# Lie theory and cohomology of relative Rota-Baxter operators

Jun Jiang  Department of Mathematics, Jilin University, Changchun 130012,
Jilin, China<EMAIL_ADDRESS>,  Yunhe Sheng  Department of
Mathematics, Jilin University, Changchun 130012, Jilin, China
<EMAIL_ADDRESS>and  Chenchang Zhu  Mathematics Institute, Georg-August-
University Gottingen, Bunsenstrasse 3-5 37073, Gottingen, Germany
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we establish a local Lie theory for relative Rota-Baxter
operators of weight $1$. First we recall the category of relative Rota-Baxter
operators of weight $1$ on Lie algebras and construct a cohomology theory for
them. We use the second cohomology group to study infinitesimal deformations
of relative Rota-Baxter operators and modified $r$-matrices. Then we introduce
a cohomology theory of relative Rota-Baxter operators on a Lie group. We
construct the differentiation functor from the category of relative Rota-
Baxter operators on Lie groups to that on Lie algebras, and extend it to the
cohomology level by proving the Van Est theorem between the two cohomology
theories. We integrate a relative Rota-Baxter operator of weight 1 on a Lie
algebra to a local relative Rota-Baxter operator on the corresponding Lie
group, and show that the local integration and differentiation are adjoint to
each other. Finally, we give two applications of our integration of Rota-
Baxter operators: one is to give an explicit formula for the factorization
problem, and the other is to provide an integration for matched pairs.

###### Key words and phrases:

Rota-Baxter operator, modified Yang-Baxter equation, cohomology, Van Est map,
differentiation, integration, Lie theory
2020 Mathematics Subject Classification. 17B38, 17B56, 22E60

###### Contents

  1. 1 Introduction
  2. 2 Relative Rota-Baxter operators on Lie algebras and their cohomology theory
    1. 2.1 The category $\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}$ of relative Rota-Baxter operators on Lie algebras
    2. 2.2 A cohomology theory in $\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}$
    3. 2.3 Cohomology theory and deformations in $\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}$ and modified $r$-matrices
  3. 3 Relative Rota-Baxter operators on Lie groups and their cohomology theory
    1. 3.1 The category $\mathsf{RB_{G}^{H}}$ of relative Rota-Baxter operators on Lie groups
    2. 3.2 A cohomology theory in $\mathsf{RB_{G}^{H}}$
  4. 4 Differentiation and Van Est theory
    1. 4.1 The differentiation functor for relative Rota-Baxter operators
    2. 4.2 The Van Est homomorphism for the cohomologies
    3. 4.3 An explicit example
  5. 5 Local integration of relative Rota-Baxter operators on Lie algebras
    1. 5.1 Local integration functor and adjointness
  6. 6 Geometric applications
    1. 6.1 Local descendent Lie group
    2. 6.2 Application in the local factorization problem
    3. 6.3 Application in the integration of matched pairs

##  1\. Introduction

The notion of Rota-Baxter operators on associative algebras was introduced by
G. Baxter and they are applied in the Connes-Kreimer’s algebraic approach to
renormalization of quantum field theory [7, 15]. In the corresponding semi-
classical world, the notion of relative Rota-Baxter operators (also called
${\mathcal{O}}$-operators) on Lie algebras was introduced in [19], and they
are closely related to various classical Yang-Baxter equations. Moreover, from
a more algebraic approach, a relative Rota-Baxter operator naturally gives
rise to a pre-Lie or a post-Lie algebra, and play important roles in
mathematical physics [2, 5].

A Rota-Baxter operator of weight 0 on a Lie algebra is naturally the operator
form of a classical skew-symmetric $r$-matrix [31]. Rota-Baxter operators of
weight 1 are in one-to-one correspondence with solutions of the modified Yang-
Baxter equation (see Remark 2.3), and give rise to factorizations of Lie
algebras. Integrating such factorizations locally to the level of Lie groups
provides natural solutions of a certain Hamiltonian system, which is of great
interest for the people coming from integrable systems [31]. In a recent work
[16], the notion of a Rota-Baxter operator on a Lie group was introduced. In
particular, one can obtain a Rota-Baxter operator of weight 1 on a Lie algebra
by taking the differentiation of a Rota-Baxter operator on a Lie group.

Rota-Baxter operators are special cases of relative Rota-Baxter operators,
where we distinguish the source and the target of the operator (see Def. 2.1
and 3.1). In this article, we further study the differentiation and
integration problem for relative Rota-Baxter operators, and explore their
cohomology theory. We find such a separation of the source and the target
makes the cohomology theory and Lie theory actually much more clear.
Differentiation on the level of cohomology, namely a version of Van-Est
theorem, is also given. We show that differentiation and local integration
work for relative Rota-Baxter operators. Moreover, these two procedures are
adjoint to each other. Our results in particular imply that we may integrate a
Rota-Baxter operator on a Lie algebra to a local Rota-Baxter operator on a Lie
group in a functorial way. This seems to provide some global geometrical
insight to the above mentioned problem in integrable systems. Moreover, via
our integration, we are able to provide an explicit formula for the
factorization problem on the level of Lie groups.

Differentiation works without much obstruction, and follows from adapted Lie
theoretical calculations. Integration, even local integration, is however, a
bit tricky: the relative Rota-Baxter operator is not a Lie algebra or a Lie
group homomorphism. To integrate such a map, we turn to its graph, which has a
relative good Lie theoretical property but not completely compatible with
exponential maps. Nevertheless, a bit luck happens, and the non-compatibleness
somehow falls in the kernel of the operator and the local integration works.
To really achieve the global integration, we expect some topological
obstruction. In fact, we have already faced this when we try to integrate an
infinite dimensional Lie algebra [26, 38], where the obstruction lies in
second cohomology groups; and when we try to integrate a Lie algebroid [9] to
a Lie groupoid. On the other hand, we might also use Malcev’s method to extend
local structures to global ones [24]. As therein, an obstruction in algebraic
terms would show up. Nevertheless, it turns out that the algebraic obstruction
therein is actually topological in nature [13]. Thus, the local integration
and some topological invariants obtained in this article should be good
ingredients for the global integration in the next step.

We notice that the cohomology theory of relative Rota-Baxter operators of
weight 0 on Lie algebras, together with those on associative algebras have
been much studied recently [10, 11, 21, 33]. Also there are some further
progresses on Rota-Baxter operators on groups in recently works [3, 4, 14].

A byproduct of our cohomology theory is to provide a classification of
deformation of modified $r$-matrices, which seems to be easier to carry out
from the operator point of view.

Now we describe our results in the time order: we first show that a relative
Rota-Baxter operator $B:\mathfrak{h}\,\rightarrow\,\mathfrak{g}$ of weight 1
on a Lie algebra $\mathfrak{g}$ with respect to an action
$\phi:\mathfrak{g}\,\rightarrow\,\mathrm{Der}(\mathfrak{h})$ gives rise to a
representation of the descendent Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{B})$ on the vector space $\mathfrak{g}$. It
turns out that the Chevalley-Eilenberg cochain complex of the descendent Lie
algebra gives rise to the controlling algebra of the relative Rota-Baxter
operator $B$ of weight 1 [34]. In principal, the controlling algebra of an
algebraic structure together with a differential coming from the algebraic
structure itself provides a good cohomology theory for this algebraic
structure. We thus define the cohomology of the relative Rota-Baxter operator
$B$ to be the Chevalley-Eilenberg cohomology of the corresponding descendent
Lie algebra. As an application or verification of this cohomology theory, we
apply the second cohomology group to study infinitesimal deformations of
relative Rota-Baxter operators of weight 1 on Lie algebras and that of
modified $r$-matrices. Then parallelly, we show that a relative Rota-Baxter
operator $\mathcal{B}:H\,\rightarrow\,G$ on a Lie group $G$ with respect to an
action $\Phi:G\,\rightarrow\,\mathrm{Aut}(H)$ gives rise to a representation
of the descendent Lie group $(H,\star)$ on the vector space $\mathfrak{g}$.
Analogously, the cohomology of a relative Rota-Baxter operator $\mathcal{B}$
is taken to be the cohomology of the corresponding descendent Lie group.
Moreover, we establish the Van Est map from the cochain complex of the
relative Rota-Baxter operator $\mathcal{B}$ on a Lie group $G$ to the cochain
complex of the relative Rota-Baxter operator $B$ on the corresponding Lie
algebra $\mathfrak{g}$. Van Est theorems hold, and this might also be viewed
as a justification of our cohomology theory for relative Rota-Baxter operators
on Lie groups.

In Section 2, we give the cohomology of a relative Rota-Baxter operator of
weight 1 on a Lie algebra $\mathfrak{g}$ with respect to an action
$\phi:\mathfrak{g}\,\rightarrow\,\mathrm{Der}(\mathfrak{h})$ using the
Chevalley-Eilenberg cohomology of the descendent Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{B})$. As applications, we study infinitesimal
deformations of a relative Rota-Baxter operator of weight 1 and modified
$r$-matrices using the second cohomology group.

In Section 3, we give the cohomology of a relative Rota-Baxter operator on a
Lie group $G$ with respect to an action $\Phi:G\,\rightarrow\,\mathrm{Aut}(H)$
using the cohomology of the descendent Lie group $(H,\star)$.

In Section 4, we first show that the Lie algebra of the descendent Lie group
$(H,\star)$ is the descendent Lie algebra $(\mathfrak{h},[\cdot,\cdot]_{B})$,
and the differentiation of the representation
$\Theta:H\,\rightarrow\,\mathfrak{gl}(\mathfrak{g})$ of the descendent Lie
group $(H,\star)$ is the representation
$\theta:\mathfrak{h}\,\rightarrow\,\mathfrak{gl}(\mathfrak{g})$ of the
descendent Lie algebra $(\mathfrak{h},[\cdot,\cdot]_{B})$. Then we achieve our
differentiation operator and consequently, we establish the Van Est map from
the cochain complex of the relative Rota-Baxter operator $\mathcal{B}$ on a
Lie group $G$ to the cochain complex of the relative Rota-Baxter operator $B$
of weight 1 on the corresponding Lie algebra $\mathfrak{g}$.

In Section 5, we introduce the notion of a local relative Rota-Baxter operator
on a Lie group $G$ and show that a relative Rota-Baxter operator
$B:\mathfrak{h}\,\rightarrow\,\mathfrak{g}$ of weight 1 on a Lie algebra can
be integrated to a local relative Rota-Baxter operator on the corresponding
connected and simply connected Lie group $G$.

In Section 6, we give two applications of the integration of Rota-Baxter
operators. One is to provide an explicit formula of local factorization
problem and the other is to provide an integration for some special matched
pairs.

Acknowledgements.  We give our warmest thanks to Jiang-Hua Lu for very helpful
discussions. This research is supported by NSFC (11922110) and DFG ZH 274/3-1.

##  2\. Relative Rota-Baxter operators on Lie algebras and their cohomology
theory

In this section, we define the cohomology of relative Rota-Baxter operators of
weight 1 on Lie algebras. As applications, we classify infinitesimal
deformations of relative Rota-Baxter operators of weight 1 on Lie algebras
using the second cohomology group.

###  2.1. The category $\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}$ of relative
Rota-Baxter operators on Lie algebras

In the sequel, $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ and
$(\mathfrak{h},[\cdot,\cdot]_{\mathfrak{h}})$ are Lie algebras. Let
$\phi:\mathfrak{g}\to\mathrm{Der}(\mathfrak{h})$ be a Lie algebra
homomorphism. We call $\phi$ an action of $\mathfrak{g}$ on $\mathfrak{h}$.

######  Definition 2.1.

Let $\phi:\mathfrak{g}\rightarrow\mathrm{Der}(\mathfrak{h})$ be an action of a
Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ on a Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{\mathfrak{h}})$. A linear map
$B:\mathfrak{h}\rightarrow\mathfrak{g}$ is called a relative Rota-Baxter
operator of weight $1$ on $\mathfrak{g}$ with respect to $(\mathfrak{h};\phi)$
if

(1)
$[B(u),B(v)]_{\mathfrak{g}}=B\Big{(}\phi(B(u))v-\phi(B(v))u+[u,v]_{\mathfrak{h}}\Big{)},\quad\forall
u,v\in\mathfrak{h}.$

In particular, if $\mathfrak{g}=\mathfrak{h}$ and the action is the adjoint
representation of $\mathfrak{g}$ on itself, then $B$ is called a Rota-Baxter
operator of weight $1$.

######  Remark 2.2.

In the context of Lie algebras, usually one considers relative Rota-Baxter
operators of weight $\lambda$, where $\lambda$ is a parameter in the front of
$[\cdot,\cdot]_{\mathfrak{h}}$ in (1). However, in the context of Lie groups,
one can only consider relative Rota-Baxter operators of weight $\pm 1$ due to
the fact that there is no linear structure on Lie groups in general. Thus, in
this paper we consider only relative Rota-Baxter operators of weight $1$, and
call them simply relative Rota-Baxter operators.

######  Remark 2.3.

A Rota-Baxter operator $B$ on $\mathfrak{g}$ one-to-one corresponds to a
solution of the modified Yang-Baxter equation, which in turn has important
applications in integrable systems. We now recall some facts from [18, 30, 31,
32]. Let $R:\mathfrak{g}\,\rightarrow\,\mathfrak{g}$ be a solution of the
modified Yang-Baxter equation:

(2)
$[R(u),R(v)]_{\mathfrak{g}}=R([R(u),v]_{\mathfrak{g}})+R([u,R(v)]_{\mathfrak{g}})-[u,v]_{\mathfrak{g}},\quad\forall
u,v\in\mathfrak{g}.$

Each such a solution (which we call a modified $r$-matrix) gives rise to a so-
called infinitesimal factorization of the Lie algebra $\mathfrak{g}$. The
authors integrated this infinitesimal factorization locally into a
factorization of the Lie group $G$ of $\mathfrak{g}$. The integrated
factorization in turn gives an elegant solution of a certain hamiltonian
system for small time $t$. The factorization itself may also be viewed as some
version of Riemann-Hilbert problem.

Under the transformation

$R={\rm{Id}}+2B,$

the operator $R$ satisfies the modified Yang-Baxter equation if and only if
the operator $B$ is a Rota-Baxter operator of weight $1$.

######  Example 2.4.

Let $\mathfrak{g}=\mathrm{up}(2;\mathbb{R})$, where
$\mathrm{up}(2;\mathbb{R})$ is the set of upper triangular matrices and
$\mathfrak{h}=\mathbb{R}$. Define
$B:\mathbb{R}\,\rightarrow\,\mathrm{up}(2;\mathbb{R})$ and
$\phi:\mathrm{up}(2;\mathbb{R})\,\rightarrow\,\mathrm{Der}(\mathbb{R})$ by

$B(r)=\left(\begin{array}[]{cc}0&r\\\ 0&0\\\
\end{array}\right),\quad\phi(\left(\begin{array}[]{cc}x&y\\\ 0&z\\\
\end{array}\right))r=xr,\quad\forall\left(\begin{array}[]{cc}x&y\\\ 0&z\\\
\end{array}\right)\in\mathrm{up}(2;\mathbb{R}),r\in\mathbb{R}.$

Then $B$ is a relative Rota-Baxter operator on the Lie algebra $\mathfrak{g}$
with respect to the action $(\mathbb{R};\phi)$.

Let $\phi:\mathfrak{g}\rightarrow\mathrm{Der}(\mathfrak{h})$ be an action of a
Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ on a Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{\mathfrak{h}})$. Define a skewsymmetric bracket
operation $[\cdot,\cdot]_{\phi}$ on $\mathfrak{g}\oplus\mathfrak{h}$ by

(3)
$[x+u,y+v]_{\phi}=[x,y]_{\mathfrak{g}}+\phi(x)v-\phi(y)u+[u,v]_{\mathfrak{h}},\quad\forall
x,y\in\mathfrak{g},~{}u,v\in\mathfrak{h}.$

Then it is straightforward to see that
$(\mathfrak{g}\oplus\mathfrak{h},[\cdot,\cdot]_{\phi})$ is a Lie algebra, and
denoted by $\mathfrak{g}\ltimes_{\phi}\mathfrak{h}$.

######  Proposition 2.5.

Let $\phi:\mathfrak{g}\rightarrow\mathrm{Der}(\mathfrak{h})$ be an action of a
Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ on a Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{\mathfrak{h}})$. Then a linear map
$B:\mathfrak{h}\rightarrow\mathfrak{g}$ is a relative Rota-Baxter operator if
and only if the graph $Gr(B)=\\{B(u)+u|u\in\mathfrak{h}\\}$ is a subalgebra of
the Lie algebra $\mathfrak{g}\ltimes_{\phi}\mathfrak{h}$.

###### Proof.

Let $B:\mathfrak{h}\rightarrow\mathfrak{g}$ be a linear map. For all
$u,v\in\mathfrak{h},$ we have

$\displaystyle[B(u)+u,B(v)+v]_{\phi}=[B(u),B(v)]_{\mathfrak{g}}+\phi(B(u))v-\phi(B(v))u+[u,v]_{\mathfrak{h}},$

which implies that the graph $Gr(B)=\\{B(u)+u|u\in\mathfrak{h}\\}$ is a
subalgebra of the Lie algebra $\mathfrak{g}\ltimes_{\phi}\mathfrak{h}$ if and
only if $B$ satisfies

$\displaystyle[B(u),B(v)]_{\mathfrak{g}}=B\Big{(}\phi(B(u))v-\phi(B(v))u+[u,v]_{\mathfrak{h}}\Big{)},$

which means that $B$ is a relative Rota-Baxter operator. ∎

Since $\mathfrak{h}$ and the graph $Gr(B)$ are isomorphic as vector spaces, we
have the following result.

