Fraser University,
6S-Lab, Nanyang Technological University, 7Princeton University,
8CSIR – Central Scientific Instruments Organisation, 9Microsoft
<EMAIL_ADDRESS>
###### Abstract

In this paper, we present the details of Women in Computer Vision Workshop -
WiCV 2023, organized alongside the hybrid CVPR 2023 in Vancouver, Canada. WiCV
aims to amplify the voices of underrepresented women in the computer vision
community, fostering increased visibility in both academia and industry. We
believe that such events play a vital role in addressing gender imbalances
within the field. The annual WiCV@CVPR workshop offers a) opportunity for
collaboration between researchers from minority groups, b) mentorship for
female junior researchers, c) financial support to presenters to alleviate
finanacial burdens and d) a diverse array of role models who can inspire
younger researchers at the outset of their careers. In this paper, we present
a comprehensive report on the workshop program, historical trends from the
past WiCV@CVPR events, and a summary of statistics related to presenters,
attendees, and sponsorship for the WiCV 2023 workshop.

##  1 Introduction

Despite remarkable progress in various computer vision research areas in
recent years, the field still grapples with a persistent lack of diversity and
inclusion. While the field of computer vision rapidly expands, female
researchers remain underrepresented in the area, constituting only a small
amount of professionals in both academia and industry. Due to this, many
female computer vision researchers can feel isolated in workspaces which
remain unbalanced due to the lack of inclusion.

The WiCV workshop is a gathering designed for all individuals, irrespective of
gender, engaged in computer vision research. It aims to appeal to researchers
at all levels, including established researchers in both industry and academia
(e.g. faculty or postdocs), graduate students pursuing a Masters or PhD, as
well as undergraduates interested in research. The overarching goal is to
enhance the visibility and recognition of female computer vision researchers
across these diverse career stages, reaching women from various backgrounds in
educational and industrial settings worldwide.

There are three key objectives of the WiCV workshop:

#### Networking and Mentoring

The first objective is to expand the WiCV network and facilitate interactions
between members of this network. This includes female students learning from
seasoned professionals who share career advice and experiences. A mentoring
banquet held alongside the workshop provides a casual environment for junior
and senior women in computer vision to meet, exchange ideas and form mentoring
or research relationships.

#### Raising Visibility

The workshop’s second objective is to elevate the visibility of women in
computer vision, both at junior and senior levels. Senior researchers are
invited to give high quality keynote talks on their research, while junior
researchers are encouraged to submit their recent or ongoing work, with many
of these being selected for oral or poster presentation through a rigorous
peer review process. This empowers junior female researchers to gain
experience presenting their work in a professional yet supportive setting. The
workshop aims for diversity not only in research topics but also in the
backgrounds of presenters. Additionally, a panel discussion provides a
platform for female colleagues to address topics of inclusion and diversity.

#### Supporting Junior Researchers

Finally, the third objective is to offer junior female researchers the
opportunity to attend a major computer vision conference that might otherwise
be financially inaccessible. This is made possible through travel grants
awarded to junior researchers who present their work during the workshop’s
poster session. These grants not only enable participation in the WiCV
workshop but also provide access to the broader CVPR conference.

##  2 Workshop Program

The workshop program featured a diverse array of sessions, including 4
keynotes, 6 oral presentations, 34 poster presentations, a panel discussion,
and a mentoring session. Consistent with previous years, our keynote speakers
were carefully selected to ensure diversity in terms of the topics that were
covered, their backgrounds, whether they work in academia or industry, and
their seniority. This deliberate choice of diverse speakers is of paramount
importance, as it offers junior researchers a multitude of potential role
models with whom they can resonate and, in turn, envision their unique career
paths.

The workshop schedule at CVPR 2023 featured a diverse range of sessions and
activities, including:

  * •

Introduction

  * •

Invited Talk 1: Angel Chang (Simon Fraser University and Canada CIFAR AI),
Connecting 3D and Language

  * •

Oral Session 1

    * –

Laia Tarrés, Sign Language Translation for Instructional Videos

    * –

Meghna Kapoor, Underwater Moving Object Detection using an End-to-End Encoder-
Decoder Architecture and GraphSage with Aggregator and Refactoring

  * •

Invited Talk 2: Devi Parikh (Generative AI Lead at Meta), Multimodal
Generative AI (AI for creativity)

  * •

Sponsors Exhibition (in person)

  * •

Oral Session 2

    * –

Deblina Bhattacharjee, Dense Multitask Learning to Reconfigure Comics

    * –

Maryam Daniali, Perception Over Time: Temporal Dynamics for Robust Image
Understanding

  * •

Invited Talk 3: Judy Hoffman (School of Interactive Computing at Georgia
Tech), Efficient and Reliable Vision Models

  * •

Sponsors Exhibition (in person)

  * •

Oral Session 3

    * –

Zoya Shafique, Nonverbal Communication Cue Recognition: A Pathway to More
Accessible Communication

    * –

Sudha Velusamy, A Light-Weight Human Eye Fixation Solution for Smartphone
Applications

  * •

Invited Talk 4: Kristen Grauman (University of Texas at Austin and Facebook AI
Research (FAIR)), Human-object interactions in first person video

  * •

Panel Discussion by Abby Stylianou, Angel Chang, Devi Parikh, Ilke Demir, Judy
Hoffman, and Kristen Grauman

  * •

Poster Session (in person as well as virtual)

  * •

Closing Remarks

  * •

Mentoring Session, Talks, and Dinner (in person)

    * –

Invited talk:

      1. 1.

Abby Stylianou (Saint Louis University and Taylor Geospatial Institute) on A
bit about my research and finding your own path to happiness and success in
this field even if it looks different

      2. 2.

Ilke Demir (Intel Corporation) on the intriguing topic Be you

    * –

In-Person mentoring session and dinner: Akshita Gupta (University of Guelph),
Dima Damen (University of Bristol), Achal Dave (Toyota Research Institute),
Federica Arrigoni (Politecnico di Milano), Hilde Kuehne (University on Bonn),
Nikhila Ravi (Meta), Katherine Liu (Toyota Research Institute), Lilly Ball
(Apple), Dian Chen (Toyota Research Institute), Shuyang Cheng (Cruise LLC),
Jeremy Jones (Apple), Zuzana Kukelova (Czech Technical University in Prague).

    * –

Virtual mentoring session on Zoom: Yifan Liu (ETH Zürich), and Dena Bazazian
(University of Plymouth).

###  2.1 Hybrid Setting

This year, our organizational approach underwent slight adjustments due to
CVPR 2023 being held in a hybrid setting, accommodating both in-person and
virtual attendance. We took deliberate steps to sure that the virtual WiCV
workshop was an engaging and interactive event. To achieve this, we took the
following steps: Talks, oral sessions, and the panel were streamed via Zoom
for virtual attendances. The poster session was repeated virtually a week
after the conference, mirroring the format of the main conference. We also
facilitated online mentoring sessions via Zoom, catering to mentors and
mentees who could only participate virtually.

##  3 Workshop Statistics

The first edition of the Women in Computer Vision (WiCV) workshop was held in
conjunction with CVPR 2015. Over the years, both the participation rate and
the quality of submissions to WiCV have steadily increased. Following the
examples from the editions held in previous years [4, 3, 6, 2, 1, 5] we have
continued to curate top-quality submissions into our workshop proceedings. By
providing oral and poster presenters the opportunity to publish their work in
the conference’s proceedings, we aim to further boost the visibility of female
researchers.

This year, the workshop was held as a half-day in-person event with hybrid
options, while the virtual component was hosted via Topia and Zoom. The in-
person gathering took place at the Vancouver Convention Centre in Canada.
Senior and junior researchers were invited to present their work, including
the poster presentations detailed in the previous Section 2.

The organizers for this year’s WiCV workshop come from diverse backgrounds in
both academia and industry, representing various institutions across different
time zones. Their diverse backgrounds and wide-ranging research areas have
enriched the organizing committee’s perspectives and contributed to a well-
rounded approach. Their broad range of research interests in computer vision
and machine learning encompass video understanding, object detection, non-
verbal communication, open-source benchmark datasets, activity recognition,
anomaly detection, autoencoders, generalization, captioning, 3D Point Cloud,
medical imaging and vision for robotics.

Figure 1: WiCV Submissions. The number of submissions over the past years of
WiCV.

This year, we received 68 high-quality submissions from a wide range of topics
and institutions, which is on par with WiCV@CVPR22. The most popular topics
included deep learning architectures and techniques followed by video action
and event recognition, segmentation and shape analysis, and medical
applications. Out of the 68 submissions, 61 underwent the review process. Six
papers were selected as oral presentations and inclusion in the CVPR23
workshop proceedings, while 34 papers were chosen for poster presentations.
The comparison with previous years is presented in Figure 1. Thanks to the
diligent efforts of an interdisciplinary program committee comprising 41
reviewers, the submitted papers received thorough evaluations and valuable
feedback. Additionally, during the mentoring session, 40 mentees received in-
person guidance from 8 mentors, and 5 mentees attended virtual sessions with 2
mentors in separate meetings via Zoom.

This year, we continued the WiCV tradition from previous workshops [1, 2, 5,
6, 7] by providing grants to assist the authors of accepted submissions in
participating in the workshop. These grants covered a range of expenses, with
the specifics varying for each attendee depending on their individual needs,
including, for example, conference registration fees, round-trip flight
itineraries, and two days of accommodation for all authors of accepted
submissions who requested funding.

The total sponsorship for this year’s workshop amounted to $30,000 USD, with
contributions of 6 sponsors, meeting our target. In Figure 2 you can find the
details with respect to the past years.

Figure 2: WiCV Sponsors. The number of sponsors and the amount of sponsorship
for WiCV. The amount is expressed in US dollar (USD).

##  4 Conclusions

WiCV at CVPR 2023 has once again proven to be a valuable opportunity for
presenters, participants, and organizers, providing a platform to unite the
community. It continues to address the persistent issue of gender balance in
our field, and we believe it has played a significant role in strengthening
the community. It provided an opportunity for people to connect from all over
the world from the comfort of their personal spaces. With a high number of
paper submissions and even greater number of attendees, we anticipate that the
workshop will continue the positive trajectory of previous years, fostering a
stronger sense of community, increased visibility, and inclusive support and
encouragement for all female researchers in academia and industry. Moreover,
WiCV Members participated in the Diversity & Inclusion Social event at CVPR.
Furthermore, WiCV also got featured by in the CVPR 2023 Magazine [8].

##  5 Acknowledgments

We express our sincere gratitude to our sponsors, including our Platinum
sponsors: Toyota Research Institute and Apple, as well as our Silver Sponsor:
Cruise, and Bronze sponsors: Deepmind, Meshcapade, and Snap Inc. Our
appreciation also extends to the San Francisco Study Center, our fiscal
sponsor, for their invaluable assistance in managing sponsorships and travel
awards. We are thankful for the support and knowledge-sharing from organizers
of previous WiCV workshops, without whom this WiCV event would not have been
possible. Finally, we extend our heartfelt thanks to the dedicated program
committee, authors, reviewers, submitters, and all participants for their
valuable contributions to the WiCV network community.

