# Cohomologies, non-abelian extensions and Wells sequences of
$\lambda$-weighted Rota-Baxter Lie coalgebras

Qinxiu Sun Corresponding author:
Sun Qinxiu (qxsun@126.com), Li Zhen (lz200002272022@163.com). Department of
Mathematics, Zhejiang University of Science and Technology, Hangzhou, 310023.
Zhen Li  Department of Mathematics, Zhejiang University of Science and
Technology, Hangzhou, 310023.

###### Abstract

In this paper, we investigate cohomologies and non-abelian extensions of
$\lambda$-weighted Rota-Baxter Lie coalgebras. First, we consider Lie
comodules and cohomologies of $\lambda$-weighted Rota-Baxter Lie coalgebras.
Next, we study non-abelian extensions of $\lambda$-weighted Rota-Baxter Lie
coalgebras and classify the non-abelian extensions in terms of non-abelian
cohomology group. Furthermore, we explore extensibility of a pair of
automorphisms about a non-abelian extension of $\lambda$-weighted Rota-Baxter
Lie coalgebras, and derive the fundamental sequences of Wells in the context
of $\lambda$-weighted Rota-Baxter Lie coalgebras. Finally, we discuss the
previous results in the case of abelian extensions of $\lambda$-weighted Rota-
Baxter Lie coalgebras.

MR Subject Classification 2020: 16T15, 17B62, 17B56, 17B10.

111Keywords: Lie coalgebra, Lie comodule, $\lambda$-weighted Rota-Baxter Lie
coalgebras, non-abelian extension, extensibility, Wells sequence

##  1 Introduction

Rota-Baxter operators on associative algebras arose initially in Baxter’s
study of the fluctuation theory in probability [5], which can be viewed as an
algebraic abstraction of the integral operators. On the other hand, Rota-
Baxter operators on Lie algebras were first considered by Kuperscmidt in the
study of classical $r$-matrices [30]. Since then, Rota-Baxter operators have
been investigated in connection with many mathematical and physical branches,
including combinatorics [22], number theory [24], operads and quantum field
theory [9].

Rota-Baxter operators with arbitrary weight (also called weighted Rota-Baxter
operators) were considered in [1, 2]. They are related with weighted
infinitesimal bialgebras, weighted Yang-Baxter equations [17], combinatorics
of rooted forests [38], post-Lie algebras and modified Yang-Baxter equations
[1]. Recently, Tang, Bai, Guo and Sheng [34] developed the deformation and
cohomology theory of $\mathcal{O}$-operators (also called relative Rota-Baxter
operators) on Lie algebras, with applications to Rota-Baxter Lie algebras in
mind. Later, Das in [11, 12] investigated the cohomologies of Rota-Baxter
operators of arbitrary weights on associative algebras and Lie algebras. Wang
and Zhou in [36] explored cohomology and homotopy theories of Rota-Baxter
algebras with any weight. There are some other related work concerning
cohomologies of Rota-Baxter operators of any weight, see [7, 25] and there
references.

The notion of a coalgebra is dual to the notion of an algebra. The theory of
coassociative coalgebras has been developed for a long time within the
framework of the theory of Hopf algebras. Lie coalgebras were investigated
earlier in [10] by W. Michaelis. It is well known that the dual of a
coassociative coalgebra is an associative algebra and the dual of a Lie
coalgebra is a Lie algebra. Regarding applications, Lie coalgebras are not
only closely related with Lie bialgebras and quantum groups, but also appeared
in various fields such as homotopy theory [33], noncommutative geometry [20].
Specifically, a bialgebra structure for Rota-Baxter Lie algebras was studied
in [3]. In a recent study, cohomologies of Lie coalgebras were examined by Du
and Tan in [16]. But so far, cohomologies of $\lambda$-weighted Rota-Baxter
Lie coalgebras are still not given. This is our first motivation for writing
the present paper.

Extensions are useful mathematical objects to understand the underlying
structures. The non-abelian extension is a relatively general one among
various extensions (e.g. central extensions, abelian extensions, non-abelian
extensions etc.). Non-abelian extensions were first developed by Eilenberg and
Maclane [18], which induce to the low dimensional non-abelian cohomology
group. Then numerous works have been devoted to non-abelian extensions of
various kinds of algebras, such as Lie (super)algebras, Leibniz algebras, Lie
2-algebras, Lie Yagamuti algebras, Rota-Baxter groups, Rota-Baxter Lie
algebras and Rota-Baxter Leibniz algebras, see [6, 8, 14, 19, 21, 23, 27, 29,
31] and their references. The abelian extensions of Lie coalgebras were
explored in [15, 16] and characterized in terms of coderivations of Lie
coalgebras. But little is known about the non-abelian extension of Lie
coalgebras particularly $\lambda$-weighted Rota-Baxter Lie coalgebras. This is
the second motivation for writing the present paper.

Another interesting study related to extensions of algebraic structures is
given by the extensibility and inducibility of a pair of automorphisms. When a
pair of automorphisms is inducible? This problem was first considered by Wells
[37] for abstract groups and further studied in [28, 32]. Since then, several
authors have studied this subject further, see [4, 23, 26, 31] and references
therein. The extensibility problem of a pair of derivations in abelian
extensions were investigated in [13, 35]. Recently, the extensibility problem
of a pair of derivations and automorphisms was extended to the context of
abelian extensions of Lie coalgebras [15]. As byproducts, the Wells short
exact sequences were obtained for various kinds of algebras [14, 15, 21, 23,
26, 28, 31], which connected various automorphism groups and the non-abelian
second cohomology. Motivated by these results, we study extensibility of a
pair of automorphisms in a non-abelian extension of $\lambda$-weighted Rota-
Baxter Lie coalgebras. This is another motivation for writing the present
paper. We give a necessary and sufficient condition for a pair of
automorphisms to be extensible. We also derive the analogue of the Wells short
exact sequences in the context of non-abelian extensions of $\lambda$-weighted
Rota-Baxter Lie coalgebras.

The paper is organized as follows. In Section 2, we introduce Lie comodules of
$\lambda$-weighted Rota-Baxter Lie coalgebras. In Section 3, we consider
cohomologies of $\lambda$-weighted Rota-Baxter Lie coalgebras. In Section 4,
we investigate non-abelian extensions and classify the non-abelian extensions
using non-abelian 2-cocycles. In Section 5, we study the problem of when a
pair of automorphisms is extensible. We address the necessary and sufficient
condition for a pair of automorphisms to be extensible. In Section 6, we build
Wells short exact sequences in the context of non-abelian extensions of
$\lambda$-weighted Rota-Baxter Lie coalgebras. Finally, we discuss these
results in the case of abelian extensions.

Throughout the paper, let $k$ be a field. Unless otherwise specified, all
vector spaces and algebras are finite dimensional over $k$.

##  2 Comodules of $\lambda$-weighted Rota-Baxter Lie coalgebras

In this section, we introduce the Lie comodules of $\lambda$-weighted Rota-
Baxter Lie coalgebras. We begin with recalling definitions of Lie coalgebras
and $\lambda$-weighted Rota-Baxter Lie coalgebras [3, 10].

######  Definition 2.1.

  1. $(i)$

A Lie coalgebra is a vector space $C$ together with a linear map
$\Delta_{C}:C\rightarrow C\otimes C$ satisfying

$\Delta_{C}=-\tau\Delta_{C},$ (2.1)
$(I\otimes\Delta_{C})\Delta_{C}-(\Delta_{C}\otimes
I)\Delta_{C}+(I\otimes\tau)(\Delta_{C}\otimes I)\Delta_{C}=0.$ (2.2)

  2. $(ii)$

Let $(C,\Delta_{C})$ be a Lie coalgebra. A linear operator $R_{C}:C\rightarrow
C$ is called a $\lambda$-weighted $(\lambda\in k)$ Rota-Baxter operator if

$\displaystyle(R_{C}\otimes R_{C})\Delta_{C}=(I\otimes R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C}R_{C}.$ (2.3)

Moreover, a Lie coalgebra $(C,\Delta_{C})$ with a $\lambda$-weighted Rota-
Baxter operator $R_{C}$ is called a $\lambda$-weighted Rota-Baxter Lie
coalgebra (Rota-Baxter Lie coalgebra of weight $\lambda$).

######  Definition 2.2.

Let $(C,\Delta_{C},R_{C})$ and
$(C^{\prime},\Delta_{C^{\prime}},R_{C^{\prime}})$ be two $\lambda$-weighted
Rota-Baxter Lie coalgebras. A homomorphism from $(C,\Delta_{C},R_{C})$ to
$(C^{\prime},\Delta_{C^{\prime}},R_{C^{\prime}})$ is a Lie coalgebra
homomorphism $\varphi:C\rightarrow C^{\prime}$ satisfying
$R_{C^{\prime}}\varphi=\varphi R_{C}$.

Let $(C,\Delta_{C},R_{C})$ be a $\lambda$-weighted Rota-Baxter Lie coalgebra.
Denote the set of all automorphisms of $(C,\Delta_{C},R_{C})$ by
$\mathrm{Aut}(C)$. Obviously, $\mathrm{Aut}(C)$ is a group. A right Lie
comodule (corepresentation) of a Lie coalgebra $(C,\Delta_{C})$ is a tuple
$(M,\rho)$, where $M$ is a vector space and $\rho:M\rightarrow M\otimes C$ is
a linear map satisfying

$\displaystyle(I\otimes\Delta_{C})\rho-(\rho\otimes
I)\rho+(I\otimes\tau)(\rho\otimes I)\rho=0.$ (2.4)

######  Definition 2.3.

Let $(C,\Delta_{C},R_{C})$ be a $\lambda$-weighted Rota-Baxter Lie coalgebra.
A right Lie comodule of $(C,\Delta_{C},R_{C})$ is a triple $(M,\rho,R_{M})$,
where $(M,\rho)$ is a right Lie comodule of $(C,\Delta_{C})$ and
$R_{M}:M\rightarrow M$ is a linear map such that

$\displaystyle(R_{M}\otimes R_{C})\rho=(R_{M}\otimes I+I\otimes
R_{C}+\lambda)\rho R_{M}.$ (2.5)

######  Example 2.1.

Any $\lambda$-weighted Rota-Baxter Lie coalgebra is a right Lie comodule of
itself, which is called the adjoint corepresentation.

######  Proposition 2.1.

Let $(C,\Delta_{C},R_{C})$ be a $\lambda$-weighted Rota-Baxter Lie coalgebra.
Then $(C^{*},[\ ,\ ]_{C^{*}},R_{C}^{*})$ is a $\lambda$-weighted Rota-Baxter
Lie algebra.

###### Proof.

It is well known that $C^{*}$ is a Lie algebra with the multiplication given
by

$\displaystyle[f,g]_{C^{*}}=(f\otimes g)\Delta_{C},~{}\forall~{}f,g\in C^{*}.$

We only need to check that $R_{C}^{*}$ is a $\lambda$-weighted Rota-Baxter
operator. Define $R_{C}^{*}(f)=fR_{C},~{}\forall~{}f\in C^{*}$. In view of
(2.3), for all $f,g\in C^{*}$, we have

$\displaystyle[R_{C}^{*}(f),R_{C}^{*}(g)]_{C^{*}}$ $\displaystyle=$
$\displaystyle[fR_{C},gR_{C}]_{C^{*}}$ $\displaystyle=$
$\displaystyle(fR_{C}\otimes gR_{C})\Delta_{C}$ $\displaystyle=$
$\displaystyle(f\otimes g)(R_{C}\otimes I+I\otimes
R_{C}+\lambda)\Delta_{C}R_{C}$ $\displaystyle=$ $\displaystyle(fR_{C}\otimes
g)\Delta_{C}R_{C}+(f\otimes gR_{C})\Delta_{C}R_{C}+\lambda(f\otimes
g)\Delta_{C}R_{C}$ $\displaystyle=$
$\displaystyle[fR_{C},g]_{C^{*}}R_{C}+[f,gR_{C}]_{C^{*}}R_{C}+\lambda[f,g]_{C^{*}}R_{C}$
$\displaystyle=$
$\displaystyle[R_{C}^{*}(f),g]_{C^{*}}R_{C}+[f,R_{C}^{*}(g)]_{C^{*}}R_{C}+\lambda[f,g]_{C^{*}}R_{C}$
$\displaystyle=$ $\displaystyle
R_{C}^{*}[R_{C}^{*}(f),g]_{C^{*}}+R_{C}^{*}[f,R_{C}^{*}(g)]_{C^{*}}+\lambda
R_{C}^{*}[f,g]_{C^{*}}$ $\displaystyle=$ $\displaystyle
R_{C}^{*}\big{(}[R_{C}^{*}(f),g]_{C^{*}}+[f,R_{C}^{*}(g)]_{C^{*}}+\lambda[f,g]_{C^{*}}\big{)}.$

This finishes the proof. ∎

######  Proposition 2.2.

Let $(C,\Delta_{C},R_{C})$ be a $\lambda$-weighted Rota-Baxter Lie coalgebra
and $(M,\rho,R_{M})$ a right Lie comodule of $(C,\Delta_{C},R_{C})$. Define
two linear maps $\tilde{\Delta}_{C}:C\rightarrow C\otimes C$ and
$\tilde{\rho}:M\rightarrow M\otimes C$ respectively by

$\displaystyle\tilde{\Delta}_{C}=(I\otimes R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C},$ (2.6) $\displaystyle\tilde{\rho}=(I\otimes
R_{C})\rho-\rho R_{M}.$ (2.7)

Then

  1. $(i)$

$(C,\tilde{\Delta}_{C},R_{C})$ is a $\lambda$-weighted Rota-Baxter Lie
coalgebra.

  2. $(ii)$

$(M,\tilde{\rho},R_{M})$ is a right Lie comodule of
$(C,\tilde{\Delta}_{C},R_{C})$.

###### Proof.

  1. $(i)$

It is obviously that (2.1) holds. We only need to check Eqs. (2.2) and (2.3)
hold for $(\tilde{\Delta}_{C},R_{C})$. In the light of Eqs. (2.6) and (2.2),

$\displaystyle(I\otimes\tilde{\Delta}_{C})\tilde{\Delta}_{C}$ $\displaystyle=$
$\displaystyle(I\otimes(I\otimes R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C})(I\otimes R_{C}+R_{C}\otimes I+\lambda)\Delta_{C}$
$\displaystyle=$ $\displaystyle(I\otimes R_{C}\otimes R_{C}+R_{C}\otimes
I\otimes R_{C}+R_{C}\otimes R_{C}\otimes R_{C}+\lambda R_{C}\otimes I\otimes
I$ $\displaystyle+\lambda I\otimes I\otimes R_{C}+I\otimes R_{C}\otimes
I+\lambda)(I\otimes\Delta_{C})\Delta_{C},$
$\displaystyle(\tilde{\Delta}_{C}\otimes I)\tilde{\Delta}_{C}$
$\displaystyle=$ $\displaystyle((I\otimes R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C}\otimes I)(I\otimes R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C}$ $\displaystyle=$ $\displaystyle(I\otimes R_{C}\otimes
R_{C}+R_{C}\otimes I\otimes R_{C}+R_{C}\otimes R_{C}\otimes I+\lambda I\otimes
I\otimes R_{C}$ $\displaystyle+\lambda R_{C}\otimes I\otimes I+\lambda
I\otimes R_{C}\otimes I+\lambda)(\Delta_{C}\otimes I)\Delta_{C},$
$\displaystyle(I\otimes\tau)(\tilde{\Delta}_{C}\otimes I)\tilde{\Delta}_{C}$
$\displaystyle=$ $\displaystyle(I\otimes\tau)(I\otimes R_{C}\otimes
R_{C}+R_{C}\otimes I\otimes R_{C}+R_{C}\otimes R_{C}\otimes I+\lambda I\otimes
I\otimes R_{C}$ $\displaystyle+\lambda R_{C}\otimes I\otimes I+\lambda
I\otimes R_{C}\otimes I+\lambda)(\Delta_{C}\otimes I)\Delta_{C}$
$\displaystyle=$ $\displaystyle(I\otimes R_{C}\otimes R_{C}+R_{C}\otimes
I\otimes R_{C}+R_{C}\otimes R_{C}\otimes I+\lambda I\otimes I\otimes R_{C}$
$\displaystyle+\lambda R_{C}\otimes I\otimes I+\lambda I\otimes R_{C}\otimes
I+\lambda)(I\otimes\tau)(\Delta_{C}\otimes I)\Delta_{C},$

which indicate that

$\displaystyle(I\otimes\tilde{\Delta}_{C})\tilde{\Delta}_{C}-(\tilde{\Delta}_{C}\otimes
I)\tilde{\Delta}_{C}+(I\otimes\tau)(\tilde{\Delta}_{C}\otimes
I)\tilde{\Delta}_{C}$ $\displaystyle=$ $\displaystyle(I\otimes R_{C}\otimes
R_{C}+R_{C}\otimes I\otimes R_{C}+R_{C}\otimes R_{C}\otimes I+\lambda I\otimes
I\otimes R_{C}$ $\displaystyle+\lambda R_{C}\otimes I\otimes I+\lambda
I\otimes R_{C}\otimes I+\lambda)$
$\displaystyle\times((I\otimes\Delta_{C})\Delta_{C}-(\Delta_{C}\otimes
I)\Delta_{C}+(I\otimes\tau)(\Delta_{C}\otimes I)\Delta_{C})$
$\displaystyle=0.$

By Eqs. (2.3) and (2.6), we can directly calculate

$\displaystyle(R_{C}\otimes R_{C})\tilde{\Delta}_{C}$ $\displaystyle=$
$\displaystyle(R_{C}\otimes R_{C})(I\otimes R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C}$ $\displaystyle=$ $\displaystyle(I\otimes
R_{C}+R_{C}\otimes I+\lambda)(R_{C}\otimes R_{C})\Delta_{C}$ $\displaystyle=$
$\displaystyle(I\otimes R_{C}+R_{C}\otimes I+\lambda)(I\otimes
R_{C}+R_{C}\otimes I+\lambda)\Delta_{C}R_{C}$ $\displaystyle=$
$\displaystyle(I\otimes R_{C}+R_{C}\otimes I+\lambda)\tilde{\Delta}_{C}R_{C}.$

Thus, $(C,\tilde{\Delta}_{C},R_{C})$ is a $\lambda$-weighted Rota-Baxter Lie
coalgebra.