######  Corollary 2.6.

Let $B:\mathfrak{h}\rightarrow\mathfrak{g}$ be a relative Rota-Baxter operator
on $\mathfrak{g}$ with respect to an action $(\mathfrak{h};\phi)$. Then
$(\mathfrak{h},[\cdot,\cdot]_{B})$ is a Lie algebra, called the descendent Lie
algebra of $B$, where

(4) $[u,v]_{B}=\phi(B(u))v-\phi(B(v))u+[u,v]_{\mathfrak{h}},\quad\forall
u,v\in\mathfrak{h}.$

Moreover, $B$ is a Lie algebra homomorphism from
$(\mathfrak{h},[\cdot,\cdot]_{B})$ to
$(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$.

######  Definition 2.7.

Let $B$ and $B^{\prime}$ be relative Rota-Baxter operators on a Lie algebra
$\mathfrak{g}$ with respect to the action $(\mathfrak{h};\phi)$. A
homomorphism from $B^{\prime}$ to $B$ consists of a Lie algebra homomorphism
$\psi_{\mathfrak{g}}:\mathfrak{g}\,\rightarrow\,\mathfrak{g}$ and a Lie
algebra homomorphism
$\psi_{\mathfrak{h}}:\mathfrak{h}\,\rightarrow\,\mathfrak{h}$ such that

(5) $\displaystyle\psi_{\mathfrak{g}}\circ B^{\prime}$ $\displaystyle=$
$\displaystyle B\circ\psi_{\mathfrak{h}},$ (6)
$\displaystyle\psi_{\mathfrak{h}}(\phi(x)v)$ $\displaystyle=$
$\displaystyle\phi(\psi_{\mathfrak{g}}(x))\psi_{\mathfrak{h}}(v),\quad\forall
x\in\mathfrak{g},v\in\mathfrak{h}.$

In particular, if both $\psi_{\mathfrak{g}}$ and $\psi_{\mathfrak{h}}$ are
invertible, $(\psi_{\mathfrak{g}},\psi_{\mathfrak{h}})$ is called an
isomorphism from $B^{\prime}$ to $B$.

In fact, (6) is equivalent that $(\psi_{\mathfrak{g}},\psi_{\mathfrak{h}})$ is
an endomorphism of the Lie algebra $\mathfrak{g}\ltimes_{\phi}\mathfrak{h}$.

It is straightforward to see that relative Rota-Baxter operators on a Lie
algebra $\mathfrak{g}$ with respect to an action $(\mathfrak{h};\phi)$
together with homomorphisms between them form a category, which we denote by
$\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}$.

###  2.2. A cohomology theory in $\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}$

We first prove some statements which are important for us to build up our
cohomology theory in a functorial way.

######  Proposition 2.8.

Let $B:\mathfrak{h}\rightarrow\mathfrak{g}$ be a relative Rota-Baxter operator
on $\mathfrak{g}$ with respect to $(\mathfrak{h};\phi)$. Define a linear map
$\theta:\mathfrak{h}\rightarrow\mathfrak{gl}(\mathfrak{g})$ by

(7) $\theta(u)x=B(\phi(x)u)+[B(u),x]_{\mathfrak{g}},\quad\forall
x\in\mathfrak{g},u\in\mathfrak{h}.$

Then $\theta$ is a representation of the descendent Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{B})$ on the vector space $\mathfrak{g}$.

###### Proof.

By the fact that $\phi(x)\in\mathrm{Der}(\mathfrak{h})$ for all
$x\in\mathfrak{g}$ and $B:\mathfrak{h}\rightarrow\mathfrak{g}$ is a relative
Rota-Baxter operator, for all $u,v\in\mathfrak{h}$, we have

$\displaystyle[\theta(u),\theta(v)]x$ $\displaystyle=$
$\displaystyle\theta(u)\theta(v)x-\theta(v)\theta(u)x$ $\displaystyle=$
$\displaystyle\theta(u)(B(\phi(x)v))+\theta(u)[B(v),x]_{\mathfrak{g}}-\theta(v)(B(\phi(x)u))-\theta(v)[B(u),x]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle
B(\phi(B(\phi(x)v))u)+[B(u),B(\phi(x)v)]_{\mathfrak{g}}+B(\phi([B(v),x]_{\mathfrak{g}})u)+[B(u),[B(v),x]_{\mathfrak{g}}]_{\mathfrak{g}}$
$\displaystyle-B(\phi(B(\phi(x)u))v)-[B(v),B(\phi(x)u)]_{\mathfrak{g}}-B(\phi([B(u),x]_{\mathfrak{g}})v)-[B(v),[B(u),x]_{\mathfrak{g}}]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle
B\Big{(}\phi(B(u))\phi(x)v-\phi(B(\phi(x)v))u+[u,\phi(x)v]_{\mathfrak{h}}\Big{)}$
$\displaystyle+B(\phi(B(\phi(x)v))u)+B(\phi([B(v),x]_{\mathfrak{g}})u)+[B(u),[B(v),x]_{\mathfrak{g}}]_{\mathfrak{g}}$
$\displaystyle-B\Big{(}\phi(B(v))\phi(x)u-\phi(B(\phi(x)u))v+[v,\phi(x)u]_{\mathfrak{h}}\Big{)}$
$\displaystyle-B(\phi(B(\phi(x)u))v)-B(\phi([B(u),x]_{\mathfrak{g}})v)-[B(v),[B(u),x]_{\mathfrak{g}}]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle
B(\phi(B(u))\phi(x)v)+B(\phi([B(v),x]_{\mathfrak{g}})u)-B(\phi(B(v))\phi(x)u)-B(\phi([B(u),x]_{\mathfrak{g}})v)$
$\displaystyle+B([u,\phi(x)v]_{\mathfrak{h}})-B([v,\phi(x)u]_{\mathfrak{h}})+[B(u),[B(v),x]_{\mathfrak{g}}]_{\mathfrak{g}}-[B(v),[B(u),x]_{\mathfrak{g}}]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle
B\Big{(}\phi(x)\phi(B(u))v-\phi(x)\phi(B(v))u+\phi(x)[u,v]_{\mathfrak{h}}\Big{)}+[[B(u),B(v)]_{\mathfrak{g}},x]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle
B(\phi(x)[u,v]_{B})+[B([u,v]_{B}),x]_{\mathfrak{g}}$ $\displaystyle=$
$\displaystyle\theta([u,v]_{B})x.$

Thus $\theta$ is a representation of the descendent Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{B})$ on the vector space $\mathfrak{g}$. ∎

######  Proposition 2.9.

Let $B:\mathfrak{h}\rightarrow\mathfrak{g}$ be a relative Rota-Baxter operator
on $\mathfrak{g}$ with respect to $(\mathfrak{h};\phi)$. Then

$\phi(x)[u,v]_{B}=[\phi(x)u,v]_{B}+[u,\phi(x)v]_{B}+\phi(\theta(v)x)u-\phi(\theta(u)x)v,\quad\forall
u,v\in\mathfrak{h},x\in\mathfrak{g}.$

###### Proof.

For any $u,v\in\mathfrak{h},x\in\mathfrak{g}$, by Proposition 2.8, we obtain

$\displaystyle\phi(x)[u,v]_{B}-([\phi(x)u,v]_{B}+[u,\phi(x)]_{B})$
$\displaystyle=$
$\displaystyle\phi(x)\phi(B(u))v-\phi(x)\phi(B(v))u+\phi(x)[u,v]_{\mathfrak{h}}$
$\displaystyle-(\phi(B(\phi(x)u))v-\phi(B(v))\phi(x)u+[\phi(x)u,v]_{\mathfrak{h}})$
$\displaystyle-(\phi(B(u))\phi(x)v-\phi(B(\phi(x)v))u+[u,\phi(x)v]_{\mathfrak{h}})$
$\displaystyle=$
$\displaystyle[\phi(B(v)),\phi(x)]u+\phi(B(\phi(x)v))u-[\phi(B(u)),\phi(x)]v-\phi(B(\phi(x)u))v$
$\displaystyle=$
$\displaystyle\phi([B(v),x]_{\mathfrak{g}}+B(\phi(x)v))u-\phi([B(u),x]_{\mathfrak{g}}+B(\phi(x)u))v$
$\displaystyle=$ $\displaystyle\phi(\theta(v)x)u-\phi(\theta(u)x)v,$

which finishes the proof. ∎

Let
$\mathrm{d}_{\mathsf{CE}}^{B}:\mathrm{Hom}(\wedge^{k}\mathfrak{h},\mathfrak{g})\rightarrow\mathrm{Hom}(\wedge^{k+1}\mathfrak{h},\mathfrak{g})$
be the corresponding Chevalley-Eilenberg coboundary operator of the descendent
Lie algebra $(\mathfrak{h},[\cdot,\cdot]_{B})$ with coefficients in
$(\mathfrak{g};\theta)$. More precisely, for all
$f\in\mathrm{Hom}(\wedge^{k}\mathfrak{h},\mathfrak{g})$ and
$u_{1},\cdots,u_{k+1}\in\mathfrak{h}$, we have

$\displaystyle\mathrm{d}_{\mathsf{CE}}^{B}f(u_{1},\cdots,u_{k+1})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k+1}(-1)^{i+1}\theta(u_{i})f(u_{1},\cdots,\hat{u_{i}},\cdots,u_{k+1})+\sum_{i<j}(-1)^{i+j}f([u_{i},u_{j}]_{B},u_{1},\cdots,\hat{u_{i}},\cdots,\hat{u_{j}},\cdots,u_{k+1})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k+1}(-1)^{i+1}B(\phi(f(u_{1},\cdots,\hat{u_{i}},\cdots,u_{k+1}))u_{i})+\sum_{i=1}^{k+1}(-1)^{i+1}[B(u_{i}),f(u_{1},\cdots,\hat{u_{i}},\cdots,u_{k+1})]_{\mathfrak{g}}$
$\displaystyle+\sum_{i<j}(-1)^{i+j}f(\phi(B(u_{i}))u_{j}-\phi(B(u_{j}))u_{i}+[u_{i},u_{j}]_{\mathfrak{h}},u_{1},\cdots,\hat{u_{i}},\cdots,\hat{u_{j}},\cdots,u_{k+1}).$

######  Definition 2.10.

Let $B:\mathfrak{h}\,\rightarrow\,\mathfrak{g}$ be a relative Rota-Baxter
operator on the Lie algebra $\mathfrak{g}$ with respect to an action
$(\mathfrak{h};\phi)$. Define the space of $k$-cochains $C^{k}(B)$ by
$C^{k}(B)=\mathrm{Hom}(\wedge^{k-1}\mathfrak{h},\mathfrak{g})$. Denote by
$C^{*}(B)=\oplus_{k=1}^{+\infty}C^{k}(B)$. The cohomology of the cochain
complex $(C^{*}(B),\mathrm{d}_{\mathsf{CE}}^{B})$ is defined to be the
cohomology for the relative Rota-Baxter operator $B$.

Denote by $H^{k}(B)$ the $k$-th cohomology group.

It implies in particular that $x\in\mathfrak{g}$ is a 1-cocycle if and only if
$B\circ\phi(x)=\mathrm{ad}_{x}\circ B,$ and
$f\in\mathrm{Hom}(\mathfrak{h},\mathfrak{g})$ is a 2-cocycle if and only if

$\displaystyle
B(\phi(f(u_{2}))u_{1})+[B(u_{1}),f(u_{2})]_{\mathfrak{g}}-B(\phi(f(u_{1}))u_{2})-[B(u_{2}),f(u_{1})]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle
f(\phi(B(u_{1}))u_{2}-\phi(B(u_{2}))u_{1}+[u_{1},u_{2}]_{\mathfrak{h}}).$

######  Remark 2.11.

The controlling algebra of relative Rota-Baxter operators of weight $1$ on Lie
algebras was given in [34]. The cohomology theory established here coincides
with the one given by the controlling algebra. As inspired by the slogan
promoted by Deligne: deformations of a given algebraic or geometric structure
are governed by a differential graded Lie algebra (DGLA), or more generally,
by an $L_{\infty}$-algebra. This DGLA or $L_{\infty}$-algebra, is called the
controlling algebra of the given algebraic or geometric structure. Moreover,
given the close relation between the cohomology theory and the deformation
theory, we may use the controlling algebra to define a cohomology theory for
the given algebraic or geometric structure.

We need the following statement to prove the functoriality of our cohomology
theory.

######  Proposition 2.12.

Let $B$ and $B^{\prime}$ be relative Rota-Baxter operators on a Lie algebra
$\mathfrak{g}$ with respect to an action $(\mathfrak{h};\phi)$ and
$(\psi_{\mathfrak{g}},\psi_{\mathfrak{h}})$ be a homomorphism from
$B^{\prime}$ to $B$.

  * (i)

$\psi_{\mathfrak{h}}$ is also a Lie algebra homomorphism from the descendent
Lie algebra $(\mathfrak{h},[\cdot,\cdot]_{B^{\prime}})$ of $B^{\prime}$ to the
descendent Lie algebra $(\mathfrak{h},[\cdot,\cdot]_{B})$ of $B$;

  * (ii)

The induced representation $(\mathfrak{g};\theta)$ of the Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{B})$ and the induced representation
$(\mathfrak{g};\theta^{\prime})$ of the Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{B^{\prime}})$ satisfy the following relation:

$\psi_{\mathfrak{g}}\circ\theta^{\prime}(u)=\theta(\psi_{\mathfrak{h}}(u))\circ\psi_{\mathfrak{g}},\quad\forall
u\in\mathfrak{h}.$

That is, for all $u\in\mathfrak{h}$, the following diagram commutes:

$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{\prime}(u)}$$\scriptstyle{\psi_{\mathfrak{g}}}$$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta(\psi_{\mathfrak{h}}(u))}$$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{\mathfrak{g}}}$$\textstyle{\mathfrak{g}.}$

###### Proof.

By (5), (6) and the fact that $\psi_{\mathfrak{h}}$ is a Lie algebra
homomorphism, we have

$\displaystyle\psi_{\mathfrak{h}}([u,v]_{B^{\prime}})$ $\displaystyle=$
$\displaystyle\psi_{\mathfrak{h}}\Big{(}\phi(B^{\prime}(u))v-\phi(B^{\prime}(v))u+[u,v]_{\mathfrak{h}}\Big{)}$
$\displaystyle=$
$\displaystyle\phi\Big{(}B(\psi_{\mathfrak{h}}(u))\Big{)}\psi_{\mathfrak{h}}(v)-\phi\Big{(}B(\psi_{\mathfrak{h}}(v))\Big{)}\psi_{\mathfrak{h}}(u)+[\psi_{\mathfrak{h}}(u),\psi_{\mathfrak{h}}(v)]_{\mathfrak{h}}$
$\displaystyle=$
$\displaystyle[\psi_{\mathfrak{h}}(u),\psi_{\mathfrak{h}}(v)]_{B},\quad\forall
u,v\in\mathfrak{h},$

which implies that $\psi_{\mathfrak{h}}$ is a homomorphism between the
descendent Lie algebras.

By (5), (6) and (7), for all $x\in\mathfrak{g},u\in\mathfrak{h}$, we have

$\displaystyle\psi_{\mathfrak{g}}(\theta^{\prime}(u)x)$ $\displaystyle=$
$\displaystyle\psi_{\mathfrak{g}}(B^{\prime}(\phi(x)u))+\psi_{\mathfrak{g}}([B^{\prime}(u),x]_{\mathfrak{g}})=B(\psi_{\mathfrak{h}}(\phi(x)u))+[\psi_{\mathfrak{g}}(B^{\prime}(u)),\psi_{\mathfrak{g}}(x)]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle
B(\phi(\psi_{g}(x))\psi_{\mathfrak{h}}(u))+[B(\psi_{\mathfrak{h}}(u)),\psi_{\mathfrak{g}}(x)]_{\mathfrak{g}}=\theta(\psi_{\mathfrak{h}}(u))\psi_{\mathfrak{g}}(x).$

We finish the proof. ∎

Let $B$ and $B^{\prime}$ be relative Rota-Baxter operators on $\mathfrak{g}$
with respect to an action $(\mathfrak{h};\phi)$, and
$(\psi_{\mathfrak{g}},\psi_{\mathfrak{h}})$ a homomorphism from $B^{\prime}$
to $B$ in which $\psi_{\mathfrak{h}}$ is invertible. Define a map
$p:C^{k}(B^{\prime})\,\rightarrow\,C^{k}(B)$ by

$p(\omega)(u_{1},\cdots,u_{k-1})=\psi_{\mathfrak{g}}(\omega(\psi_{\mathfrak{h}}^{-1}(u_{1}),\cdots,\psi_{\mathfrak{h}}^{-1}(u_{k-1}))),\quad\forall
u_{i}\in\mathfrak{h}.$

######  Theorem 2.13.

With above notations, $p$ is a cochain map from the cochain complex
$(C^{*}(B^{\prime}),\mathrm{d}_{\mathsf{CE}}^{B^{\prime}})$ to the cochain
complex $(C^{*}(B),\mathrm{d}_{\mathsf{CE}}^{B})$. Consequently, it induces a
homomorphism $p_{*}$ from the cohomology group $H^{k}(B^{\prime})$ to
$H^{k}(B)$ for all $k\geq 1$.