##  6 Contact

Website: https://sites.google.com/view/wicvcvpr2023/home
E-mail<EMAIL_ADDRESS>
Facebook: https://www.facebook.com/WomenInComputerVision/
Twitter: https://twitter.com/wicvworkshop
Google group<EMAIL_ADDRESS>

## References

  * [1] Z. Akata, D. Bazazian, Y. Hasson, A. Kanazawa, H. Kuehne, and G. Varol.  WiCV at ECCV2018: The Fifth Women in Computer Vision Workshop.  In Proceedings of European Conference on Computer Vision Workshops, 2018.
  * [2] I. Amerini, E. Balashova, S. Ebrahimi, K. Leonard, A. Nagrani, and A. Salvador.  WiCV 2019: The Sixth Women In Computer Vision Workshop.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, June 2019.
  * [3] D. Antensteiner, S. Bucci, A. Goel, M. Halawa, N. Kalavakonda, T. Kasarla, M. Liu, N. Samet, and I. Sheth.  Wicv 2022: The tenth women in computer vision workshop.  arXiv preprint arXiv:2208.11388, 2022.
  * [4] D. Antensteiner, M. Halawa, A. Aslam, I. Sheth, S. Herath, Z. Huang, S. S. Y. Kim, A. Akula, and X. Wang.  Wicv 2023: The eleventh women in computer vision workshop, 2023.  https://sites.google.com/view/wicvcvpr2023/home.
  * [5] I. Demir, D. Bazazian, A. Romero, V. Sharmanska, and L. Tchapmi.  WiCV 2018: The Fourth Women In Computer Vision Workshop.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, pages 1860–1862, 2018.
  * [6] H. Doughty, N. Karessli, K. Leonard, B. Li, C. Martinez, A. Mobasher, A. Nagrani, and S. Yadav.  Wicv 2020: The seventh women in computer vision workshop.  arXiv preprint arXiv:2101.03787, 2021.
  * [7] A. Goel, N. Kalavakonda, N. Karessli, T. Kasarla, K. Leonard, B. Li, N. Samet, , and G. Zamzmi.  Wicv 2021: The eighth women in computer vision workshop, 2022.
  * [8] R. Vision.  WiCV@CVPR2023 Magazine.  https://www.rsipvision.com/CVPR2023-Thursday/18/, 2023.


111Corresponding author

#  Cohomology and Deformation theory of Modified Rota-Baxter Leibniz algebras

Yizheng Li  School of Mathematical Sciences, Qufu Normal University, Qufu
273165, P. R. of China<EMAIL_ADDRESS>and  $\mbox{Dingguo\ Wang}^{1}$
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, P. R. of
China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we define the cohomology of a modified Rota-Baxter Leibniz
algebra with coefficients in a suitable representation. As applications of our
cohomology, we study formal one-parameter deformations and abelian extensions
of modified Rota-Baxter Leibniz algebras.

###### Key words and phrases:

Modified Rota-Baxter Leibniz algebra, cohomology, deformation, Abelian
extension

###### 2010 Mathematics Subject Classification:

16E40 16S80 17A30

###### Contents

  1. 1 Leibniz algebras, representations and their cohomology theory
  2. 2 Representations of modified Rota-Baxter Leibniz algebras
  3. 3 Cohomology theory of modified Rota-Baxter Leibniz algebras
    1. 3.1 Cohomology of modified Rota-Baxter operators
    2. 3.2 Cohomology of modified Rota-Baxter Leibniz algebras
  4. 4 Formal deformations of modified Rota-Baxter Leibniz algebras
  5. 5 Abelian extensions of modified Rota-Baxter Leibniz algebras

## Introduction

In 1960, Baxter [2] introduced the notion of Rota-Baxter operators on
associative algebras in his study of fluctuation theory in probability. Rota-
Baxter operators have been found many applications, including in Connes-
Kreimer’s algebraic approach to the renormalization in perturbative quantum
field theory [3]. More precisely. For a fixed constant $\lambda$, a Rota-
Baxter operator of weight $\lambda$ is a linear operator $P$ on an associative
algebra $R$ that satisfies the Rota-Baxter equation:

(1) $\displaystyle P(x)P(y)=P(P(x)y)+P(xP(y))+\lambda P(xy),\forall x,y\in R.$

For more details on the Rota-Baxter operator, see [8].

Semonov-Tian-Shansky [12] found that, under suitable conditions, the operator
form of the classical Yang-Baxter equation is precisely the Rota-Baxter
operator of weight 0 on a Lie algebra. As a modified form of the operator form
of the classical Yang-Baxter equation, he also introduced in that paper the
modified classical Yang-Baxter equation whose solutions are called modified
$r$-matrices, modified $r$-matrices have been found many applications,
including applications in the study of generalized Lax pairs and affine
geometry on Lie groups. Recently, the authors have considered formal
deformations of modified $r$-matrices motivated by the deformations of
classical $r$-matrices in [9].

A Rota-Baxter operator gives a modified Rota-Baxter operator through a linear
transformation [7]. Guo et al. studied the Rota-Baxter type algebra and
constructed the free object uniformly in that category in [15]. Recently, the
author has obtained the cohomology and formal deformations of modified Rota-
Baxter algebras in [6].

The concept of Leibniz algebras was introduced by Loday [10, 11] in the study
of the algebraic $K$-theory. Relative Rota-Baxter operators on Leibniz
algebras were studied in [14], which is the main ingredient in the study of
the twisting theory and the bialgebra theory for Leibniz algebras [13].
Recently, the author has considered weighted relative Rota-Baxter operators on
Leibniz algebras in [5]. Our aim in this paper is to consider modified Rota-
Baxter Leibniz algebras, and define the cohomology of a modified Rota-Baxter
Leibniz algebra with coefficients in a suitable representation. As
applications of our cohomology, we study formal one-parameter deformations and
abelian extensions of modified Rota-Baxter Leibniz algebras.

The paper is organized as follows. In Section 1, we recall some basic
definitions about Leibniz algebras and their cohomology. In Section 2, we
consider modified Rota-Baxter Leibniz algebras and introduce their
representations. In Section 3, we define the cohomology of a modified Rota-
Baxter Leibniz algebra with coefficients in a suitable representation. In
Section 4, we study formal deformation theory and rigidity of modified Rota-
Baxter Leibniz algebras. Finally, abelian extensions are considered in Section
5.

Throughout this paper, let ${\bf k}$ be a field of characteristic $0$. Except
specially stated, vector spaces are ${\bf k}$-vector spaces and all tensor
products are taken over ${\bf k}$.

##  1\. Leibniz algebras, representations and their cohomology theory

We start with the background of Leibniz algebras and their cohomology that we
refer the reader to [4, 11] for more details.

######  Definition 1.1.

A Leibniz algebra is a vector space ${\mathfrak{g}}$ together with a bilinear
operation (called bracket)
$[\cdot,\cdot]_{\mathfrak{g}}:{\mathfrak{g}}\otimes{\mathfrak{g}}\rightarrow{\mathfrak{g}}$
satisfying

$\displaystyle[x,[y,z]_{\mathfrak{g}}]_{\mathfrak{g}}=[[x,y]_{\mathfrak{g}},z]_{\mathfrak{g}}+[y,[x,z]_{\mathfrak{g}}]_{\mathfrak{g}},~{}\text{
for }x,y,z\in{\mathfrak{g}}.$

A Leibniz algebra as above may be denoted by the pair
$({\mathfrak{g}},[\cdot,\cdot])$ or simply by ${\mathfrak{g}}$ when no
confusion arises. A Leibniz algebra whose bilinear bracket is skewsymmetric is
nothing but a Lie algebra. Thus, Leibniz algebras are the non-skewsymmetric
analogue of Lie algebras.

######  Definition 1.2.

A representation of a Leibniz algebra
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ consists of a triple
$(V,\rho^{L},\rho^{R})$ in which $V$ is a vector space and
$\rho^{L},\rho^{R}:{\mathfrak{g}}\rightarrow gl(V)$ are linear maps satisfying
for $x,y\in{\mathfrak{g}}$,

$\displaystyle\begin{cases}\rho^{L}([x,y]_{\mathfrak{g}})=\rho^{L}(x)\rho^{L}(y)-\rho^{L}(y)\rho^{L}(x),\\\
\rho^{R}([x,y]_{\mathfrak{g}})=\rho^{L}(x)\rho^{R}(y)-\rho^{R}(y)\rho^{L}(x),\\\
\rho^{R}([x,y]_{\mathfrak{g}})=\rho^{L}(x)\rho^{R}(y)+\rho^{R}(y)\rho^{R}(x).\end{cases}$

It follows that any Leibniz algebra ${\mathfrak{g}}$ is a representation of
itself with

$\displaystyle\rho^{L}(x)=L_{x}=[x,~{}]_{\mathfrak{g}}~{}~{}\text{ and
}~{}~{}\rho^{R}(x)=R_{x}=[~{},x]_{\mathfrak{g}},\text{ for
}x\in{\mathfrak{g}}.$

Here $L_{x}$ and $R_{x}$ denotes the left and right multiplications by $x$,
respectively. This is called the regular representation.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ be a Leibniz algebra and
$(V,\rho^{L},\rho^{R})$ be a representation of it. The cohomology of the
Leibniz algebra ${\mathfrak{g}}$ with coefficients in $V$ is the cohomology of
the cochain complex $\\{C^{*}({\mathfrak{g}},V),\delta\\}$, where
$C^{n}(\mathfrak{g},V)=\mathrm{Hom}({\mathfrak{g}}^{\otimes n},V)$ for $n\geq
0$, and the coboundary operator $\delta:C^{n}({\mathfrak{g}},V)\rightarrow
C^{n+1}({\mathfrak{g}},V)$ given by

$\displaystyle(\delta^{n}f)(x_{1},\ldots,x_{n+1})$
$\displaystyle=\sum^{n}_{i=1}(-1)^{i+1}\rho^{L}(x_{i})f(x_{1},\ldots,\hat{x}_{i},\ldots,x_{n+1})+(-1)^{n+1}\rho^{R}(x_{n+1})f(x_{1},\ldots,x_{n})$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i}f(x_{1},\ldots,\hat{x}_{i},\ldots,x_{j-1},[x_{i},x_{j}]_{\mathfrak{g}},x_{j+1},\ldots,x_{n+1}),$

for $x_{1},\ldots,x_{n+1}\in{\mathfrak{g}}$. The corresponding cohomology
groups are denoted by $H^{*}_{\mathrm{Leib}}({\mathfrak{g}},V).$ This
cohomology has been first appeared in [4] and rediscover by Loday and
Pirashvili [11]. This cohomology is also the Loday-Pirashvili cohomology.

##  2\. Representations of modified Rota-Baxter Leibniz algebras

In this section, we consider modified Rota-Baxter Leibniz algebras and
introduce their representations. We also provide various examples and new
constructions.

######  Definition 2.1.

Let $\lambda\in{\bf k}$. Given a Leibniz algebra
$({\mathfrak{g}},\mu=[\cdot,\cdot]_{\mathfrak{g}})$, it is called a modified
Rota-Baxter Leibniz algebra of weight $\lambda$, if there is a modified Rota-
Baxter operator $K:{\mathfrak{g}}\rightarrow{\mathfrak{g}}$ of weight
$\lambda$ such that

(2) $\displaystyle[K(x),K(y)]_{\mathfrak{g}}$ $\displaystyle=$ $\displaystyle
K\big{(}[K(x),y]_{\mathfrak{g}}+[x,K(y)]_{\mathfrak{g}})+\lambda[x,y]_{\mathfrak{g}}$

for any $x,y\in{\mathfrak{g}}$.

A modified Rota-Baxter Leibniz algebra of weight $\lambda$ is a triple
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$ consisting of a Leibniz
algebra $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ together with a
modified Rota-Baxter operator $K$ of weight $\lambda$ on it.