  2. $(ii)$

Take the same procedure as the proof of $(i)$.

∎

##  3 Cohomologies of $\lambda$-weighted Rota-Baxter Lie coalgebras

In this section, we are devoted to studying cohomologies of $\lambda$-weighted
Rota-Baxter Lie coalgebras.

In the following, we denote

$a\otimes^{(1)}(b_{1}\otimes b_{2}\otimes b_{3})=a\otimes b_{1}\otimes
b_{2}\otimes b_{3},$ $b_{1}\otimes a\otimes b_{2}\otimes
b_{3}=a\otimes^{(2)}(b_{1}\otimes b_{2}\otimes b_{3}),$
$a\otimes^{(k)}(b_{1}\otimes\cdots\otimes b_{k}\otimes\cdots\otimes
b_{n-1})=b_{1}\otimes\cdots\otimes a\otimes b_{k}\otimes\cdots\otimes
b_{n-1},$ $b_{1}\otimes\cdots\otimes b_{n-1}\otimes
a=a\otimes^{(n)}(b_{1}\otimes\cdots\otimes b_{n-1}),$ $(a_{1}\otimes
a_{2})\otimes^{(2)}(b_{1}\otimes b_{2}\otimes b_{3})=b_{1}\otimes(a_{1}\otimes
a_{2})\otimes b_{2}\otimes b_{3},$

for all $a,a_{1},a_{2},b_{i}\in C~{}(i=1,2\cdots)$.

At first, we recall cohomologies of Lie coalgebras studied in [16].

Let $(C,\Delta_{C})$ be a Lie coalgebra. Suppose that $(M,\rho)$ is a right
Lie comodule of $(C,\Delta_{C})$. Denote the set of $n$-cochains ($n\geq 0$)
by $C^{n}(M,C)$, where

$C^{n}(M,C)=\mathrm{Hom}(M,\wedge^{n}C),~{}n\geq 0,$

and the coboundary operator $\partial^{n}:C^{n}(M,C)\rightarrow C^{n+1}(M,C)$
is given by, for any $h\in C^{n}(M,C)$,

$\displaystyle\partial^{0}(h)=(h\otimes I)\rho,$ (3.1)
$\displaystyle\partial^{n}(h)=\frac{1}{2}\sum_{k=1}^{n}(-1)^{k}\mathrm{Alt}(\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})h+(-1)^{n-1}\mathrm{Alt}(h\otimes
I)\rho,\ n\geq 1,$ (3.2)

where $\mathrm{Alt}:\otimes^{n}C\longrightarrow\otimes^{n}C$ is given by

$\mathrm{Alt}(c_{1}\otimes\cdots\otimes
c_{n})=\frac{1}{n!}\sum_{\sigma\in\mathfrak{S}_{n}}\mathrm{sgn}(\sigma)\psi_{\sigma}(c_{1}\otimes\cdots\otimes
c_{n}),~{}\forall~{}c_{i}\in C.$

Denote the set of all n-cocycles and n-coboundaries respectively by
$\mathrm{Z}^{n}(M,C)$ and $\mathrm{B}^{n}(M,C)$. Define
$\mathrm{H}^{n}(M,C)=\mathrm{Z}^{n}(M,C)/\mathrm{B}^{n}(M,C),$ which is called
the n-cohomology group of $(C,\Delta_{C})$ with coefficients in $(M,\rho)$.

Moreover, let $(C,\Delta_{C},R_{C})$ be a $\lambda$-weighted Rota-Baxter Lie
coalgebra and $(M,\rho,R_{M})$ a right Lie comodule of it. Proposition 2.2
indicates that $(M,\tilde{\rho},R_{M})$ is a right Lie comodule of the
$\lambda$-weighted Rota-Baxter Lie coalgebra $(C,\tilde{\Delta}_{C},R_{C})$.
Consider the cohomology of $(C,\tilde{\Delta}_{C})$ with coefficients in
$(M,\tilde{\rho})$. Denote the set of $n$-cochains by

$\tilde{C}^{n}(M,C)=\mathrm{Hom}(M,\wedge^{n}C),$

and a coboundary map
$\tilde{\partial}^{n}:\tilde{C}^{n}(M,C)\rightarrow\tilde{C}^{n+1}(M,C)$ given
by, for any $h\in\tilde{C}^{n}(M,C)$,

$\displaystyle\tilde{\partial}^{0}(h)=$ $\displaystyle(h\otimes
I)\tilde{\rho}=(h\otimes R_{C})\rho-(h\otimes I)\rho R_{M},$ (3.3)
$\displaystyle\tilde{\partial}^{n}(h)=$
$\displaystyle\frac{1}{2}\sum_{k=1}^{n}(-1)^{k}\mathrm{Alt}((I\otimes
R_{C}+R_{C}\otimes I+\lambda)\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})h$
$\displaystyle+(-1)^{n-1}\mathrm{Alt}(h\otimes
R_{C})\rho-(-1)^{n-1}\mathrm{Alt}(h\otimes I)\rho R_{M},\ n\geq 1.$ (3.4)

Then $\\{\tilde{C}^{*}(M,C),\tilde{\partial}^{*}\\}$ is a cochain complex. The
corresponding n-cohomology group is

$\displaystyle\mathrm{\tilde{H}}^{n}(M,C)=\mathrm{\tilde{Z}}^{n}(M,C)/\mathrm{\tilde{B}}^{n}(M,C),$

where
$\mathrm{\tilde{Z}}^{n}(M,C)=\mathrm{Ker}(\tilde{\partial}^{n}),~{}\mathrm{\tilde{B}}^{n}(M,C)=\mathrm{Im}(\tilde{\partial}^{n-1}).$

Denote

$R_{C}^{(i)_{n}}=\underbrace{(I\otimes\cdots\otimes R_{C}\otimes\cdots\otimes
R_{C}\otimes\cdots\otimes I)}_{\hbox{where $R_{C}$ appears i
times}},~{}\hbox{we sum up over all possible variants}.$

######  Proposition 3.1.

For all $c,c_{1},c_{2},c_{i}\in C$, we have

  1. $(i)$

$\mathrm{Alt}(c)=c,~{}\mathrm{Alt}(c_{1}\otimes
c_{2})=\frac{1}{2}(c_{1}\otimes c_{2}-c_{2}\otimes
c_{1}),~{}\mathrm{Alt}(\Delta_{C})=\Delta_{C}.$

  2. $(ii)$

$\mathrm{Alt}\big{(}(R_{C}\otimes R_{C})(c_{1}\otimes
c_{2})\big{)}=(R_{C}\otimes R_{C})\mathrm{Alt}(c_{1}\otimes
c_{2})=\frac{1}{2}((R_{C}\otimes R_{C})(c_{1}\otimes c_{2})-(R_{C}\otimes
R_{C})(c_{2}\otimes c_{1}))$.

  3. $(iii)$

$\mathrm{Alt}((R_{C}\otimes I+I\otimes R_{C})(c_{1}\otimes
c_{2}))=\frac{1}{2}\big{(}(R_{C}\otimes I)(c_{1}\otimes c_{2})-(I\otimes
R_{C})(c_{2}\otimes c_{1})+(I\otimes R_{C})(c_{1}\otimes c_{2})-(R_{C}\otimes
I)(c_{2}\otimes c_{1})\big{)}=(R_{C}\otimes I+I\otimes
R_{C})\mathrm{Alt}(c_{1}\otimes c_{2})$.

  4. $(iv)$

$\mathrm{Alt}(R_{C}^{\otimes n}(c_{1}\otimes\cdots\otimes
c_{n}))=R_{C}^{\otimes n}\mathrm{Alt}(c_{1}\otimes\cdots\otimes c_{n})$.

  5. $(v)$

$\mathrm{Alt}(R^{(i)_{n}}_{C}(c_{1}\otimes\cdots\otimes
c_{n}))=R^{(i)_{n}}_{C}\mathrm{Alt}(c_{1}\otimes\cdots\otimes c_{n})$.

###### Proof.

Items $(i)$-$(iv)$ can be obtained easily.

$(v)$ For any $c_{i}\in C~{}(i=1,2,3\cdots)$, due to
$\psi_{\sigma}(R^{(i)_{n}}_{C})=R^{(i)_{n}}_{C}$,

$\displaystyle\mathrm{Alt}(R^{(i)_{n}}_{C}(c_{1}\otimes\cdots\otimes c_{n}))$
$\displaystyle=$
$\displaystyle\frac{1}{n!}\sum_{\sigma}\mathrm{sgn}(\sigma)\psi_{\sigma}(R^{(i)_{n}}_{C}(c_{1}\otimes\cdots\otimes
c_{n}))$ $\displaystyle=$
$\displaystyle\frac{1}{n!}\sum_{\sigma}\mathrm{sgn}(\sigma)\psi_{\sigma}(R^{(i)_{n}}_{C})\psi_{\sigma}(c_{1}\otimes\cdots\otimes
c_{n})$ $\displaystyle=$
$\displaystyle\frac{1}{n!}\sum_{\sigma}\mathrm{sgn}(\sigma)R^{(i)_{n}}_{C}\psi_{\sigma}(c_{1}\otimes\cdots\otimes
c_{n})$ $\displaystyle=$ $\displaystyle
R^{(i)_{n}}_{C}\mathrm{Alt}(c_{1}\otimes\cdots\otimes c_{n}).$

∎

In the following, we characterize the relationship between the two cochain
complexes $\\{C^{*}(M,C),\partial^{*}\\}$ and
$\\{\tilde{C}^{*}(M,C),\tilde{\partial}^{*}\\}$.

######  Proposition 3.2.

The collection of maps
$\\{\delta^{n}:C^{n}(M,C)\rightarrow\tilde{C}^{n}(M,C)\\}_{n\geq 0}$ defined
by, for any $h\in C^{n}(M,C)$,

$\displaystyle\delta^{0}(h)=h,$ (3.5)
$\displaystyle\delta^{n}(h)=R_{C}^{\otimes
n}h-\sum^{n-1}_{i=0}\lambda^{n-i-1}R_{C}^{(i)_{n}}hR_{M},$ (3.6)

is a homomorphism of cochain complexes from $\\{C^{*}(M,C),\partial^{*}\\}$ to
$\\{\tilde{C}^{*}(M,C),\tilde{\partial}^{*}\\}$, that is,

$\displaystyle\delta^{n+1}\partial^{n}=\tilde{\partial}^{n}\delta^{n}.$

###### Proof.

Using Eqs. (3.2), (3.4) and (3.6), for all $m^{*}\in M^{*}$,

$\displaystyle\tilde{\partial}^{0}\delta^{0}(m^{*})(m)-\delta^{1}\partial^{0}(m^{*})(m)$
$\displaystyle=$
$\displaystyle\tilde{\partial}^{0}(m^{*})(m)-\delta^{1}((m^{*}\otimes
I)\rho)(m)$ $\displaystyle=$ $\displaystyle(m^{*}\otimes
R_{C})\rho(m)-(m^{*}\otimes I)\rho R_{M}(m)-R_{C}(m^{*}\otimes
I)\rho(m)+(m^{*}\otimes I)\rho R_{M}(m)$ $\displaystyle=$ $\displaystyle
m^{*}(m_{0})R_{C}(m_{1})-m^{*}(m_{0})R_{C}(m_{1})$ $\displaystyle=$
$\displaystyle 0,$

and by (3.1), for any $f\in C^{1}(M,C)$, we have

$\displaystyle\tilde{\partial}^{1}\delta^{1}(f)-\delta^{2}\partial^{1}(f)$
$\displaystyle=$
$\displaystyle\tilde{\partial}^{1}(R_{C}f-fR_{M})-\delta^{2}(-\frac{1}{2}\mathrm{Alt}\Delta_{C}f+\mathrm{Alt}(f\otimes
I)\rho)$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\mathrm{Alt}(I\otimes
R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C}(R_{C}f-fR_{M})+\mathrm{Alt}((R_{C}f-fR_{M})\otimes
R_{C})\rho$ $\displaystyle-\mathrm{Alt}((R_{C}f-fR_{M})\otimes I)\rho
R_{M}-(R_{C}\otimes
R_{C})(-\frac{1}{2}\mathrm{Alt}\Delta_{C}f+\mathrm{Alt}(f\otimes I)\rho)$
$\displaystyle+(R_{C}\otimes I+I\otimes
R_{C}+\lambda)(-\frac{1}{2}\mathrm{Alt}\Delta_{C}f+\mathrm{Alt}(f\otimes
I)\rho)R_{M}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\mathrm{Alt}(R_{C}\otimes
R_{C})\Delta_{C}f+\frac{1}{2}\mathrm{Alt}(I\otimes R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C}fR_{M}+\mathrm{Alt}(R_{C}f\otimes R_{C})\rho$
$\displaystyle-\mathrm{Alt}(fR_{M}\otimes
R_{C})\rho-\mathrm{Alt}(R_{C}f\otimes I)\rho R_{M}+\mathrm{Alt}(fR_{M}\otimes
I)\rho R_{M}$ $\displaystyle+\frac{1}{2}(R_{C}\otimes
R_{C})\mathrm{Alt}\Delta_{C}f-(R_{C}\otimes R_{C})\mathrm{Alt}(f\otimes
I)\rho$ $\displaystyle-\frac{1}{2}(R_{C}\otimes I+I\otimes
R_{C}+\lambda)\mathrm{Alt}\Delta_{C}fR_{M}+(R_{C}\otimes I+I\otimes
R_{C}+\lambda)\mathrm{Alt}(f\otimes I)\rho R_{M}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(R_{C}\otimes
R_{C})\mathrm{Alt}\Delta_{C}f+\frac{1}{2}(I\otimes R_{C}+R_{C}\otimes
I+\lambda)\mathrm{Alt}\Delta_{C}fR_{M}$ $\displaystyle+(R_{C}\otimes
R_{C})\mathrm{Alt}(f\otimes I)\rho-\mathrm{Alt}(I\otimes R_{C}+R_{C}\otimes
I+\lambda)(f\otimes I)\rho R_{M}$ $\displaystyle+\frac{1}{2}(R_{C}\otimes
R_{C})\mathrm{Alt}\Delta_{C}f-(R_{C}\otimes R_{C})\mathrm{Alt}(f\otimes
I)\rho$ $\displaystyle-\frac{1}{2}(R_{C}\otimes I+I\otimes
R_{C}+\lambda)\mathrm{Alt}\Delta_{C}fR_{M}+(R_{C}\otimes I+I\otimes
R_{C}+\lambda)\mathrm{Alt}(f\otimes I)\rho R_{M}$ $\displaystyle=0.$