###### Proof.

For all $\omega\in C^{k}(B^{\prime})$, by Proposition 2.12, we have

$\displaystyle\mathrm{d}_{\mathsf{CE}}^{B}(p(\omega))(u_{1},u_{2},\cdots,u_{k})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}(-1)^{i+1}\theta(u_{i})p(\omega)(u_{1},\cdots,\hat{u_{i}},\cdots,u_{k})$
$\displaystyle+\sum_{i<j}(-1)^{i+j}p(\omega)([u_{i},u_{j}]_{B},u_{1},\cdots,\hat{u_{i}},\cdots,\hat{u_{j}},\cdots,u_{k})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}(-1)^{i+1}\theta(u_{i})\psi_{\mathfrak{g}}\big{(}\omega(\psi_{\mathfrak{h}}^{-1}(u_{1}),\cdots,\hat{\psi_{\mathfrak{h}}^{-1}(u_{i})},\cdots,\psi_{\mathfrak{h}}^{-1}(u_{k}))\big{)}$
$\displaystyle+\sum_{i<j}(-1)^{i+j}\psi_{\mathfrak{g}}\Big{(}\omega(\psi_{\mathfrak{h}}^{-1}([u_{i},u_{j}]_{B}),\psi_{\mathfrak{h}}^{-1}(u_{1}),\cdots,\hat{\psi_{\mathfrak{h}}^{-1}(u_{i})},\cdots,\hat{\psi_{\mathfrak{h}}^{-1}(u_{j})},\cdots,\psi_{\mathfrak{h}}^{-1}(u_{k}))\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}(-1)^{i+1}\psi_{\mathfrak{g}}\big{(}\theta^{\prime}(\psi_{\mathfrak{h}}^{-1}(u_{i}))\omega(\psi_{\mathfrak{h}}^{-1}(u_{1}),\cdots,\hat{\psi_{\mathfrak{h}}^{-1}(u_{i})},\cdots,\psi_{\mathfrak{h}}^{-1}(u_{k}))\big{)}$
$\displaystyle+\sum_{i<j}(-1)^{i+j}\psi_{\mathfrak{g}}\Big{(}\omega([\psi_{\mathfrak{h}}^{-1}(u_{i}),\psi_{\mathfrak{h}}^{-1}(u_{j})]_{B^{{}^{\prime}}},\psi_{\mathfrak{h}}^{-1}(u_{1}),\cdots,\hat{\psi_{\mathfrak{h}}^{-1}(u_{i})},\cdots,\hat{\psi_{\mathfrak{h}}^{-1}(u_{j})},\cdots,\psi_{\mathfrak{h}}^{-1}(u_{k}))\Big{)}$
$\displaystyle=$
$\displaystyle\psi_{\mathfrak{g}}\big{(}\mathrm{d}_{\mathsf{CE}}^{B^{\prime}}\omega(\psi_{\mathfrak{h}}^{-1}(u_{1}),\cdots,\psi_{\mathfrak{h}}^{-1}(u_{k}))\big{)}$
$\displaystyle=$ $\displaystyle
p(\mathrm{d}_{\mathsf{CE}}^{B^{\prime}}\omega)(u_{1},\cdots,u_{k}),\quad\forall
u_{i}\in\mathfrak{h}.$

Thus $p$ is a cochain map. Consequently it induces a homomorphism $p_{*}$ from
the cohomology group $H^{k}(B^{\prime})$ to $H^{k}(B)$ for all $k\geq 1$. ∎

###  2.3. Cohomology theory and deformations in
$\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}$ and modified $r$-matrices

In this subsection, we show that the second cohomology group classifies the
infinitesimal deformations of relative Rota-Baxter operators. This may be
viewed as a justification of the cohomology theory that we established in the
previous subsection. In the end, we use our cohomology theory to study the
deformations of modified $r$-matrices as an application.

######  Definition 2.14.

Let $B:\mathfrak{h}\,\rightarrow\,\mathfrak{g}$ be a relative Rota-Baxter
operator on $\mathfrak{g}$ with respect to an action $(\mathfrak{h};\phi)$ and
$\hat{B}:\mathfrak{h}\rightarrow\mathfrak{g}$ be a linear map. If
$B_{t}=B+t\hat{B}$ is still a relative Rota-Baxter operator on the Lie algebra
$\mathfrak{g}$ with respect to the action $(\mathfrak{h};\phi)$ for all $t$,
we say that $\hat{B}$ generates a one-parameter infinitesimal deformation of
the relative Rota-Baxter operator $B$.

Since $B_{t}=B+t\hat{B}$ is a relative Rota-Baxter operator on the Lie algebra
$\mathfrak{g}$ with respect to the representation $(\mathfrak{h};\phi)$ for
all $t$, we have

(9)
$[\hat{B}(u),\hat{B}(v)]_{\mathfrak{g}}=\hat{B}(\phi(\hat{B}(u))v)-\hat{B}(\phi(\hat{B}(v))u),$

and

$\displaystyle[\hat{B}(u),B(v)]_{\mathfrak{g}}+[B(u),\hat{B}(v)]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle
B(\phi(\hat{B}(u))v)+\hat{B}(\phi(B(u))v)-B(\phi(\hat{B}(v))u)-\hat{B}(\phi(B(v))u)+\hat{B}([u,v]_{\mathfrak{h}}).$

Note that equation (9) states that $\hat{B}$ is a relative Rota-Baxter
operator of wight $0$ on the Lie algebra $\mathfrak{g}$ with respect to the
representation $(\mathfrak{h};\phi)$. The equation (2.3) states that $\hat{B}$
is a $2$-cocycle of the relative Rota-Baxter operator $B$.

######  Definition 2.15.

Let $B:\mathfrak{h}\rightarrow\mathfrak{g}$ be a relative Rota-Baxter operator
on $\mathfrak{g}$ with respect to $(\mathfrak{h};\phi)$. Two one-parameter
infinitesimal deformations $B_{t}^{1}=B+t\hat{B}_{1}$ and
$B_{t}^{2}=B+t\hat{B}_{2}$ are said to be equivalent if there exists an
$x\in\mathfrak{g}$ such that
$({\rm{Id}}_{\mathfrak{g}}+t\mathrm{ad}_{x},{\rm{Id}}_{\mathfrak{h}}+t\phi(x))$
is a homomorphism from $B_{t}^{1}$ to $B_{t}^{2}$. In particular, a one-
parameter infinitesimal deformation $B_{t}$ of a relative Rota-Baxter operator
$B$ is said to be trivial if there exists an $x\in\mathfrak{g}$ such that
$({\rm{Id}}_{\mathfrak{g}}+t\mathrm{ad}_{x},{\rm{Id}}_{\mathfrak{h}}+t\phi(x))$
is a homomorphism from $B_{t}$ to $B$.

Let
$({\rm{Id}}_{\mathfrak{g}}+t\mathrm{ad}_{x},{\rm{Id}}_{\mathfrak{h}}+t\phi(x))$
be a homomorphism from $B_{t}^{1}$ to $B_{t}^{2}$. Then we have

$({\rm{Id}}_{\mathfrak{g}}+t\mathrm{ad}_{x})(B+t\hat{B}_{1})u=(B+t\hat{B}_{2})({\rm{Id}}_{\mathfrak{h}}+t\phi(x))u,\quad\forall
u\in\mathfrak{h},$

which implies

$\displaystyle[x,\hat{B}_{1}(u)]_{\mathfrak{g}}$ $\displaystyle=$
$\displaystyle\hat{B}_{2}(\phi(x)u),$ (11)
$\displaystyle\hat{B}_{1}(u)-\hat{B}_{2}(u)$ $\displaystyle=$ $\displaystyle
B(\phi(x)u)+[B(u),x]_{\mathfrak{g}}\quad\forall
x\in\mathfrak{g},u\in\mathfrak{h}.$

By (2.3), we have

$\hat{B}_{1}-\hat{B}_{2}=\mathrm{d}_{\mathsf{CE}}^{B}x,$

where $\mathrm{d}_{\mathsf{CE}}^{B}$ is given by (2.2). Thus we have

######  Theorem 2.16.

Let $B:\mathfrak{h}\rightarrow\mathfrak{g}$ be a relative Rota-Baxter operator
on $\mathfrak{g}$ with respect to $(\mathfrak{h};\phi)$. If two one-parameter
infinitesimal deformations $B_{t}^{1}=B+t\hat{B}_{1}$ and
$B_{t}^{2}=B+t\hat{B}_{2}$ are equivalent, then $\hat{B}_{1}$ and
$\hat{B}_{2}$ are in the same cohomology class of $H^{2}(B)$ defined in
Definition 2.10.

Since modified $r$-matrices and Rota-Baxter operators of weight $1$ are
related via the formula $R={\rm{Id}}+2B$ (see Remark 2.3), it follows that our
cohomology theory can also control the infinitesimal deformations of modified
$r$-matrices. More details will be provided in a future work.

######  Definition 2.17.

Given a Lie algebra $\mathfrak{g}$, let $R$ be a modified $r$-matrix
satisfying Eq. (2). If $R_{t}=R+t\hat{R}$ is still a modified $r$-matrix for
all $t$, we say that $\hat{R}$ generates a one-parameter infinitesimal
deformation of $R$.

######  Definition 2.18.

Let $R_{t}^{1}=R+t\hat{R}_{1}$ and $R_{t}^{2}=R+t\hat{R}_{2}$ be two
infinitesimal deformations of a modified $r$-matrix $R$. They are said to be
equivalent if there exists an $x\in\mathfrak{g}$ such that
${\rm{Id}}_{\mathfrak{g}}+t\mathrm{ad}_{x}$ is a homomorphism from $R_{t}^{1}$
to $R_{t}^{2}$, i.e. ${\rm{Id}}_{\mathfrak{g}}+t\mathrm{ad}_{x}$ is a Lie
algebra endomorphism and

$({\rm{Id}}_{\mathfrak{g}}+t\mathrm{ad}_{x})\circ
R_{t}^{1}=R_{t}^{2}\circ({\rm{Id}}_{\mathfrak{g}}+t\mathrm{ad}_{x}).$

######  Theorem 2.19.

Let $R_{t}^{1}=R+t\hat{R}_{1}$ and $R_{t}^{2}=R+t\hat{R}_{2}$ be two
equivalent infinitesimal deformations of a modified $r$-matrix $R$. Then
$\hat{R}_{1}$ and $\hat{R}_{2}$ are in the same cohomology class of $H^{2}(B)$
defined in Definition 2.10 for $B=\frac{1}{2}(R-{\rm{Id}})$.

###### Proof.

It is obvious that an infinitesimal deformation of a Rota-Baxter operator
$B_{t}=B+t\hat{B}$ gives rise to an infinitesimal deformation of a modified
$r$-matrix $R_{t}=R+t(2\hat{B})$, with $R={\rm{Id}}+2B$. Conversely, an
infinitesimal deformation of a modified $r$-matrix $R_{t}=R+t\hat{R}$ gives
rise to an infinitesimal deformation of a Rota-Baxter operator
$B_{t}=B+t(\frac{\hat{R}}{2})$, with $B=\frac{1}{2}(R-{\rm{Id}})$. Therefore
$\hat{R}$ is a 2-cocycle in $(C^{*}(B),\mathrm{d}^{B}_{\mathsf{CE}})$ by
Theorem 2.16. Similarly, $R_{t}^{1}=R+t\hat{R}_{1}$ and
$R_{t}^{2}=R+t\hat{R}_{2}$ are equivalent infinitesimal deformations if and
only if their corresponding Rota-Baxter operators are equivalent. Therefore by
Theorem 2.16, both $\hat{R}_{1}$ and $\hat{R}_{2}$ are cocycles in
$(C^{*}(B),\mathrm{d}^{B}_{\mathsf{CE}})$ and they are differed by a
coboundary therein. ∎

##  3\. Relative Rota-Baxter operators on Lie groups and their cohomology
theory

In this section, we introduce the cohomology theory of relative Rota-Baxter
operators on Lie groups. In the sequel, $(G,e_{G},\cdot_{G})$ and
$(H,e_{H},\cdot_{H})$ are always Lie groups.

###  3.1. The category $\mathsf{RB_{G}^{H}}$ of relative Rota-Baxter operators
on Lie groups

######  Definition 3.1.

Let $\Phi:G\rightarrow\mathrm{Aut}(H)$ be an action of $G$ on $H$. A smooth
map $\mathcal{B}:H\rightarrow G$ is called a relative Rota-Baxter operator if

(12)
$\mathcal{B}(h_{1})\cdot_{G}\mathcal{B}(h_{2})=\mathcal{B}(h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))h_{2}),\quad\forall
h_{1},h_{2}\in H.$

In particular if the action $\Phi$ is the adjoint action $\mathrm{Ad}$ of the
Lie group $G$ on itself, it reduces to a Rota-Baxter operator introduced in
[16].

######  Example 3.2.

([16]) Let $G$ be a Lie group and $G_{+},G_{-}$ be two subgroups such that
$G=G_{+}G_{-}$ and $G_{+}\cap G_{-}=\\{e\\}$. Define $\mathcal{B}:G\to G$ by

$\mathcal{B}(g)=g_{-}^{-1},\quad\forall~{}g=g_{+}g_{-},\quad\mbox{where
}g_{+}\in G_{+},g_{-}\in G_{-}.$

Then $\mathcal{B}$ is a Rota-Baxter operator.

Let $G$ be a connected compact real Lie group and $(\mathbb{R},\Phi)$ be a
one-dimensional representation. Thus for any $g\in G$, it follows that
$|\Phi(g)|=1$. Moreover, $G$ is a connected Lie group, thus $\Phi(g)=1$ for
any $g\in G$. Thus a smooth map $\mathcal{B}:\mathbb{R}\,\rightarrow\,G$ is a
relative Rota-Baxter operator with respect to the action $\Phi$ if and only if
$\mathcal{B}$ is a group homomorphism from $\mathbb{R}$ to $G$. This sort of
phenomenon also extends to the non-compact case in the following example:

######  Example 3.3.

Let $(G,e_{G},\cdot_{G})=(\mathrm{UP}(2;\mathbb{R}),I,\cdot)$, where
$\mathrm{UP}(2;\mathbb{R})$ is the set of invertible upper triangular matrices
and $H=\mathbb{R}$. Define
$\mathcal{B}:\mathbb{R}\,\rightarrow\,\mathrm{UP}(2;\mathbb{R})$ and
$\Phi:\mathrm{UP}(2;\mathbb{R})\,\rightarrow\,\mathrm{Aut}(\mathbb{R})$ by

$\mathcal{B}(r)=\left(\begin{array}[]{cc}1&r\\\ 0&1\\\
\end{array}\right),\quad\Phi(\left(\begin{array}[]{cc}a&b\\\ 0&c\\\
\end{array}\right))r=ar,\quad\forall\left(\begin{array}[]{cc}a&b\\\ 0&c\\\
\end{array}\right)\in\mathrm{UP}(2;\mathbb{R}),r\in\mathbb{R}.$

Then $\mathcal{B}$ is a relative Rota-Baxter operator on the Lie group $G$
with respect to the action $(\mathbb{R};\Phi)$.

Given an action $\Phi:G\to\mathrm{Aut}(H)$, we have the semi-direct product
Lie group $G\ltimes_{\Phi}H$, with multiplication $\cdot_{\Phi}$ given by

$(g_{1},h_{1})\cdot_{\Phi}(g_{2},h_{2})=(g_{1}\cdot_{G}g_{2},h_{1}\cdot_{H}\Phi(g_{1})h_{2}),\quad\forall
g_{i}\in G,h_{i}\in H,i=1,2.$

######  Proposition 3.4.

Let $\Phi:G\to\mathrm{Aut}(H)$ be an action of a Lie group $G$ on a Lie group
$H$. Then a smooth map $\mathcal{B}:H\rightarrow G$ is a relative Rota-Baxter
operator if and only if the graph $Gr(\mathcal{B})=\\{(\mathcal{B}(h),h)|h\in
H\\}$ is a Lie subgroup of the Lie group $G\ltimes_{\Phi}H$.

###### Proof.

Let $\mathcal{B}:H\rightarrow G$ be a smooth map. For all $h_{1},h_{2}\in H,$
we have

$\displaystyle(\mathcal{B}(h_{1}),h_{1})\cdot_{\Phi}(\mathcal{B}(h_{2}),h_{2})=(\mathcal{B}(h_{1})\cdot_{G}\mathcal{B}(h_{2}),h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))h_{2}),$

which implies that the graph $Gr(\mathcal{B})$ is a Lie subgroup of the Lie
group $G\ltimes_{\Phi}H$ if and only if $\mathcal{B}$ satisfies

$\displaystyle\mathcal{B}(h_{1})\cdot_{G}\mathcal{B}(h_{2})=\mathcal{B}(h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))h_{2}),$

which means that $\mathcal{B}$ is a relative Rota-Baxter operator. ∎

For any $g\in G$, we have
$\Phi(g)e_{H}=\Phi(g)(e_{H}\cdot_{H}e_{H})=\Phi(g)e_{H}\cdot_{H}\Phi(g)e_{H}$,
thus $\Phi(g)e_{H}=e_{H}$.

######  Proposition 3.5.

Let $\mathcal{B}:H\rightarrow G$ be a relative Rota-Baxter operator on $G$
with respect to an action $(H;\Phi)$. Then $(H,e_{H},\star)$ is a Lie group,
called the descendent Lie group of $\mathcal{B}$ and denoted by
$H_{\mathcal{B}}$, where

(13) $h_{1}\star
h_{2}=h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))h_{2},\quad\forall h_{1},h_{2}\in
H.$

For $h\in H$, its inverse $h^{\dagger}$ with respect to the multiplication
$\star$ is given by

(14) $h^{\dagger}=\Phi(\mathcal{B}(h)^{-1})h^{-1},$

where $h^{-1}$ is the inverse element of $h$ in $(H,e_{H},\cdot_{H})$.

Moreover, $\mathcal{B}:(H,e_{H},\star)\rightarrow(G,e_{G},\cdot_{G})$ is a Lie
group homomorphism.