######  Example 2.2.

Consider the three-dimensional Leibniz algebra
$(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ given with respect to a basis
$\\{e_{1},e_{2},e_{3}\\}$ whose nonzero products are given as follows:

$\displaystyle[e_{1},e_{1}]=e_{3}.$

Then $K=\left(\begin{array}[]{ccc}a_{11}&a_{12}&a_{13}\\\
a_{21}&a_{22}&a_{23}\\\ a_{31}&a_{32}&a_{33}\\\ \end{array}\right)$ is a
modified Rota-Baxter operator of weight $\lambda$ on
$(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ if and only if

$\displaystyle[Ke_{i},Ke_{j}]_{\mathfrak{g}}=K([Ke_{i},e_{j}]_{\mathfrak{g}}+[e_{i},Ke_{j}]_{\mathfrak{g}})+\lambda[e_{i},e_{j}]_{\mathfrak{g}},~{}~{}~{}~{}~{}\forall
i,j=1,2,3.$

We have
$[Ke_{1},Ke_{1}]_{\mathfrak{g}}=[a_{11}e_{1}+a_{21}e_{2}+a_{31}e_{3},a_{11}e_{1}+a_{21}e_{2}+a_{31}e_{3}]_{\mathfrak{g}}=a^{2}_{11}e_{3}$
and

$\displaystyle
K([Ke_{1},e_{1}]_{\mathfrak{g}}+[e_{1},Ke_{1}]_{g})+\lambda[e_{1},e_{1}]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle
K([a_{11}e_{1}+a_{21}e_{2}+a_{31}e_{3},e_{1}]_{\mathfrak{g}}+[e_{1},a_{11}e_{1}+a_{21}e_{2}+a_{31}e_{3}]_{\mathfrak{g}})+\lambda
e_{3}$ $\displaystyle=$ $\displaystyle 2a_{11}Ke_{3}+\lambda e_{3}$
$\displaystyle=$ $\displaystyle
2a_{11}a_{13}e_{1}+2a_{11}a_{23}e_{2}+2a_{11}a_{33}e_{3}+\lambda e_{3}.$

Thus, by
$[Ke_{1},Ke_{1}]_{\mathfrak{g}}=K([Ke_{1},e_{1}]_{\mathfrak{g}}+[e_{1},Ke_{1}]_{\mathfrak{g}})+\lambda[e_{1},e_{1}]_{\mathfrak{g}}$,
we have

$\displaystyle
a_{11}a_{13}=0,a_{11}a_{23}=0,a^{2}_{11}=2a_{11}a_{33}+\lambda.$

Similarly, we obtain

$\displaystyle
a_{11}a_{12}=a_{12}a_{33},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{12}a_{13}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{12}a_{23}=0;$
$\displaystyle
a_{11}a_{13}=a_{13}a_{33},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{13}a_{13}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{13}a_{23}=0;$
$\displaystyle
a_{12}a_{11}=a_{12}a_{33},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{12}a_{13}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{12}a_{23}=0;$
$\displaystyle
a_{13}a_{11}=a_{13}a_{33},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{13}a_{13}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{13}a_{23}=0,$
$\displaystyle
a^{2}_{12}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a^{2}_{13}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{12}a_{13}=0.$

Summarize the above discussion, we have the following two cases:

(1) If $a_{11}=a_{12}=a_{13}=0$, then any
$K=\left(\begin{array}[]{ccc}0&0&0\\\ a_{21}&a_{22}&a_{23}\\\
a_{31}&a_{32}&a_{33}\\\ \end{array}\right)$ is a modified Rota-Baxter operator
of weight 0 on $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$.

(2) If $a_{12}=a_{13}=a_{23}=0$ and $a_{11}\neq 0$, then any
$K=\left(\begin{array}[]{ccc}a_{11}&0&0\\\ a_{21}&a_{22}&0\\\
a_{31}&a_{32}&a_{33}\\\ \end{array}\right)$ is a modified Rota-Baxter operator
of weight $(a^{2}_{11}-2a_{11}a_{33})$ on
$(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$.

######  Definition 2.3.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$ and
$({\mathfrak{g}}^{\prime},[\cdot,\cdot]_{{\mathfrak{g}}^{\prime}},K^{\prime})$
be two modified Rota-Baxter Leibniz algebras of weight $\lambda$. A morphism
from $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$ to
$({\mathfrak{g}}^{\prime},[\cdot,\cdot]_{{\mathfrak{g}}^{\prime}},K^{\prime})$
is a Leibniz algebra homomorphism
$\phi:{\mathfrak{g}}\rightarrow{\mathfrak{g}}^{\prime}$ satisfying $\phi\circ
K=K^{\prime}\circ\phi$.

Denote by $\mathsf{MLeib}$ the category of modified Rota-Baxter Leibniz
algebras of weight $\lambda$ with obvious morphisms.

Recall that a linear map $T:{\mathfrak{g}}\rightarrow{\mathfrak{g}}$ is said
to be a Rota-Baxter operator of weight $\lambda$ on the Leibniz algebra
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ if $T$ satisfies

$\displaystyle[T(x),T(y)]_{\mathfrak{g}}=T([T(x),y]_{\mathfrak{g}}+[x,T(y)]_{\mathfrak{g}}+\lambda[x,y]_{\mathfrak{g}}),\text{~{}for~{}}x,y\in{\mathfrak{g}}.$

A Rota-Baxter Leibniz algebra of weight $\lambda$ is a triple
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},T)$ consisting of a Leibniz
algebra $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ together with a Rota-
Baxter operator $T$ of weight $\lambda$ on it.

Furthermore, we have

$\displaystyle[(2T+\lambda\mathrm{id_{\mathfrak{g}}})(x),(2T+\lambda\mathrm{id_{\mathfrak{g}}})(y)]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle[2T(x)+\lambda x,2T(y)+\lambda
y]_{\mathfrak{g}}$ $\displaystyle=$ $\displaystyle
4[T(x),T(y)]_{\mathfrak{g}}+2\lambda[T(x),y]_{\mathfrak{g}}+2\lambda[x,T(y)]_{\mathfrak{g}}+\lambda^{2}[x,y]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle
4T([T(x),y]_{\mathfrak{g}}+[x,T(y)]_{\mathfrak{g}}+\lambda[x,y]_{\mathfrak{g}})$
$\displaystyle+2\lambda[T(x),y]_{\mathfrak{g}}+2\lambda[x,T(y)]_{\mathfrak{g}}+\lambda^{2}[x,y]_{\mathfrak{g}}$
$\displaystyle=$
$\displaystyle(2T+\lambda\mathrm{id_{\mathfrak{g}}})\big{(}[(2T+\lambda\mathrm{id_{\mathfrak{g}}})(x),y]_{\mathfrak{g}}+[x,(2T+\lambda\mathrm{id_{\mathfrak{g}}})(y)]_{\mathfrak{g}})-\lambda^{2}[x,y]_{\mathfrak{g}}.$

Thus,
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},2T+\lambda\mathrm{id_{\mathfrak{g}}})$
is a modified Rota-Baxter Leibniz algebra of $-\lambda^{2}$.

According to the above discussion, we have

######  Lemma 2.4.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ be a Leibniz algebra. Then
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},T)$ is a Rota-Baxter Leibniz
algebra of weight $\lambda$ if and only if
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},2T+\lambda\mathrm{id_{\mathfrak{g}}})$
is a modified Rota-Baxter Leibniz algebra of weight $-\lambda^{2}$.

######  Definition 2.5.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$ be a modified Rota-
Baxter Leibniz algebra of weight $\lambda$ and $(V,\rho^{L},\rho^{R})$ be a
representation over Leibniz algebra
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$. We say that
$(V,\rho^{L},\rho^{R},K_{V})$ is a representation over modified Rota-Baxter
Leibniz algebra $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$ of weight
$\lambda$ if $(V,\rho^{L},\rho^{R})$ is endowed with a linear operator
$K_{V}:V\rightarrow V$ such that the following equations

$\displaystyle\rho^{L}(K(x))K_{V}(v)=K_{V}\big{(}\rho^{L}(K(x))v+\rho^{L}(x)K_{V}(v)\big{)}+\lambda\rho^{L}(x)v$
(3)
$\displaystyle\rho^{R}(K(x))K_{V}(v)=K_{V}\big{(}\rho^{R}(K(x))v+\rho^{R}(x)K_{V}(v)\big{)}+\lambda\rho^{R}(x)v$

hold for any $x\in{\mathfrak{g}}$ and $v\in V$.

######  Example 2.6.

It is ready to see that $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$
itself is a representation over the modified Rota-Baxter Leibniz algebra
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$ of weight $\lambda$.

######  Definition 2.7.

[5] Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},T)$ be a Rota-Baxter
Leibniz algebra of weight $\lambda$ and $(V,\rho^{L},\rho^{R})$ be a
representation over Leibniz algebra
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$. We say that
$(V,\rho^{L},\rho^{R},T_{V})$ is a representation over Rota-Baxter Leibniz
algebra $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},T)$ of weight $\lambda$
if $(V,\rho^{L},\rho^{R})$ is endowed with a linear operator
$T_{V}:V\rightarrow V$ such that the following equations

$\displaystyle\rho^{L}(T(x))T_{V}(v)=T_{V}\big{(}\rho^{L}(T(x))v+\rho^{L}(x)T_{V}(v)+\lambda\rho^{L}(x)v\big{)}$
(4)
$\displaystyle\rho^{R}(T(x))T_{V}(v)=T_{V}\big{(}\rho^{R}(T(x))v+\rho^{R}(x)T_{V}(v)+\lambda\rho^{R}(x)v\big{)}$

hold for any $x\in{\mathfrak{g}}$ and $v\in V$.

The following result finds the relation between representation over Rota-
Baxter Leibniz algebras and over modified Rota-Baxter Leibniz algebras.

######  Proposition 2.8.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},T)$ be a Rota-Baxter Leibniz
algebra of weight $\lambda$ and $(V,\rho^{L},\rho^{R},T_{V})$ be a
representation over Leibniz algebra
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$. Then
$(V,\rho^{L},\rho^{R},2T_{V}+\lambda\mathrm{id}_{M})$ is a representation over
the modified Rota-Baxter Leibniz algebras
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},2T+\lambda\mathrm{id_{\mathfrak{g}}})$
of weight $-\lambda^{2}$.

###### Proof.

For any $x\in{\mathfrak{g}}$ and $v\in V$, we have

$\displaystyle\rho^{L}(2T(x)+\lambda x)(2T_{V}(v)+\lambda v)$ $\displaystyle=$
$\displaystyle
4\rho^{L}(T(x))T_{V}(v)+2\lambda(\rho^{L}(T(x))v+\rho^{L}(x)T_{V}(v))+\lambda^{2}\rho^{L}(x)v$
$\displaystyle=$ $\displaystyle
4T_{V}\big{(}\rho^{L}(T(x))v+\rho^{L}(x)T_{V}(v)+\lambda\rho^{L}(x)v\big{)}$
$\displaystyle+2\lambda(\rho^{L}(T(x))v+\rho^{L}(x)T_{V}(v))+\lambda^{2}\rho^{L}(x)v$
$\displaystyle=$
$\displaystyle(2T_{V}+\lambda\mathrm{id}_{M})\big{(}\rho^{L}((2T+\lambda\mathrm{id_{\mathfrak{g}}})(x))v+\rho^{L}(x)(2T_{V}+\lambda\mathrm{id}_{M})(v)\big{)}-\lambda^{2}\rho^{L}(x)v.$

Similarly, we can show that

$\displaystyle\rho^{R}(2T(x)+\lambda x)(2T_{V}(v)+\lambda v)$ $\displaystyle=$
$\displaystyle(2T_{V}+\lambda\mathrm{id}_{M})\big{(}\rho^{R}((2T+\lambda\mathrm{id_{\mathfrak{g}}})(x))v+\rho^{R}(x)(2T_{V}+\lambda\mathrm{id}_{M})(v)\big{)}-\lambda^{2}\rho^{R}(x)v.$

And completed the proof. ∎

In the following, we construct the semidirect product in the context of
modified Rota-Baxter Leibniz algebras of any weight.

######  Proposition 2.9.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$ be a modified Rota-
Baxter Leibniz algebra of weight $\lambda$ and $(V,\rho^{L},\rho^{R},K_{V})$
be a representation of it. Then $({\mathfrak{g}}\oplus V,K\oplus K_{V})$ is a
modified Rota-Baxter Leibniz algebra, where the Leibniz bracket on
${\mathfrak{g}}\oplus V$ is given by the semidirect product

(5) $\displaystyle[x+u,y+v]_{{\mathfrak{g}}\oplus V}$ $\displaystyle=$
$\displaystyle[x,y]_{\mathfrak{g}}+\rho^{L}(x)v+\rho^{R}(y)u,$

for any $x,y\in{\mathfrak{g}}$ and $u,v\in V$.