For each $h\in C^{n}(M,C)~{}(n\geq 2)$, according to Proposition 3.1,

$\displaystyle\delta^{n+1}\partial^{n}(h)$ $\displaystyle=$ $\displaystyle
R_{C}^{\otimes(n+1)}\partial^{n}h-\sum^{n}_{i=0}\lambda^{n-i}R_{C}^{(i)_{n+1}}\partial^{n}hR_{M}$
$\displaystyle=$ $\displaystyle
R_{C}^{\otimes(n+1)}(\frac{1}{2}\sum_{k=1}^{n}(-1)^{k}\mathrm{Alt}(\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})h+(-1)^{n-1}\mathrm{Alt}(h\otimes
I)\rho)$
$\displaystyle-\sum^{n}_{i=0}\lambda^{n-i}R_{C}^{(i)_{n+1}}(\frac{1}{2}\sum_{k=1}^{n}(-1)^{k}\mathrm{Alt}(\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})h+(-1)^{n-1}\mathrm{Alt}(h\otimes
I)\rho)R_{M}$ $\displaystyle=$
$\displaystyle\underset{A_{1}}{\underbrace{\frac{1}{2}\sum_{k=1}^{n}(-1)^{k}\mathrm{Alt}R_{C}^{\otimes(n+1)}(\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})h}}+\underset{A_{2}}{\underbrace{(-1)^{n-1}\mathrm{Alt}R_{C}^{\otimes(n+1)}(h\otimes
I)\rho}}$
$\displaystyle\underset{A_{3}}{\underbrace{-\frac{1}{2}\sum_{k=1}^{n}\sum^{n}_{i=0}(-1)^{k}\lambda^{n-i}\mathrm{Alt}R_{C}^{(i)_{n+1}}(\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})hR_{M}}}$
$\displaystyle\underset{A_{4}}{\underbrace{-(-1)^{n-1}\sum^{n}_{i=0}\lambda^{n-i}\mathrm{Alt}R_{C}^{(i)_{n+1}}(h\otimes
I)\rho R_{M}}},$

and

$\displaystyle\tilde{\partial}^{n}\delta^{n}(h)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{k=1}^{n}(-1)^{k}\mathrm{Alt}((I\otimes
R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})\delta^{n}h$
$\displaystyle+(-1)^{n-1}\mathrm{Alt}(\delta^{n}h\otimes
R_{C})\rho-(-1)^{n-1}\mathrm{Alt}(\delta^{n}h\otimes I)\rho R_{M}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{k=1}^{n}(-1)^{k}\mathrm{Alt}((I\otimes
R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})(R_{C}^{\otimes
n}h-\sum^{n-1}_{i=0}\lambda^{n-i-1}R_{C}^{(i)_{n}}hR_{M})$
$\displaystyle+(-1)^{n-1}\mathrm{Alt}((R_{C}^{\otimes
n}h-\sum^{n-1}_{i=0}\lambda^{n-i-1}R_{C}^{(i)_{n}}hR_{M})\otimes R_{C})\rho$
$\displaystyle-(-1)^{n-1}\mathrm{Alt}((R_{C}^{\otimes
n}h-\sum^{n-1}_{i=0}\lambda^{n-i-1}R_{C}^{(i)_{n}}hR_{M})\otimes I)\rho R_{M}$
$\displaystyle=$
$\displaystyle\underset{B_{1}}{\underbrace{\frac{1}{2}\sum_{k=1}^{n}(-1)^{k}\mathrm{Alt}R_{C}^{\otimes(n+1)}(\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})h}}$
$\displaystyle\underset{B_{2}}{\underbrace{-\frac{1}{2}\sum_{k=1}^{n}\sum^{n-1}_{i=0}(-1)^{k}\lambda^{n-i-1}\mathrm{Alt}((I\otimes
R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})R_{C}^{(i)_{n}}hR_{M}}}$
$\displaystyle\underset{B_{3}}{\underbrace{-(-1)^{n-1}\sum^{n-1}_{i=0}\lambda^{n-i-1}\mathrm{Alt}(R_{C}^{(i)_{n}}hR_{M}\otimes
R_{C})\rho}}$
$\displaystyle\underset{B_{4}}{\underbrace{+(-1)^{n-1}\sum^{n-1}_{i=0}\lambda^{n-i-1}\mathrm{Alt}(R_{C}^{(i)_{n}}hR_{M}\otimes
I)\rho R_{M}}}$
$\displaystyle+\underset{B_{5}}{\underbrace{(-1)^{n-1}\mathrm{Alt}(R_{C}^{\otimes
n}h\otimes
R_{C})\rho}}\underset{B_{6}}{\underbrace{-(-1)^{n-1}\mathrm{Alt}(R_{C}^{\otimes
n}h\otimes I)\rho R_{M}}}.$

We only need to check that

$\displaystyle A_{1}+A_{2}+A_{3}+A_{4}=B_{1}+B_{2}+B_{3}+B_{4}+B_{5}+B_{6}.$

Since $A_{1}=B_{1}$ and $A_{2}=B_{5}$, we only need to prove respectively

$\displaystyle A_{3}$ $\displaystyle=B_{2},$ $\displaystyle A_{4}$
$\displaystyle=B_{3}+B_{4}+B_{6}.$

Indeed, using (2.3), we can calculate directly

$\displaystyle A_{3}=$
$\displaystyle-\frac{1}{2}\sum_{k=1}^{n}\sum^{n-1}_{i=0}(-1)^{k}\lambda^{n-i}\mathrm{Alt}((I\otimes
I)\otimes^{(k)}R_{C}^{(i)_{n-1}})(\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})hR_{M}$
$\displaystyle-\frac{1}{2}\sum_{k=1}^{n}\sum^{n-1}_{i=0}(-1)^{k}\lambda^{n-i-1}\mathrm{Alt}((R_{C}\otimes
I)\otimes^{(k)}R_{C}^{(i)_{n-1}})(\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})hR_{M}$
$\displaystyle-\frac{1}{2}\sum_{k=1}^{n}\sum^{n-1}_{i=0}(-1)^{k}\lambda^{n-i-1}\mathrm{Alt}((I\otimes
R_{C})\otimes^{(k)}R_{C}^{(i)_{n-1}})(\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})hR_{M}$
$\displaystyle-\frac{1}{2}\sum_{k=1}^{n}\sum^{n-2}_{i=0}(-1)^{k}\lambda^{n-i-2}\mathrm{Alt}((R_{C}\otimes
R_{C})\otimes^{(k)}R_{C}^{(i)_{n-1}})(\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})hR_{M}$
$\displaystyle=$
$\displaystyle\underset{A_{31}}{\underbrace{-\frac{1}{2}\sum_{k=1}^{n}\sum^{n-1}_{i=0}(-1)^{k}\lambda^{n-i}\mathrm{Alt}((I\otimes
I)\otimes^{(k)}R_{C}^{(i)_{n-1}})(\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})hR_{M}}}$
$\displaystyle\underset{A_{32}}{\underbrace{-\frac{1}{2}\sum_{k=1}^{n}\sum^{n-1}_{i=0}(-1)^{k}\lambda^{n-i-1}\mathrm{Alt}((R_{C}\otimes
I)\otimes^{(k)}R_{C}^{(i)_{n-1}})(\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})hR_{M}}}$
$\displaystyle\underset{A_{33}}{\underbrace{-\frac{1}{2}\sum_{k=1}^{n}\sum^{n-1}_{i=0}(-1)^{k}\lambda^{n-i-1}\mathrm{Alt}((I\otimes
R_{C})\otimes^{(k)}R_{C}^{(i)_{n-1}})(\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})hR_{M}}}$
$\displaystyle\underset{A_{34}}{\underbrace{-\frac{1}{2}\sum_{k=1}^{n}\sum^{n-2}_{i=0}(-1)^{k}\lambda^{n-i-2}\mathrm{Alt}((R_{C}\otimes
I+I\otimes
R_{C}+\lambda)\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})(R_{C}\otimes^{(k)}R_{C}^{(i)_{n-1}})hR_{M}}},$

and

$\displaystyle B_{2}=$
$\displaystyle-\frac{1}{2}\sum_{k=1}^{n}\sum^{n-1}_{i=0}(-1)^{k}\lambda^{n-i-1}\mathrm{Alt}((I\otimes
R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})R_{C}^{(i)_{n}}hR_{M}$
$\displaystyle=$
$\displaystyle\underset{B_{21}}{\underbrace{-\frac{1}{2}\sum_{k=1}^{n}\sum^{n-1}_{i=0}(-1)^{k}\lambda^{n-i-1}\mathrm{Alt}((I\otimes
R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})(I\otimes^{(k)}R_{C}^{(i)_{n}})hR_{M}}}$
$\displaystyle\underset{B_{22}}{\underbrace{-\frac{1}{2}\sum_{k=1}^{n}\sum^{n-2}_{i=0}(-1)^{k}\lambda^{n-i-2}\mathrm{Alt}((I\otimes
R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C}\otimes^{(k)}I^{\otimes(n-1)})(R_{C}\otimes^{(k)}R_{C}^{(i)_{n}})hR_{M}}}.$

Thus, $A_{31}+A_{32}+A_{33}=B_{21}$ and $A_{34}=B_{22}$, that is,
$A_{3}=B_{2}$.

According to (2.5),

$\displaystyle A_{4}=$
$\displaystyle-(-1)^{n-1}\sum^{n}_{i=0}\lambda^{n-i}\mathrm{Alt}R_{C}^{(i)_{n+1}}(h\otimes
I)\rho R_{M}$ $\displaystyle=$
$\displaystyle-(-1)^{n-1}\sum^{n}_{i=0}\lambda^{n-i}\mathrm{Alt}(R_{C}^{(i)_{n}}h\otimes
I)\rho
R_{M}-(-1)^{n-1}\sum^{n-1}_{i=0}\lambda^{n-i-1}\mathrm{Alt}(R_{C}^{(i)_{n}}h\otimes
R_{C})\rho R_{M}$ $\displaystyle=$
$\displaystyle-(-1)^{n-1}\mathrm{Alt}(R_{C}^{\otimes n}h\otimes I)\rho
R_{M}-(-1)^{n-1}\sum^{n-1}_{i=0}\lambda^{n-i-1}\lambda\mathrm{Alt}(R_{C}^{(i)_{n}}h\otimes
I)\rho R_{M}$
$\displaystyle-(-1)^{n-1}\sum^{n-1}_{i=0}\lambda^{n-i-1}\mathrm{Alt}(R_{C}^{(i)_{n}}h\otimes
R_{C})\rho R_{M}$ $\displaystyle=$ $\displaystyle
B_{6}-(-1)^{n-1}\sum^{n-1}_{i=0}\lambda^{n-i-1}\lambda\mathrm{Alt}(R_{C}^{(i)_{n}}h\otimes
I)\rho R_{M}$
$\displaystyle-(-1)^{n-1}\sum^{n-1}_{i=0}\lambda^{n-i-1}\mathrm{Alt}(R_{C}^{(i)_{n}}h\otimes
R_{C})\rho R_{M},$

and

$\displaystyle B_{3}=$
$\displaystyle-(-1)^{n-1}\sum^{n-1}_{i=0}\lambda^{n-i-1}\mathrm{Alt}(R_{C}^{(i)_{n}}h\otimes
I)(R_{M}\otimes R_{C})\rho$ $\displaystyle=$
$\displaystyle-(-1)^{n-1}\sum^{n-1}_{i=0}\lambda^{n-i-1}\mathrm{Alt}(R_{C}^{(i)_{n}}h\otimes
I)(R_{M}\otimes I+I\otimes R_{C}+\lambda)\rho R_{M}$ $\displaystyle=$
$\displaystyle-B_{4}+A_{4}-B_{6}.$

Therefore, $A_{4}=B_{3}+B_{4}+B_{6}$. The proof is completed. ∎

Let

$C^{n}_{RB}(M,C)=\left\\{\begin{aligned} &C^{0}(M,C)=M^{*},&n=0,\\\
&C^{1}(M,C)\oplus\tilde{M}^{*},&n=1,\\\
&C^{n}(M,C)\oplus\tilde{C}^{n-1}(M,C),&n\geq 1.\end{aligned}\right.$

Define a linear map $\partial^{n}_{RB}:C^{n}_{RB}(M,C)\rightarrow
C^{n+1}_{RB}(M,C)$ by

$\displaystyle\partial_{RB}^{0}(m^{*})=(\partial^{0}(m^{*}),-\frac{1}{2}\delta^{0}(m^{*})),~{}~{}\forall~{}m^{*}\in
M^{*},$
$\displaystyle\partial_{RB}^{n}(f,g)=(\partial^{n}(f),-\tilde{\partial}^{n-1}(g)-\frac{1}{2}\delta^{n}(f)),~{}~{}\forall~{}f\in
C^{n}(M,C),~{}g\in\tilde{C}^{n-1}(M,C).$

######  Proposition 3.3.

$\\{C^{*}_{RB}(M,C),\partial^{*}_{RB}\\}$ is a cochain complex , that is,

$\partial^{n+1}_{RB}\partial^{n}_{RB}=0~{}(n\geq 0).$

###### Proof.

  1. $(i)$

When $n=0$, for any $m^{*}\in M^{*}$,

$\partial^{1}_{RB}\partial^{0}_{RB}(m^{*})(m)=\big{(}\partial^{1}(\partial^{0}(m^{*})),-\tilde{\partial}^{0}(-\frac{1}{2}\delta^{0}(m^{*}))-\frac{1}{2}\delta^{1}(\partial^{0}(m^{*}))\big{)}=0.$

  2. $(ii)$

When $n=1$, for any $f\in C^{1}(M,C)$,

$\partial^{2}_{RB}\partial^{1}_{RB}(f,m^{*})=(\partial^{2}(\partial^{1}(f)),-\tilde{\partial}^{1}(-\tilde{\partial}^{0}(m^{*})-\frac{1}{2}\delta^{1}(f))-\frac{1}{2}\delta^{2}(\partial^{1}(f)))=0.$

  3. $(iii)$

When $n\geq 2$, for all $f\in C^{n}(M,C),~{}g\in C^{n-1}(M,C)$,

$\partial^{n+1}_{RB}\partial^{n}_{RB}(f,g)=(\partial^{n+1}\partial^{n}(f),-\tilde{\partial}^{n}(-\tilde{\partial}^{n-1}(g)-\frac{1}{2}\delta^{n}(f))-\frac{1}{2}\delta^{n+1}\partial^{n}(f))=0.$

∎

######  Definition 3.1.

The cohomology group of the cochain complex
$\\{C^{*}_{RB}(M,C),\partial^{*}_{RB}\\}$ is called the cohomology group of
the $\lambda$-weighted Rota-Baxter Lie coalgebra $(C,\Delta_{C},R_{C})$ with
coefficients in $(M,\rho,R_{M})$. Denote it by

$\displaystyle\mathrm{H}^{n}_{RB}(M,C)=\mathrm{Z}^{n}_{RB}(M,C)/\mathrm{B}^{n}_{RB}(M,C),$

where
$\mathrm{Z}^{n}_{RB}(M,C)=\mathrm{ker}(\partial^{n}_{RB}),~{}\mathrm{B}^{n}_{RB}(M,C)=\mathrm{Im}(\partial^{n-1}_{RB}).$

In the last section, we will need a certain subcomplex of the cochain complex
$\\{C^{*}_{RB}(M,C),\partial^{*}_{RB}\\}$ given by

$\bar{C}^{n}_{RB}(M,C)=\left\\{\begin{aligned} &C^{0}(M,C)=M^{*},&n=0,\\\
&C^{1}(M,C),&n=1,\\\ &C^{n}(M,C)\oplus\tilde{C}^{n-1}(M,C),&n\geq
2,\end{aligned}\right.$

and $\bar{\partial}^{n}_{RB}=\partial^{n}_{RB}|_{\bar{C}^{n}_{RB}(M,C)}$. The
corresponding n-cohomology group is denoted by
$\mathrm{\bar{H}}^{n}_{RB}(M,C)=\mathrm{\bar{Z}}^{n}_{RB}(M,C)/\mathrm{\bar{B}}^{n}_{RB}(M,C)$,
which is called the reduced cohomology group of $(C,\Delta_{C},R_{C})$ with
coefficients in $(M,\rho,R_{M})$. Obviously,
$\mathrm{\bar{H}}^{n}_{RB}(M,C)=\mathrm{H}^{n}_{RB}(M,C)$ when $n\geq 3$.

By direct computations,

  1. $(i)$

$(\mathrm{\bar{H}}^{1}_{RB})$.