###### Proof.

It is obvious that $e_{H}\star h=\Phi(e_{H})h=h$ and $h\star
e_{H}=h\cdot_{H}\Phi(\mathcal{B}(h))e_{H}=h$. Thus $e_{H}$ is the unit in
$(H,e_{H},\star)$.

For any $h\in H$, by (12), we have
$\mathcal{B}(h)^{-1}=\mathcal{B}(\Phi(\mathcal{B}(h)^{-1})h^{-1})$. By (13),
we have

$\displaystyle\Phi(\mathcal{B}(h)^{-1})h^{-1}\star h$ $\displaystyle=$
$\displaystyle\Phi(\mathcal{B}(h)^{-1})h^{-1}\cdot_{H}\Phi(\mathcal{B}(\Phi(\mathcal{B}(h)^{-1})h^{-1}))h$
$\displaystyle=$
$\displaystyle\Phi(\mathcal{B}(\Phi(\mathcal{B}(h)^{-1})h^{-1}))h^{-1}\cdot_{H}\Phi(\mathcal{B}(\Phi(\mathcal{B}(h)^{-1})h^{-1}))h$
$\displaystyle=$
$\displaystyle\Phi(\mathcal{B}(\Phi(\mathcal{B}(h)^{-1})h^{-1}))(h^{-1}\cdot_{H}h)$
$\displaystyle=$ $\displaystyle e_{H}.$

In a similar way, we have $h\star\Phi(\mathcal{B}(h)^{-1})h^{-1}=e_{H}.$ Thus
$\Phi(\mathcal{B}(h)^{-1})h^{-1}$ is the inverse element of $h$ in
$(H,e_{H},\star)$.

Finally, we prove that $\star$ is associative. For all $h_{1},h_{2}$ and
$h_{3}\in H$, by (12) and (13), we have

$\displaystyle(h_{1}\star h_{2})\star h_{3}$ $\displaystyle=$
$\displaystyle(h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))h_{2})\star h_{3}$
$\displaystyle=$ $\displaystyle
h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))h_{2}\cdot_{H}\Phi(\mathcal{B}(h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))h_{2}))h_{3}$
$\displaystyle=$ $\displaystyle
h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))h_{2}\cdot_{H}\Phi(\mathcal{B}(h_{1})\cdot_{G}\mathcal{B}(h_{2}))h_{3}$
$\displaystyle=$ $\displaystyle
h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))(h_{2}\cdot_{H}\Phi(\mathcal{B}(h_{2}))h_{3})$
$\displaystyle=$ $\displaystyle h_{1}\star(h_{2}\star h_{3}).$

Thus $(H,e_{H},\star)$ is a Lie group. By (12) and $\mathcal{B}(e_{H})=e_{G}$,
$\mathcal{B}$ is a Lie group homomorphism from $(H,e_{H},\star)$ to
$(G,e_{G},\cdot_{G})$. ∎

######  Definition 3.6.

Let $\mathcal{B}$ and $\mathcal{B}^{\prime}$ be relative Rota-Baxter operators
on a Lie group $G$ with respect to an action $(H;\Phi)$. A homomorphism from
$\mathcal{B}^{\prime}$ to $\mathcal{B}$ consists of a Lie group homomorphism
$\Psi_{G}:G\,\rightarrow\,G$ and a Lie group homomorphism
$\Psi_{H}:H\,\rightarrow\,H$ such that

(17) $\displaystyle\mathcal{B}\circ\Psi_{H}$ $\displaystyle=$
$\displaystyle\Psi_{G}\circ\mathcal{B}^{\prime},$ (18)
$\displaystyle\Psi_{H}(\Phi(g)h)$ $\displaystyle=$
$\displaystyle\Phi(\Psi_{G}(g))\Psi_{H}(h),\quad\forall g\in G,h\in H.$

In particular, if both $\Psi_{G}$ and $\Psi_{H}$ are invertible,
$(\Psi_{G},\Psi_{H})$ is called an isomorphism from $\mathcal{B}^{\prime}$ to
$\mathcal{B}$.

In fact, (18) is equivalent to the fact that $(\Psi_{G},\Psi_{H})$ is an
endomorphism of the Lie group $G\ltimes_{\Phi}H$.

It is clear that relative Rota-Baxter operators on a Lie group $G$ with
respect to an action $(H;\Phi)$ together with homomorphisms between them form
a category, which we denote by $\mathsf{RB_{G}^{H}}$.

###  3.2. A cohomology theory in $\mathsf{RB_{G}^{H}}$

To establish a parallel cohomology theory in $\mathsf{RB_{G}^{H}}$, we need to
first find an action of the descendent Lie group on the vector space
$\mathfrak{g}$.

######  Lemma 3.7.

Let $\Phi:G\rightarrow\mathrm{Aut}(H)$ be an action of $G$ on $H$. Then for
all $h_{1},h_{2}\in H$, we have

$\Phi(g)(h_{1}\star
h_{2})=\Phi(g)h_{1}\cdot_{H}\Phi(g\cdot_{G}\mathcal{B}(h_{1}))h_{2}.$

###### Proof.

It is straightforward. ∎

######  Theorem 3.8.

Let $\mathcal{B}$ be a relative Rota-Baxter operator on $G$ with respect to an
action $(H;\Phi)$. Define $\Theta:H\rightarrow\text{Diff}(G)$ by

(19)
$\Theta(h)g=(\mathcal{B}(\Phi(g)h^{{\dagger}}))^{-1}\cdot_{G}g\cdot_{G}\mathcal{B}(h^{{\dagger}}),\quad\forall
g\in G,h\in H.$

Then $\Theta$ is an action of the descendent Lie group $(H,e_{H},\star)$ on
the manifold $G$.

###### Proof.

For all $h_{1},h_{2}\in H$ and $g\in G$, by Lemma 3.7, we have

$\displaystyle\Theta(h_{1})\Theta(h_{2})g$ $\displaystyle=$
$\displaystyle\Theta(h_{1})\Big{(}\mathcal{B}(\Phi(g)h_{2}^{{\dagger}})^{-1}\cdot_{G}g\cdot_{G}\mathcal{B}(h_{2}^{{\dagger}})\Big{)}$
$\displaystyle=$
$\displaystyle\mathcal{B}(\Phi\Big{(}\mathcal{B}(\Phi(g)h_{2}^{{\dagger}})^{-1}\cdot_{G}g\cdot_{G}\mathcal{B}(h_{2}^{{\dagger}})\Big{)}h_{1}^{{\dagger}})^{-1}\cdot_{G}\mathcal{B}(\Phi(g)h_{2}^{{\dagger}})^{-1}\cdot_{G}g\cdot_{G}\mathcal{B}(h_{2}^{{\dagger}})\cdot_{G}\mathcal{B}(h_{1}^{{\dagger}})$
$\displaystyle=$
$\displaystyle\Big{(}\mathcal{B}(\Phi(g)h_{2}^{{\dagger}})\cdot_{G}\mathcal{B}(\Phi\Big{(}\mathcal{B}(\Phi(g)h_{2}^{{\dagger}})^{-1}\cdot_{G}g\cdot_{G}\mathcal{B}(h_{2}^{{\dagger}})\Big{)}h_{1}^{{\dagger}})\Big{)}^{-1}\cdot_{G}g\cdot_{G}\mathcal{B}(h_{2}^{{\dagger}}\star
h_{1}^{{\dagger}})$ $\displaystyle=$
$\displaystyle\mathcal{B}\Big{(}\Phi(g)h_{2}^{{\dagger}}\cdot_{H}\Phi(\mathcal{B}(\Phi(g)h_{2}^{{\dagger}}))\Phi\Big{(}\mathcal{B}(\Phi(g)h_{2}^{{\dagger}})^{-1}\cdot_{G}g\cdot_{G}\mathcal{B}(h_{2}^{{\dagger}})\Big{)}h_{1}^{{\dagger}}\Big{)}^{-1}\cdot_{G}g\cdot_{G}\mathcal{B}(h_{2}^{{\dagger}}\star
h_{1}^{{\dagger}})$ $\displaystyle=$
$\displaystyle\mathcal{B}\Big{(}\Phi(g)h_{2}^{{\dagger}}\cdot_{H}\Phi\Big{(}g\cdot_{G}\mathcal{B}(h_{2}^{{\dagger}})\Big{)}h_{1}^{{\dagger}}\Big{)}^{-1}\cdot_{G}g\cdot_{G}\mathcal{B}(h_{2}^{{\dagger}}\star
h_{1}^{{\dagger}})$ $\displaystyle=$
$\displaystyle\Big{(}\mathcal{B}(\Phi(g)(h_{2}^{{\dagger}}\star
h_{1}^{{\dagger}}))\Big{)}^{-1}\cdot_{G}g\cdot_{G}\mathcal{B}(h_{2}^{{\dagger}}\star
h_{1}^{{\dagger}})$ $\displaystyle=$ $\displaystyle\Theta(h_{1}\star h_{2})g.$

Thus, $\Theta$ is an action of the descendent Lie group $(H,e_{H},\star)$ on
the manifold $G$. ∎

Since $\Theta(h)\in\text{Diff}(G)$ for all $h\in H$ and
$\Theta(h)e_{G}=e_{G}$, then
$\Theta(h)_{*e_{G}}:\mathfrak{g}\rightarrow\mathfrak{g}$ is an isomorphism of
vector spaces. By $\Theta(h_{1}\star h_{2})=\Theta(h_{1})\Theta(h_{2})$, we
have $\Theta(h_{1}\star
h_{2})_{*e_{G}}=\Theta(h_{1})_{*e_{G}}\Theta(h_{2})_{*e_{G}}$. Thus we obtain
a Lie group homomorphism from the descendent Lie group $(H,e_{H},\star)$ to
$GL(\mathfrak{g})$, which is also denoted by
$\Theta:H\,\rightarrow\,GL(\mathfrak{g})$. Consequently, we have the following
result which plays important roles in our definition of the cohomology of
relative Rota-Baxter operators on Lie groups.

######  Lemma 3.9.

With the above notations, $\Theta:H\,\rightarrow\,GL(\mathfrak{g})$ is an
action of the descendent Lie group $(H,e_{H},\star)$ on the vector space
$\mathfrak{g}$.

Now we are ready to define a cohomology theory for relative Rota-Baxter
operators on Lie groups. First we recall a standard version of the smooth
cohomology of a Lie group $G$ with coefficients in a $G$-module $A$ with a
$G$-action $\Pi$ (see e.g. [36]). An $n$-cochain is a smooth map

$\alpha_{n}:\underbrace{G\times\cdots\times G}_{n}\,\rightarrow\,A.$

The set of $n$-cochains forms an abelian group, which will be denoted by
$C^{n}(G,A)$. The differential $\mathrm{d}:C^{n}(G,A)\rightarrow C^{n+1}(G,A)$
is defined by

$\displaystyle\mathrm{d}(\alpha_{n})(g_{1},\cdots,g_{n},g_{n+1})$
$\displaystyle=$
$\displaystyle\Pi(g_{1})\alpha_{n}(g_{2},\cdots,g_{n},g_{n+1})+(-1)^{n+1}\alpha_{n}(g_{1},\cdots,g_{n})$
$\displaystyle+\sum_{i=1}^{n}(-1)^{i}\alpha_{n}(g_{1},\cdots,g_{i-1},g_{i}\cdot_{G}g_{i+1},g_{i+2},\cdots
g_{n+1}).$

We consider the Lie group $(H,e_{H},\star)$ and its module $\mathfrak{g}$ via
the action $\Theta:H\,\rightarrow\,GL(\mathfrak{g})$ given in Lemma 3.9.
Denote by $C^{k}(\mathcal{B})=C^{k-1}(H,\mathfrak{g})$ and by
$\mathrm{d}^{\mathcal{B}}$ the group cohomology differential. Parallel to
Definition 2.10, we have

######  Definition 3.10.

Let $\mathcal{B}$ be a relative Rota-Baxter operator on $(G,e_{G},\cdot_{G})$
with respect to an action $(H;\Phi)$. The cohomology of the cochain complex
$(C^{*}(\mathcal{B})=\oplus_{k=1}^{+\infty}C^{k}(\mathcal{B}),\mathrm{d}^{\mathcal{B}})$
is defined to be the cohomology for the relative Rota-Baxter operator
$\mathcal{B}$.

Denote by $\mathcal{H}^{k}({\mathcal{B}})$ the $k$-th cohomology group.

######  Remark 3.11.

Our cohomology theory in $\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}$ is well
rooted via the theory of controlling algebras and is tested and justified via
the deformation theory. The above cohomology theory in $\mathsf{RB_{G}^{H}}$
on the level of Lie groups will be justified via the Van-Est theory, which is
the content of the next section.

The following statements are important for the functoriality of our cohomology
theory.

######  Proposition 3.12.

Let $\mathcal{B}$ and $\mathcal{B}^{\prime}$ be relative Rota-Baxter operators
on a Lie group $G$ with respect to an action $(H;\Phi)$ and
$(\Psi_{G},\Psi_{H})$ be a homomorphism from $\mathcal{B}^{\prime}$ to
$\mathcal{B}$. Then $\Psi_{H}$ is a Lie group homomorphism from the descendent
Lie group $(H,e_{H},\star_{\mathcal{B}^{\prime}})$ of $\mathcal{B}^{\prime}$
to the descendent Lie group $(H,e_{H},\star_{\mathcal{B}})$ of $\mathcal{B}$.

###### Proof.

By (17), (18) and the fact that $\Psi_{H}$ is a Lie group homomorphism, we
have

$\displaystyle\Psi_{H}(h_{1}\star_{\mathcal{B}^{\prime}}h_{2})$
$\displaystyle=$
$\displaystyle\Psi_{H}(h_{1}\cdot_{H}\mathcal{B}^{\prime}(\Phi(h_{1}))h_{2})=\Psi_{H}(h_{1})\cdot_{H}\Psi(\mathcal{B}^{\prime}(\Phi(h_{1}))h_{2})$
$\displaystyle=$
$\displaystyle\Psi_{H}(h_{1})\cdot_{H}\Phi\Big{(}\mathcal{B}(\Psi_{H}(h_{1}))\Big{)}\Psi_{H}(h_{2})=\Psi_{H}(h_{1})\star_{\mathcal{B}}\Psi_{H}(h_{2}),$

which implies that $\Psi_{H}$ is a homomorphism between the descendent Lie
groups. ∎

######  Proposition 3.13.

Let $\mathcal{B}$ and $\mathcal{B}^{\prime}$ be relative Rota-Baxter operators
on $G$ with respect to an action $(H;\Phi)$ and $(\Psi_{G},\Psi_{H})$ be a
homomorphism from $\mathcal{B}^{\prime}$ to $\mathcal{B}$. Then the induced
action $\Theta^{\prime}$ of the Lie group $(H,\star_{\mathcal{B}^{\prime}})$
on $G$ and the induced action $\Theta$ of the Lie group
$(H,\star_{\mathcal{B}})$ on $G$ satisfy the following relation:

$\Psi_{G}\circ\Theta^{\prime}(h)=\Theta(\Psi_{H}(h))\circ\Psi_{G},\quad\forall
h\in H.$

That is for all $h\in H$, the following diagram commutes:

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta^{\prime}(h)}$$\scriptstyle{\Psi_{G}}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta(\Psi_{H}(h))}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{G}}$$\textstyle{G.}$

###### Proof.

By (17), (18), (19) and Proposition 3.12, for all $g\in G,h\in H$, we have

$\displaystyle\Psi_{G}(\Theta^{\prime}(h)g)$ $\displaystyle=$
$\displaystyle\Psi_{G}((\mathcal{B}^{\prime}(\Phi(g)h^{{\dagger}^{\prime}}))^{-1}\cdot_{G}g\cdot_{G}\mathcal{B}^{\prime}(h^{{\dagger}^{\prime}}))$
$\displaystyle=$
$\displaystyle\Psi_{G}((\mathcal{B}^{\prime}(\Phi(g)h^{{\dagger}^{\prime}}))^{-1})\cdot_{G}\Psi_{G}(g)\cdot_{G}\Psi_{G}(\mathcal{B}^{\prime}(h^{{\dagger}^{\prime}}))$
$\displaystyle=$
$\displaystyle\Psi_{G}(\mathcal{B}^{\prime}(\Phi(g)h^{{\dagger}^{\prime}}))^{-1}\cdot_{G}\Psi_{G}(g)\cdot_{G}\mathcal{B}(\Psi_{H}(h^{{\dagger}^{\prime}}))$
$\displaystyle=$
$\displaystyle\mathcal{B}(\Psi_{H}(\Phi(g)h^{{\dagger}}))^{-1}\cdot_{G}\Psi_{G}(g)\cdot_{G}\mathcal{B}((\Psi_{H}(h))^{{\dagger}})$
$\displaystyle=$
$\displaystyle\mathcal{B}(\Phi(\Psi_{G}(g))(\Psi_{H}(h))^{{\dagger}})^{-1}\cdot_{G}\Psi_{G}(g)\cdot_{G}\mathcal{B}((\Psi_{H}(h))^{{\dagger}})$
$\displaystyle=$ $\displaystyle\Theta(\Psi_{H}(h))\Psi_{G}(g).$

We finish the proof. ∎

Taking differentiation, we get

######  Corollary 3.14.

Let $\mathcal{B}$ and $\mathcal{B}^{\prime}$ be relative Rota-Baxter operators
on $G$ with respect to an action $(H;\Phi)$, and $(\Psi_{G},\Psi_{H})$ be a
homomorphism from $\mathcal{B}^{\prime}$ to $\mathcal{B}$. Then the action
$\Theta^{\prime}$ of the descendent Lie group
$(H,e_{H},\star_{\mathcal{B}^{\prime}})$ on $\mathfrak{g}$ and the action
$\Theta$ of the descendent Lie group $(H,e_{H},\star_{\mathcal{B}})$ on
$\mathfrak{g}$ satisfy the following relation:

$\Theta(\Psi_{H}(h))\circ(\Psi_{G})_{*e_{G}}=(\Psi_{G})_{*e_{G}}\circ\Theta^{\prime}(h),\quad\forall
h\in H.$

Now we prove the functoriality of this cohomology theory. Let $\mathcal{B}$
and $\mathcal{B}^{\prime}$ be relative Rota-Baxter operators on $G$ with
respect to an action $(H;\Phi)$. Let $(\Psi_{G},\Psi_{H})$ be a homomorphism
from $\mathcal{B}^{\prime}$ to $\mathcal{B}$ in which $\Psi_{H}$ is
invertible. Define a map
$P:C^{k}(\mathcal{B}^{\prime})\,\rightarrow\,C^{k}(\mathcal{B})$ by

$P(\omega)(h_{1},\cdots,h_{k-1})=(\Psi_{G})_{*e_{G}}\omega(\Psi_{H}^{-1}(h_{1}),\cdots,\Psi_{H}^{-1}(h_{k-1})),\quad\forall
h_{i}\in H.$

######  Theorem 3.15.

With the above notations, $P$ is cochain map from the cochain complex
$(C^{*}(\mathcal{B}^{\prime}),\mathrm{d}^{\mathcal{B}^{\prime}})$ to
$(C^{*}(\mathcal{B}),\mathrm{d}^{\mathcal{B}})$, which induces a homomorphism
$P_{*}$ from the cohomology group $\mathcal{H}^{k}(\mathcal{B}^{\prime})$ to
$\mathcal{H}^{k}(\mathcal{B})$ for all $k\geq 1$.