###### Proof.

For any $x,y\in{\mathfrak{g}}$ and $u,v\in V$, we have

$\displaystyle[(K\oplus K_{V})(x+u),(K\oplus
K_{V})(y+v)]_{{\mathfrak{g}}\oplus V}$ $\displaystyle=$
$\displaystyle[K(x),K(y)]_{\mathfrak{g}}+\rho^{L}(K(x))K_{V}(v)+\rho^{R}(K(y))K_{V}(u)$
$\displaystyle=$ $\displaystyle
K\big{(}[K(x),y]_{\mathfrak{g}}+[x,K(y)]_{\mathfrak{g}}\big{)}+\lambda[x,y]_{\mathfrak{g}}+K_{V}\big{(}\rho^{L}(K(x))v+\rho^{L}(x)K_{V}(v)\big{)}$
$\displaystyle+\lambda\rho^{L}(x)v+K_{V}\big{(}\rho^{R}(K(y))u+\rho^{R}(y)K_{V}(u)\big{)}+\lambda\rho^{R}(y)u$
$\displaystyle=$ $\displaystyle(K\oplus K_{V})\big{(}[(K\oplus
K_{V})(x+u),y+v]_{{\mathfrak{g}}\oplus V}+[x+u,(K\oplus
K_{V})(y+v)]_{{\mathfrak{g}}\oplus V}\big{)}$
$\displaystyle+\lambda[x+u,y+v]_{{\mathfrak{g}}\oplus V}$

This shows that $K\oplus K_{V}$ is a modified Rota-Baxter operator of weight
$\lambda$ on the semidirect product Leibniz algebra. Hence the result follows.
∎

######  Proposition 2.10.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$ be a modified Rota-
Baxter Leibniz algebra of weight $\lambda$. Define a new binary operation as:

(6) $\displaystyle[x,y]_{K}:$ $\displaystyle=$
$\displaystyle[K(x),y]_{\mathfrak{g}}+[x,K(y)]_{\mathfrak{g}},$

for any $x,y\in{\mathfrak{g}}$. Then

  * (i)

$({\mathfrak{g}},[\cdot,\cdot]_{K})$ is a new Leibniz algebra, we denote this
Leibniz algebra by ${\mathfrak{g}}_{K}$;

  * (ii)

the triple $({\mathfrak{g}}_{K},[\cdot,\cdot]_{K},K)$ is a modified Rota-
Baxter Leibniz algebra of weight $\lambda$.

###### Proof.

(i) For any $x,y,z\in{\mathfrak{g}}$, we have

$\displaystyle[x,[y,z]_{K}]_{K}$
$\displaystyle=[x,[K(y),z]_{\mathfrak{g}}+[y,K(z)]_{\mathfrak{g}}]_{K}$
$\displaystyle=[K(x),[K(y),z]_{\mathfrak{g}}]_{\mathfrak{g}}+[x,K[K(y),z]_{\mathfrak{g}}]_{\mathfrak{g}}+[K(x),[y,K(z)]_{\mathfrak{g}}]_{\mathfrak{g}}+[x,K[y,K(z)]_{\mathfrak{g}}]_{\mathfrak{g}}$
$\displaystyle=[K(x),[K(y),z]_{\mathfrak{g}}]_{\mathfrak{g}}+[K(x),[y,K(z)]_{\mathfrak{g}}]_{\mathfrak{g}}+[x,[K(y),K(z)]_{\mathfrak{g}}]_{\mathfrak{g}}-\lambda[x,[y,z]_{\mathfrak{g}}]_{\mathfrak{g}}$
$\displaystyle=[[K(x),K(y)]_{\mathfrak{g}},z]_{\mathfrak{g}}+[K(y),[K(x),z]_{\mathfrak{g}}]_{\mathfrak{g}}+[[K(x),y]_{\mathfrak{g}},K(z)]_{\mathfrak{g}}+[y,[K(x),K(z)]_{\mathfrak{g}}]_{\mathfrak{g}}$
$\displaystyle+[[x,K(y)]_{\mathfrak{g}},K(z)]_{\mathfrak{g}}+[K(y),[x,K(z)]_{\mathfrak{g}}]_{\mathfrak{g}}-\lambda[[x,y]_{\mathfrak{g}}z]_{\mathfrak{g}}-\lambda[y,[x,z]_{\mathfrak{g}}]_{\mathfrak{g}}$
$\displaystyle=[[K(x),y]_{\mathfrak{g}}+[x,K(y)]_{\mathfrak{g}},z]_{\mathfrak{g}}+[K(y),[K(x),z]_{\mathfrak{g}}]_{\mathfrak{g}}+[[K(x),y]_{\mathfrak{g}},K(z)]_{\mathfrak{g}}$
$\displaystyle+[y,[K(x),z]_{\mathfrak{g}}+[x,K(z)]_{\mathfrak{g}}]_{\mathfrak{g}}+[[x,K(y)]_{\mathfrak{g}},K(z)]_{\mathfrak{g}}+[K(y),[x,K(z)]_{\mathfrak{g}}]_{\mathfrak{g}}$
$\displaystyle=[[x,y]_{K},z]_{K}+[y,[x,z]_{K}]_{K}.$

This shows that $({\mathfrak{g}},[\cdot,\cdot]_{K})$ is a Leibniz algebra.

(ii) For any $x,y\in{\mathfrak{g}}$, we have

$\displaystyle[K(x),K(y)]_{K}$
$\displaystyle=[K^{2}(x),K(y)]_{\mathfrak{g}}+[K(x),K^{2}(y)]_{\mathfrak{g}}$
$\displaystyle=K\big{(}[K^{2}(x),y]_{\mathfrak{g}}+[K(x),K(y)]_{\mathfrak{g}})+\lambda[K(x),y]_{\mathfrak{g}}$
$\displaystyle+K\big{(}[K(x),K(y)]_{\mathfrak{g}}+[x,K^{2}(y)]_{\mathfrak{g}})+\lambda[x,K(y)]_{\mathfrak{g}}$
$\displaystyle=K([K(x),y]_{K}+[x,K(y)]_{K})+\lambda[x,y]_{K}.$

This shows that the triple $({\mathfrak{g}}_{K},[\cdot,\cdot]_{K},K)$ is a
modified Rota-Baxter Leibniz algebra of weight $\lambda$. ∎

The following result will be useful in the next section to construct the
cohomology of modified Rota-Baxter Leibniz algebras of weight $\lambda$.

######  Theorem 2.11.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$ be a modified Rota-
Baxter Leibniz algebra of weight $\lambda$ and $(V,\rho^{L},\rho^{R},K_{V})$
be a representation of it. Define maps
$\rho_{K}^{L},\rho_{K}^{R}:{\mathfrak{g}}\rightarrow$ End$(V)$ by

$\displaystyle\rho^{L}_{K}(x)v:=\rho^{L}(K(x))v-K_{V}(\rho^{L}(x)v),$ (7)
$\displaystyle\rho^{R}_{K}(x)v:=\rho^{R}(K(x))v-K_{V}(\rho^{R}(x)v)$

for any $x\in{\mathfrak{g}},v\in V$. Then $\rho^{L}_{K},\rho^{R}_{K}$ defines
a representation of the Leibniz algebra
$({\mathfrak{g}}_{K},[\cdot,\cdot]_{K})$ on $V$. Moreover,
$(V,\rho_{K}^{L},\rho_{K}^{R},K_{V})$ is a representation of the modified
Rota-Baxter Leibniz algebra $({\mathfrak{g}}_{K},[\cdot,\cdot]_{K},K)$ of
weight $\lambda$.

###### Proof.

One can show that $\rho_{K}^{L},\rho_{K}^{R}$ define a representation of the
Leibniz algebra $({\mathfrak{g}}_{K},[\cdot,\cdot]_{K})$ on $V$. In fact, we
have

$\displaystyle\rho_{K}^{L}(x)\rho_{K}^{L}(y)v-\rho_{K}^{L}(y)\rho_{K}^{L}(x)v$
$\displaystyle=$
$\displaystyle\rho_{K}^{L}(x)(\rho^{L}(K(y))v-K_{V}(\rho^{L}(y)v))-\rho_{K}^{L}(y)(\rho^{L}(K(x))v-K_{V}(\rho^{L}(x)v))$
$\displaystyle=$
$\displaystyle\rho^{L}(K(x))(\rho^{L}(K(y))v-K_{V}(\rho^{L}(y)v))-K_{V}(\rho^{L}(x)(\rho^{L}(K(y))v-\rho^{L}(x)K_{V}(\rho^{L}(y)v))$
$\displaystyle\rho^{L}(K(y))(\rho^{L}(K(x))v-K_{V}(\rho^{L}(x)v))-K_{V}(\rho^{L}(y)(\rho^{L}(K(x))v-\rho^{L}(y)K_{V}(\rho^{L}(x)v))$
$\displaystyle=$
$\displaystyle\rho^{L}(K(x))\rho^{L}(K(y))v-K_{V}(\rho^{L}(K(x))\rho^{L}(y)v+\rho^{L}(x)K_{V}(\rho^{L}(y)v)-K_{V}(\rho^{L}(x)\rho^{L}(y)v))$
$\displaystyle-
K_{V}(\rho^{L}(x)\rho^{L}(K(y))v-\rho^{L}(x)K_{V}(\rho^{L}(y)v))-\rho^{L}(K(y))\rho^{L}(K(x))v$
$\displaystyle+K_{V}(\rho^{L}(K(y))\rho^{L}(x)v+\rho^{L}(y)K_{V}(\rho^{L}(x)v)-K_{V}(\rho^{L}(y)\rho^{L}(x)v))$
$\displaystyle-
K_{V}(\rho^{L}(y)\rho^{L}(K(x))v-\rho^{L}(y)K_{V}(\rho^{L}(x)v))$
$\displaystyle=$
$\displaystyle\rho^{L}([K(x),K(y)]_{\mathfrak{g}})v-K_{V}(\rho^{L}([K(x),y]_{\mathfrak{g}}+[x,K(y)]_{\mathfrak{g}})v)-\lambda\rho^{L}([x,y]_{\mathfrak{g}})$
$\displaystyle=$
$\displaystyle\rho^{L}([x,y]_{K})v-K_{V}(\rho^{L}([x,y]_{K})v)$
$\displaystyle=$ $\displaystyle\rho^{L}_{K}([x,y]_{K})v.$

Similar to prove that

$\displaystyle\rho_{K}^{R}([x,y]_{K})=\rho_{K}^{L}(x)\rho_{K}^{R}(y)-\rho_{K}^{R}(y)\rho_{K}^{L}(x).$

Furthermore, we have

$\displaystyle\rho_{K}^{R}(y)\rho_{K}^{L}(x)v+\rho_{K}^{R}(y)\rho_{K}^{R}(x)v$
$\displaystyle=$
$\displaystyle\rho_{K}^{R}(y)(\rho^{L}(K(x))v-K_{V}(\rho^{L}(x)v))+\rho_{K}^{R}(y)(\rho^{R}(K(x))v-K_{V}(\rho^{R}(x)v))$
$\displaystyle=$
$\displaystyle\rho^{R}(K(y))(\rho^{L}(K(x))v-K_{V}(\rho^{L}(x)v))-K_{V}(\rho^{R}(y)(\rho^{L}(K(x))v-K_{V}(\rho^{L}(x)v)))$
$\displaystyle+\rho^{R}(K(y))(\rho^{R}(K(x))v-K_{V}(\rho^{R}(x)v))-K_{V}(\rho^{R}(y)(\rho^{R}(K(x))v-K_{V}(\rho^{R}(x)v)))$
$\displaystyle=$ $\displaystyle 0.$

Moreover, we have

$\displaystyle\rho^{L}_{K}(K(x))K_{V}(v)$ $\displaystyle=$
$\displaystyle\rho^{L}(K^{2}(x))K_{V}(v)-K_{V}(\rho^{L}(K(x))K_{V}(V))$
$\displaystyle=$ $\displaystyle
K_{V}\big{(}\rho^{L}(K^{2}(x))v+\rho^{L}(N(x))N_{V}(v)\big{)}+\lambda\rho^{L}(K(x))v$
$\displaystyle-K^{2}_{V}\big{(}\rho^{L}(K(x))v+\rho^{L}(x)K_{V}(v)\big{)}-\lambda
K_{V}(\rho^{L}(x)v)$ $\displaystyle=$ $\displaystyle
K_{V}\big{(}\rho^{L}_{K}(K(x))v+\rho^{L}_{K}(x)K_{V}(v)\big{)}+\lambda\rho^{L}_{K}(x)v.$

Therefore, $(V,\rho_{K}^{L},\rho_{K}^{R},K_{V})$ is a representation of the
modified Rota-Baxter Leibniz algebra
$({\mathfrak{g}}_{K},[\cdot,\cdot]_{K},K)$. ∎

##  3\. Cohomology theory of modified Rota-Baxter Leibniz algebras

In this section, we will define a cohomology theory for modified Rota-Baxter
Leibniz algebras.