$\displaystyle\mathrm{\bar{B}}^{1}_{RB}(M,C)=\\{\big{(}(m^{*}\otimes
I)\rho,-\frac{1}{2}m^{*}\big{)}|m^{*}\in M^{*}\\},$
$\displaystyle\mathrm{\bar{Z}}^{1}_{RB}(M,C)=\left\\{f\in
C^{1}_{RB}(M,C)\left|\begin{aligned} &(f\otimes I)\rho-(I\otimes
f)\tau\rho=\Delta_{C}f,\\\ &fR_{M}=R_{C}f\end{aligned}\right.\right\\}.$ (3.7)

  2. $(ii)$

$(\mathrm{\bar{H}}^{2}_{RB})$.

$\displaystyle\bar{B}^{2}_{RB}(M,C)=$
$\displaystyle\left\\{(\mu,\nu)\left|\begin{aligned} &\mu=\frac{1}{2}(f\otimes
I)\rho-\frac{1}{2}(I\otimes f)\tau\rho-\frac{1}{2}\Delta_{C}f,\\\
&\nu=-\frac{1}{2}R_{C}f+\frac{1}{2}fR_{M},f\in\bar{C}^{1}_{RB}(M,C)\end{aligned}\right.\right\\}.$
(3.8)

Since

$\displaystyle\partial^{2}(f)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathrm{Alt}(I\otimes\Delta_{C})h-\frac{1}{2}\mathrm{Alt}(\Delta_{C}\otimes
I)h-\mathrm{Alt}(f\otimes I)\rho,$ $\displaystyle=$
$\displaystyle(I\otimes\Delta)h+(I\otimes\tau)(\Delta\otimes
I)h+(I\otimes\tau)(h\otimes I)\rho-(I\otimes h)\tau\rho$
$\displaystyle-(\Delta\otimes I)h-(h\otimes I)\rho$ $\displaystyle=$
$\displaystyle 0,$ (3.9)
$\displaystyle-\tilde{\partial}^{1}(g)-\frac{1}{2}\delta^{2}(f)$
$\displaystyle=$ $\displaystyle\frac{1}{2}\mathrm{Alt}(I\otimes
R_{C}+R_{C}\otimes I+\lambda)\Delta_{C}g-\mathrm{Alt}(g\otimes
R_{C})\rho+\mathrm{Alt}(g\otimes I)\rho R_{M}$
$\displaystyle-\frac{1}{2}(R_{C}\otimes R_{C})f+\frac{1}{2}(I\otimes
R_{C}+R_{C}\otimes I+\lambda)fR_{M}$ $\displaystyle=$ $\displaystyle(g\otimes
I)\rho R_{M}-\tau(g\otimes I)\rho R_{M}+(I\otimes R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C}g$ $\displaystyle-(g\otimes R_{C})\rho+\tau(g\otimes
R_{C})\rho-(R_{C}\otimes R_{C})f+(I\otimes R_{C}+R_{C}\otimes
I+\lambda)fR_{M}$ $\displaystyle=$ $\displaystyle 0.$ (3.10)

Thus,

$\mathrm{\bar{Z}}^{2}_{RB}(M,C)=\\{(f,g)\in\bar{C}^{2}_{RB}(M,C)|(f,g)~{}satisfies~{}Eqs.~{}(\ref{Z21})-(\ref{Z22})\\}.$

######  Theorem 3.1.

We have the following long exact sequence of cohomology groups

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{\tilde{H}}^{n-1}(M,C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[i]}$$\textstyle{\mathrm{H}^{n}_{RB}(M,C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[p]}$$\textstyle{\mathrm{H}^{n}(M,C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[-\frac{1}{2}\delta^{n}]}$$\textstyle{\mathrm{\tilde{H}}^{n}(M,C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

where

$[g]\in\mathrm{\tilde{H}}^{n-1}(M,C),~{}~{}[i][g]=[i(g)]=[(0,g)],$
$[(f,g)]\in\mathrm{H}^{n}_{RB}(M,C),~{}~{}[p][(f,g)]=[p(f,g)]=[f],$
$f\in\mathrm{H}^{n}(M,C),~{}~{}[-\frac{1}{2}\delta^{n}][f]=[-\frac{1}{2}\delta^{n}(f)].$

##  4 Non-abelian extensions of $\lambda$-weighted Rota-Baxter Lie coalgebras

In this section, we investigate non-abelian extensions of $\lambda$-weighted
Rota-Baxter Lie coalgebras, define the non-abelian second cohomology groups,
and verify that the non-abelian extensions can be classified by the second
non-abelian cohomology groups.

######  Definition 4.1.

Let $(C,\Delta_{C},R_{C})$ and $(M,\Delta_{M},R_{M})$ be two
$\lambda$-weighted Rota-Baxter Lie coalgebras. A non-abelian extension of
$(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$ is a $\lambda$-weighted Rota-
Baxter Lie coalgebra $(E,\Delta_{E},R_{E})$, which fits into a short exact
sequence of $\lambda$-weighted Rota-Baxter Lie coalgebras

(4.3)

When $(M,\Delta_{M},R_{M})$ is an abelian $\lambda$-weighted Lie coalgebra,
the $\mathcal{E}$ is called an abelian extension of $(C,\Delta_{C},R_{C})$ by
$(M,\Delta_{M},R_{M})$. Denote an extension as above simply by
$(E,\Delta_{E},R_{E})$ or $\mathcal{E}$.

A retraction of a non-abelian extension $(E,\Delta_{E},R_{E})$ of
$(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$ is a linear map
$t:E\rightarrow C$ such that $tf=I_{C}$.

######  Definition 4.2.

Let $(E_{1},\Delta_{E_{1}},R_{E_{1}})$ and $(E_{2},\Delta_{E_{2}},R_{E_{2}})$
be two non-abelian extensions of $(C,\Delta_{C},R_{C})$ by
$(M,\Delta_{M},R_{M})$. They are said to be equivalent if there is an
isomomorphism $\theta:E_{1}\rightarrow E_{2}$ of $\lambda$-weighted Rota-
Baxter Lie coalgebras such that the following commutative diagram holds:

(4.8)

We denote the equivalent classes of non-abelian extensions by
$\mathrm{Ext}_{nab}(M,C)$.

######  Definition 4.3.

Let $(C,\Delta_{C},R_{C})$ and $(M,\Delta_{M},R_{M})$ be two
$\lambda$-weighted Rota-Baxter Lie coalgebras. A non-abelian 2-cocycle on
$(C,\Delta_{C},R_{C})$ with values in $(M,\Delta_{M},R_{M})$ is a triple
$(h,\rho,\phi)$ of linear maps $h:M\rightarrow C\otimes C,~{}\rho:M\rightarrow
M\otimes C$ and $\phi:M\rightarrow C$, satisfying the following identities:

$\tau h+h=0,$ (4.9) $(I\otimes\Delta_{C})h-(\Delta_{C}\otimes
I)h+(I\otimes\tau)(\Delta_{C}\otimes I)h=(h\otimes I)\rho+(I\otimes
h)\tau\rho-(I\otimes\tau)(h\otimes I)\rho,$ (4.10)
$(I\otimes\Delta_{C})\rho+(I\otimes h)\Delta_{M}=(\rho\otimes
I)\rho-(I\otimes\tau)(\rho\otimes I)\rho,$ (4.11) $(\Delta_{M}\otimes
I)\rho=(I\otimes\tau)(\rho\otimes I)\Delta_{M}+(I\otimes\rho)\Delta_{M},$
(4.12) $\displaystyle(\phi\otimes
R_{C})\rho-(R_{C}\otimes\phi)\tau\rho-(\phi\otimes I)\rho
R_{M}+(I\otimes\phi)\tau\rho R_{M}$
$\displaystyle+(\phi\otimes\phi)\Delta_{M}-(I\otimes R_{C}+R_{C}\otimes
I+\lambda)\Delta_{C}\phi$ $\displaystyle=(R_{C}\otimes I+I\otimes
R_{C}+\lambda)hR_{M}-(R_{C}\otimes R_{C})h,$ (4.13)
$(R_{M}\otimes\phi)\Delta_{M}+(R_{M}\otimes
R_{C})\rho=(I\otimes\phi)\Delta_{M}R_{M}+(I\otimes R_{C}+R_{M}\otimes
I+\lambda)\rho R_{M}.$ (4.14)

######  Definition 4.4.

Two non-abelian 2-cocycles $(h,\rho,\phi)$ and
$(h^{\prime},\rho^{\prime},\phi^{\prime})$ on $(C,\Delta_{C},R_{C})$ with
values in $(M,\Delta_{M},R_{M})$ are said to be equivalent, if there exists a
linear map $\varphi:M\rightarrow C$ such that the following equalities hold:

$\displaystyle h^{\prime}-h$ $\displaystyle=(\varphi\otimes
I)\rho-\tau(\varphi\otimes
I)\rho+(\varphi\otimes\varphi)\Delta_{M}-\Delta_{C}\varphi,$ (4.15)
$\displaystyle\rho^{\prime}-\rho$ $\displaystyle=(I\otimes\varphi)\Delta_{M},$
(4.16) $\displaystyle\phi^{\prime}-\phi$ $\displaystyle=\varphi
R_{M}-R_{C}\varphi.$ (4.17)

Denote the set of all non-abelian 2-cocycles on $(C,\Delta_{C},R_{C})$ with
values in $(M,\Delta_{M},R_{M})$ by $\mathrm{Z}_{nab}^{2}(M,C)$. The non-
abelian second cohomology group $\mathrm{H}^{2}_{nab}(M,C)$ is the quotient of
$\mathrm{Z}_{nab}^{2}(M,C)$ by this equivalence relation. Denote the
equivalent class of non-abelian 2-cocycle $(h,\rho,\phi)$ by
$[(h,\rho,\phi)]$.

Using the above notations, we define a linear map

$\Delta_{(h,\rho,\phi)}:C\oplus M\longrightarrow(C\oplus M)\otimes(C\oplus M)$

by

$\Delta_{(h,\rho)}(c+m)=\Delta_{C}(c)+h(m)+\Delta_{M}(m)+\rho(m)-\tau\rho(m),~{}~{}\forall~{}c\in
C,m\in M,$ (4.18)

and a linear map

$R_{\phi}:C\oplus M\longrightarrow C\oplus M$

by

$R_{\phi}(c+m)=R_{C}(c)+R_{M}(m)+\phi(m),~{}~{}\forall~{}c\in C,m\in M.$
(4.19)

######  Proposition 4.1.

With the above notations, $(C\oplus M,\Delta_{(h,\rho)},R_{\phi})$ is a
$\lambda$-weighted Rota-Baxter Lie coalgebra if and only if $(h,\rho,\phi)$ is
a non-abelian 2-cocycles on $(C,\Delta_{C},R_{C})$ with values in
$(M,\Delta_{M},R_{M})$. Denote the $\lambda$-weighted Rota-Baxter Lie
coalgebra $(C\oplus M,\Delta_{(h,\rho)},R_{\phi})$ simply by
$C\oplus_{(h,\rho,\phi)}M$.

###### Proof.

$(C\oplus M,\Delta_{(h,\rho)},R_{\phi})$ is a $\lambda$-weighted Rota-Baxter
Lie coalgebra if and only if Eqs. (2.1)-(2.3) hold for
$(\Delta_{(h,\rho)},R_{\phi})$. It is easy to be proved that (2.1) holds if
and only if (4.9) holds. For any $c\in C$, Eqs. (2.2), (2.3) hold if and only
if $(C,\Delta_{C},R_{C})$ is a $\lambda$-weighted Rota-Baxter Lie coalgebra.
According to Eqs. (2.2), (2.3) and (4.18), for all $m\in M$,

$\displaystyle(I\otimes\Delta_{(h,\rho)})\Delta_{(h,\rho)}(m)$
$\displaystyle=$
$\displaystyle(I\otimes\Delta_{(h,\rho)})(\Delta_{M}(m)+\rho(m)-\tau\rho(m)+h(m))$
$\displaystyle=$
$\displaystyle(I\otimes(\cancel{\Delta_{M}(m)}+\rho(m)-\tau\rho(m)+h(m)))\Delta_{M}(m)+(I\otimes\Delta_{C})\rho(m)$
$\displaystyle-(I\otimes(\Delta_{M}(m)+\rho(m)-\tau\rho(m)+h(m)))\tau\rho(m)+(I\otimes\Delta_{C})h(m),$
$\displaystyle(\Delta_{(h,\rho)}\otimes I)\Delta_{(h,\rho)}(m)$
$\displaystyle=$ $\displaystyle(\Delta_{(h,\rho)}\otimes
I)(\Delta_{M}(m)+\rho(m)-\tau\rho(m)+h(m))$ $\displaystyle=$
$\displaystyle(\Delta_{C}\otimes
I)h(m)+(\cancel{(\Delta_{M}(m)}+\rho(m)-\tau\rho(m)+h(m))\otimes
I)\Delta_{M}(m)$ $\displaystyle-(\Delta_{C}\otimes
I)\tau\rho(m)+((\Delta_{M}(m)+\rho(m)-\tau\rho(m)+h(m))\otimes I)\rho(m),$
$\displaystyle(I\otimes\tau)(\Delta_{(h,\rho)}\otimes I)\Delta_{(h,\rho)}(m)$
$\displaystyle=$ $\displaystyle(I\otimes\tau)(\Delta_{C}\otimes
I)h(m)+(I\otimes\tau)((\cancel{\Delta_{M}(m)}+\rho(m)-\tau\rho(m)+h(m))\otimes
I)\Delta_{M}(m)$ $\displaystyle-(I\otimes\tau)(\Delta_{C}\otimes
I)\tau\rho(m)+(I\otimes\tau)((\Delta_{M}(m)+\rho(m)-\tau\rho(m)+h(m))\otimes
I)\rho(m).$

Thus Eq. (2.2) holds if and only if Eqs. (4.10)-(4.12) hold. By the same
token, Eq. (2.3) holds for $(\Delta_{(h,\rho)},R_{\phi})$ if and only if Eqs.
(4.13),(4.14) hold. The proof is completed. ∎

Let $(C,\Delta_{C},R_{C})$ and $(M,\Delta_{M},R_{M})$ be two
$\lambda$-weighted Rota-Baxter Lie coalgebras. Suppose that

is a non-abelian extension of $(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$
with a retraction $t$ of $(E,\Delta_{E},R_{E})$. For all $m\in M$, since $g$
is surjective, there exists an element $e\in E$, such that $m=g(e)$. Define
linear maps $h_{t}:M\rightarrow C\otimes C,~{}\rho_{t}:M\rightarrow M\otimes
C$ and $\phi_{t}:M\rightarrow C$ respectively by

$\displaystyle h_{t}(m)=h_{t}(g(e))=(t\otimes t)\Delta_{E}(e)-\Delta_{C}t(e),$
(4.20) $\displaystyle\rho_{t}(m)=\rho_{t}(g(e))=(g\otimes t)\Delta_{E}(e),$
(4.21) $\displaystyle\phi_{t}(m)=\phi_{t}(g(e))=tR_{E}(e)-R_{C}t(e).$ (4.22)

For all $m\in M$, if $m=g(e_{1})=g(e_{2}),~{}e_{1},e_{2}\in E$, since
$\mathrm{Ker}g=\mathrm{Im}f$, there exists an element $c\in C$ such that
$f(c)=e_{1}-e_{2}$. Using $(f\otimes f)\Delta_{C}=\Delta_{C}f$ and $tf=I_{C}$,
we have

$\displaystyle((t\otimes t)\Delta_{E}-\Delta_{C}t)(e_{1}-e_{2})$
$\displaystyle=$ $\displaystyle(t\otimes t)\Delta_{E}f(c)-\Delta_{C}tf(c)$
$\displaystyle=$ $\displaystyle(tf\otimes tf)\Delta_{C}(c)-\Delta_{C}(c)$
$\displaystyle=$ $\displaystyle 0,$

which implies that $h$ is independent on the choice of $e$. Similarly, we can
prove that $\rho,\phi$ are independent on the choice of $e$.

######  Proposition 4.2.

With the above notations, $(h_{t},\rho_{t},\phi_{t})$ is a non-abelian
2-cocycles on $(C,\Delta_{C},R_{C})$ with values in $(M,\Delta_{M},R_{M})$. We
call it the non-abelian 2-cocycle corresponding to the extension $\mathcal{E}$
induced by the retraction $t$. Moreover, $(C\oplus
M,\Delta_{(h_{t},\rho_{t})},R_{\phi_{t}})$ is a $\lambda$-weighted Rota-Baxter
Lie coalgebra. Denote this $\lambda$-weighted Rota-Baxter Lie coalgebra simply
by $C\oplus_{(h_{t},\rho_{t},\phi_{t})}M$.