###### Proof.

For all $\omega\in C^{k}(\mathcal{B}^{\prime})$, by Corollary 3.14, we have

$\displaystyle\mathrm{d}^{\mathcal{B}}(P\omega)(h_{1},h_{2},\cdots,h_{k})$
$\displaystyle=$
$\displaystyle\Theta(h_{1})(\Psi_{G})_{*e_{G}}\omega(\Psi_{H}^{-1}(h_{2}),\cdots,\Psi_{H}^{-1}(h_{k}))$
$\displaystyle+\sum_{i=1}^{k-1}(-1)^{i}P(\omega)(h_{1},\cdots,h_{i-1},h_{i}\star_{\mathcal{B}}h_{i+1},h_{i+2},\cdots,h_{k})$
$\displaystyle+(-1)^{k}P(\omega)(h_{1},\cdots,h_{k-1})$ $\displaystyle=$
$\displaystyle(\Psi_{G})_{*e_{G}}\Theta^{\prime}(\Psi_{H}^{-1}(h_{1}))\omega(\Psi_{H}^{-1}(h_{2}),\cdots,\Psi_{H}^{-1}(h_{k}))$
$\displaystyle+\sum_{i=1}^{k-1}(-1)^{i}(\Psi_{G})_{*e_{G}}\omega(\Psi_{H}^{-1}(h_{1}),\cdots,\Psi_{H}^{-1}(h_{i-1}),\Psi_{H}^{-1}(h_{i})\star_{\mathcal{B}^{\prime}}\Psi_{H}^{-1}(h_{i+1}),\Psi_{H}^{-1}(h_{i+2}),\cdots,\Psi_{H}^{-1}(h_{k}))$
$\displaystyle+(-1)^{k}(\Psi_{G})_{*e_{G}}\omega(\Psi_{H}^{-1}(h_{1}),\cdots,\Psi_{H}^{-1}(h_{k-1}))$
$\displaystyle=$
$\displaystyle(\Psi_{G})_{*e_{G}}\mathrm{d}^{\mathcal{B}^{\prime}}\omega(\Psi_{H}^{-1}(h_{1}),\cdots,\Psi_{H}^{-1}(h_{k}))$
$\displaystyle=$ $\displaystyle
P(\mathrm{d}^{\mathcal{B}^{\prime}}\omega)(h_{1},\cdots,h_{k}),\quad\forall
h_{i}\in H.$

Thus $P$ is a cochain map. Consequently it induces a homomorphism $P_{*}$ from
the cohomology group $\mathcal{H}^{k}(B^{\prime})$ to $\mathcal{H}^{k}(B)$ for
all $k\geq 1$. ∎

##  4\. Differentiation and Van Est theory

In this section, we establish the differentiation functor for relative Rota-
Baxter operators and the Van Est homomorphism on the cohomology level. The
classical Van Est isomorphism [35] gives the relation between the smooth
cohomology of Lie groups and the cohomology of Lie algebras. See [1, 6, 8, 17,
22, 25, 28, 37] for various Van Est type theorems.

###  4.1. The differentiation functor for relative Rota-Baxter operators

Let $\Phi$ be an action of a Lie group $(G,e_{G},\cdot_{G})$ on a manifold
$M$. Define $\Delta:G\times M\rightarrow M$ by $\Delta(g,m)=\Phi(g)m$. Then
$\Delta(e_{G},m)=m$ and
$\Delta(g_{1}\cdot_{G}g_{2},m)=\Delta(g_{1},\Delta(g_{2},m))$, for all
$g_{1},g_{2}\in G,m\in M$. Moreover, we have the following Leibniz rule,

$\displaystyle\frac{d}{dt}\bigg{|}_{t=0}\Delta(\gamma_{1}(t),\gamma_{2}(t))$
$\displaystyle=$
$\displaystyle\frac{d}{dt}\bigg{|}_{t=0}\Delta(\gamma_{1}(0),\gamma_{2}(t))+\frac{d}{dt}\bigg{|}_{t=0}\Delta(\gamma_{1}(t),\gamma_{2}(0))$
$\displaystyle=$
$\displaystyle\frac{d}{dt}\bigg{|}_{t=0}\Phi(\gamma_{1}(t))\gamma_{2}(t)$
$\displaystyle=$
$\displaystyle\frac{d}{dt}\bigg{|}_{t=0}\Phi(\gamma_{1}(0))\gamma_{2}(t)+\frac{d}{dt}\bigg{|}_{t=0}\Phi(\gamma_{1}(t))\gamma_{2}(0),$

where $\gamma_{1}(t)$ is a path in $G$ with $\gamma_{1}(0)=g$ and
$\gamma_{2}(t)$ is a path in $M$ with $\gamma_{2}(0)=m$.

Let $(G,e_{G},\cdot_{G})$ and $(H,e_{H},\cdot_{H})$ be Lie groups and denote
by $\exp_{G}$ and $\exp_{H}$ the exponential maps for the Lie groups
$(G,e_{G},\cdot_{G})$ and $(H,e_{H},\cdot_{H})$ respectively. Let
$\Phi:G\rightarrow\mathrm{Aut}(H)$ be an action of $G$ on $H$. Since
$\Phi(g)\in\mathrm{Aut}(H)$ for all $g\in G$, then
$\Phi(g)_{*e_{H}}:\mathfrak{h}\rightarrow\mathfrak{h}$ is a Lie algebra
isomorphism. By $\Phi(g_{1}\cdot_{G}g_{2})=\Phi(\mathfrak{g}_{1})\Phi(g_{2})$,
we have
$\Phi(g_{1}\cdot_{G}g_{2})_{*e_{H}}=\Phi(g_{1})_{*e_{H}}\Phi(g_{2})_{*e_{H}}$.
Thus we obtain a Lie group homomorphism from $G$ to
$\mathrm{Aut}(\mathfrak{h})$, which we denote by
$\tilde{\Phi}:G\,\rightarrow\,\mathrm{Aut}(\mathfrak{h})$. Then taking the
differential, we obtain a Lie algebra homomorphism
$\phi:=\tilde{\Phi}_{*e_{G}}$ from the Lie algebra $\mathfrak{g}$ to
$\mathrm{Der}(\mathfrak{h})$. We call $\phi$ the differentiated action of
$\Phi$. In fact, Lie II theorem tells us that
$\mathrm{Aut}(H)\cong\mathrm{Aut}(\mathfrak{h})$, if $H$ is connected and
simply connected. Therefore when both $G$ and $H$ are connected and simply
connected, given a Lie algebra action
$\phi:\mathfrak{g}\to\mathrm{Der}(\mathfrak{h})$, there is a unique integrated
action $\Phi:G\to\mathrm{Aut}(H)$ whose differentiation is $\phi$. This
procedure can be well explained by the following diagram:

(20)
$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathrm{Aut}(H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$differentiation$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\Phi}}$differentiation$\textstyle{\mathrm{Aut}(\mathfrak{h})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$differentiation$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi=\tilde{\Phi}_{*e_{G}}}$$\textstyle{\mathrm{Der}(\mathfrak{h}).}$

######  Theorem 4.1.

Let $\mathcal{B}:H\,\rightarrow\,G$ be a relative Rota-Baxter operator on $G$
with respect to an action $(H;\Phi)$. Define
$B:\mathfrak{h}\rightarrow\mathfrak{g}$ by

(21) $B=\mathcal{B}_{*e_{H}},$

which is the tangent map of $\mathcal{B}$ at the identity $e_{H}$. Then $B$ is
a relative Rota-Baxter operator on $\mathfrak{g}$ with respect to the action
$(\mathfrak{h};\phi)$, where $\phi$ is the differentiated action of $\Phi$
defined in (20).

###### Proof.

For all $u,v\in\mathfrak{h}$, we have

$\displaystyle[B(u),B(v)]_{\mathfrak{g}}$ $\displaystyle=$
$\displaystyle\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\exp_{G}(tB(u))\cdot_{G}\exp_{G}(sB(v))\cdot_{G}\exp_{G}(-tB(u))$
$\displaystyle=$
$\displaystyle\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\mathcal{B}(\exp_{H}(tu))\cdot_{G}\mathcal{B}(\exp_{H}(sv))\cdot_{G}(\mathcal{B}(\exp_{H}(tu)))^{-1}$
$\displaystyle=$
$\displaystyle\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\mathcal{B}(\exp_{H}(tu))\cdot_{G}\mathcal{B}(\exp_{H}(sv))\cdot_{G}\mathcal{B}\Big{(}\Phi(\mathcal{B}(\exp_{H}(tu))^{-1})\exp_{H}(-tu)\Big{)}$
$\displaystyle=$
$\displaystyle\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\mathcal{B}\Big{(}\exp_{H}(tu)\cdot_{H}\Phi(\mathcal{B}(\exp_{H}(tu)))\exp_{H}(sv)\Big{)}\cdot_{G}\mathcal{B}\Big{(}\Phi(\mathcal{B}(\exp_{H}(tu))^{-1})\exp_{H}(-tu)\Big{)}$
$\displaystyle=$
$\displaystyle\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\mathcal{B}\Big{(}\mathrm{Ad}_{\exp_{H}(tu)}(\Phi(\mathcal{B}(\exp_{H}(tu)))\exp_{H}(sv))\cdot_{H}(\exp_{H}(tu)\cdot_{H}\Phi(\mathrm{Ad}_{\mathcal{B}(\exp_{H}(tu))}\mathcal{B}(\exp_{H}(sv)))\exp_{H}(-tu))\Big{)}$
$\displaystyle=$
$\displaystyle\mathcal{B}_{*e_{G}}\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\mathrm{Ad}_{\exp_{H}(tu)}(\Phi(\mathcal{B}(\exp_{H}(tu)))\exp_{H}(sv))\cdot_{H}(\exp_{H}(tu)\cdot_{H}\Phi(\mathrm{Ad}_{\mathcal{B}(\exp_{H}(tu))}\mathcal{B}(\exp_{H}(sv)))\exp_{H}(-tu))$
$\displaystyle=$
$\displaystyle\mathcal{B}_{*e_{G}}\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\Big{(}\mathrm{Ad}_{\exp_{H}(tu)}\Phi(\mathcal{B}(\exp_{H}(tu)))\exp_{H}(sv)\Big{)}$
$\displaystyle+\mathcal{B}_{*e_{G}}\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\exp_{H}(tu)\cdot_{H}\Phi(\mathrm{Ad}_{\mathcal{B}(\exp_{H}(tu))}\mathcal{B}(\exp_{H}(sv)))\exp_{H}(-tu)$
$\displaystyle=$
$\displaystyle\mathcal{B}_{*e_{G}}\Big{(}\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\mathrm{Ad}_{\exp_{H}(tu)}\exp_{H}(sv)+\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\Phi(\mathcal{B}(\exp_{H}(tu)))\exp_{H}(sv)\Big{)}$
$\displaystyle+\mathcal{B}_{*e_{G}}\frac{d^{2}}{dsdt}\bigg{|}_{t,s=0}\exp_{H}(tu)\cdot_{H}\Phi(\mathrm{Ad}_{\mathcal{B}(\exp_{H}(tu))}\mathcal{B}(\exp_{H}(sv)))\exp_{H}(-tu)$
$\displaystyle=$
$\displaystyle\mathcal{B}_{*e_{G}}\Big{(}[u,v]_{\mathfrak{h}}+\phi(\mathcal{B}_{*e_{G}}(u))v\Big{)}+\mathcal{B}_{*e_{G}}\frac{d^{2}}{dsdt}\bigg{|}_{t,s=0}\Phi(\mathrm{Ad}_{\mathcal{B}(\exp_{H}(tu))}\mathcal{B}(\exp_{H}(sv)))\exp_{H}(-tu)$
$\displaystyle=$
$\displaystyle\mathcal{B}_{*e_{G}}\Big{(}[u,v]_{\mathfrak{h}}+\phi(\mathcal{B}_{*e_{G}}(u))v\Big{)}+\mathcal{B}_{*e_{G}}\frac{d^{2}}{dsdt}\bigg{|}_{t,s=0}\Phi(\mathcal{B}(\exp_{H}(sv)))\exp_{H}(-tu)$
$\displaystyle=$
$\displaystyle\mathcal{B}_{*e_{G}}\Big{(}[u,v]_{\mathfrak{h}}+\phi(\mathcal{B}_{*e_{G}}(u))v\Big{)}-\mathcal{B}_{*e_{G}}(\phi(\mathcal{B}_{*e_{G}}(v))u)$
$\displaystyle=$ $\displaystyle
B([u,v]_{\mathfrak{h}}+\phi(B(u))v-\phi(B(v))u).$

Thus $B$ is a relative Rota-Baxter operator. ∎

The above result extends to the level of morphisms:

######  Proposition 4.2.

Let $\mathcal{B}$ and $\mathcal{B}^{\prime}$ be relative Rota-Baxter operators
on a Lie group $G$ with respect to an action $(H;\Phi)$, and
$(\Psi_{G},\Psi_{H})$ be a homomorphism from $\mathcal{B}^{\prime}$ to
$\mathcal{B}$. Then $((\Psi_{G})_{*e_{G}},(\Psi_{H})_{*e_{H}})$ is a
homomorphism from the relative Rota-Baxter operator
$\mathcal{B}^{\prime}_{*e_{H}}$ to $\mathcal{B}_{*e_{H}}$.

###### Proof.

We denote $((\Psi_{G})_{*e_{G}},(\Psi_{H})_{*e_{H}})$ by
$(\psi_{\mathfrak{g}},\psi_{\mathfrak{h}})$ and
$\mathcal{B}^{\prime}_{*e_{H}}=B^{\prime},\mathcal{B}_{*e_{H}}=B$. Since
$\Psi_{G}:G\,\rightarrow\,G$ and $\Psi_{H}:H\,\rightarrow\,H$ are Lie group
homomorphisms, thus
$\psi_{\mathfrak{g}}:\mathfrak{g}\,\rightarrow\,\mathfrak{g}$ and
$\psi_{\mathfrak{h}}:\mathfrak{h}\,\rightarrow\,\mathfrak{h}$ are Lie algebra
homomorphisms. By (17), we have
$B\circ\psi_{\mathfrak{h}}=\psi_{\mathfrak{g}}\circ B^{\prime}$. Then we have

$\displaystyle\frac{d}{ds}\frac{d}{dt}\bigg{|}_{s,t=0}\Psi_{H}(\Phi(\exp_{G}(sx))\exp_{H}(tu))=\psi_{\mathfrak{h}}\frac{d}{ds}\frac{d}{dt}\bigg{|}_{t,s=0}\Phi(\exp_{G}(sx))\exp_{H}(tu)=\psi_{\mathfrak{h}}(\phi(x)u)$

and

$\displaystyle\frac{d}{ds}\frac{d}{dt}\bigg{|}_{t,s=0}\Phi(\Psi_{G}(\exp_{G}(sx)))\Psi_{H}(\exp_{H}(tu))$
$\displaystyle=$
$\displaystyle\frac{d}{ds}\frac{d}{dt}\bigg{|}_{t,s=0}\Phi(\exp_{G}(s\psi_{\mathfrak{g}}(x)))\exp_{H}(t\psi_{\mathfrak{h}}(u))$
$\displaystyle=$
$\displaystyle\phi(\psi_{\mathfrak{g}}(x))\psi_{\mathfrak{h}}(u),\quad\forall
x\in\mathfrak{g},u\in\mathfrak{h}.$

By (18), we have
$\psi_{\mathfrak{h}}(\phi(x)u)=\phi(\psi_{\mathfrak{g}}(x))\psi_{\mathfrak{h}}(u)$,
which implies that $((\Psi_{G})_{*e_{G}},(\Phi_{H})_{*e_{H}})$ is a
homomorphism from $\mathcal{B}^{\prime}_{*e_{H}}$ to $\mathcal{B}_{*e_{H}}$. ∎

Given Lie groups $G,H$ and their Lie algebras $\mathfrak{g},\mathfrak{h}$
respectively, then the above differentiation procedure gives us a functor

(22)
$\mathbf{\mathfrak{D}}:\mathsf{RB_{G}^{H}}\to\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}.$

Actually the set of homomorphisms between relative Rota-Baxter operators on
Lie groups and the set of homomorphisms between the differentiated relative
Rota-Baxter operators on the corresponding Lie algebras are isomorphic. First
we need a lemma:

######  Lemma 4.3.

Let $G$ and $H$ be connected Lie groups whose Lie algebras are $\mathfrak{g}$
and $\mathfrak{h}$ respectively. Let $\Phi:G\rightarrow\mathrm{Aut}(H)$ be an
action $G$ on $H$ and
$\phi:\mathfrak{g}\,\rightarrow\,\mathrm{Der}(\mathfrak{h})$ be the induced
action of $\mathfrak{g}$ on $\mathfrak{h}$. Let $\Psi_{G}:G\,\rightarrow\,G$
and $\Psi_{H}:H\,\rightarrow\,H$ be Lie groups homomorphisms and
$\psi_{\mathfrak{g}}:\mathfrak{g}\,\rightarrow\,\mathfrak{g}$ and
$\psi_{\mathfrak{h}}:\mathfrak{h}\,\rightarrow\,\mathfrak{h}$ be the induced
Lie algebra homomorphisms. If $\psi_{\mathfrak{g}}$ and $\psi_{\mathfrak{h}}$
satisfies (6). Then $\Psi_{G}$ and $\Psi_{H}$ satisfy (18), i.e.