###  3.1. Cohomology of modified Rota-Baxter operators

Firstly, let’s study the cohomology of modified Rota-Baxter operators.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$ be a modified Rota-
Baxter Leibniz algebra and $(V,\rho^{L},\rho^{R},K_{V})$ be a representation
over it. Recall that Proposition 2.10 and Proposition 2.11 give a new Leibniz
algebra ${\mathfrak{g}}_{K}$ and a new representation $V$ over
${\mathfrak{g}}_{K}$. Consider the cochain complex of ${\mathfrak{g}}_{K}$
with coefficients in $V$:

${\mathrm{C}}^{\bullet}_{{\mathrm{Leib}}}({\mathfrak{g}}_{K},V)=\bigoplus\limits_{n=0}^{\infty}{\mathrm{C}}^{n}_{{\mathrm{Leib}}}({\mathfrak{g}}_{K},V).$

More precisely, for $n\geqslant 0$,
${\mathrm{C}}^{n}_{{\mathrm{Leib}}}({\mathfrak{g}}_{K},V)={\rm
Hom}(\mathfrak{g}^{\otimes n},V)$ and its differential

$\partial^{n}:{\mathrm{C}}^{n}_{\mathrm{Leib}}({\mathfrak{g}}_{K},\
V)\rightarrow{\mathrm{C}}^{n+1}_{\mathrm{Leib}}({\mathfrak{g}}_{K},V)$

is defined as:

$\displaystyle(\partial^{n}f)(x_{1},\ldots,x_{n+1})$
$\displaystyle=\sum^{n}_{i=1}(-1)^{i+1}\rho^{L}_{K}(x_{i})f(x_{1},\ldots,\hat{x}_{i},\ldots,x_{n+1})+(-1)^{n+1}\rho^{R}_{K}(x_{n+1})f(x_{1},\ldots,x_{n})$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i}f(x_{1},\ldots,\hat{x}_{i},\ldots,x_{j-1},[x_{i},x_{j}]_{K},x_{j+1},\ldots,x_{n+1}),$

for $x_{1},\ldots,x_{n+1}\in{\mathfrak{g}}$.

######  Definition 3.1.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$ be a modified Rota-
Baxter Leibniz algebra and $(V,\rho^{L},\rho^{R},K_{V})$ be a representation
over it. Then the cochain complex
$({\mathrm{C}}^{\bullet}_{\mathrm{Leib}}({\mathfrak{g}}_{K},V),\partial)$ is
called the cochain complex of modified Rota-Baxter operator $K$ with
coefficients in $(V,\rho^{L}_{K},\rho^{R}_{K},K_{V})$, denoted by
$C_{\mathsf{MO}}^{\bullet}({\mathfrak{g}},V)$. The cohomology of
$C_{\mathsf{MO}}^{\bullet}({\mathfrak{g}},V)$, denoted by
$\mathrm{H}_{\mathsf{MO}}^{\bullet}({\mathfrak{g}},V)$, are called the
cohomology of modified Rota-Baxter operator $K$ with coefficients in
$(V,\rho^{L}_{K},\rho^{R}_{K},K_{V})$.

When $(V,\rho^{L},\rho^{R},K_{V})$ is the regular representation
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$, we denote
${\mathrm{C}}^{\bullet}_{\mathsf{MO}}({\mathfrak{g}},{\mathfrak{g}})$ by
${\mathrm{C}}^{\bullet}_{\mathsf{MO}}({\mathfrak{g}})$ and call it the cochain
complex of modified Rota-Baxter operator $K$, and denote
${\mathrm{H}}^{\bullet}_{\mathsf{MO}}({\mathfrak{g}},{\mathfrak{g}})$ by
${\mathrm{H}}^{\bullet}_{\mathsf{MO}}({\mathfrak{g}})$ and call it the
cohomology of modified Rota-Baxter operator $K$.

###  3.2. Cohomology of modified Rota-Baxter Leibniz algebras

In this subsection, we will combine the cohomology of Leibniz algebras and the
cohomology of modified Rota-Baxter operators to define a cohomology theory for
modified Rota-Baxter Leibniz algebras.

Let $V=(V,\rho^{L},\rho^{R},K_{V})$ be a representation over a modified Rota-
Baxter Leibniz algebra
${\mathfrak{g}}=({\mathfrak{g}},\mu=[\cdot,\cdot]_{\mathfrak{g}},K)$. Now,
let’s construct a chain map

$\Phi^{\bullet}:{\mathrm{C}}^{\bullet}_{{\mathrm{Leib}}}({\mathfrak{g}},V)\rightarrow
C_{\mathsf{MO}}^{\bullet}({\mathfrak{g}},V),$

i.e., the following commutative diagram:

$\textstyle{{\mathrm{C}}^{0}_{{\mathrm{Leib}}}({\mathfrak{g}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{0}}$$\scriptstyle{\Phi^{0}}$$\textstyle{{\mathrm{C}}^{1}_{{\mathrm{Leib}}}({\mathfrak{g}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi^{1}}$$\textstyle{{\mathrm{C}}^{n}_{{\mathrm{Leib}}}({\mathfrak{g}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{n}}$$\scriptstyle{\Phi^{n}}$$\textstyle{{\mathrm{C}}^{n+1}_{{\mathrm{Leib}}}({\mathfrak{g}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi^{n+1}}$$\textstyle{{\mathrm{C}}^{0}_{\mathsf{MO}}({\mathfrak{g}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial^{0}}$$\textstyle{{\mathrm{C}}^{1}_{\mathsf{MO}}({\mathfrak{g}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{C}}^{n}_{\mathsf{MO}}({\mathfrak{g}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial^{n}}$$\textstyle{{\mathrm{C}}^{n+1}_{\mathsf{MO}}({\mathfrak{g}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$.

Define $\Phi^{0}={\rm Id}_{{\rm Hom}(k,V)}={\rm Id}_{V}$, and for $n\geqslant
1$ and $f\in{\mathrm{C}}^{n}_{{\mathrm{Leib}}}({\mathfrak{g}},V)$, define
$\Phi^{n}(f)\in{\mathrm{C}}^{n}_{\mathsf{MO}}({\mathfrak{g}},V)$ as:

$\displaystyle\Phi^{n}(f)(x_{1},\ldots,x_{n})=$ $\displaystyle
f(K(x_{1}),\ldots,K(x_{n}))$ $\displaystyle-\sum_{1\leqslant
i_{1}<\cdots<i_{r}\leqslant
n,~{}~{}r~{}~{}\mathrm{odd}}(-\lambda)^{\frac{r-1}{2}}K_{V}\circ
f(K(x_{1}),\ldots,x_{i_{1}},\ldots,x_{i_{2}},\ldots,x_{i_{r}},\ldots,K(x_{n}))$
$\displaystyle-\sum_{1\leqslant i_{1}<\cdots<i_{r}\leqslant
n,~{}~{}r~{}~{}\mathrm{even}}(-\lambda)^{\frac{r}{2}+1}K_{V}\circ
f(K(x_{1}),\ldots,x_{i_{1}},\ldots,x_{i_{2}},\ldots,x_{i_{r}},\ldots,K(x_{n})).$

######  Proposition 3.2.

The map
$\Phi^{\bullet}:{\mathrm{C}}^{\bullet}_{\mathrm{Leib}}({\mathfrak{g}},V)\rightarrow{\mathrm{C}}^{\bullet}_{\mathsf{MO}}({\mathfrak{g}},V)$
is a chain map.

######  Definition 3.3.

Let $V=(V,\rho^{L},\rho^{R},K_{V})$ be a representation over a modified Rota-
Baxter Leibniz algebra
${\mathfrak{g}}=({\mathfrak{g}},\mu=[\cdot,\cdot]_{\mathfrak{g}},K)$. We
define the cochain complex
$({\mathrm{C}}^{\bullet}_{\mathsf{MLeib}}({\mathfrak{g}},V),d^{\bullet})$ of
modified Rota-Baxter Leibniz algebras $({\mathfrak{g}},\mu,K)$ with
coefficients in $(V,\rho^{L},\rho^{R},K_{V})$ to the negative shift of the
mapping cone of $\Phi^{\bullet}$, that is, let

${\mathrm{C}}^{0}_{\mathsf{MLeib}}({\mathfrak{g}},V)={\mathrm{C}}^{0}_{\mathrm{Leib}}({\mathfrak{g}},V)\quad\mathrm{and}\quad{\mathrm{C}}^{n}_{\mathsf{MLeib}}({\mathfrak{g}},V)={\mathrm{C}}^{n}_{\mathrm{Leib}}({\mathfrak{g}},V)\oplus{\mathrm{C}}^{n-1}_{\mathsf{MO}}({\mathfrak{g}},V),\forall
n\geqslant 1,$

and the differential
$d^{n}:{\mathrm{C}}^{n}_{\mathsf{MLeib}}({\mathfrak{g}},V)\rightarrow{\mathrm{C}}^{n+1}_{\mathsf{MLeib}}({\mathfrak{g}},V)$
is given by

$d^{n}(f,g)=(\delta^{n}(f),-\partial^{n-1}(g)-\Phi^{n}(f))$

for any $f\in{\mathrm{C}}^{n}_{\mathrm{Leib}}({\mathfrak{g}},V)$ and
$g\in{\mathrm{C}}^{n-1}_{\mathsf{MO}}({\mathfrak{g}},V)$. The cohomology of
$({\mathrm{C}}^{\bullet}_{\mathsf{MLeib}}({\mathfrak{g}},V),d^{\bullet})$,
denoted by ${\mathrm{H}}_{\mathsf{MLeib}}^{\bullet}({\mathfrak{g}},V)$, is
called the cohomology of the modified Rota-Baxter Leibniz algebra
$({\mathfrak{g}},\mu,K)$ with coefficients in $(V,\rho^{L},\rho^{R},K_{V})$.
When $(V,\rho^{L},\rho^{R},K_{V})=({\mathfrak{g}},\mu,K)$, we just denote
${\mathrm{C}}^{\bullet}_{\mathsf{MLeib}}({\mathfrak{g}},{\mathfrak{g}}),{\mathrm{H}}^{\bullet}_{\mathsf{MLeib}}({\mathfrak{g}},{\mathfrak{g}})$
by
${\mathrm{C}}^{\bullet}_{\mathsf{MLeib}}({\mathfrak{g}}),{\mathrm{H}}_{\mathsf{MLeib}}^{\bullet}({\mathfrak{g}})$
respectively, and call them the cochain complex, the cohomology of modified
Rota-Baxter Leibniz algebra $({\mathfrak{g}},\mu,T)$ respectively.