###### Proof.

It can obtained by direct calculation. ∎

######  Lemma 4.1.

Let $(h_{i},\rho_{i},\phi_{i})$ be the non-abelian 2-cocycle corresponding to
the extension

induced by retraction $t_{i}$ (i=1,2). Then $(h_{1},\rho_{1},\phi_{1})$ and
$(h_{2},\rho_{2},\phi_{2})$ are equivalent, that is, the equivalent classes of
non-abelian 2-cocycles corresponding to a non-abelian extension induced by a
retraction are independent on the choice of retractions.

###### Proof.

Let $(E,\Delta_{E},R_{E})$ be a non-abelian extension of
$(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$. Suppose that $t_{1},t_{2}$
are different retractions of the extension $\mathcal{E}$,
$(h_{1},\rho_{1},\phi_{1})$ and $(h_{2},\rho_{2},\phi_{2})$ are the
corresponding non-abelian 2-cocycles respectively. Since $g$ is surjective,
there is an element $e\in E$ for all $m\in M$ such that $m=g(e)$. So we can
define a linear map $\varphi:M\rightarrow C$ by

$\varphi(m)=\varphi(g(e))=t_{2}(e)-t_{1}(e),~{}~{}\forall~{}m\in M.$ (4.23)

In the light of Eqs. (4.20)-(4.22) and (4.23), for all $m\in M$, we have

$\displaystyle h_{2}(m)-h_{1}(m)$ $\displaystyle=$ $\displaystyle(t_{2}\otimes
t_{2})\Delta_{E}(e)-\Delta_{C}t_{2}(e)-(t_{1}\otimes
t_{1})\Delta_{E}(e)+\Delta_{C}t_{1}(e)$ $\displaystyle=$
$\displaystyle((\varphi g+t_{1})\otimes(\varphi
g+t_{1}))\Delta_{E}(e)-\Delta_{C}(\varphi g+t_{1})(e)-(t_{1}\otimes
t_{1})\Delta_{E}(e)+\Delta_{C}t_{1}(e)$ $\displaystyle=$
$\displaystyle(\varphi g\otimes\varphi g)\Delta_{E}(e)+(t_{1}\otimes\varphi
g)\Delta_{E}(e)+(\varphi g\otimes t_{1})\Delta_{E}(e)-\Delta_{C}\varphi(m)$
$\displaystyle=$
$\displaystyle(\varphi\otimes\varphi)\Delta_{M}(m)-\tau(\varphi\otimes
I)\rho(m)+(\varphi\otimes I)\rho(m)-\Delta_{C}\varphi(m).$

Analogously,

$\displaystyle\rho_{2}(m)-\rho_{1}(m)=(I\otimes\varphi)\Delta_{M}(m),$
$\displaystyle\phi_{2}(m)-\phi_{1}(m)=\varphi R_{M}(m)-R_{C}\varphi(m).$

Thus $(h_{1},\rho_{1},\phi_{1})$ and $(h_{2},\rho_{2},\phi_{2})$ are
equivalent non-abelian 2-cocycles via a linear map $\varphi$. ∎

According to Proposition 4.1 and Proposition 4.2, given a non-abelian
extension

of $(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$ with a retraction $t$, we
have a non-abelian 2-cocycle $(h_{t},\rho_{t},\phi_{t})$ and a
$\lambda$-weighted Rota-Baxter Lie coalgebras $(C\oplus
M,\Delta_{(h_{t},\rho_{t})},R_{\phi_{t}})$. It follows that

is a non-abelian extension of $(C,\Delta_{C},R_{C})$ by
$(M,\Delta_{M},R_{M})$. Define a linear map

$\theta:E\longrightarrow
C\oplus_{(h_{t},\rho_{t},\phi_{t})}M,~{}\theta(w)=g(w)+t(w),~{}~{}\forall~{}w\in
E.$

We claim that $\theta$ is an isomorphism of $\lambda$-weighted Rota-Baxter Lie
coalgebras (the proof can be found in Lemma 5.1 ) such that the following
commutative diagram holds:

which indicates that the non-abelian extensions $\mathcal{E}$ and
$\mathcal{E}_{(h_{t},\rho_{t},\phi_{t})}$ of $(C,\Delta_{C},R_{C})$ by
$(M,\Delta_{M},R_{M})$ are equivalent. On the other hand, if $(h,\rho,\phi)$
is a non-abelian 2-cocycle on $(C,\Delta_{C},R_{C})$ with values in
$(M,\Delta_{M},R_{M})$, there is a $\lambda$-weighted Rota-Baxter Lie
coalgebra $(C\oplus M,\Delta_{(h,\rho)},R_{\phi})$, which yields the following
non-abelian extension of $(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$:

where $i$ is the inclusion and $\pi$ is the projection.

In the sequel, we characterize the relationship between non-abelian 2-cocycles
and non-abelian extensions.

######  Theorem 4.1.

Let $(C,\Delta_{C},R_{C})$ and $(M,\Delta_{M},R_{M})$ be two
$\lambda$-weighted Rota-Baxter Lie coalgebras. Then the equivalent classes of
non-abelian extensions of $(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$ are
classified by the equivalent classes of non-abelian 2-cocycles. In other
words,

$\mathrm{Ext}_{nab}(M,C)\cong\mathrm{H}^{2}_{nab}(M,C).$

###### Proof.

Define a linear map

$\Phi:\mathrm{Ext}_{nab}(M,C)\longrightarrow\mathrm{H}^{2}_{nab}(M,C),~{}$

where $\Phi$ assigns an equivalent class of non-abelian extensions to the
classes of corresponding non-abelian 2-cocycles. Firstly, we prove that $\Phi$
is well-defined. Assume that two non-abelian extensions
$(E_{1},\Delta_{E_{1}},R_{E_{1}})$ and $(E_{2},\Delta_{E_{2}},R_{E_{2}})$ of
$(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$ are equivalent via an
isomorphism $\theta$, that is, the commutative diagram (4.8) holds. Let
$t_{2}$ be a retraction of $(E_{2},\Delta_{E_{2}},R_{E_{2}})$. Thanks to
$t_{2}\theta f_{1}=t_{2}f_{2}=I_{C}$, we have $t_{1}=t_{2}\theta$ is a
retraction of $(E_{1},\Delta_{E_{1}},R_{E_{1}})$. Let
$(h_{1},\rho_{1},\phi_{1})$ and $(h_{2},\rho_{2},\phi_{2})$ be the
corresponding non-abelian 2-cocycles induced by retractions $t_{1},t_{2}$
respectively. In view of (4.20), for any $m\in M$, we get

$\displaystyle h_{1}(m)$ $\displaystyle=h_{1}(g_{1}(e))=(t_{1}\otimes
t_{1})\Delta_{E_{1}}(e)-\Delta_{C}t_{1}(e)$ $\displaystyle=(t_{2}\theta\otimes
t_{2}\theta)\Delta_{E_{1}}(e)-\Delta_{C}t_{2}\theta(e)$
$\displaystyle=(t_{2}\otimes
t_{2})\Delta_{E_{2}}\theta(e)-\Delta_{C}t_{2}\theta(e)$
$\displaystyle=h_{2}(g_{2}\theta(e))=h_{2}(g_{1}(e))=h_{2}(m).$

By the same token,

$\displaystyle\rho_{1}(m)=\rho_{2}(m),~{}~{}\phi_{1}(m)=\phi_{2}(m).$

These indicate that $(h_{1},\rho_{1},\phi_{1})=(h_{2},\rho_{2},\phi_{2})$.
Thus, $\Phi$ is well-defined. Secondly, we check that $\Phi$ is bijective. In
fact, assume that $\Phi([\mathcal{E}_{1}])=[(h_{1},\rho_{1},\phi_{1})]$ and
$\Phi([\mathcal{E}_{2}])=[(h_{2},\rho_{2},\phi_{2})]$. If
$[(h_{1},\rho_{1},\phi_{1})]=[(h_{2},\rho_{2},\phi_{2})]$, we get that the two
non-abelian 2-cocycles $(h_{1},\rho_{1},\phi_{1})$ and
$(h_{2},\rho_{2},\phi_{2})$ are equivalent via a linear map
$\varphi:M\rightarrow C$, satisfying Eqs. (4.15)-(4.17). Define a linear map

$\theta:C\oplus_{(h_{1},\rho_{1},\phi_{1})}M\rightarrow
C\oplus_{(h_{2},\rho_{2},\phi_{1})}M$

by

$\theta(c+m)=c+\varphi(m)+m,~{}~{}\forall~{}c\in C,m\in M.$ (4.24)

Clearly, $\theta$ is bijective. In the following, we state that $\theta$ is a
homomorphism of $\lambda$-weighted Rota-Baxter Lie coalgebras. In fact, using
Eqs. (4.18), (4.15), (4.16) and (4.24), we obtain

$\displaystyle(\theta\otimes\theta)\Delta_{(h_{1},\rho_{1})}(c+m)-\Delta_{(h_{2},\rho_{2})}\theta(c+m)$
$\displaystyle=$
$\displaystyle(\theta\otimes\theta)(\Delta_{c}(c)+h_{1}(m)+\Delta_{M}(m)+\rho_{1}(m)-\tau\rho_{1}(m))-\Delta_{(h_{2},\rho_{2})}(c+m+\varphi(m))$
$\displaystyle=$
$\displaystyle\Delta_{C}(c)+h_{1}(m)+\Delta_{M}(m)+(\varphi\otimes\varphi)\Delta_{M}(m)+(I\otimes\varphi)\Delta_{M}(m)+(\varphi\otimes
I)\Delta_{M}(m)$ $\displaystyle+\rho_{1}(m)+(\varphi\otimes
I)\rho_{1}(m)-\tau\rho_{1}(m)-\tau(\varphi\otimes
I)\rho_{1}(m)-\Delta_{C}(c)-h_{2}(m)-\Delta_{C}\varphi(m)$
$\displaystyle-\Delta_{M}(m)-\rho_{2}(m)+\tau\rho_{2}(m)$ $\displaystyle=$
$\displaystyle
h_{1}(m)-h_{2}(m)+(\varphi\otimes\varphi)\Delta_{M}(m)+(\varphi\otimes
I)\rho_{1}(m)-\Delta_{C}\varphi(m)-\tau(\varphi\otimes I)\rho_{1}(m)$
$\displaystyle+\rho_{1}(m)-\rho_{2}(m)+(I\otimes\varphi)\Delta_{M}(m)-\tau\rho_{1}(m)+\tau\rho_{2}(m)+(I\otimes\varphi)\Delta_{M}(m)$
$\displaystyle=$ $\displaystyle 0,$

which yields that

$\displaystyle(\theta\otimes\theta)\Delta_{(h_{1},\rho_{1})}=\Delta_{(h_{2},\rho_{2})}\theta.$

By Eqs. (4.17), (4.19) and (4.24), we get

$\displaystyle(\theta R_{\phi_{1}}-R_{\phi_{2}}\theta)(c+m)$
$\displaystyle=R_{C}(c)+R_{M}(m)+\phi_{1}(m)+\varphi R_{M}(m)$ $\displaystyle-
R_{C}(c)-R_{C}\varphi(m)-R_{M}(m)-\phi_{2}(m)$
$\displaystyle=\phi_{1}(m)-\phi_{2}(m)+\varphi R_{M}(m)-R_{C}\varphi(m)$
$\displaystyle=0,$

which indicates that $\theta R_{\phi_{1}}=R_{\phi_{2}}\theta$. In all,
$\theta$ is an isomorphism of $\lambda$-weighted Rota-Baxter Lie coalgebras.
It is easy to check that the following diagram

is commutative. Therefore,
$[\mathcal{E}_{(h_{1},\rho_{1},\phi_{1})}]=[\mathcal{E}_{(h_{2},\rho_{2},\phi_{2})}]$,
which indicates that $\Phi$ is injective. For any equivalent class of non-
abelian 2-cocycles $[(h,\rho)]$, by Proposition 4.1, there is a non-abelian
extension of $(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$:

Therefore, $\Phi([\mathcal{E}_{(h,\rho,\phi)}])=[(h,\rho,\phi)]$, which
follows that $\Phi$ is surjective. In all, $\Phi$ is bijective. This proof is
completed. ∎

##  5 Extensibility of a pair of automorphisms

In this section, we study extensibility of a pair of automorphisms of
$\lambda$-weighted Rota-Baxter Lie coalgebras.

In the following, we always suppose that

$\textstyle{\mathcal{E}:0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

is a fixed non-abelian extension of the $\lambda$-weighted Rota-Baxter Lie
coalgebra $(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$, and $t$ is its
retraction. Denote
$\mathrm{Aut}_{C}(E)=\\{\gamma\in\mathrm{Aut}(E)|\gamma(C)=C\\}$.

######  Definition 5.1.

A pair $(\alpha,\beta)\in\mathrm{Aut}(C)\times\mathrm{Aut}(M)$ is said to be
extensible with respect to the non-abelian extension $\mathcal{E}$ if there
exists an element $\gamma\in\mathrm{Aut}_{C}(E)$ such that

$f\alpha=\gamma f,~{}\beta g=g\gamma,$ (5.1)

that is, the following commutative diagram holds:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{f}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{g}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

It is natural to ask the following question:

When is a pair of $\lambda$-weighted Rota-Baxter Lie coalgebra isomorphisms
$(\alpha,\beta)\in\mathrm{Aut}(C)\times\mathrm{Aut}(M)$ extensible? We discuss
this theme in the sequel.

######  Lemma 5.1.

Let $(h,\rho,\phi)$ be the non-abelian 2-cocycle corresponding to the non-
abelian extension $\mathcal{E}$ induced by the retraction $t$. Then the non-
abelian extension $\mathcal{E}$ is equivalent to the non-abelian extension

$\textstyle{\mathcal{E}_{(h,\rho,\phi)}:0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{C}}$$\textstyle{C\oplus_{(h,\rho,\phi)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{M}}$$\textstyle{M,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $i_{C}(resp.~{}\pi_{M})$ is the canonical injection
$(resp.~{}projection)$.

###### Proof.

In the light of $(h,\rho,\phi)$ being a non-abelian 2-cocycle, by Proposition
4.1, $(C\oplus M,\Delta_{(h,\rho)},R_{\phi})$ is a $\lambda$-weighted Rota-
Baxter Lie coalgebra. Define

$\theta:E\longrightarrow
C\oplus_{(h,\rho,\phi)}M,~{}~{}\hbox{by}~{}~{}\theta(e)=t(e)+g(e),\forall~{}e\in
E.$ (5.2)

It is easy to verify that the following diagram is commutative:

(5.7)

By Short Five Lemma, $\theta$ is bijective. Using Eqs. (4.20), (4.21), (4.18)
and (5.2), we obtain

$\displaystyle(\theta\otimes\theta)\Delta_{E}(e)$ $\displaystyle=$
$\displaystyle(t\otimes t)\Delta_{E}(e)+(t\otimes g)\Delta_{E}(e)+(g\otimes
t)\Delta_{E}(e)+(g\otimes g)\Delta_{E}(e)$ $\displaystyle=$ $\displaystyle
h(g(e))+\Delta_{C}(t(e))-\tau\rho(g(e))+\rho(g(e))+\Delta_{M}(g(e))$
$\displaystyle=$ $\displaystyle\Delta_{(h,\rho)}(t(e)+g(e))$ $\displaystyle=$
$\displaystyle\Delta_{(h,\rho)}\theta(e).$

By Eqs. (4.22), (4.19) and (5.2),

$\displaystyle R_{\phi}\theta(e)$ $\displaystyle=$ $\displaystyle
R_{\phi}(t(e)+g(e))$ $\displaystyle=$ $\displaystyle
R_{C}(t(e))+\phi(g(e))+R_{C}(g(e))$ $\displaystyle=$ $\displaystyle
tR_{E}(e)+gR_{E}(e)$ $\displaystyle=$ $\displaystyle\theta R_{E}(e).$

Thus, $\theta$ is an isomorphism of $\lambda$-weighted Rota-Baxter Lie
coalgebras. Therefore, the non-abelian extensions
$\mathcal{E}_{(h,\rho,\phi)}$ and $\mathcal{E}$ are equivalent via the map
$\theta$. ∎

######  Lemma 5.2.

Assume that $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ are two equivalent non-
abelian extensions of $(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$. Let
$(\alpha,\beta)\in\mathrm{Aut}(C)\times\mathrm{Aut}(M)$. Then $(\alpha,\beta)$
is extensible with respect to $\mathcal{E}_{1}$ if and only if
$(\alpha,\beta)$ is extensible with respect to $\mathcal{E}_{2}$.