$\Phi(\Psi_{G}(g))\Psi_{H}(h)=\Psi_{H}(\Phi(g)h),\quad\forall g\in G,h\in H.$

###### Proof.

We first show that (18) holds for $g$ and $h$ in a small neighborhood of
identities. In fact, by writing $g=\exp_{G}x$ and $h=\exp_{H}u$ and using (6)
and Diagram (20), we have

$\displaystyle\Psi_{H}(\Phi(\exp_{G}(x))\exp_{H}(u))$ $\displaystyle=$
$\displaystyle\Psi_{H}(\exp_{H}(\tilde{\Phi}(\exp_{G}(x))u))=\Psi_{H}(\exp_{H}(\exp_{\mathrm{Aut}(\mathfrak{h})}(\phi(x))u))$
$\displaystyle=$
$\displaystyle\exp_{H}(\psi_{\mathfrak{h}}(\exp_{\mathrm{Aut}(\mathfrak{h})}(\phi(x))u))=\exp_{H}(\exp_{\mathrm{Aut}(\mathfrak{h})}(\phi(\psi_{\mathfrak{g}}(x)))\psi_{\mathfrak{h}}(u))$
$\displaystyle=$
$\displaystyle\exp_{H}(\tilde{\Phi}(\exp_{G}(\psi_{\mathfrak{g}}(x)))\psi_{\mathfrak{h}}(u))=\Phi(\Psi_{G}(\exp_{G}(x)))\exp_{H}(\psi_{\mathfrak{h}}(u))$
$\displaystyle=$
$\displaystyle\Phi(\Psi_{G}(\exp_{G}(x)))\Psi_{H}(\exp_{H}(u)).$

Since $G$ and $H$ are connected, we can write
$G=\cup_{n\in\mathbb{N}}U_{G}^{n}$ and $H=\cup_{n\in\mathbb{N}}U_{H}^{n}$, for
some small open set $U_{G}\subseteq G$ and $U_{H}\subseteq H$ containing
$e_{G}$ and $e_{H}$ respectively. Then $\Psi_{H}$ being a Lie group
homomorphism and $\Phi$ being an action of Lie groups imply that for
$g=\exp_{G}x$, and $h=h_{1}\cdot\dots\cdot h_{k}$, but each $h_{j}$ very
small, (18) still holds; finally, the same facts imply that for
$g=g_{1}\cdot\dots\cdot g_{l}$, with each $g_{j}$ very small, and any $h\in
H$, (18) still holds. Thus we have shown that (18) holds in general. ∎

######  Theorem 4.4.

Let $G$ and $H$ be connected, simply connected Lie groups,
$\mathcal{B}^{\prime}$ and $\mathcal{B}$ be relative Rota-Baxter operators on
$G$ with respect to the action $(H;\Phi)$. Then

$\mathrm{Hom}_{\mathsf{RB_{G}^{H}}}(\mathcal{B}^{\prime},\mathcal{B})\cong\mathrm{Hom}_{\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}}(\mathbf{\mathfrak{D}}(\mathcal{B}^{\prime}),\mathbf{\mathfrak{D}}(\mathcal{B})).$

###### Proof.

For any
$(\Psi_{G},\Psi_{H})\in\mathrm{Hom}_{\mathsf{RB_{G}^{H}}}(\mathcal{B}^{\prime},\mathcal{B})$,
we obtain
$((\Psi_{G})_{*e_{G}},(\Psi_{H})_{*e_{H}})\in\mathrm{Hom}_{\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}}(\mathbf{\mathfrak{D}}(\mathcal{B}^{\prime}),\mathbf{\mathfrak{D}}(\mathcal{B}))$
via Proposition 4.2.

Conversely, for any
$(\psi_{\mathfrak{g}},\psi_{\mathfrak{h}})\in\mathrm{Hom}_{\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}}(\mathbf{\mathfrak{D}}(\mathcal{B}^{\prime}),\mathbf{\mathfrak{D}}(\mathcal{B}))$,
$(\psi_{\mathfrak{g}},\psi_{\mathfrak{h}})$ is a Lie algebra endomorphism of
the semi-direct product Lie algebra $\mathfrak{g}\ltimes_{\phi}\mathfrak{h}$
and $(\psi_{\mathfrak{g}},\psi_{\mathfrak{h}})(Gr(B^{\prime}))\subseteq
Gr(B)$. By Lemma 4.3, there exist unique Lie group homomorphisms
$\Psi_{G}:G\,\rightarrow\,G$ and $\Psi_{H}:H\,\rightarrow\,H$ such that
$(\Psi_{G},\Psi_{H})$ is a Lie group endomorphism of the semi-direct product
Lie group $G\ltimes_{\Phi}H$. Since the image
$(\Psi_{G},\Psi_{H})Gr(\mathcal{B}^{\prime})$ is a Lie subgroup of
$G\ltimes_{\Phi}H$ and its Lie algebra is
$(\psi_{\mathfrak{g}},\psi_{\mathfrak{h}})(Gr(B^{\prime}))\subseteq Gr(B)$, it
follows111It is obvious that the inclusion happens locally near identity, the
global statement follows again using the fact that both
$(\Psi_{G},\Psi_{H})Gr(\mathcal{B}^{\prime})$ and $Gr(\mathcal{B})$ are
connected and they can be written as products of elements near identity.  that
$(\Psi_{G},\Psi_{H})Gr(\mathcal{B}^{\prime})\subseteq Gr(\mathcal{B})$. This
in turn implies that
$\Psi_{G}\circ\mathcal{B}^{\prime}=\mathcal{B}\circ\Psi_{H}$. Thus
$(\Psi_{G},\Psi_{H})\in\mathrm{Hom}_{\mathsf{RB_{G}^{H}}}(\mathcal{B}^{\prime},\mathcal{B})$,
and
$\mathrm{Hom}_{\mathsf{RB_{G}^{H}}}(\mathcal{B}^{\prime},\mathcal{B})\cong\mathrm{Hom}_{\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}}(\mathbf{\mathfrak{D}}(\mathcal{B}^{\prime}),\mathbf{\mathfrak{D}}(\mathcal{B})).$
∎

###  4.2. The Van Est homomorphism for the cohomologies

Now we extend the above differentiation to the level of cohomology. For this,
let us consider the relation between the descendent Lie group of the relative
Rota-Baxter operator $\mathcal{B}$ and the descendent Lie algebra of the
relative Rota-Baxter operator $\mathbf{\mathfrak{D}}(\mathcal{B})$.

######  Proposition 4.5.

Let $\mathcal{B}$ be a relative Rota-Baxter operator on $G$ with respect to
the action $(H;\Phi)$. Then the Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{\star})$ of the descendent Lie group
$(H,e_{H},\star)$ is the descendent Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{\mathbf{\mathfrak{D}}(\mathcal{B})})$ of the
relative Rota-Baxter operator $\mathbf{\mathfrak{D}}(\mathcal{B})$ on the
level of Lie algebras.

###### Proof.

We denote by $\mathfrak{exp}$ the exponential map for the Lie group
$(H,e_{H},\star)$. For all $u,v\in\mathfrak{h}$, with the help of Proposition
3.5, we have

$\displaystyle[u,v]_{\star}$ $\displaystyle=$
$\displaystyle\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\mathfrak{exp}(tu)\star\mathfrak{exp}(sv)\star\mathfrak{exp}(-tu)$
$\displaystyle=$
$\displaystyle\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\Big{(}\mathfrak{exp}(tu))\cdot_{H}\Phi(\mathcal{B}(\mathfrak{exp}(sv)))\mathfrak{exp}(sv)\Big{)}\star\mathfrak{exp}(-tu)$
$\displaystyle=$
$\displaystyle\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\Big{(}\mathfrak{exp}(tu))\cdot_{H}\Phi(\mathcal{B}(\mathfrak{exp}(sv)))\mathfrak{exp}(sv)\Big{)}\star\Phi(\mathcal{B}(\mathfrak{exp}(tu))^{-1})(\mathfrak{exp}(tu))^{-1}$
$\displaystyle=$
$\displaystyle\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\Big{(}\mathfrak{exp}(tu)\cdot_{H}\Phi(\mathcal{B}(\mathfrak{exp}(tu)))\mathfrak{exp}(sv)\Big{)}\cdot_{H}\Phi(\mathrm{Ad}_{\mathcal{B}(\mathfrak{exp}(tu))}\mathcal{B}(\mathfrak{exp}(sv)))(\mathfrak{exp}(tu))^{-1}$
$\displaystyle=$
$\displaystyle\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\Big{(}\mathrm{Ad}_{\mathfrak{exp}(tu)}(\Phi(\mathcal{B}(\mathfrak{exp}(tu)))\mathfrak{exp}(sv))\Big{)}\cdot_{H}\Big{(}\mathfrak{exp}(tu)\cdot_{H}\Phi(\mathrm{Ad}_{\mathcal{B}(\mathfrak{exp}(tu))}\mathcal{B}(\mathfrak{exp}(sv)))(\mathfrak{exp}(tu))^{-1}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\Big{(}\mathrm{Ad}_{\mathfrak{exp}(tu)}\Phi(\mathcal{B}(\mathfrak{exp}(tu)))\mathfrak{exp}(sv)\Big{)}$
$\displaystyle+\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\mathfrak{exp}(tu)\cdot_{H}\Phi(\mathrm{Ad}_{\mathcal{B}(\mathfrak{exp}(tu))}\mathcal{B}(\mathfrak{exp}(sv)))(\mathfrak{exp}(tu))^{-1}$
$\displaystyle=$
$\displaystyle\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\mathrm{Ad}_{\mathfrak{exp}(tu)}\mathfrak{exp}(sv)+\frac{d^{2}}{dtds}\bigg{|}_{t,s=0}\Phi(\mathcal{B}(\mathfrak{exp}(tu)))\mathfrak{exp}(sv)$
$\displaystyle+\frac{d^{2}}{dsdt}\bigg{|}_{t,s=0}\mathfrak{exp}(tu)\cdot_{H}\Phi(\mathrm{Ad}_{\mathcal{B}(\mathfrak{exp}(tu))}\mathcal{B}(\mathfrak{exp}(sv)))(\mathfrak{exp}(tu))^{-1}$
$\displaystyle=$
$\displaystyle[u,v]_{\mathfrak{h}}+\phi(\mathcal{B}_{*e_{H}}(u))v+\frac{d^{2}}{dsdt}\bigg{|}_{t,s=0}\Phi(\mathrm{Ad}_{\mathcal{B}(\mathfrak{exp}(tu))}\mathcal{B}(\mathfrak{exp}(sv)))(\mathfrak{exp}(tu))^{-1}$
$\displaystyle=$
$\displaystyle[u,v]_{\mathfrak{h}}+\phi(\mathcal{B}_{*e_{H}}(u))v+\frac{d^{2}}{dsdt}\bigg{|}_{t,s=0}\Phi(\mathcal{B}(\mathfrak{exp}(sv)))(\mathfrak{exp}(tu))^{-1}$
$\displaystyle=$
$\displaystyle[u,v]_{\mathfrak{h}}+\phi(\mathcal{B}_{*e_{H}}(u))v-\phi(\mathcal{B}_{*e_{H}}(v))u$

Therefore,
$[\cdot,\cdot]_{\star}=[\cdot,\cdot]_{\mathbf{\mathfrak{D}}(\mathcal{B})}$,
which implies that the Lie algebra of the descendent Lie group
$(H,e_{H},\star)$ is the descendent Lie algebra of the relative Rota-Baxter
operator $\mathbf{\mathfrak{D}}(\mathcal{B})$. ∎

By Theorem 3.8, we know that $\Theta$ is an action of the descendent Lie group
$(H,e_{H},\star)$ on $G$. By Lemma 3.9,
$\Theta:H\,\rightarrow\,GL(\mathfrak{g})$ is an action of the descendent Lie
group $(H,e_{H},\star)$ on the vector space $\mathfrak{g}$. Taking the
differentiation, we have the following result.

######  Proposition 4.6.

Let $\mathcal{B}:H\,\rightarrow\,G$ be a relative Rota-Baxter operator on $G$
with respect to an action $(H;\Phi)$. Then the differentiation of the action
$\Theta:H\,\rightarrow\,GL(\mathfrak{g})$ of the descendent Lie group
$(H,e_{H},\star)$ on $\mathfrak{g}$ is exactly the representation
$\theta:\mathfrak{h}\,\rightarrow\,\mathfrak{gl}(\mathfrak{g})$ of the
descendent Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{\mathbf{\mathfrak{D}}(\mathcal{B})})$ on
$\mathfrak{g}$ given in Proposition 2.8. That is

$\Theta_{*e_{H}}(u)x=\mathbf{\mathfrak{D}}(\mathcal{B})(\phi(x)u)+[\mathbf{\mathfrak{D}}(\mathcal{B})(u),x]_{\mathfrak{g}},\quad\forall
x\in\mathfrak{g},u\in\mathfrak{h}.$

###### Proof.

By the definition of $\Theta$, and use the above notations in Proposition 4.2
and Proposition 4.5, we have

$\displaystyle\Theta_{*e_{H}}(u)x$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\frac{d}{ds}\bigg{|}_{t,s=0}\Theta(\mathfrak{exp}(tu))\exp_{G}(sx)$
$\displaystyle=$
$\displaystyle\frac{d}{dt}\frac{d}{ds}\bigg{|}_{t,s=0}\mathcal{B}(\Phi(\exp_{G}(sx))\mathfrak{exp}(-tu))^{-1}\cdot_{G}\exp_{G}(sx)\cdot_{G}\mathcal{B}(\mathfrak{exp}(-tu))$
$\displaystyle=$
$\displaystyle\frac{d}{dt}\frac{d}{ds}\bigg{|}_{t,s=0}\mathcal{B}(\mathfrak{exp}(-tu))^{-1}\cdot_{G}\exp_{G}(sx)\cdot_{G}\mathcal{B}(\mathfrak{exp}(-tu))$
$\displaystyle+\frac{d}{dt}\frac{d}{ds}\bigg{|}_{t,s=0}\mathcal{B}(\Phi(\exp_{G}(sx))\mathfrak{exp}(-tu))^{-1}\cdot_{G}\mathcal{B}(\mathfrak{exp}(-tu))$
$\displaystyle=$
$\displaystyle[\mathbf{\mathfrak{D}}(\mathcal{B})(u),x]_{\mathfrak{g}}+\frac{d}{ds}\frac{d}{dt}\bigg{|}_{t,s=0}\mathcal{B}(\Phi(\exp_{G}(sx))\mathfrak{exp}(-tu))^{-1}$
$\displaystyle=$
$\displaystyle[\mathbf{\mathfrak{D}}(\mathcal{B})(u),x]_{\mathfrak{g}}+\mathbf{\mathfrak{D}}(\mathcal{B})(\phi(x)u),$

which implies that $\Theta_{*e_{H}}=\theta$. ∎

Let $\mathcal{B}:H\,\rightarrow\,G$ be a relative Rota-Baxter operator on $G$
with respect to an action $(H;\Phi)$, we define
$\mathsf{VE}:C^{k}(\mathcal{B})\,\rightarrow\,C^{k}(\mathbf{\mathfrak{D}}(\mathcal{B}))$
by

$\displaystyle\mathsf{VE}(F)(u_{1},\cdots,u_{k-1})$ $\displaystyle=$
$\displaystyle\sum_{s\in\mathsf{S}(k-1)}(-1)^{|s|}\frac{d}{dt_{s(1)}}\cdots\frac{d}{dt_{s(k-1)}}\bigg{|}_{t_{s(1)}=t_{s(2)}=\cdots=t_{s(k-1)}=0}F(\mathfrak{exp}(t_{s(1)}u_{s(1)}),\cdots,\mathfrak{exp}(t_{s(k-1)}u_{s(k-1)})),$

for all $F\in C^{k}(\mathcal{B}),~{}u_{1},\cdots,u_{k-1}\in\mathfrak{h}$.