There is an obvious short exact sequence of complexes:

(8) $\displaystyle
0\to{\mathrm{C}}^{\bullet}_{\mathsf{MO}}({\mathfrak{g}},V)\to{\mathrm{C}}^{\bullet}_{\mathsf{MLeib}}({\mathfrak{g}},V)\to{\mathrm{C}}^{\bullet}_{{\mathrm{Leib}}}({\mathfrak{g}},V)\to
0$

which induces a long exact sequence of cohomology groups

$0\to{\mathrm{H}}^{0}_{\mathsf{MLeib}}({\mathfrak{g}},V)\to\mathrm{H}^{0}_{\mathrm{Leib}}({\mathfrak{g}},V)\to{\mathrm{H}}^{0}_{\mathsf{MO}}({\mathfrak{g}},V)\to{\mathrm{H}}^{1}_{\mathsf{MLeib}}({\mathfrak{g}},V)\to\mathrm{H}^{1}_{\mathrm{Leib}}({\mathfrak{g}},V)\to\cdot
s$
$\cdots\to\mathrm{H}^{p}_{\mathrm{Leib}}({\mathfrak{g}},V)\to{\mathrm{H}}^{p}_{\mathsf{MO}}({\mathfrak{g}},V)\to{\mathrm{H}}^{p+1}_{\mathsf{MLeib}}({\mathfrak{g}},V)\to\mathrm{H}^{p+1}_{\mathrm{Leib}}({\mathfrak{g}},V)\to\cdots$

##  4\. Formal deformations of modified Rota-Baxter Leibniz algebras

In this section, we will study formal deformations of modified Rota-Baxter
Leibniz algebras and interpret them via lower degree cohomology groups of
modified Rota-Baxter Leibniz algebras defined in last section.

Let $({\mathfrak{g}},\mu,K)$ be a modified Rota-Baxter Leibniz algebra.
Consider a 1-parameterized family:

$\mu_{t}=\sum_{i=0}^{\infty}\mu_{i}t^{i},\
\mu_{i}\in{\mathrm{C}}^{2}_{\mathrm{Leib}}({\mathfrak{g}}),\quad
K_{t}=\sum_{i=0}^{\infty}K_{i}t^{i},\
K_{i}\in{\mathrm{C}}^{1}_{\mathsf{MO}}({\mathfrak{g}}).$

######  Definition 4.1.

A 1-parameter formal deformation of modified Rota-Baxter Leibniz algebra
$({\mathfrak{g}},\mu,K)$ is a pair $(\mu_{t},K_{t})$ which endows the flat
${\bf k}[[t]]$-module ${\mathfrak{g}}[[t]]$ with a modified Rota-Baxter
Leibniz algebra structure over ${\bf k}[[t]]$ such that
$(\mu_{0},K_{0})=(\mu,K)$.

Power series $\mu_{t}$ and $K_{t}$ determine a 1-parameter formal deformation
of modified Rota-Baxter Leibniz algebra $({\mathfrak{g}},\mu,K)$ if and only
if for any $x,y,z\in{\mathfrak{g}}$, the following equations hold :

$\displaystyle\mu_{t}(x,\mu_{t}(y,z))=\mu_{t}(\mu_{t}(x,y),z)+\mu_{t}(y,\mu_{t}(x,z))$
$\displaystyle\mu_{t}(K_{t}(x),K_{t}(y))=K_{t}(\mu_{t}(K_{t}(x),y)+\mu_{t}(x,K_{t}(y))\big{)}+\lambda\mu_{t}(x,y).$

By expanding these equations and comparing the coefficient of $t^{n}$, we
obtain that $\\{\mu_{i}\\}_{i\geqslant 0}$ and $\\{K_{i}\\}_{i\geqslant 0}$
have to satisfy: for any $n\geqslant 0$,

(9)
$\displaystyle\sum_{i+j=n}\mu_{i}(x,\mu_{j}(y,z))=\sum_{i+j=n}\mu_{i}(\mu_{j}(x,y),z)+\mu_{i}(y,\mu_{j}(x,z)),$
(10)
$\displaystyle\sum_{i+j+k=n}\mu_{i}(K_{j}(x),K_{k}(y))=\sum_{i+j+k=n}K_{i}\big{(}\mu_{j}(K_{k}(x),y)+\mu_{j}(x,K_{k}(y))\big{)}+\lambda\mu_{k}(x,y).$

Obviously, when $n=0$, the above conditions are exactly the associativity of
$\mu=\mu_{0}$ and Equation (2) which is the defining relation of a modified
Rota-Baxter operator $K=K_{0}$.

######  Proposition 4.2.

Let $({\mathfrak{g}}[[t]],\mu_{t},K_{t})$ be a 1-parameter formal deformation
of a modified Rota-Baxter Leibniz algebra $({\mathfrak{g}},\mu,K)$. Then
$(\mu_{1},K_{1})$ is a 1-cocycle in the cochain complex
$C_{\mathsf{MLeib}}^{\bullet}({\mathfrak{g}})$.

###### Proof.

When $n=1$, Equations (9) and (10) become

$\displaystyle[x,\mu_{1}(y,z)]_{\mathfrak{g}}+\mu_{1}(x,[y,z]_{\mathfrak{g}})=[\mu_{1}(x,y),z]_{\mathfrak{g}}+[y,\mu_{1}(x,z)]_{\mathfrak{g}}+\mu_{1}([x,y]_{\mathfrak{g}},z)+\mu_{1}(y,[x,z]_{\mathfrak{g}}),$

and

$\displaystyle[K_{1}(x),K(y)]_{\mathfrak{g}}+[K(x),K_{1}(y)]_{\mathfrak{g}}+\mu_{1}(K(x),K(y))$
$\displaystyle=K_{1}\big{(}[K(x),y]_{\mathfrak{g}}+[x,K(y)]_{\mathfrak{g}}\big{)}+\lambda[x,y]_{\mathfrak{g}}+K\big{(}\mu_{1}(K(x),y)+\mu_{1}(x,K(y))\big{)}+\lambda[x,y]_{\mathfrak{g}}$
$\displaystyle+K\big{(}[K_{1}(x),y]_{\mathfrak{g}}+[x,K_{1}(y)]_{\mathfrak{g}}\big{)}+\lambda\mu_{1}(x,y).$

Note that the first equation is exactly
$\delta^{2}(\mu_{1})=0\in{\mathrm{C}}^{\bullet}_{{\mathrm{Leib}}}({\mathfrak{g}})$
and that second equation is exactly to

$\Phi^{2}(\mu_{1})=-\partial^{1}(K_{1})\in{\mathrm{C}}^{\bullet}_{\mathsf{MO}}({\mathfrak{g}}).$

So $(\mu_{1},K_{1})$ is a 2-cocycle in
${\mathrm{C}}^{\bullet}_{\mathsf{MLeib}}({\mathfrak{g}})$. ∎

######  Definition 4.3.

The 1-cocycle $(\mu_{1},K_{1})$ is called the infinitesimal of the 1-parameter
formal deformation $({\mathfrak{g}}[[t]],\mu_{t},K_{t})$ of a modified Rota-
Baxter Leibniz algebra $({\mathfrak{g}},\mu,K)$.

######  Definition 4.4.

Let $({\mathfrak{g}}[[t]],\mu_{t},K_{t})$ and
$({\mathfrak{g}}[[t]],\mu_{t}^{\prime},K_{t}^{\prime})$ be two 1-parameter
formal deformations of a modified Rota-Baxter Leibniz algebra
$({\mathfrak{g}},\mu,K)$. A formal isomorphism from
$({\mathfrak{g}}[[t]],\mu_{t}^{\prime},K_{t}^{\prime})$ to
$({\mathfrak{g}}[[t]],\mu_{t},K_{t})$ is a power series
$\psi_{t}=\sum_{i=0}\psi_{i}t^{i}:{\mathfrak{g}}[[t]]\rightarrow{\mathfrak{g}}[[t]]$,
where $\psi_{i}:{\mathfrak{g}}\rightarrow{\mathfrak{g}}$ are linear maps with
$\psi_{0}={\rm Id}_{\mathfrak{g}}$, such that:

(11) $\displaystyle\psi_{t}\circ\mu_{t}^{\prime}$ $\displaystyle=$
$\displaystyle\mu_{t}\circ(\psi_{t}\otimes\psi_{t}),$ (12)
$\displaystyle\psi_{t}\circ K_{t}^{\prime}$ $\displaystyle=$ $\displaystyle
K_{t}\circ\psi_{t}.$

In this case, we say that the two 1-parameter formal deformations
$({\mathfrak{g}}[[t]],\mu_{t},K_{t})$ and
$({\mathfrak{g}}[[t]],\mu_{t}^{\prime},K_{t}^{\prime})$ are equivalent.

Given a modified Rota-Baxter Leibniz algebra $({\mathfrak{g}},\mu,K)$, the
power series $\mu_{t},K_{t}$ with
$\mu_{i}=\delta_{i,0}\mu,K_{i}=\delta_{i,0}K$ make
$({\mathfrak{g}}[[t]],\mu_{t},K_{t})$ into a $1$-parameter formal deformation
of $({\mathfrak{g}},\mu,K)$. Formal deformations equivalent to this one are
called trivial.

######  Theorem 4.5.

The infinitesimals of two equivalent 1-parameter formal deformations of
$({\mathfrak{g}},\mu,K)$ are in the same cohomology class in
${\mathrm{H}}^{\bullet}_{\mathsf{MLeib}}({\mathfrak{g}})$.

###### Proof.

Let
$\psi_{t}:({\mathfrak{g}}[[t]],\mu_{t}^{\prime},K_{t}^{\prime})\rightarrow({\mathfrak{g}}[[t]],\mu_{t},K_{t})$
be a formal isomorphism. Expanding the identities and collecting coefficients
of $t$, we get from Equations (11) and (12):

$\displaystyle\mu_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\mu_{1}+\mu\circ({\rm
Id}\otimes\psi_{1})-\psi_{1}\circ\mu+\mu\circ(\psi_{1}\otimes{\rm Id}),$
$\displaystyle K_{1}^{\prime}$ $\displaystyle=$ $\displaystyle
K_{1}+K\circ\psi_{1}-\psi_{1}\circ K,$

that is, we have

$(\mu_{1}^{\prime},K_{1}^{\prime})-(\mu_{1},K_{1})=(\delta^{1}(\psi_{1}),-\Phi^{1}(\psi_{1}))=d^{1}(\psi_{1},0)\in{\mathrm{C}}^{\bullet}_{\mathsf{MLeib}}({\mathfrak{g}}).$

∎

######  Definition 4.6.

A modified Rota-Baxter Leibniz algebra $({\mathfrak{g}},\mu,K)$ is said to be
rigid if every 1-parameter formal deformation is trivial.

######  Theorem 4.7.

Let $({\mathfrak{g}},\mu,K)$ be a modified Rota-Baxter Leibniz algebra. If
${\mathrm{H}}^{2}_{\mathsf{MLeib}}({\mathfrak{g}})=0$, then
$({\mathfrak{g}},\mu,K)$ is rigid.

###### Proof.

Let $({\mathfrak{g}}[[t]],\mu_{t},K_{t})$ be a $1$-parameter formal
deformation of $({\mathfrak{g}},\mu,K)$. By Proposition 4.2, $(\mu_{1},K_{1})$
is a $2$-cocycle. By ${\mathrm{H}}^{2}_{\mathsf{MLeib}}({\mathfrak{g}})=0$,
there exists a $1$-cochain

$(\psi_{1}^{\prime},x)\in{\mathrm{C}}^{1}_{\mathsf{MLeib}}({\mathfrak{g}})=C^{1}_{{\mathrm{Leib}}}({\mathfrak{g}})\oplus{\rm
Hom}(k,{\mathfrak{g}})$

such that $(\mu_{1},K_{1})=d^{1}(\psi_{1}^{\prime},x),$ that is,
$\mu_{1}=\delta^{1}(\psi_{1}^{\prime})$ and
$K_{1}=-\partial^{0}(x)-\Phi^{1}(\psi_{1}^{\prime})$. Let
$\psi_{1}=\psi_{1}^{\prime}+\delta^{0}(x)$. Then
$\mu_{1}=\delta^{1}(\psi_{1})$ and $K_{1}=-\Phi^{1}(\psi_{1})$, as it can be
readily seen that $\Phi^{1}(\delta^{0}(x))=\partial^{0}(x)$.