###### Proof.

One can take the same procedure of abelian extensions of Lie coalgebras, see
[15]. ∎

######  Theorem 5.1.

Let $(h,\rho,\phi)$ be a non-abelian 2-cocycle corresponding to the non-
abelian extension $\mathcal{E}$ induced by the retraction $t$. Then
$(\alpha,\beta)\in\mathrm{Aut}(C)\times\mathrm{Aut}(M)$ is extensible with
respective to $\mathcal{E}$ if and only if there is a linear map
$\varphi:M\rightarrow C$ satisfies

$h\beta-(\alpha\otimes\alpha)h=(\varphi\otimes\alpha)\rho-\tau(\varphi\otimes\alpha)\rho-\Delta_{C}\varphi+(\varphi\otimes\varphi)\Delta_{M},$
(5.8) $\rho\beta-(\beta\otimes\alpha)\rho=(\beta\otimes\varphi)\Delta_{M},$
(5.9) $\phi\beta-\alpha\phi=\varphi R_{M}-R_{C}\varphi.$ (5.10)

###### Proof.

Assume that $(\alpha,\beta)$ is extensible with respective to $\mathcal{E}$,
then there is an automorphism $\gamma\in\mathrm{Aut}_{C}(E)$ such that (5.1)
holds. Since $g$ is surjective, for all $m\in M$, there is an element $e\in
E$, such that $m=g(e)$. Define a linear map $\varphi:M\rightarrow C$ by

$\varphi(m)=\varphi g(e)=t\gamma(e)-\alpha t(e),~{}~{}\forall~{}m\in M.$
(5.11)

We should check that $\varphi$ doesn’t depend on the choice of $e\in E$. In
fact, for all $m\in M$, if $m=g(e_{1})=g(e_{2}),~{}e_{1},e_{2}\in E$, due to
$\mathrm{Ker}g=\mathrm{Im}f$, there exists an element $c\in C$ such that
$f(c)=e_{1}-e_{2}$. Thanks to (5.1) and $tf=I_{C}$, we have

$t\gamma(e_{1}-e_{2})-\alpha t(e_{1}-e_{2})=t\gamma f(c)-\alpha
tf(c)=tf\alpha(c)-\alpha(c)=0,$

which follows that $\varphi$ doesn’t depend on the choice of $e\in E$. For all
$m\in M$, $\beta(m)=\beta g(e)=g\gamma(e)$. Thus, according to Eqs. (4.20),
(4.21) and (5.11), we get

$\displaystyle h\beta(m)-(\alpha\otimes\alpha)h(m)$ $\displaystyle=$
$\displaystyle(t\otimes t)\Delta_{E}(\gamma(e))-\Delta_{C}(t\gamma(e))-(\alpha
t\otimes\alpha t)\Delta_{E}(e)+(\alpha\otimes\alpha)\Delta_{C}(t(e))$
$\displaystyle=$ $\displaystyle(t\gamma\otimes
t\gamma)\Delta_{E}(e)-\Delta_{C}(t\gamma(e))-(\alpha t\otimes\alpha
t)\Delta_{E}(e)+\Delta_{C}(\alpha t(e))$ $\displaystyle=$
$\displaystyle\Big{(}(\varphi g+\alpha t)\otimes(\varphi g+\alpha
t)\Big{)}\Delta_{E}(e)-\Delta_{C}((\varphi g+\alpha t)(e))-(\alpha
t\otimes\alpha t)\Delta_{E}(e)+\Delta_{C}(\alpha t(e))$ $\displaystyle=$
$\displaystyle(\varphi g\otimes\varphi g)\Delta_{E}(e)+(\alpha t\otimes\alpha
t)\Delta_{E}(e)+(\varphi g\otimes\alpha t)\Delta_{E}(e)+(\alpha
t\otimes\varphi g)\Delta_{E}(e)$ $\displaystyle-\Delta_{C}(\varphi
g(e))-\Delta_{C}(\alpha t(e))-(\alpha t\otimes\alpha
t)\Delta_{E}(e)+\Delta_{C}(\alpha t(e))$ $\displaystyle=$
$\displaystyle(\varphi\otimes\varphi)\Delta_{M}(m)+(\varphi\otimes\alpha)\rho(m)-\tau(\varphi\otimes\alpha)\rho(m)-\Delta_{C}(\varphi(m)),$

which yields that (5.8) holds. Analogously, we can check that Eqs. (5.9)-
(5.10) hold. Conversely, in the light of Lemma 5.1 and Lemma 5.2, we only need
to prove that $(\alpha,\beta)\in\mathrm{Aut}(C)\times\mathrm{Aut}(M)$ is
extensible with respect to the non-abelian extension

$\textstyle{\mathcal{E}_{(h,\rho,\phi)}:0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{C}}$$\textstyle{C\oplus_{(h,\rho,\phi)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{M}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

In fact, take $\gamma=\begin{bmatrix}\alpha&\varphi\\\ 0&\beta\end{bmatrix}$,
that is,

$\gamma(c+m)=\alpha(c)+\varphi(m)+\beta(m),~{}~{}\forall~{}c\in C,m\in M.$
(5.12)

It is clear that $\gamma$ is a bijection,
$\gamma|_{C}=\alpha,~{}i_{C}\alpha=\gamma i_{C}$ and
$\pi_{M}\gamma=\beta\pi_{M}$. In the remaining part, we only need to verify
that $\gamma$ is a homomorphism of $\lambda$-weighted Rota-Baxter Lie
coalgebras.

For all $c\in C,m\in M$, in view of Eqs. (4.18), (5.8), (5.9) and (5.12), by
direct computations, we have

$\displaystyle(\gamma\otimes\gamma)\Delta_{(h,\rho)}(c+m)$
$\displaystyle=(\gamma\otimes\gamma)(\Delta_{C}(c)+h(m)+\rho(m)-\tau\rho(m)+\Delta_{M}(m))$
$\displaystyle=$
$\displaystyle(\alpha\otimes\alpha)\Delta_{C}(c)+(\alpha\otimes\alpha)h(m)+(\beta\otimes\alpha)\rho(m)+(\varphi\otimes\alpha)\rho(m)$
$\displaystyle-\tau(\beta\otimes\alpha)\rho(m)-\tau(\varphi\otimes\alpha)\rho(m)+(\beta\otimes\beta)\Delta_{M}(m)$
$\displaystyle+(\beta\otimes\varphi)\Delta_{M}(m)+(\varphi\otimes\beta)\Delta_{M}(m)+(\varphi\otimes\varphi)\Delta_{M}(m)$
$\displaystyle=$
$\displaystyle\Delta_{C}(\alpha(c))+\Delta_{C}(\varphi(m))+(h+\rho-\tau\rho)(\beta(m))+\Delta_{M}(\beta(m))$
$\displaystyle=$
$\displaystyle\Delta_{(h,\rho)}(\alpha(c)+\varphi(m)+\beta(m))$
$\displaystyle=$ $\displaystyle\Delta_{(h,\rho)}\gamma(c+m),$

which implies that
$\Delta_{(h,\rho)}\gamma=(\gamma\otimes\gamma)\Delta_{(h,\rho)}$. By the same
token, $R_{\phi}\gamma=\gamma R_{\phi}$. Hence $\gamma$ is a homomorphism of
$\lambda$-weighted Rota-Baxter Lie coalgebras. In all,
$(\alpha,\beta)\in\mathrm{Aut}(C)\times\mathrm{Aut}(M)$ is extensible with
respect to the non-abelian extension $\mathcal{E}_{(h,\rho,\phi)}$. We
complete the proof. ∎

Let $(h,\rho,\phi)$ be a non-abelian 2-cocycle corresponding to the non-
abelian extension $\mathcal{E}$ induced by the retraction $t$. For any
$(\alpha,\beta)\in\mathrm{Aut}(C)\times\mathrm{Aut}(M)$, we define a triple
$(h_{(\alpha,\beta)},\rho_{(\alpha,\beta)},\phi_{(\alpha,\beta)})$ of linear
maps $h_{(\alpha,\beta)}:M\rightarrow C\otimes
C,~{}\rho_{(\alpha,\beta)}:M\rightarrow M\otimes C$ and
$\phi_{(\alpha,\beta)}:M\rightarrow C$ respectively by

$h_{(\alpha,\beta)}=(\alpha\otimes\alpha)h\beta^{-1},~{}\rho_{(\alpha,\beta)}=(\beta\otimes\alpha)\rho\beta^{-1},~{}\phi_{(\alpha,\beta)}=\alpha\phi\beta^{-1}.$
(5.13)

######  Proposition 5.1.

With the above notations, the triple
$(h_{(\alpha,\beta)},\rho_{(\alpha,\beta)},\phi_{(\alpha,\beta)})$ is a non-
abelian 2-cocycle.

###### Proof.

According to Eqs. (4.10) and (5.13),

$\displaystyle(I\otimes\Delta)h_{(\alpha,\beta)}-(I\otimes
h_{(\alpha,\beta)})\tau\rho_{(\alpha,\beta)}-(\Delta\otimes
I)h_{(\alpha,\beta)}-(h_{(\alpha,\beta)}\otimes I)\rho_{(\alpha,\beta)}$
$\displaystyle+(I\otimes\tau)(\Delta\otimes
I)h_{(\alpha,\beta)}+(I\otimes\tau)(h_{(\alpha,\beta)}\otimes
I)\rho_{(\alpha,\beta)}$ $\displaystyle=$
$\displaystyle(I\otimes\Delta)(\alpha\otimes\alpha)h\beta^{-1}-(I\otimes(\alpha\otimes\alpha)h\beta^{-1})\tau(\beta\otimes\alpha)\rho\beta^{-1}-(\Delta\otimes
I)(\alpha\otimes\alpha)h\beta^{-1}$
$\displaystyle-((\alpha\otimes\alpha)h\beta^{-1}\otimes
I)(\beta\otimes\alpha)\rho\beta^{-1}+(I\otimes\tau)(\Delta\otimes
I)(\alpha\otimes\alpha)h\beta^{-1}$
$\displaystyle+(I\otimes\tau)((\alpha\otimes\alpha)h\beta^{-1}\otimes
I)(\beta\otimes\alpha)\rho\beta^{-1}$ $\displaystyle=$
$\displaystyle(\alpha\otimes\Delta\alpha)h\beta^{-1}-(\alpha\otimes(\alpha\otimes\alpha)h)\tau\rho\beta^{-1}-(\Delta\alpha\otimes\alpha)h\beta^{-1}$
$\displaystyle-((\alpha\otimes\alpha)h\otimes\alpha)\rho\beta^{-1}+(I\otimes\tau)(\Delta\alpha\otimes\alpha)h\beta^{-1}+(I\otimes\tau)((\alpha\otimes\alpha)h\otimes\alpha)\rho\beta^{-1}$
$\displaystyle=$
$\displaystyle(\alpha\otimes\alpha\otimes\alpha)(I\otimes\Delta)h\beta^{-1}-(\alpha\otimes\alpha\otimes\alpha)(I\otimes
h)\tau\rho\beta^{-1}-(\alpha\otimes\alpha\otimes\alpha)(\Delta\otimes
I)h\beta^{-1}$ $\displaystyle-(\alpha\otimes\alpha\otimes\alpha)(h\otimes
I)\rho\beta^{-1}+(\alpha\otimes\alpha\otimes\alpha)(I\otimes\tau)(\Delta\otimes
I)h\beta^{-1}$
$\displaystyle+(\alpha\otimes\alpha\otimes\alpha)(I\otimes\tau)(h\otimes
I)\rho\beta^{-1}$ $\displaystyle=$
$\displaystyle(\alpha\otimes\alpha\otimes\alpha)((I\otimes\Delta)h-(I\otimes
h)\tau\rho-(\Delta\otimes I)h-(h\otimes I)\rho+(I\otimes\tau)(\Delta\otimes
I)h$ $\displaystyle+(I\otimes\tau)(h\otimes I)\rho)\beta^{-1}$
$\displaystyle=$ $\displaystyle 0,$

which yields that Eq. (4.10) holds for
$(h_{(\alpha,\beta)},\rho_{(\alpha,\beta)})$. To check that Eqs. (4.11)-(4.14)
holds for $(h_{(\alpha,\beta)},\rho_{(\alpha,\beta)},\phi_{(\alpha,\beta)})$,
one can take the same procedure. Therefore,
$(h_{(\alpha,\beta)},\rho_{(\alpha,\beta)},\phi_{(\alpha,\beta)})$ is a non-
abelian 2-cocycle. ∎

######  Theorem 5.2.

Let $(h,\rho,\phi)$ be a non-abelian 2-cocycle corresponding to the non-
abelian extension $\mathcal{E}$ induced by the retraction $t$. Then
$(\alpha,\beta)\in\mathrm{Aut}(C)\times\mathrm{Aut}(M)$ is extensible with
respect to $\mathcal{E}$ if and only if
$(h_{(\alpha,\beta)},\rho_{(\alpha,\beta)},\phi_{(\alpha,\beta)})$ and
$(h,\rho,\phi)$ are equivalent non-abelian 2-cocycles.

###### Proof.

Suppose that $(\alpha,\beta)\in\mathrm{Aut}(C)\times\mathrm{Aut}(M)$ is
extensible with respect to $\mathcal{E}$, by Theorem 5.1, there is a linear
map $\varphi:M\rightarrow C$ satisfying Eqs. (5.8)-(5.10). For all $m\in M$,
there is $m_{0}\in M$ such that $m=\beta(m_{0})$. Since $g$ is surjective,
there exist elements $e,e_{0}\in E$, such that $m=g(e),m_{0}=g(e_{0})$.
Combining Eqs. (5.8)-(5.9) and (5.13),

$\displaystyle h(m)-h_{(\alpha,\beta)}(m)$ $\displaystyle=$ $\displaystyle
h(m)-(\alpha\otimes\alpha)h(\beta^{-1}(m))$ $\displaystyle=$ $\displaystyle
h(\beta(m_{0}))-(\alpha\otimes\alpha)h(m_{0})$ $\displaystyle=$
$\displaystyle(\varphi\otimes\alpha)\rho(m_{0})-\tau(\varphi\otimes\alpha)\rho(m_{0})-\Delta_{C}(\varphi(m_{0}))+(\varphi\otimes\varphi)\Delta_{M}(m_{0})$
$\displaystyle=$
$\displaystyle\Big{(}\varphi\otimes\alpha-\tau(\varphi\otimes\alpha)\Big{)}\rho(\beta^{-1}(m_{0}))-\Delta_{C}(\varphi\beta^{-1}(m_{0}))+(\varphi\otimes\varphi)\Delta_{M}(\beta^{-1}(m_{0}))$
$\displaystyle=$ $\displaystyle(\varphi\beta^{-1}\otimes
I)(\beta\otimes\alpha)\rho(\beta^{-1}(m))-\tau(\varphi\beta^{-1}\otimes
I)(\beta\otimes\alpha)\rho(\beta^{-1}(m))$
$\displaystyle-\Delta_{C}(\varphi\beta^{-1}(m))+(\varphi\beta^{-1}\otimes\varphi\beta^{-1})\Delta_{M}(m)$
$\displaystyle=$ $\displaystyle\Big{(}\varphi\beta^{-1}\otimes
I-\tau(\varphi\beta^{-1}\otimes
I)\Big{)}\rho_{(\alpha,\beta)}(m)-\Delta_{C}(\varphi\beta^{-1}(m))+(\varphi\beta^{-1}\otimes\varphi\beta^{-1})\Delta_{M}(m).$

By the same token,

$\displaystyle\rho(m)-\rho_{(\alpha,\beta)}(m)=(I\otimes\varphi\beta^{-1})\Delta_{M}(m),$
$\displaystyle\phi(m)-\phi_{(\alpha,\beta)}(m)=(\varphi\beta^{-1})R_{M}-R_{C}(\varphi\beta^{-1}).$

Thus, $(h,\rho,\phi)$ and
$(h_{(\alpha,\beta)},\rho_{(\alpha,\beta)},\phi_{(\alpha,\beta)})$ are
equivalent non-abelian 2-cocycles via a linear map $\varphi\beta^{-1}$. The
converse part can be checked similarly. ∎

##  6 Wells exact sequences for $\lambda$-weighted Rota-Baxter Lie coalgebras

In this section, we always suppose that

$\textstyle{\mathcal{E}:0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

is a fixed non-abelian extension of the $\lambda$-weighted Rota-Baxter Lie
coalgebra $(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$, and $t$ is its
retraction. Then there is a linear map $s:M\rightarrow E$ such that

$ft+sg=I_{E}.$ (6.1)

Let $(h,\rho,\phi)$ be a non-abelian 2-cocycle corresponding to the non-
abelian extension $\mathcal{E}$ induced by the retraction $t$. Define a map
$W:\mathrm{Aut}(C)\times\mathrm{Aut}(M)\rightarrow\mathrm{H}^{2}_{nab}(M,C)$
by

$W(\alpha,\beta)=[(h_{(\alpha,\beta)},\rho_{(\alpha,\beta)},\phi_{(\alpha,\beta)})-(h,\rho,\phi)].$
(6.2)

The map $W$ is called the Wells map.