######  Theorem 4.7.

With the above notations, $\mathsf{VE}$ is a cochain map, i.e. we have the
following commutative diagram

$\textstyle{\cdots\longrightarrow
C^{k-1}(\mathcal{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{VE}}$$\scriptstyle{\quad\mathrm{d}^{\mathcal{B}}}$$\textstyle{C^{k}(\mathcal{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{VE}}$$\scriptstyle{\quad\mathrm{d}^{\mathcal{B}}}$$\textstyle{C^{k+1}(\mathcal{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{VE}}$$\textstyle{\cdots}$$\textstyle{\cdots\longrightarrow
C^{k-1}(\mathbf{\mathfrak{D}}(\mathcal{B}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\mathrm{d}_{\mathsf{CE}}^{\mathbf{\mathfrak{D}}(\mathcal{B})}}$$\textstyle{C^{k}(\mathbf{\mathfrak{D}}(\mathcal{B}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\mathrm{d}_{\mathsf{CE}}^{\mathbf{\mathfrak{D}}(\mathcal{B})}}$$\textstyle{C^{k+1}(\mathbf{\mathfrak{D}}(\mathcal{B}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots.}$

Consequently, $\mathsf{VE}$ induces a homomorphism $\mathsf{VE}_{*}$ from the
cohomology group $\mathcal{H}^{k}(\mathcal{B})$ to
$H^{k}(\mathbf{\mathfrak{D}}(\mathcal{B}))$. The map $\mathsf{VE}$ is called
the Van Est map.

###### Proof.

By Proposition 4.5, the differentiation of the descendent Lie group
$(H,e_{H},\star)$ is the descendent Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{\mathbf{\mathfrak{D}}(\mathcal{B})})$. By
Proposition 4.6, the differentiation of the action $\Theta$ of the descendent
Lie group $(H,e_{H},\star)$ is the representation $\theta$ of the descendent
Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{\mathbf{\mathfrak{D}}(\mathcal{B})})$. Since the
cochains $C^{k}(\mathcal{B})$ and $C^{k}(\mathbf{\mathfrak{D}}(\mathcal{B}))$
are in fact exactly those of the descendent Lie group $(H,e_{H},\star)$ and
the descendent Lie algebra
$(\mathfrak{h},[\cdot,\cdot]_{\mathbf{\mathfrak{D}}(\mathcal{B})})$
respectively, the conclusion follows from the classical argument for the
cohomologies of Lie groups and Lie algebras.∎

Just as the classical situation, under certain conditions, the cohomology
group $\mathcal{H}^{k}(\mathcal{B})$ and
$H^{k}(\mathbf{\mathfrak{D}}(\mathcal{B}))$ are isomorphic.

######  Theorem 4.8.

If the Lie group $(H,e_{H},\cdot_{H})$ is connected and its homotopy groups
are trivial in $1,2,\cdots,n$, then for $1\leq k\leq n$, the cohomology group
$\mathcal{H}^{k}(\mathcal{B})$ of the relative Rota-Baxter operator
$\mathcal{B}$ on the Lie group $G$ is isomorphic to the cohomology group
$H^{k}(\mathbf{\mathfrak{D}}(\mathcal{B}))$ of the relative Rota-Baxter
operator on the level of Lie algebras.

###### Proof.

As for the previous theorem, the conclusion also follows from the classical
argument for the cohomologies of Lie groups and Lie algebras. ∎

###  4.3. An explicit example

At the end of this section, we give a concrete example to demonstrate the
differentiation procedure, and we also calculate a second cohomology group in
this example.

######  Example 4.9.

Consider the Euclidean Lie group

$G=\\{X|X=\left(\begin{array}[]{cc}A&\alpha\\\ 0&1\\\
\end{array}\right),A\in\mathrm{SO}(n),\alpha\in\mathbb{R}^{n}\\}.$

Then we have $G=G_{1}G_{2}$, where $G_{1}=\\{\left(\begin{array}[]{cc}A&0\\\
0&1\\\ \end{array}\right)|A\in\mathrm{SO}(n)\\}$ and
$G_{2}=\\{\left(\begin{array}[]{cc}I_{n\times n}&\alpha\\\ 0&1\\\
\end{array}\right)|\alpha\in\mathbb{R}^{n}\\}$. Since any
$\left(\begin{array}[]{cc}A&\alpha\\\ 0&1\\\ \end{array}\right)$ can be
written as $\left(\begin{array}[]{cc}A&\alpha\\\ 0&1\\\
\end{array}\right)=\left(\begin{array}[]{cc}A&0\\\ 0&1\\\
\end{array}\right)\left(\begin{array}[]{cc}I_{n\times n}&A^{T}\alpha\\\ 0&1\\\
\end{array}\right),$ by Example 3.2, the map $\mathcal{B}:G\,\rightarrow\,G$
defined by

$\mathcal{B}(\left(\begin{array}[]{cc}A&\alpha\\\ 0&1\\\
\end{array}\right))=\left(\begin{array}[]{cc}I_{n\times n}&-A^{T}\alpha\\\
0&1\\\ \end{array}\right),\quad\forall\left(\begin{array}[]{cc}A&\alpha\\\
0&1\\\ \end{array}\right)\in G$

is a Rota-Baxter operator on $G$. By Proposition 3.5, the multiplication
$\star$ in the descendent Lie group $(G,\star)$ is given by

$\left(\begin{array}[]{cc}A&\alpha\\\ 0&1\\\
\end{array}\right)\star\left(\begin{array}[]{cc}C&\beta\\\ 0&1\\\
\end{array}\right)=\left(\begin{array}[]{cc}AC&ACA^{T}\alpha+A\beta\\\ 0&1\\\
\end{array}\right)$

and the inverse is given by $\left(\begin{array}[]{cc}A&\alpha\\\ 0&1\\\
\end{array}\right)^{\dagger}=\left(\begin{array}[]{cc}A^{T}&-(A^{T})^{2}\alpha\\\
0&1\\\ \end{array}\right)$. It is straightforward to deduce that $G$ is the
direct product of $G_{1}$ and $G_{2}$, i.e. $G=G_{1}\star G_{2}$, where
$G_{1}=\\{Y|Y=\left(\begin{array}[]{cc}C&0\\\ 0&1\\\
\end{array}\right),C\in\mathrm{SO}(n)\\}$ and
$G_{2}=\\{Z|Z=\left(\begin{array}[]{cc}I&\beta\\\ 0&1\\\
\end{array}\right),\beta\in\mathbb{R}^{n}\\}$. By Theorem 3.8, the action
$\Theta$ of the descendent Lie group $(G,\star)$ on the manifold $G$ is given
by

$\displaystyle\Theta(\left(\begin{array}[]{cc}A&\alpha\\\ 0&1\\\
\end{array}\right))\left(\begin{array}[]{cc}C&\beta\\\ 0&1\\\
\end{array}\right)$ $\displaystyle=$
$\displaystyle\Big{(}\mathcal{B}(\left(\begin{array}[]{cc}C&\beta\\\ 0&1\\\
\end{array}\right)\left(\begin{array}[]{cc}A^{T}&-(A^{T})^{2}\alpha\\\ 0&1\\\
\end{array}\right)\left(\begin{array}[]{cc}C^{T}&-C^{T}\beta\\\ 0&1\\\
\end{array}\right))\Big{)}^{-1}\left(\begin{array}[]{cc}C&\beta\\\ 0&1\\\
\end{array}\right)\mathcal{B}(\left(\begin{array}[]{cc}A^{T}&-(A^{T})^{2}\alpha\\\
0&1\\\ \end{array}\right))$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}C&CAC^{T}\beta\\\ 0&1\\\
\end{array}\right),$

and the induced representation, which we use the same notation
$\Theta:G\,\rightarrow\,GL(\mathfrak{g})$ is given by

$\Theta(\left(\begin{array}[]{cc}A&\alpha\\\ 0&1\\\
\end{array}\right))\left(\begin{array}[]{cc}x&u\\\ 0&0\\\
\end{array}\right)=\frac{d}{dt}\bigg{|}_{t=0}\Theta(\left(\begin{array}[]{cc}A&\alpha\\\
0&1\\\ \end{array}\right))\left(\begin{array}[]{cc}\exp(tx)&tu\\\ 0&1\\\
\end{array}\right)=\left(\begin{array}[]{cc}x&Au\\\ 0&0\\\
\end{array}\right),$

where the Lie algebra $\mathfrak{g}$ of the Euclidean Lie group $G$ is given
by

$\mathfrak{g}=\\{\left(\begin{array}[]{cc}x&u\\\ 0&0\\\
\end{array}\right)|x\in\mathfrak{so}(n),u\in\mathbb{R}^{n}\\}.$

Then the differentiation $\mathbf{\mathfrak{D}}(\mathcal{B})$ is given by

$\mathbf{\mathfrak{D}}(\mathcal{B})(\left(\begin{array}[]{cc}x&u\\\ 0&0\\\
\end{array}\right)))=\frac{d}{dt}\bigg{|}_{t=0}\mathcal{B}(\left(\begin{array}[]{cc}\exp(tx)&tu\\\
0&1\\\ \end{array}\right))=\left(\begin{array}[]{cc}0&-u\\\ 0&0\\\
\end{array}\right),$

for all $\left(\begin{array}[]{cc}x&u\\\ 0&0\\\
\end{array}\right)\in\mathfrak{g}$. By Proposition 4.5, the descendent Lie
algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathbf{\mathfrak{D}}(\mathcal{B})})$ is
given by

$\displaystyle[\left(\begin{array}[]{cc}x&u\\\ 0&0\\\
\end{array}\right),\left(\begin{array}[]{cc}y&v\\\ 0&0\\\
\end{array}\right)]_{\mathbf{\mathfrak{D}}(\mathcal{B})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}x&u\\\ 0&0\\\
\end{array}\right)\left(\begin{array}[]{cc}y&v\\\ 0&0\\\
\end{array}\right)-\left(\begin{array}[]{cc}y&v\\\ 0&0\\\
\end{array}\right)\left(\begin{array}[]{cc}x&u\\\ 0&0\\\ \end{array}\right)$
$\displaystyle+[\mathbf{\mathfrak{D}}(\mathcal{B})(\left(\begin{array}[]{cc}x&u\\\
0&0\\\ \end{array}\right)),\left(\begin{array}[]{cc}y&v\\\ 0&0\\\
\end{array}\right)]-[\mathbf{\mathfrak{D}}(\mathcal{B})(\left(\begin{array}[]{cc}y&v\\\
0&0\\\ \end{array}\right)),\left(\begin{array}[]{cc}x&u\\\ 0&0\\\
\end{array}\right)]$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}xy-yx&xv-yu\\\ 0&0\\\
\end{array}\right)+\left(\begin{array}[]{cc}0&yu-xv\\\ 0&0\\\
\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}xy-yx&0\\\ 0&0\\\ \end{array}\right).$

Therefore, the descendent Lie algebra
$(\mathfrak{g},[\cdot,\cdot]_{\mathbf{\mathfrak{D}}(\mathcal{B})})$ is the
direct sum of the Lie algebra $\mathfrak{so}(n)$ and the abelian Lie algebra
$\mathbb{R}^{n}$, and obviously $\mathfrak{so}(n)$ is an ideal. By Proposition
4.6, the representation
$\theta:\mathfrak{g}\,\rightarrow\,\mathfrak{gl}(\mathfrak{g})$ of the
descendent Lie algebra
$(\mathfrak{g},[\cdot,\cdot]_{\mathbf{\mathfrak{D}}(\mathcal{B})})$ on
$\mathfrak{g}$ is given by

$\theta(\left(\begin{array}[]{cc}x&u\\\ 0&0\\\
\end{array}\right))\left(\begin{array}[]{cc}y&v\\\ 0&0\\\
\end{array}\right)=\frac{d}{dt}\bigg{|}_{t=0}\Theta(\left(\begin{array}[]{cc}\mathfrak{exp}(tx)&tu\\\
0&1\\\ \end{array}\right))\left(\begin{array}[]{cc}y&v\\\ 0&0\\\
\end{array}\right)=\left(\begin{array}[]{cc}0&xv\\\ 0&0\\\
\end{array}\right).$

Denote by
$\mathfrak{g}^{\mathfrak{so}(n)}=\\{\xi\in\mathfrak{g}|\theta(\left(\begin{array}[]{cc}x&0\\\
0&0\\\ \end{array}\right))\xi=0,\forall x\in\mathfrak{so}(n)\\}.$ Then it is
obvious that
$\mathfrak{g}^{\mathfrak{so}(n)}=\\{\left(\begin{array}[]{cc}\mathfrak{so}(n)&0\\\
0&0\\\ \end{array}\right)\\}.$ With above preparations, using the exact
sequences of low degree terms in the Hochschild-Serre spectral sequence, we
obtain

$0\,\rightarrow\,H^{1}(\mathfrak{g}/{\mathfrak{so}(n)},\mathfrak{g}^{{\mathfrak{so}(n)}})\,\rightarrow\,H^{1}(\mathfrak{g},\mathfrak{g})\,\rightarrow\,H^{1}({\mathfrak{so}(n)},\mathfrak{g})^{\mathfrak{g}/{\mathfrak{so}(n)}}\,\rightarrow\,H^{2}(\mathfrak{g}/{\mathfrak{so}(n)},\mathfrak{g}^{{\mathfrak{so}(n)}})\,\rightarrow\,H^{2}(\mathfrak{g},\mathfrak{g}).$

Since $\mathfrak{so}(n)$ is a semisimple Lie algebra, we have
$H^{1}({\mathfrak{so}(n)},\mathfrak{g})^{\mathfrak{g}/{\mathfrak{so}(n)}}=0$,
thus
$H^{1}(\mathfrak{g}/{\mathfrak{so}(n)},\mathfrak{g}^{{\mathfrak{so}(n)}})\cong
H^{1}(\mathfrak{g},\mathfrak{g})$. Since
$\mathfrak{g}/{\mathfrak{so}(n)}=\mathbb{R}^{n}$ is abelian and the
representation is trivial, it follows that

$H^{1}(\mathfrak{g}/{\mathfrak{so}(n)},\mathfrak{g}^{{\mathfrak{so}(n)}})=\mathrm{Hom}(\mathbb{R}^{n},{\mathfrak{so}(n)})\cong\mathbb{R}^{\frac{n^{2}(n-1)}{2}}.$

Therefore, we have

$H^{2}(\mathbf{\mathfrak{D}}(\mathcal{B}))=H^{1}(\mathfrak{g},\mathfrak{g})\cong\mathbb{R}^{\frac{n^{2}(n-1)}{2}}.$

##  5\. Local integration of relative Rota-Baxter operators on Lie algebras

In this section, we introduce the notion of a local relative Rota-Baxter
operator on a Lie group and show that any relative Rota-Baxter operator on a
Lie algebra can be integrated to a local relative Rota-Baxter operator on the
Lie group integrating the Lie algebra $\mathfrak{g}.$

######  Definition 5.1.

Let $\Phi:G\,\rightarrow\,\mathrm{Aut}(H)$ be an action of $G$ on $H$. If
there exists an open neighborhood $U$ of $e_{H}$ in $H$ and a smooth map
$\mathcal{B}:U\,\rightarrow\,G$ such that

(32)
$\mathcal{B}(h_{1})\cdot_{G}\mathcal{B}(h_{2})=\mathcal{B}(h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))h_{2}),\quad
h_{1},h_{2}\in U,$

whenever the element $h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))h_{2}$ is also in
$U$, then $\mathcal{B}$ is called a local relative Rota-Baxter operator on $G$
with respect to the action $(H;\Phi)$.

In particular, if $H=G$ and the action is the adjoint action of $G$ on itself,
then we call $\mathcal{B}$ a local Rota-Baxter operator.

Two local relative Rota-Baxter operators $\mathcal{B}:U\,\rightarrow\,G$ and
$\mathcal{B}^{\prime}:U^{\prime}\,\rightarrow\,G$ are defined to be equal if
there exists an open neighborhood $\bar{U}\subset U\cap U^{\prime}$ of
$e_{H}$, such that $B|_{\bar{U}}=B^{\prime}|_{\bar{U}}.$

######  Definition 5.2.

Let $\mathcal{B}:U\,\rightarrow\,G$ and
$\mathcal{B}^{\prime}:U^{\prime}\,\rightarrow\,G$ be two local relative Rota-
Baxter operators on $G$ with respect to an action $(H;\Phi)$. A homomorphism
from $\mathcal{B}^{\prime}$ to $\mathcal{B}$ consists of Lie group
homomorphisms $\Psi_{G}:G\,\rightarrow\,G$ and $\Psi_{H}:H\,\rightarrow\,H$
such that (17) holds in an open neighborhood $\bar{U}\subset U\cap U^{\prime}$
of $e_{H}$ and (18) holds.

###  5.1. Local integration functor and adjointness

It is clear that local relative Rota-Baxter operators on a Lie group $G$ with
respect to an action $(H;\Phi)$ together with homomorphisms between them also
form a category, which we denote by $\mathsf{LRB_{G}^{H}}$. Since the
differentiation functor established in the previous section depends only on
the local information near the identity element, it induces a differentiation
functor,

(33)
$\mathbf{\mathfrak{D}}:\mathsf{LRB_{G}^{H}}\to\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}},$

which we use the same letter by abusing the notations.

Let $B:\mathfrak{h}\,\rightarrow\,\mathfrak{g}$ be a relative Rota-Baxter
operator on $\mathfrak{g}$ with respect to an action $(\mathfrak{h};\phi)$.
Let $G$ and $H$ be Lie groups (not necessarily connected and simply connected)
integrating $\mathfrak{g}$ and $\mathfrak{h}$ respectively. If there is an
action $\Phi:G\to\mathrm{Aut}(H)$ whose differentiation is $\phi$ (see (20)),
and a local relative Rota-Baxter operator $\mathcal{B}$ on $G$ with respect to
$\Phi$ such that $\mathbf{\mathfrak{D}}(\mathcal{B})=B$, then we call that
$\mathcal{B}$ integrates $B$.