Setting $\psi_{t}={\rm Id}_{\mathfrak{g}}-\psi_{1}t$, we have a deformation
$({\mathfrak{g}}[[t]],\overline{\mu}_{t},\overline{K}_{t})$, where

$\overline{\mu}_{t}=\psi_{t}^{-1}\circ\mu_{t}\circ(\psi_{t}\times\psi_{t})$

and

$\overline{K}_{t}=\psi_{t}^{-1}\circ K_{t}\circ\psi_{t}.$

It can be easily verify that $\overline{\mu}_{1}=0,\overline{K}_{1}=0$. Then

$\begin{array}[]{rcl}\overline{\mu}_{t}&=&\mu+\overline{\mu}_{2}t^{2}+\cdots,\\\
K_{t}&=&K+\overline{K}_{2}t^{2}+\cdots.\end{array}$

By Equations (9) and (10), we see that $(\overline{\mu}_{2},\overline{K}_{2})$
is still a $2$-cocyle, so by induction, we can show that
$({\mathfrak{g}}[[t]],\mu_{t},K_{t})$ is equivalent to the trivial extension
$({\mathfrak{g}}[[t]],\mu,K).$ Thus, $({\mathfrak{g}},\mu,K)$ is rigid.

∎

##  5\. Abelian extensions of modified Rota-Baxter Leibniz algebras

In this section, we study abelian extensions of modified Rota-Baxter Leibniz
algebras and show that they are classified by the first cohomology, as one
would expect of a good cohomology theory.

Notice that a vector space $V$ together with a linear transformation
$K_{V}:V\to V$ is naturally a modified Rota-Baxter Leibniz algebra where the
multiplication on $V$ is defined to be $[u,v]_{V}=0$ for all $u,v\in V.$

######  Definition 5.1.

An abelian extension of modified Rota-Baxter Leibniz algebras is a short exact
sequence of morphisms of modified Rota-Baxter Leibniz algebras

(13) $\displaystyle 0\to(V,[\cdot,\cdot]_{V},K_{V})\stackrel{{\scriptstyle
i}}{{\to}}(\hat{{\mathfrak{g}}},[\cdot,\cdot]_{\hat{{\mathfrak{g}}}},\hat{K})\stackrel{{\scriptstyle
p}}{{\to}}({\mathfrak{g}},[\cdot,\cdot]_{{\mathfrak{g}}},K)\to 0,$

that is, there exists a commutative diagram:

$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>{V}@>{i}>{}>\hat{{\mathfrak{g}}}@>{p}>{}>{\mathfrak{g}}@>{}>{}>0\\\
@V{{K_{V}}}V{}V@V{{\hat{K}}}V{}V@V{K}V{}V\\\
0@>{}>{}>{V}@>{i}>{}>\hat{{\mathfrak{g}}}@>{p}>{}>{\mathfrak{g}}@>{}>{}>0,\end{CD}$

where the modified Rota-Baxter Leibniz algebra $(V,[\cdot,\cdot]_{V},K_{V})$
satisfies $[\cdot,v]_{V}=0$ for all $v\in V.$

We will call
$(\hat{{\mathfrak{g}}},[\cdot,\cdot]_{\hat{{\mathfrak{g}}}},\hat{K})$ an
abelian extension of $({\mathfrak{g}},[\cdot,\cdot]_{{\mathfrak{g}}},K)$ by
$(V,[\cdot,\cdot]_{V},K_{V})$.

######  Definition 5.2.

Let
$(\hat{{\mathfrak{g}}}_{1},[\cdot,\cdot]_{\hat{{\mathfrak{g}}}_{1}},\hat{K}_{1})$
and
$(\hat{{\mathfrak{g}}}_{2},[\cdot,\cdot]_{\hat{{\mathfrak{g}}}_{2}},\hat{K}_{2})$
be two abelian extensions of
$({\mathfrak{g}},[\cdot,\cdot]_{{\mathfrak{g}}},K)$ by
$(V,[\cdot,\cdot]_{V},K_{V})$. They are said to be isomorphic if there exists
an isomorphism of modified Rota-Baxter Leibniz algebras
$\zeta:(\hat{{\mathfrak{g}}}_{1},\hat{K}_{1})\rightarrow(\hat{{\mathfrak{g}}}_{2},\hat{K}_{2})$
such that the following commutative diagram holds:

(14)
$\displaystyle\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>{(V,[\cdot,\cdot]_{V},K_{V})}@>{i}>{}>(\hat{{\mathfrak{g}}}_{1},[\cdot,\cdot]_{\hat{{\mathfrak{g}}}_{1}},{\hat{K}_{1}})@>{p}>{}>({\mathfrak{g}},[\cdot,\cdot]_{{\mathfrak{g}}},K)@>{}>{}>0\\\
\Big{\|}@V{\zeta}V{}V\Big{\|}\\\
0@>{}>{}>{(V,[\cdot,\cdot]_{V},K_{V})}@>{i}>{}>(\hat{{\mathfrak{g}}}_{2},[\cdot,\cdot]_{\hat{{\mathfrak{g}}}_{2}},{\hat{K}_{2}}<EMAIL_ADDRESS>
A section of an abelian extension
$(\hat{{\mathfrak{g}}},[\cdot,\cdot]_{\hat{{\mathfrak{g}}}},{\hat{K}})$ of
$({\mathfrak{g}},[\cdot,\cdot]_{{\mathfrak{g}}},K)$ by
$(V,[\cdot,\cdot]_{V},K_{V})$ is a linear map
$s:{\mathfrak{g}}\rightarrow\hat{{\mathfrak{g}}}$ such that $p\circ s={\rm
Id}_{\mathfrak{g}}$.

We will show that isomorphism classes of abelian extensions of
$({\mathfrak{g}},[\cdot,\cdot]_{{\mathfrak{g}}},K)$ by
$(V,[\cdot,\cdot]_{V},K_{V})$ are in bijection with the second cohomology
group ${{\mathrm{H}}}_{\mathsf{MLeib}}^{2}({\mathfrak{g}},V)$.

Let $(\hat{{\mathfrak{g}}},[\cdot,\cdot]_{\hat{{\mathfrak{g}}}},\hat{K})$ be
an abelian extension of $({\mathfrak{g}},[\cdot,\cdot]_{{\mathfrak{g}}},K)$ by
$(V,[\cdot,\cdot]_{V},K_{V})$ having the form Equation (13). Choose a section
$s:{\mathfrak{g}}\rightarrow\hat{{\mathfrak{g}}}$, and define
$\rho^{L},\rho^{R}:{\mathfrak{g}}\rightarrow\mathfrak{gl}(V)$

$\displaystyle\rho^{L}(x)v:=[s(x),i(v)]_{\hat{{\mathfrak{g}}}}~{}~{}~{}~{}\rho^{R}(x)v:=[i(v),s(x)]_{\hat{{\mathfrak{g}}}}.$

for all $x\in{\mathfrak{g}},v\in V$.

######  Proposition 5.3.

With the above notations, $(V,\rho^{L},\rho^{R},K_{V})$ is a representation
over $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$.

###### Proof.

It is straightforward to check that $(V,\rho^{L},\rho^{R})$ is a
representation over $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$. Moreover,
${\hat{K}}(s(x))-s(K(x))\in V$ means that
$\rho^{L}({\hat{K}}(s(x))v=\rho^{L}(s(K(x)))v$ and
$\rho^{R}({\hat{K}}(s(x))v=\rho^{R}(s(K(x)))v$. Thus we have

$\displaystyle\rho^{L}(K(x))K_{V}(v)$ $\displaystyle=\rho^{L}(sK(x))K_{V}(v)$
$\displaystyle=K_{V}\big{(}\rho^{L}(\hat{K}(s(x)))v+\rho^{L}(s(x))K_{V}(v)\big{)}+\lambda\rho^{L}(s(x))v$
$\displaystyle=K_{V}\big{(}\rho^{L}(K(x))v+\rho^{L}(x)K_{V}(v)\big{)}+\lambda\rho^{L}(x)v.$

Similar to prove that

$\displaystyle\rho^{R}(K(x))K_{V}(v)$
$\displaystyle=K_{V}\big{(}\rho^{R}(K(x))v+\rho^{R}(x)K_{V}(v)\big{)}+\lambda\rho^{R}(x)v.$

Hence, $(V,\rho^{L},\rho^{R},K_{V})$ is a representation over
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$. ∎

We further define linear maps
$\psi:{\mathfrak{g}}\otimes{\mathfrak{g}}\rightarrow V$ and
$\chi:{\mathfrak{g}}\rightarrow V$ respectively by

$\displaystyle\psi(x,y)$
$\displaystyle=[s(x),s(y)]_{\hat{{\mathfrak{g}}}}-s([x,y]_{\mathfrak{g}}),\quad\forall
x,y\in{\mathfrak{g}},$ $\displaystyle\chi(x)$
$\displaystyle={\hat{K}}(s(x))-s(K(x)),\quad\forall x\in{\mathfrak{g}}.$

######  Proposition 5.4.

The pair $(\psi,\chi)$ is a 2-cocycle of modified Rota-Baxter Leibniz algebras
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$ with coefficients in the
representation $(V,\rho^{L},\rho^{R},K_{V})$ introduced in Proposition 5.3.

The proof is by direct computations, so it is left to the reader.

The choice of the section $s$ in fact determines a splitting

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\hat{{\mathfrak{g}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{t}$$\textstyle{{\mathfrak{g}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{0}$

subject to $t\circ i={\rm Id}_{V},t\circ s=0$ and $it+sp={\rm
Id}_{\hat{{\mathfrak{g}}}}$. Then there is an induced isomorphism of vector
spaces

$\left(\begin{array}[]{cc}p&t\end{array}\right):\hat{{\mathfrak{g}}}\cong{\mathfrak{g}}\oplus
V:\left(\begin{array}[]{c}s\\\ i\end{array}\right).$

We can transfer the modified Rota-Baxter Leibniz algebra structure on
$\hat{{\mathfrak{g}}}$ to ${\mathfrak{g}}\oplus V$ via this isomorphism. It is
direct to verify that this endows ${\mathfrak{g}}\oplus V$ with a Leibniz
bracket $[\cdot,\cdot]_{\psi}$ and a modified Rota-Baxter operator $K_{\chi}$
defined by

(15) $\displaystyle[x+u,y+v]_{\psi}$
$\displaystyle=[x,y]_{\mathfrak{g}}+\rho^{L}(x)v+\rho^{R}(y)u+\psi(x,y),$ (16)
$\displaystyle K_{\chi}(x+u)$ $\displaystyle=K(x)+\chi(x)+K_{V}(u),$

for any $x,y\in{\mathfrak{g}},\,u,v\in V$. Moreover, we get an abelian
extension

$0\to(V,[\cdot,\cdot]_{V},K_{V})\stackrel{{\scriptstyle\left(\begin{array}[]{cc}s&i\end{array}\right)}}{{\to}}({\mathfrak{g}}\oplus
V,K_{\chi})\stackrel{{\scriptstyle\left(\begin{array}[]{c}p\\\
t\end{array}\right)}}{{\to}}({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)\to
0$

which is easily seen to be isomorphic to the original one (13).

Now we investigate the influence of different choices of sections.

######  Proposition 5.5.

  * (i)

Different choices of the section $s$ give the same representation on
$(V,[\cdot,\cdot]_{V},K_{V})$;

  * (ii)

the cohomological class of $(\psi,\chi)$ does not depend on the choice of
sections.

###### Proof.

Let $s_{1}$ and $s_{2}$ be two distinct sections of $p$. We define
$\gamma:{\mathfrak{g}}\rightarrow V$ by $\gamma(x)=s_{1}(x)-s_{2}(x)$.