######  Proposition 6.1.

The Well maps $W$ does not depend on the choice of retractions.

###### Proof.

For all $m\in M$, there is a $m_{0}\in M$ such that $m=\beta(m_{0})$. Since
$g$ is surjective, there exist elements $e,e_{0}\in E$, such that
$m=g(e),m_{0}=g(e_{0})$. Assume that $t^{\prime}$ is another retraction and
$(h^{\prime},\rho^{\prime},\phi^{\prime})$ is the corresponding induced non-
abelian 2-cocycle. Then $(h^{\prime},\rho^{\prime},\phi^{\prime})$ and
$(h,\rho,\phi)$ are equivalent non-abelian 2-cocycles via a linear map

$\varphi(m)=\varphi(g(e))=t^{\prime}(e)-t(e),~{}~{}\forall~{}m\in M.$ (6.3)

Using Eqs. (4.20)-(4.21), (5.13) and (6.3), we have

$\displaystyle h^{\prime}_{\alpha,\beta}(m)-h_{\alpha,\beta}(m)$
$\displaystyle=$
$\displaystyle(\alpha\otimes\alpha)h^{\prime}(\beta^{-1}(m))-(\alpha\otimes\alpha)h(\beta^{-1}(m))$
$\displaystyle=$
$\displaystyle(\alpha\otimes\alpha)h^{\prime}(m_{0})-(\alpha\otimes\alpha)h(m_{0})$
$\displaystyle=$
$\displaystyle(\alpha\otimes\alpha)h^{\prime}(g(e_{0}))-(\alpha\otimes\alpha)h(g(e_{0}))$
$\displaystyle=$ $\displaystyle(\alpha\otimes\alpha)((t^{\prime}\otimes
t^{\prime})\Delta_{E}(e_{0})-\Delta_{C}t^{\prime}(e_{0})-(t\otimes
t)\Delta_{E}(e_{0})+\Delta_{C}t(e_{0}))$ $\displaystyle=$
$\displaystyle(\alpha\otimes\alpha)\Big{(}((\varphi g+t)\otimes(\varphi
g+t))\Delta_{E}(e_{0})-(t\otimes t)\Delta_{E}(e_{0})-\Delta_{C}\varphi
g(e_{0})\Big{)}$ $\displaystyle=$ $\displaystyle(\alpha\otimes\alpha)((\varphi
g\otimes\varphi g)\Delta_{E}(e_{0})+(\varphi g\otimes
t)\Delta_{E}(e_{0})+(t\otimes\varphi g)\Delta_{E}(e_{0})-\Delta_{C}\varphi
g(e_{0}))$ $\displaystyle=$
$\displaystyle(\alpha\varphi\otimes\alpha\varphi)\Delta_{M}(m_{0})+(\alpha\varphi\otimes\alpha)\rho(m_{0})-(\alpha\otimes\alpha\varphi)\tau\rho(m_{0})-\Delta_{C}(\alpha\varphi(m_{0}))$
$\displaystyle=$
$\displaystyle(\alpha\varphi\beta^{-1}\otimes\alpha\varphi\beta^{-1})\Delta_{M}(m)+\Big{(}(\alpha\varphi\otimes\alpha)-(\alpha\varphi\otimes\alpha)\tau\Big{)}\rho\beta^{-1}(m)-\Delta_{C}(\alpha\varphi\beta^{-1}(m))$
$\displaystyle=$
$\displaystyle(\alpha\varphi\beta^{-1}\otimes\alpha\varphi\beta^{-1})\Delta_{M}(m)+\Big{(}(\alpha\varphi\beta^{-1}\otimes
I)-\tau(\alpha\varphi\beta^{-1}\otimes I)\Big{)}\rho_{(\alpha,\beta)}(m)$
$\displaystyle-\Delta_{C}(\alpha\varphi\beta^{-1}(m)).$

Analogously,

$\displaystyle\rho^{\prime}_{(\alpha,\beta)}(m)-\rho_{(\alpha,\beta)}(m)=(I\otimes\alpha\varphi\beta^{-1})\Delta_{M}(m),$
$\displaystyle\phi^{\prime}_{(\alpha,\beta)}(m)-\phi_{(\alpha,\beta)}(m)=(\alpha\varphi\beta^{-1})R_{M}(m)-R_{C}(\alpha\varphi\beta^{-1})(m).$

Thus,
$(h^{\prime}_{(\alpha,\beta)},\rho^{\prime}_{(\alpha,\beta)},\phi^{\prime}_{(\alpha,\beta)})$
and $(h_{(\alpha,\beta)},\rho_{(\alpha,\beta)},\phi_{(\alpha,\beta)})$ are
equivalent non-abelian 2-cocycles via the linear map
$\alpha\varphi\beta^{-1}$. Combining Lemma 4.1, we get that
$(h^{\prime}_{(\alpha,\beta)},\rho^{\prime}_{(\alpha,\beta)},\phi^{\prime}_{(\alpha,\beta)})-(h^{\prime},\rho^{\prime},\phi^{\prime})$
and
$(h_{(\alpha,\beta)},\rho_{(\alpha,\beta)},\phi_{(\alpha,\beta)})-(h,\rho,\phi)$
are equivalent via the linear map $\alpha\varphi\beta^{-1}-\varphi$. ∎

######  Lemma 6.1.

The following linear map $K$ is well defined:

$K:\mathrm{Aut}_{C}(E)\longrightarrow\mathrm{Aut}(C)\times\mathrm{Aut}(M),~{}K(\gamma)=(\alpha,\beta),~{}~{}\forall~{}\gamma\in\mathrm{Aut}_{C}(E),$
(6.4)

where

$\alpha(c)=t\gamma f(c),~{}\beta(m)=g\gamma(e),~{}~{}\forall~{}c\in C,m\in
M\hbox{ and }m=g(e),e\in E.$ (6.5)

###### Proof.

It is similar to Lemma 6.1 [15]. ∎

######  Theorem 6.1.

There is an exact sequence:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Aut}_{C}^{M}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\textstyle{\mathrm{Aut}_{C}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{K}$$\textstyle{\mathrm{Aut}(C)\times\mathrm{Aut}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{W}$$\textstyle{\mathrm{H}^{2}_{nab}(M,C)}$

where
$\mathrm{Aut}_{C}^{M}(E)=\\{\gamma\in\mathrm{Aut}_{C}(E)|K(\gamma)=(I_{C},I_{M})\\}$.

###### Proof.

It is obviously, $\mathrm{Ker}K=\mathrm{Im}T$ and $T$ is injective. We only
need to prove that $\mathrm{Ker}W=\mathrm{Im}K$. In view of Theorem 5.2, for
all $(\alpha,\beta)\in\mathrm{Ker}W$, we know that $(\alpha,\beta)$ is
extensible with respect to the non-abelian extension $\mathcal{E}$, that is,
there is a $\gamma\in\mathrm{Aut}_{C}^{M}(E)$, such that $f\alpha=\gamma
f,~{}\beta g=g\gamma$, which follows that $\alpha=tf\alpha=t\gamma
f,~{}\beta(m)=\beta g(e)=g\gamma(e).$ Thus, $(\alpha,\beta)\in\mathrm{Im}K$.
On the other hand, for any $(\alpha,\beta)\in\mathrm{Im}K$, there is an
isomorphism $\gamma\in\mathrm{Aut}_{C}(E)$, such that (6.5) holds. Combining
(6.1) and $\mathrm{Im}f=\mathrm{Ker}g$, we have $f\alpha=ft\gamma
f=(I_{E}-sg)\gamma f=\gamma f$ and $\beta g=g\gamma$. Thus, $(\alpha,\beta)$
is extensible with respect to the non-abelian extension $\mathcal{E}$.
According to Theorem 5.2, $(\alpha,\beta)\in\mathrm{Ker}W$. So
$\mathrm{Ker}W=\mathrm{Im}K$. ∎

Suppose that

$\displaystyle\mathrm{Z}_{nab}^{1}(M,C)$
$\displaystyle=\left\\{\varphi:M\rightarrow C\left|\begin{aligned}
&(\varphi\otimes I)\rho-\tau(\varphi\otimes
I)\rho=\Delta_{C}\varphi-(\varphi\otimes\varphi)\Delta_{M},\\\
&(I\otimes\varphi)\Delta_{M}=0,\varphi
R_{M}=R_{C}\varphi\end{aligned}\right.\right\\}.$ (6.6)

It is easy to check that $\mathrm{Z}_{nab}^{1}(M,C)$ is an abelian group,
which is called a non-abelian 1-cocycle.

######  Proposition 6.2.

(i) The linear map $\chi:\mathrm{KerK}\rightarrow\mathrm{Z}_{nab}^{1}(M,C)$
given by

$\chi(\gamma)=\varphi_{\gamma},~{}~{}\forall~{}\gamma\in\mathrm{Ker}K,$

is a homomorphism of groups, where

$\varphi_{\gamma}(m)=t\gamma(e)-t(e),~{}~{}\forall~{}m\in M,g(e)=m~{}\hbox{for
some}~{}e\in E.$ (6.7)

(ii) $\chi$ is an isomorphism, that is,
$\mathrm{KerK}\cong\mathrm{Z}_{nab}^{1}(M,C)$.

###### Proof.

(i) For all $m\in M$, if there are $e_{1},e_{2}\in E$ such that
$m=g(e_{1})=g(e_{2})$. Due to $\mathrm{Ker}g=\mathrm{Im}f$, there is a $c\in
C$ satisfying $f(c)=e_{1}-e_{2}$, then

$\displaystyle t\gamma(e_{1}-e_{2})-t(e_{1}-e_{2})=t\gamma f(c)-tf(c)=0.$

Thus, $\varphi_{\gamma}$ is independent on the choice of $e$. Using Eqs.
(4.20)-(4.21), (6.5) and (6.7), for all $m\in M$, we have

$\displaystyle(\varphi_{\gamma}\otimes I)\rho(m)-\tau(\varphi_{\gamma}\otimes
I)\rho(m)-\Delta_{C}(\varphi_{\gamma}(m))+(\varphi_{\gamma}\otimes\varphi_{\gamma})\Delta_{M}(m)$
$\displaystyle=$ $\displaystyle(\varphi_{\gamma}g\otimes
t)\Delta_{E}(e)-\tau(\varphi_{\gamma}g\otimes
t)\Delta_{E}(e)-\Delta_{C}(\varphi_{\gamma}g(e))+(\varphi_{\gamma}\otimes\varphi_{\gamma})\Delta_{M}(g(e))$
$\displaystyle=$ $\displaystyle(t\gamma\otimes t)\Delta_{E}(e)-(t\otimes
t)\Delta_{E}(e)+(t\otimes t\gamma)\Delta_{E}(e)-(t\otimes t)\Delta_{E}(e)$
$\displaystyle-\Delta_{C}(t\gamma(e))+\Delta_{C}(t(e))+(t\gamma\otimes
t\gamma)\Delta_{E}(e)-(t\gamma\otimes t)\Delta_{E}(e)$
$\displaystyle-(t\otimes t\gamma)\Delta_{E}(e)+(t\otimes t)\Delta_{E}(e)$
$\displaystyle=$ $\displaystyle\Delta_{C}(t(e))-(t\otimes
t)\Delta_{E}(e)+(t\gamma\otimes t\gamma)\Delta_{E}(e)-\Delta_{C}(t\gamma(e))$
$\displaystyle=$ $\displaystyle h(g\gamma(e))-h(g(e))$ $\displaystyle=$
$\displaystyle 0.$

Analogously,
$(I\otimes\varphi_{\gamma})\Delta_{M}=0,~{}~{}\varphi_{\gamma}R_{M}-R_{C}\varphi_{\gamma}=0$.
Therefore, $\varphi_{\gamma}\in\mathrm{Z}_{nab}^{1}(M,C)$ and $\chi$ is well-
defined. For any $\gamma_{1},\gamma_{2}\in\mathrm{Ker}K$ and $m\in M$, we
obtain

$\displaystyle\chi(\gamma_{1}\gamma_{2})(m)$
$\displaystyle=t\gamma_{1}\gamma_{2}(e)-t(e)$
$\displaystyle=(\varphi_{\gamma_{1}}g+t)\gamma_{2}(e)-t(e)$
$\displaystyle=\varphi_{\gamma_{1}}g(e)+\varphi_{\gamma_{2}}(m)$
$\displaystyle=\varphi_{\gamma_{1}}(m)+\varphi_{\gamma_{2}}(m)$
$\displaystyle=\chi(\gamma_{1})(m)+\chi(\gamma_{2})(m),$

which means that $\chi$ is a homomorphism of groups.

(ii) For all $\gamma\in\mathrm{Ker}K$, then we have $K(\gamma)=0$, that is,
$t\gamma f=I_{C},g=g\gamma.$ It follows that $\gamma(e)-e\in\mathrm{Ker}g$.
Combining $\mathrm{Imf}=\mathrm{Ker}g$, there is an element $c\in C$ such that
$\gamma(e)-e=f(c)$. If $\chi(\gamma)=0$, then
$\chi(\gamma)(m)=t\gamma(e)-t(e)=0.$ Thus, $c=tf(c)=t\gamma(e)-t(e)=0$. Then
$\gamma(e)-e=0$ and thus $\gamma=I_{E}$, which indicates that $\chi$ is
injective. Secondly, we check that $\chi$ is surjective. For any
$\varphi\in\mathrm{Z}_{nab}^{1}(M,C)$, give a linear map $\gamma:E\rightarrow
E$ by

$\gamma(e)=f\varphi g(e)+e,~{}~{}\forall~{}e\in E.$ (6.8)

Then $\gamma$ is a homomorphism of $\lambda$-weighted Rota-Baxter Lie
coalgebras. Indeed, by Eqs. (4.21),(6.1) and (6.6), we have for all $e\in E$,

$\displaystyle(f\varphi g\otimes I)\Delta_{E}(e)$ $\displaystyle=$
$\displaystyle(f\varphi g\otimes ft)\Delta_{E}(e)+(f\varphi g\otimes
sg)\Delta_{E}(e)$ $\displaystyle=$ $\displaystyle(f\otimes f)(\varphi\otimes
I)(g\otimes t)\Delta_{E}(e)+(f\otimes s)(\varphi\otimes I)\Delta_{M}(g(e))$
$\displaystyle=$ $\displaystyle(f\otimes f)(\varphi\otimes
I)\rho(g(e))+(f\otimes s)(\varphi\otimes I)\Delta_{M}(g(e))$ $\displaystyle=$
$\displaystyle(f\otimes f)(\varphi\otimes I)\rho(g(e)),$ (6.9)

and by the same token,

$(I\otimes f\varphi g)\Delta_{E}(e)=-\tau(f\otimes f)(\varphi\otimes
I)\rho(g(e)).$ (6.10)

Using Eqs. (6.8)-(6.10), we have for all $e\in E$,

$\displaystyle(\gamma\otimes\gamma)\Delta_{E}(e)$ $\displaystyle=$
$\displaystyle(f\varphi g\otimes I)\Delta_{E}(e)+(I\otimes f\varphi
g)\Delta_{E}(e)+(I\otimes I)\Delta_{E}(e)+(f\varphi g\otimes f\varphi
g)\Delta_{E}(e)$ $\displaystyle=$ $\displaystyle(f\otimes f)(\varphi\otimes
I)\rho(g(e))-\tau(f\otimes f)(\varphi\otimes I)\rho(m)+(f\varphi\otimes
f\varphi)\Delta_{M}(g(e))+\Delta_{E}(e)$ $\displaystyle=$
$\displaystyle(f\otimes f)\Delta_{C}\varphi(g(e))+\Delta_{E}(e)$
$\displaystyle=$ $\displaystyle\Delta_{C}(f\varphi g(e)+e)$ $\displaystyle=$
$\displaystyle\Delta_{E}(\gamma(e)).$

Analogously, $\gamma R_{E}=R_{E}\gamma$. Therefore, $\gamma$ is a homomorphism
of $\lambda$-weighted Rota-Baxter Lie coalgebras. In the sequel, we state that
$\gamma$ is bijective. If $\gamma(e)=f\varphi g(e)+e=0$, then $0=ftf\varphi
g(e)+e=-ft(e)+e$. Combining $gf=0$, we get $f\varphi g(e)=f\varphi gft(e)=0$,
which follows that $e=-f\varphi g(e)=0$. So $\gamma$ is injective. For all
$e\in E$, due to $gf=0$, we have $\gamma(e-f\varphi g(e))=e-f\varphi
g(e)+f\varphi g(e-f\varphi g(e))=e$, which yields that $\gamma$ is bijective.
Since $gf=0$, $\gamma f(c)=f\varphi gf(c)+f(c)=f(c),~{}\forall~{}c\in C$. In
all, $\gamma\in\mathrm{Aut}_{C}(E)$. Combining $gf=0,tf=I_{C}$ and (6.8), for
all $c\in C,m\in M$, we have

$\displaystyle\alpha(c)=t\gamma f(c)=t(f\varphi gf(c)+f(c))=tf(c)=c,$
$\displaystyle\beta(m)=g\gamma(e)=g(f\varphi g(e)+e)=g(e)=m,$

which imply that $\alpha=I_{C},\beta=I_{M}$, thus $\gamma\in\mathrm{Ker}K$.
Therefore, $\chi$ is bijective. So
$\mathrm{Ker}K\simeq\mathrm{Z}_{nab}^{1}(M,C)$. ∎

Combining Theorem 6.1 and Proposition 6.2, we have

######  Theorem 6.2.

There is an exact sequence:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
25.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
25.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Z}_{nab}^{1}(M,C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
82.45085pt\raise 5.30833pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.30833pt\hbox{$\scriptstyle{i}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 96.65665pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
96.65665pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Aut}_{C}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
143.51732pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{K}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 159.74023pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
159.74023pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Aut}(C)\times\mathrm{Aut}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
247.04726pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{W}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 263.83893pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
263.83893pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{H}^{2}_{nab}(M,C)}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

##  7 Particular case: abelian extensions of $\lambda$-weighted Rota-Baxter
Lie coalgebras

In this section, we investigate the results of previous section in particular
case. Let $(C,\Delta_{C},R_{C})$ and $(M,\Delta_{M},R_{M})$ be two
$\lambda$-weighted Rota-Baxter Lie coalgebras. Let

$\textstyle{\mathcal{E}:0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

be an abelian extension of $(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$.
Denote the set of all equivalent classes of abelian extensions
$(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$ by $\mathrm{Ext}_{ab}(M,C)$.

######  Proposition 7.1.

The triple $(M,\rho,R_{M})$ is a right Lie comodule of $(C,\Delta_{C},R_{C})$,
where $\rho$ is given by (4.21).

###### Proof.

Since $g$ is surjective, for all $m\in M$, there is an element $e\in E$ such
that $g(e)=m$. By direct computations,

$\displaystyle(I\otimes\Delta_{C})\rho(m)-(\rho\otimes
I)\rho(m)+(I\otimes\tau)(\rho\otimes I)\rho(m)$ $\displaystyle=$
$\displaystyle(I\otimes\Delta_{C})\rho g(e)-(\rho\otimes I)\rho
g(e)+(I\otimes\tau)(\rho\otimes I)\rho g(e)$ $\displaystyle=$
$\displaystyle(g\otimes\Delta_{C}t)\Delta_{E}(e)-(\rho g\otimes
t)\Delta_{E}(e)+(I\otimes\tau)(\rho g\otimes t)\Delta_{E}(e)$ $\displaystyle=$
$\displaystyle(g\otimes\Delta_{C}t)\Delta_{E}(e)-((g\otimes
t)\Delta_{E}\otimes t)\Delta_{E}(e)+(I\otimes\tau)((g\otimes
t)\Delta_{E}\otimes t)\Delta_{E}(e)$ $\displaystyle=$
$\displaystyle(g\otimes\Delta_{C}t)\Delta_{E}(e)-(g\otimes(t\otimes
t)\Delta_{E})\Delta_{E}(e)$ $\displaystyle=$
$\displaystyle(g\otimes(\Delta_{C}t-(t\otimes t)\Delta_{E}))\Delta_{E}(e)$
$\displaystyle=$ $\displaystyle(g\otimes-
hg)\Delta_{E}(e)=(I\otimes-h)\Delta_{M}(m)=0.$

By the same token, (2.5) holds. Thus, $(M,\rho,R_{M})$ is a right Lie comodule
of $(C,\Delta_{C},R_{C})$. ∎

######  Remark 7.1.

In the case of non-abelian extensions, $(M,\rho,R_{M})$ is not a right Lie
comodule of $(C,\Delta_{C},R_{C})$.

######  Theorem 7.1.

  1. $(i)$

The triple $(C\oplus M,\Delta_{(h,\rho)},R_{\phi})$ is a $\lambda$-weighted
Rota-Baxter Lie coalgebra if and only if $(h,\phi)$ is a 2-cocycle of
$(C,\Delta_{C},R_{C})$ with coefficients in the Lie comodule $(M,\rho,R_{M})$.

  2. $(ii)$

Abelian extensions of a $\lambda$-weighted Rota-Baxter Lie coalgebra
$(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$ are classified by the second
cohomology group $\mathrm{\bar{H}}_{RB}^{2}(M,C)$ of $(C,\Delta_{C},R_{C})$
with coefficients in $(M,\rho,R_{M})$.

###### Proof.

  1. $(i)$

It is similar to the proof of Proposition 4.1.

  2. $(ii)$

By the same token as in Theorem 4.1, we can get the statement.

∎

######  Theorem 7.2.

Let $\mathcal{E}:0\rightarrow C\stackrel{{\scriptstyle
f}}{{\rightarrow}}E\stackrel{{\scriptstyle g}}{{\rightarrow}}M\rightarrow 0$
be an abelian extension of a $\lambda$-weighted Rota-Baxter Lie coalgebra
$(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$ and $t$ be its retraction.
Assume that $(h,\phi)$ is a 2-cocycle and $(M,\rho,R_{M})$ is a right Lie
comodule of $(C,\Delta_{C},R_{C})$ associated to $\mathcal{E}$. A pair
$(\alpha,\beta)\in\mathrm{Aut}(C)\times\mathrm{Aut}(M)$ is extensible with
respect to the abelian extension $\mathcal{E}$ if and only if there is a
linear map $\varphi:M\rightarrow C$ satisfying the following conditions:

$h\beta-(\alpha\otimes\alpha)h=(\varphi\otimes\alpha)\rho-\tau(\varphi\otimes\alpha)\rho-\Delta_{C}\varphi,$
$\rho\beta=(\beta\otimes\alpha)\rho,$ $\phi\beta-\alpha\phi=\varphi
R_{M}-R_{C}\varphi.$

###### Proof.

It can be get directly from Theorem 5.1. ∎

Let $\mathcal{E}:0\rightarrow C\stackrel{{\scriptstyle
f}}{{\rightarrow}}E\stackrel{{\scriptstyle g}}{{\rightarrow}}M\rightarrow 0$
be an abelian extension of the$\lambda$-weighted Rota-Baxter Lie coalgebra
$(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$ and $t$ be its retraction.
Assume that $(h,\phi)$ is a 2-cocycle and $(M,\rho,R_{M})$ is a right Lie
comodule of $(C,\Delta_{C},R_{C})$ associated to $\mathcal{E}$.

The space

$C_{\rho}=\\{(\alpha,\beta)\in\mathrm{Aut}(C)\times\mathrm{Aut}(M)|\rho\beta=(\beta\otimes\alpha)\rho,~{}~{}\forall~{}m\in
V\\}$

is called the space of compatible pairs of automorphisms. It is easy to verify
that $C_{\rho}$ is a subgroup of $\mathrm{Aut}(C)\times\mathrm{Aut}(M)$. For
all $(\alpha,\beta)\in\mathrm{Aut}(C)\times\mathrm{Aut}(M)$,
$(h_{(\alpha,\beta)},\phi_{(\alpha,\beta)})$ may not be a 2-cocycle. Indeed,
we have

######  Proposition 7.2.

The tuple $(h_{(\alpha,\beta)},\phi_{(\alpha,\beta)})$ is a 2-cocycle
corresponding to the abelian extension $\mathcal{E}$ if $(\alpha,\beta)\in
C_{\rho}$.

###### Proof.

Take the same procedure as the proof of Proposition 5.1. ∎

######  Theorem 7.3.

Let $\mathcal{E}:0\rightarrow C\stackrel{{\scriptstyle
f}}{{\rightarrow}}E\stackrel{{\scriptstyle g}}{{\rightarrow}}M\rightarrow 0$
be an abelian extension of a $\lambda$-weighted Rota-Baxter Lie coalgebra
$(C,\Delta_{C},R_{C})$ by $(M,\Delta_{M},R_{M})$ and $(h,\phi)$ be a 2-cocycle
associated to $\mathcal{E}$. A pair
$(\alpha,\beta)\in\mathrm{Aut}(C)\times\mathrm{Aut}(M)$ is extensible with
respect to the abelian extension $\mathcal{E}$ if and only if the following
conditions hold:

  1. $(i)$

$(\alpha,\beta)\in C_{\rho}$.

  2. $(ii)$

$(h,\phi)$ and $(h_{(\alpha,\beta)},\phi_{(\alpha,\beta)})$ are in the same
cohomological class.

###### Proof.

Combining Theorem 5.2 and Proposition 7.2, we get the statement. ∎

In the case of abelian extensions, $\mathrm{Z}^{1}_{nab}(M,C)$ defined by
(6.6) turns to $\mathrm{\bar{Z}}_{RB}^{1}(M,C)$ given by (3.7). In the light
of Theorem 6.2 and Theorem 7.3, we have the following exact sequence:

######  Theorem 7.4.

There is an exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{\bar{Z}}_{RB}^{1}(M,C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathrm{Aut}_{C}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{K}$$\textstyle{C_{\rho}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{W}$$\textstyle{\mathrm{\bar{H}}^{2}_{RB}(M,C).}$

Acknowledgments

Project supported by the National Natural Science Foundation of China
(11871421), the Natural Science Foundation of Zhejiang Province of China
(LY19A010001) and the Science and Technology Planning Project of Zhejiang
Province (2022C01118).

Statements and Declarations

All datasets underlying the conclusions of the paper are available to readers.
No conflict of interest exits in the submission of this manuscript.

## References

  * [1] C. Bai, L. Guo, X. Ni, Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras, Comm. Math. Phys. 297 (2010), 553-596.
  * [2] C. Bai, L. Guo, X. Ni, Relative Rota-Baxter algebras and tridendriform algebras, J. Algebra Appl. 12 (2013), 1350027.
  * [3] C. Bai, L. Guo, G. Liu, T. Ma, Rota-Baxter Lie bialgebras, classical Yang-Baxter equations and special L-dendriform bialgebras, arXiv.2207.08703.
  * [4] V. G. Bardakov, M. Singh, Extensions and automorphisms of Lie algebras, J. Algebra Appl. 16 (2017), 1750162.
  * [5] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10 (1960), 731-742.
  * [6] J. M. Casas, E. Khmaladze, M. Ladra, Low-dimensional non-abelian Leibniz cohomology, Forum Math. 25 (3) (2013), 443–469.
  * [7] S. Chen, Q. Lou, Q. Sun, Cohomologies of Rota-Baxter Lie triple systems and applications, Commun. Algebra 51 (10) (2023), 4299-4315.
  * [8] S. Chen, Y. Sheng, Z. Zheng, Non-abelian extensions of Lie 2-algebras, Sci. China Math. 55 (8) (2012), 1655–1668.
  * [9] A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210 (2000), 249-273.
  * [10] W. Michaelis, Lie coalgebras, Adv. in Math. 38 (1) (1980), 1-54.
  * [11] A. Das, Cohomology and deformations of weighted Rota-Baxter operators, J. Math. Phys. 63 (2022), 091703.
  * [12] A. Das, Cohomology of weighted Rota-Baxter Lie algebras and Rota-Baxter paired operators, arXiv.2109.01972.
  * [13] A. Das, A. Mandal, Extensions, deformations and categorifications of AssDer pairs, arXiv: 2002.11415.
  * [14] A. Das, N. Ratheeb, Extensions and automorphisms of Rota-Baxter groups, J. Algebra 636 (2023), 626–665.
  * [15] L. Du, Y. Tan, Wells sequences for abelian extensions of Lie coalgebras, J. Algebra Appl. 20 (8) (2021), 2150149.
  * [16] L. Du, Y. Tan, Coderivations, abelian extensions and cohomology of Lie coalgebras, Comm. Algebra 49 (10) (2021), 4519-4542.
  * [17] K. Ebrahimi-Fard, Loday-type algebras and the Rota-Baxter relation, Lett. Math. Phys. 61 (2002), 139-147.
  * [18] S. Eilenberg, S. Maclane, Cohomology theory in abstract groups. II. Group extensions with non-abelian kernel, Ann. Math. 48 (1947), 326-341.
  * [19] Y. Frégier, Non-abelian cohomology of extensions of Lie algebras as Deligne groupoid, J. Algebra 398 (2014), 243–257.
  * [20] W. L. Gan, T. Schedler, The necklace Lie coalgebra and renormalization algebras, J. Noncommut. Geom. 2 (2008), 195–214.
  * [21] S. Goswamia, S. K. Mishraa, G. Mukherjee, Automorphisms of extensions of Lie-Yamaguti algebras and inducibility problem, J. Algebra 641 (2024), 268–306.
  * [22] N. S. Gu, L. Guo, Generating functions from the viewpoint of Rota-Baxter algebras, Discrete Math. 338 (2015), 536-554.
  * [23] Y. Guo, B. Hou, Crossed modules and non-abelian extensions of Rota-Baxter Leibniz algebras, J. Geom. Phys. 191 (2023), 104906.
  * [24] L. Guo, B. Zhang, Renormalization of multiple zeta values, J. Algebra 319 (2008), 3770-3809.
  * [25] S. Guo, Y. Qin, K. Wang, G. Zhou, Deformations and cohomology theory of Rota-Baxter 3-Lie algebras of arbitrary weights, J. Geom. Phys. 183 (2023), 104704.
  * [26] S. K. Hazra, A. Habib, Wells exact sequence and automorphisms of extensions of Lie superalgebras, J. Lie Theory 30 (2020), 179–199.
  * [27] N. Inassaridze, E. Khmaladze, M. Ladra, Non-abelian cohomology and extensions of Lie algebras, J. Lie Theory 18 (2008), 413–432.
  * [28] P. Jin, H. Liu, The Wells exact sequence for the automorphism group of a group extension, J. Algebra 324 (2010), 1219–1228.
  * [29] J. Liu, Y. Sheng, Q. Wang, On non-abelian extensions of Leibniz algebras, Commun. Algebra 46 (2) (2018), 574–587.
  * [30] B. A. Kupershmidt, What a classical $r$-matrix really is, J. Nonlinear Math. Phys. 6 (4) (1999), 448-488.
  * [31] S. K. Mishra, A. Das, S. K. Hazra, Non-abelian extensions of Rota-Baxter Lie algebras and inducibility of automorphisms, Linear. Algebra. Appl. 669 (2023), 147-174.
  * [32] I. B. S. Passi, M. Singh, M. K. Yadav, Automorphisms of abelian group extensions, J. Algebra 324 (2010), 820–830.
  * [33] D. Sinha, B. Walter, Lie coalgebras and rational homotopy theory, II: Hopf invari ants, Trans. Amer. Math. Soc. 365 (2013), 861–883.
  * [34] R. Tang, C. Bai, L. Guo, Y. Sheng, Deformations and their controlling cohomologies of $\mathcal{O}$-operators, Comm. Math. Phys. 368 (2) (2019), 665-700.
  * [35] R. Tang, Y. Frégier, Y. Sheng, Cohomologies of a Lie algebra with a derivation and applications, J. Algebra. 534 (2) (2019), 65-99.
  * [36] K. Wang, G. Zhou, Deformations and homotopy theory of Rota-Baxter algebras of any weight, arXiv:2108.06744.
  * [37] C. Wells, Automorphisms of group extensions, Trans. Amer. Math. Soc. 155 (1971), 189-194.
  * [38] Y. Zhang, D. Chen, X. Gao, Y. Luo, Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles, Pacific J. Math. 302 (2019), 741-766.