Since the modified Rota-Baxter Leibniz algebra $(V,[\cdot,\cdot]_{V},K_{V})$
satisfies $[u,v]_{V}=0$ for all $u,v\in V$,

$\rho^{L}(s_{1}(x))u=\rho^{L}(s_{2}(x)+\gamma(x))u=\rho^{L}(s_{2}(x))u.$

So different choices of the section $s$ give the same representation on
$(V,[\cdot,\cdot]_{V},K_{V})$;

We show that the cohomological class of $(\psi,\chi)$ does not depend on the
choice of sections. Then

$\displaystyle\psi_{1}(x,y)$
$\displaystyle=[s_{1}(x),s_{1}(y)]_{\hat{{\mathfrak{g}}}}-s_{1}([x,y]_{\mathfrak{g}})$
$\displaystyle=[s_{2}(x)+\gamma(x),s_{2}(y)+\gamma(y)]_{\hat{{\mathfrak{g}}}}-(s_{2}([x,y]_{\mathfrak{g}})+\gamma([x,y]_{\mathfrak{g}}))$
$\displaystyle=([s_{2}(x),s_{2}(y)]_{\hat{{\mathfrak{g}}}}-s_{2}([x,y]_{\mathfrak{g}}))+[s_{2}(x),\gamma(y)]_{\hat{{\mathfrak{g}}}}+[\gamma(x),s_{2}(y)]_{\hat{{\mathfrak{g}}}}-\gamma([x,y]_{\mathfrak{g}})$
$\displaystyle=([s_{2}(x),s_{2}(y)]_{\hat{{\mathfrak{g}}}}-s_{2}([x,y]_{\mathfrak{g}}))+[x,\gamma(y)]_{\hat{{\mathfrak{g}}}}+[\gamma(x),y]_{\hat{{\mathfrak{g}}}}-\gamma([x,y]_{\mathfrak{g}})$
$\displaystyle=\psi_{2}(x,y)+\delta(\gamma)(x,y)$

and

$\displaystyle\chi_{1}(x)$ $\displaystyle={\hat{K}}(s_{1}(x))-s_{1}(K(x))$
$\displaystyle={\hat{K}}(s_{2}(x)+\gamma(x))-(s_{2}(K(x))+\gamma(K(x)))$
$\displaystyle=({\hat{K}}(s_{2}(x))-s_{2}(K(x)))+{\hat{K}}(\gamma(x))-\gamma(K(x))$
$\displaystyle=\chi_{2}(x)+K_{V}(\gamma(x))-\gamma(K(x))$
$\displaystyle=\chi_{2}(x)-\Phi^{1}(\gamma)(x).$

That is, $(\psi_{1},\chi_{1})=(\psi_{2},\chi_{2})+d^{1}(\gamma)$. Thus
$(\psi_{1},\chi_{1})$ and $(\psi_{2},\chi_{2})$ form the same cohomological
class in ${\mathrm{H}}_{\mathsf{MLeib}}^{2}({\mathfrak{g}},V)$.

∎

We show now the isomorphic abelian extensions give rise to the same cohomology
classes.

######  Proposition 5.6.

Let $V$ be a vector space and $K_{V}\in{\rm End}_{\bf k}(V)$. Then
$(V,[\cdot,\cdot]_{V},K_{V})$ is a modified Rota-Baxter Leibniz algebra with
trivial Leibniz bracket. Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$
be a modified Rota-Baxter Leibniz algebra. Two isomorphic abelian extensions
of modified Rota-Baxter Leibniz algebras
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$ by
$(V,[\cdot,\cdot]_{V},K_{V})$ give rise to the same cohomology class in
${\mathrm{H}}_{\mathsf{MLeib}}^{2}({\mathfrak{g}},V)$.

###### Proof.

Assume that
$(\hat{{\mathfrak{g}}}_{1},[\cdot,\cdot]_{\hat{{\mathfrak{g}}}_{1}},{\hat{K}_{1}})$
and
$(\hat{{\mathfrak{g}}}_{2},[\cdot,\cdot]_{\hat{{\mathfrak{g}}}_{2}},{\hat{K}_{2}})$
are two isomorphic abelian extensions of
$({\mathfrak{g}},[\cdot,\cdot]_{{\mathfrak{g}}},K)$ by
$(V,[\cdot,\cdot]_{V},K_{V})$ as is given in (14). Let $s_{1}$ be a section of
$(\hat{{\mathfrak{g}}}_{1},[\cdot,\cdot]_{\hat{{\mathfrak{g}}}_{1}},{\hat{K}_{1}})$.
As $p_{2}\circ\zeta=p_{1}$, we have

$p_{2}\circ(\zeta\circ s_{1})=p_{1}\circ s_{1}={\rm Id}_{{\mathfrak{g}}}.$

Therefore, $\zeta\circ s_{1}$ is a section of
$(\hat{{\mathfrak{g}}}_{2},[\cdot,\cdot]_{\hat{{\mathfrak{g}}}_{2}},{\hat{K}_{2}})$.
Denote $s_{2}:=\zeta\circ s_{1}$. Since $\zeta$ is a homomorphism of modified
Rota-Baxter Leibniz algebras such that $\zeta|_{V}=\mathrm{id}_{V}$,
$\zeta(\rho^{L}(x)m)=\zeta(\rho^{L}(s_{1}(x))m)=\rho^{L}(s_{2}(x))m=\rho^{L}(x)m$,
so $\zeta|_{V}:V\to V$ is compatible with the induced representation. We have

$\displaystyle\psi_{2}(x,y)$
$\displaystyle=[s_{2}(x),s_{2}(y)]_{\hat{{\mathfrak{g}}}_{2}}-s_{2}([x,y]_{{\mathfrak{g}}}$
$\displaystyle=[\zeta(s_{1}(x)),\zeta(s_{1}(y))]_{\hat{{\mathfrak{g}}}_{2}}-\zeta(s_{1}([x,y]_{{\mathfrak{g}}})$
$\displaystyle=\zeta([s_{1}(x),s_{1}(y)]_{\hat{{\mathfrak{g}}}_{1}}-s_{1}([x,y]_{{\mathfrak{g}}})$
$\displaystyle=\zeta(\psi_{1}(x,y))$ $\displaystyle=\psi_{1}(x,y)$

and

$\displaystyle\chi_{2}(x)$
$\displaystyle={\hat{K}_{2}}(s_{2}(x))-s_{2}(K(x))={\hat{K}_{2}}(\zeta(s_{1}(x)))-\zeta(s_{1}(K(x)))$
$\displaystyle=\zeta({\hat{K}_{1}}(s_{1}(x))-s_{1}(K(x)))=\zeta(\chi_{1}(x))$
$\displaystyle=\chi_{1}(x).$

Consequently, two isomorphic abelian extensions give rise to the same element
in ${\mathrm{H}}_{\mathsf{MLeib}}^{2}({\mathfrak{g}},V)$. ∎

Now we consider the reverse direction.

Let $(V,\rho^{L},\rho^{R},K_{V})$ be a representation over a modified Rota-
Baxter Leibniz algebra $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$,
given two linear maps $\psi:{\mathfrak{g}}\otimes{\mathfrak{g}}\rightarrow V$
and $\chi:{\mathfrak{g}}\rightarrow V$, one can define a Leibniz bracket
$[\cdot,\cdot]_{\psi}$ and an operator $K_{\chi}$ on ${\mathfrak{g}}\oplus V$
by Equations (15) and (16). The following fact is important:

######  Proposition 5.7.

The triple $({\mathfrak{g}}\oplus V,[\cdot,\cdot]_{\psi},K_{\chi})$ is a
modified Rota-Baxter Leibniz algebra if and only if $(\psi,\chi)$ is a
2-cocycle of the modified Rota-Baxter Leibniz algebra
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)$ with coefficients in
$(V,\rho^{L},\rho^{R},K_{V})$. In this case, we obtain an abelian extension

$0\to(V,\rho^{L},\rho^{R},K_{V})\stackrel{{\scriptstyle\left(\begin{array}[]{cc}0&\mathrm{id}\end{array}\right)}}{{\to}}({\mathfrak{g}}\oplus
V,K_{\chi})\stackrel{{\scriptstyle\left(\begin{array}[]{c}\mathrm{id}\\\
0\end{array}\right)}}{{\to}}({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},K)\to
0,$

and the canonical section
$s=\left(\begin{array}[]{cc}\mathrm{id}&0\end{array}\right):({\mathfrak{g}},K)\to({\mathfrak{g}}\oplus
V,K_{\chi})$ endows $V$ with the original representation.

###### Proof.

If $({\mathfrak{g}}\oplus V,[\cdot,\cdot]_{\psi},K_{\chi})$ is a modified
Rota-Baxter Leibniz algebra, then $[\cdot,\cdot]_{\psi}$ implies

(17)
$\displaystyle\psi(x,[y,z]_{\mathfrak{g}})+\rho^{L}(x)\psi(y,z)=\psi([x,y]_{\mathfrak{g}},z)+\rho^{L}(y)\psi(x,z)+\rho^{R}(z)\psi(x,y),$

which means $\delta^{2}(\psi)=0$ in
${\mathrm{C}}^{\bullet}({\mathfrak{g}},V)$. Since $K_{\chi}$ is a modified
Rota-Baxter operator, for any $x,y\in{\mathfrak{g}},u,v\in V$, we have

$\displaystyle[K_{\chi}(x+u),K_{\chi}(y+v)]_{\psi}=K_{\chi}\big{(}[K_{\chi}(x+u),y+v]_{\psi}+[x+u,K_{\chi}(y+v)]_{\psi}\big{)}+\lambda[x+u,y+v]_{\psi}.$

That is,

$\partial^{1}(\chi)+\Phi^{2}(\psi)=0.$

Hence, $(\psi,\chi)$ is a 2-cocycle.

Conversely, if $(\psi,\chi)$ is a 2-cocycle, one can easily check that
$({\mathfrak{g}}\oplus V,[\cdot,\cdot]_{\psi},K_{\chi})$ is a modified Rota-
Baxter Leibniz algebra.

The last statement is clear. ∎

Finally, we show the following result:

######  Proposition 5.8.

Two cohomologous $2$-cocyles give rise to isomorphic abelian extensions.

###### Proof.

Given two 2-cocycles $(\psi_{1},\chi_{1})$ and $(\psi_{2},\chi_{2})$, we can
construct two abelian extensions $({\mathfrak{g}}\oplus
V,[\cdot,\cdot]_{\psi_{1}},K_{\chi_{1}})$ and $({\mathfrak{g}}\oplus
V,[\cdot,\cdot]_{\psi_{2}},K_{\chi_{2}})$ via Equations (15) and (16). If they
represent the same cohomology class in
${\mathrm{H}}_{\mathsf{MLeib}}^{2}({\mathfrak{g}},V)$, then there exists two
linear maps $\gamma_{0}:k\rightarrow V,\gamma_{1}:{\mathfrak{g}}\to V$ such
that

$(\psi_{1},\chi_{1})=(\psi_{2},\chi_{2})+(\delta^{1}(\gamma_{1}),-\Phi^{1}(\gamma_{1})-\partial^{0}(\gamma_{0})).$

Notice that $\partial^{0}=\Phi^{1}\circ\delta^{0}$. Define
$\gamma:{\mathfrak{g}}\rightarrow V$ to be
$\gamma_{1}+\delta^{0}(\gamma_{0})$. Then $\gamma$ satisfies

$(\psi_{1},\chi_{1})=(\psi_{2},\chi_{2})+(\delta^{1}(\gamma),-\Phi^{1}(\gamma)).$

Define $\zeta:{\mathfrak{g}}\oplus V\rightarrow{\mathfrak{g}}\oplus V$ by

$\zeta(x+u):=x-\gamma(x)+u.$

Then $\zeta$ is an isomorphism of these two abelian extensions
$({\mathfrak{g}}\oplus V,[\cdot,\cdot]_{\psi_{1}},K_{\chi_{1}})$ and
$({\mathfrak{g}}\oplus N,[\cdot,\cdot]_{\psi_{2}},K_{\chi_{2}})$. ∎

Acknowledgments. The work is supported by Natural Science Foundation of China
(Grant Nos. 11871301, 12271292).

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