######  Theorem 5.3.

Let $B:\mathfrak{h}\,\rightarrow\,\mathfrak{g}$ be a relative Rota-Baxter
operator on a Lie algebra $\mathfrak{g}$ with respect to an action
$(\mathfrak{h};\phi)$. Let $G$ and $H$ be the connected and simply connected
Lie groups integrating $\mathfrak{g}$ and $\mathfrak{h}$ respectively, and
$\Phi:G\to\mathrm{Aut}(H)$ be the integrated action (see the paragraph above
(20)). Then there is a unique local relative Rota-Baxter operator
$\mathcal{B}$ on $G$ with respect to the action $(H;\Phi)$ integrating $B$.

###### Proof.

We consider the semi-direct product Lie group $G\ltimes_{\Phi}H$, with
multiplication $\cdot_{\Phi}$ given by

$(g_{1},h_{1})\cdot_{\Phi}(g_{2},h_{2})=(g_{1}\cdot_{G}g_{2},h_{1}\cdot_{H}\Phi(g_{1})h_{2}),\quad\forall
g_{i}\in G,h_{i}\in H,i=1,2.$

Its Lie algebra is the semi-direct product Lie algebra
$\mathfrak{g}\ltimes_{\phi}\mathfrak{h}$, and by Proposition 2.5, $Gr(B)$ is a
Lie subalgebra of $\mathfrak{g}\ltimes_{\phi}\mathfrak{h}$. Thus there exists
a connected Lie subgroup $E$ of $G\ltimes_{\Phi}H$ such that its Lie algebra
is $Gr(B)$.

Define a smooth map $P_{H}:E\,\rightarrow\,H$ by $P_{H}(g,h)=h$ for all
$(g,h)\in E.$ Then its tangent map at the identity
${P_{H}}_{*}:Gr(B)\,\rightarrow\,\mathfrak{h}$ is given by

${P_{H}}_{*}(B(u),u)=u,\quad\forall u\in\mathfrak{h},$

which is an isomorphism from the vector space $Gr(B)$ to the vector space
$\mathfrak{h}$. Thus there exists an open set $V\subseteq E$ such that
$P_{H}|_{V}$ is an isomorphism of manifolds, which implies that there is an
open set $U$ of $H$ which contains $e_{H}$ and a smooth map
$\mathcal{B}:U\,\rightarrow\,G$, such that $V\cong Gr(\mathcal{B})$.

For all $h_{1},h_{2}\in U$, we have

$(\mathcal{B}(h_{1}),h_{1})\cdot_{\Phi}(\mathcal{B}(h_{2}),h_{2})=(\mathcal{B}(h_{1})\cdot_{G}\mathcal{B}(h_{2}),h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))h_{2}).$

Therefore, as soon as $h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))h_{2}\in U$, we
have

$(\mathcal{B}(h_{1})\cdot_{G}\mathcal{B}(h_{2}),h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))h_{2})\in
Gr({\mathcal{B}}),$

which implies that
$\mathcal{B}(h_{1})\cdot_{G}\mathcal{B}(h_{2})=\mathcal{B}(h_{1}\cdot_{H}\Phi(\mathcal{B}(h_{1}))h_{2})$.
Therefore, $\mathcal{B}:U\,\rightarrow\,G$ is a local relative Rota-Baxter
operator. Furthermore, since $Gr(\mathcal{B})\subset E$, and the Lie algebra
of $E$ is $Gr(B)$, it follows that $\mathcal{B}_{*}=B$, and $\mathcal{B}$ is
an integration of $B$.

Now we show the uniqueness of $\mathcal{B}$ through an explicit formula.
Denote by $\mathrm{Exp}$ the exponential map for the Lie group
$G\ltimes_{\Phi}H$, and by $P_{H}$ the projection $G\ltimes_{\Phi}H\to H$. For
all $x\in\mathfrak{g},u\in\mathfrak{h}$, it is obvious that

$\mathrm{Exp}(x,u)=(\exp_{G}x,P_{H}\mathrm{Exp}(x,u)).$

Since the Lie algebra of the Lie subgroup $E$ is $Gr(B)$, it follows that
locally $\mathrm{Exp}(B(u),u)\in Gr(\mathcal{B})\subseteq E$. Therefore,

(34) $\mathcal{B}(P_{H}(\mathrm{Exp}(B(u),u)))=\exp_{G}B(u).$

Since $P_{H}\circ\mathrm{Exp}:Gr(B)\to H$ is a local isomorphism in the
neighborhood of the identity222The map itself is also defined locally
therein., the uniqueness of $\mathcal{B}$ naturally follows. ∎

Now we extend the integration to the level of morphisms and establish a
functor

$\mathfrak{I}:\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}\to\mathsf{LRB_{G}^{H}}.$

######  Theorem 5.4.

Let $B$ and $B^{\prime}$ be relative Rota-Baxter operators on a Lie algebra
$\mathfrak{g}$ with respect to an action $(\mathfrak{h};\phi)$ and
$\psi=(\psi_{\mathfrak{g}},\psi_{\mathfrak{h}})$ be a homomorphism from
$B^{\prime}$ to $B$. Let $G$ and $H$ be connected and simply connected Lie
groups integrating $\mathfrak{g}$ and $\mathfrak{h}$ respectively, and let
$\mathcal{B}^{\prime}:U^{\prime}\to G$ and $\mathcal{B}:U\to G$ be the
integrated local relative Rota-Baxter operators of $B^{\prime}$ and $B$
respectively, as in the previous theorem. Let $\Psi_{G}$ and $\Psi_{H}$ be the
Lie group homomorphisms integrating the Lie algebra homomorphisms
$\psi_{\mathfrak{g}}$ and $\psi_{\mathfrak{h}}$ respectively. Then
$(\Psi_{G},\Psi_{H})$ is a homomorphism from $\mathcal{B}^{\prime}$ to
$\mathcal{B}$. Consequently, we obtain a functor

(35)
$\mathfrak{I}:\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}\to\mathsf{LRB_{G}^{H}}.$

###### Proof.

To prove that $(\Psi_{G},\Psi_{H})$ is a homomorphism from
$\mathcal{B}^{\prime}$ to $\mathcal{B}$, we need to show Eq. (18), and a local
version of Eq. (17). Eq. (18) does not involve relative Rota-Baxter operators
and follows directly from Lemma 4.3.

Now to show Eq. (17) holds locally, we need to show that
$\mathcal{B}\circ\Psi_{H}=\Psi_{G}\circ\mathcal{B}^{\prime}$ holds in an open
neighborhood $\bar{U}\subset U\cap U^{\prime}$ of $e_{H}$. This follows from
the explicit formula (34), the corresponding infinitesimal condition (5), and
the fact that $(\Psi_{G},\Psi_{H}):G\ltimes_{\Phi}H\to G\ltimes_{\Phi}H$ is
the integrated homomorphism of the Lie algebra homomorphism
$(\psi_{\mathfrak{g}},\psi_{\mathfrak{h}}):\mathfrak{g}\ltimes_{\phi}\mathfrak{h}\to\mathfrak{g}\ltimes_{\phi}\mathfrak{h}$
(i.e. Lemma 4.3). More precisely, we have

$\displaystyle\Psi_{G}\Big{(}\mathcal{B}^{\prime}(P_{H}(\mathrm{Exp}(B^{\prime}(u),u)))\Big{)}$
$\displaystyle=$
$\displaystyle\Psi_{G}(\exp_{G}B^{\prime}(u))=\exp_{G}(\psi_{\mathfrak{g}}B^{\prime}(u))=\exp_{G}(B(\psi_{\mathfrak{h}}(u)))$
$\displaystyle=$
$\displaystyle\mathcal{B}(P_{H}(\mathrm{Exp}(B(\psi_{\mathfrak{h}}(u)),\psi_{\mathfrak{h}}(u))))=\mathcal{B}(P_{H}(\mathrm{Exp}(\psi_{\mathfrak{g}}(B^{\prime}(u)),\psi_{\mathfrak{h}}(u))))$
$\displaystyle=$
$\displaystyle\mathcal{B}(P_{H}((\Psi_{G},\Psi_{H})\mathrm{Exp}(B^{\prime}(u),u)))=\mathcal{B}\Psi_{H}(P_{H}(\mathrm{Exp}(B^{\prime}(u),u))).$

Thus we obtain

$\Psi_{G}\circ\mathcal{B}^{\prime}=\mathcal{B}\circ\Psi_{H},$

which implies that $(\Psi_{G},\Psi_{H})$ is a homomorphism from
$\mathcal{B}^{\prime}$ to $\mathcal{B}$. Then it is straightforward to see
that $\mathfrak{I}$ is a functor. ∎

######  Theorem 5.5.

The integration functor $\mathfrak{I}$ in (35) and the differentiation functor
$\mathbf{\mathfrak{D}}$ in (33) are adjoint functors. More precisely,
$\mathfrak{I}$ is left adjoint to $\mathbf{\mathfrak{D}}$ and
$\mathbf{\mathfrak{D}}$ is right adjoint to $\mathfrak{I}$.

###### Proof.

We need to show that, there is an isomorphism
$\alpha_{B^{\prime}\mathcal{B}}$, such that

(36)
$\alpha_{B^{\prime}\mathcal{B}}:\mathrm{Hom}_{\mathsf{LRB_{G}^{H}}}(\mathfrak{I}(B^{\prime}),\mathcal{B})\xrightarrow{\cong}\mathrm{Hom}_{\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}}(B^{\prime},\mathbf{\mathfrak{D}}(\mathcal{B})),\quad\forall
B^{\prime}\in\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}},\mathcal{B}\in\mathsf{LRB_{G}^{H}},$

and $\alpha$ is bi-natural in $B^{\prime}$ and in $\mathcal{B}$, that is, when
fixing $\mathcal{B}$, $\alpha_{-\mathcal{B}}$ is a natural isomorphism between
functors $\mathrm{Hom}_{\mathsf{LRB_{G}^{H}}}(\mathfrak{I}(-),\mathcal{B})$
and
$\mathrm{Hom}_{\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}}(-,\mathbf{\mathfrak{D}}(\mathcal{B}))$;
and when fixing $B^{\prime}$, $\alpha_{B^{\prime}-}$ is a natural isomorphism
$\mathrm{Hom}_{\mathsf{LRB_{G}^{H}}}(\mathfrak{I}(B^{\prime}),-)\xrightarrow{\cong}\mathrm{Hom}_{\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}}(B^{\prime},\mathbf{\mathfrak{D}}(-))$.

We see that by the definition of $\mathfrak{I}$ in Theorem 5.4, $G$ and $H$
are required to be connected and simply connected. Given a morphism
$\Psi:=(\Psi_{G},\Psi_{H})\in\mathrm{Hom}_{\mathsf{LRB_{G}^{H}}}(\mathfrak{I}(B^{\prime}),\mathcal{B})$,
it is not hard to verify that
$\alpha_{B^{\prime}\mathcal{B}}:\Psi\mapsto\mathbf{\mathfrak{D}}(\Psi)$, as
constructed in Proposition 4.2, is an isomorphism with the inverse constructed
in Theorem 5.4. The bi-naturality of $\alpha$ follows from the fact that
$\alpha_{B^{\prime}\mathcal{B}}$ preserves the composition, which in turn
follows from the functoriality of $\mathbf{\mathfrak{D}}$. We demonstrate in
the following the naturality in $B^{\prime}$ and leave the other to interested
readers.

Let $f^{\prime}:B^{\prime}_{1}\to B^{\prime}_{2}$ be a morphism in
$\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}$. The naturality of
$\alpha_{-\mathcal{B}}$ follows from the commutativity of the diagram,

(37)
$\textstyle{\mathrm{Hom}_{\mathsf{LRB_{G}^{H}}}(\mathfrak{I}(B^{\prime}_{1}),\mathcal{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{B^{\prime}_{1}\mathcal{B}}}$$\textstyle{\mathrm{Hom}_{\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}}(B^{\prime}_{1},\mathbf{\mathfrak{D}}(\mathcal{B}))}$$\textstyle{\mathrm{Hom}_{\mathsf{LRB_{G}^{H}}}(\mathfrak{I}(B^{\prime}_{2}),\mathcal{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{B^{\prime}_{2}\mathcal{B}}}$$\scriptstyle{\mathfrak{I}(f^{\prime})^{*}}$$\textstyle{\mathrm{Hom}_{\mathsf{RB_{\mathfrak{g}}^{\mathfrak{h}}}}(B^{\prime}_{2},\mathbf{\mathfrak{D}}(\mathcal{B}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{f^{\prime*}}$

where $\;{}^{*}$ denotes the precomposition (or pullback). In fact, for all
$\Psi\in\mathrm{Hom}_{\mathsf{LRB_{G}^{H}}}(\mathfrak{I}(B^{\prime}_{2}),\mathcal{B})$,
we have

(38)
$\alpha_{B^{\prime}_{1}\mathcal{B}}\mathfrak{I}(f^{\prime})^{*}(\Psi)=\alpha_{B^{\prime}_{1}\mathcal{B}}(\Psi\circ\mathfrak{I}(f^{\prime}))=\mathbf{\mathfrak{D}}(\Psi\circ\mathfrak{I}(f^{\prime}))=\mathbf{\mathfrak{D}}(\Psi)\circ
f^{\prime}=f^{\prime*}\alpha_{B^{\prime}_{2}\mathcal{B}}(\Psi),$

which implies that $\alpha_{-\mathcal{B}}$ is natural. ∎

##  6\. Geometric applications

In this section, we give some geometric applications including the explicit
expression of the local factorization of a Lie group given in [31] and the
integration of matched pairs of Lie algebras given by Rota-Baxter operators.

###  6.1. Local descendent Lie group

As in the global case, a local Rota-Baxter operator also determines a local
descendent Lie group. This turns out to be important to our geometric
application. Thus we make this construction here in this subsection.

While the definition of a Lie group is standard, the definition of a local Lie
group varies slightly. Thus we first fix our definition which is taken mostly
from the classical one in [29], and however with a slight modification similar
to the more recent paper [27].

######  Definition 6.1.

([29, 27]) A local Lie group $L$ is a manifold equipped with an open set
$V\subset L$ together with

  1. (i)

a multiplication map $m:V\times V\to L$, which satisfies
$g_{1}(g_{2}g_{3})=(g_{1}g_{2})g_{3}$, as long as
$g_{1},g_{2},g_{3},g_{1}g_{2},g_{2}g_{3}\in V$ (here we write $m(g_{1},g_{2})$
as $g_{1}g_{2}$ for short),

  2. (ii)

an identity element $e\in V$, which satisfies $eg=ge=g$, for all $g\in V$,

  3. (iii)

an inverse map $i:V\to V$, such that $i(V)=V$, and $g^{-1}g=e=gg^{-1}$ (here
we write $i(g)$ as $g^{-1}$ for short).

######  Theorem 6.2.

Let $\mathcal{B}:U\to G$ be a local Rota-Baxter operator on the Lie group
$(G,e,\cdot)$. Then it gives rise to a local Lie group structure on $G:$

  * $\bullet$

the multiplication $\star$ is given by

(39) $g_{1}\star g_{2}:=g_{1}\cdot\mathcal{B}(g_{1})\cdot
g_{2}\cdot\mathcal{B}(g_{1})^{-1},$

  * $\bullet$

the identity element is $e$, which is the identity element of the Lie group
$G$,

  * $\bullet$

the inverse $g^{\dagger}$ of $g$ is given by

(40) $g^{\dagger}:=\mathcal{B}(g)^{-1}\cdot g^{-1}\cdot\mathcal{B}(g).$

###### Proof.

Notice that the inverse is defined from $U$ to $G$, thus we may take $V:=U\cap
U^{\dagger}$. Clearly $V$ is an open set of $G$ containing $e$. Note that
$\star:U\times U\to G$ is well-defined, thus $\star$ is well-defined also on
$V\times V$.

If $g_{1},g_{2},g_{3},g_{1}\star g_{2},g_{2}\star g_{3}\in V$, then the same
calculation (3.1) for proving the associativity in Proposition 3.5 still goes
through because all the relevant elements are in $V$ which is within the
definition domain of $\mathcal{B}$.

By (32), for all $g\in U$, we have
$\mathcal{B}(g)\cdot\mathcal{B}(g^{\dagger})=\mathcal{B}(g\cdot\mathcal{B}(g)\cdot
g^{\dagger}\cdot\mathcal{B}(g)^{-1})=\mathcal{B}(e)=e$. This shows that
$\mathcal{B}(g^{\dagger})=\mathcal{B}(g)^{-1}$. Thus

(41)
$(g^{\dagger})^{\dagger}=\mathcal{B}(g)\cdot(g^{\dagger})^{-1}\cdot\mathcal{B}(g)^{-1}=g.$

Therefore, $(U^{\dagger})^{\dagger}=U$, which implies that $(V)^{\dagger}=V$.
Moreover, for all $g\in V$, $g^{\dagger}\star g=e=g\star g^{\dagger}$ follows
from the same calculation (3.1) in Proposition 3.5 because
$\mathcal{B}(g^{\dagger})=\mathcal{B}(g)^{-1}$ still holds for $g\in U$. Thus
$(V\subset G,\star,{\dagger})$ is a local Lie group. ∎

This local Lie group is called the descendent local Lie group of
$\mathcal{B}$.

######  Remark 6.3.

Similar to the case of the descendent Lie group, it is clear that

  * •

the Lie algebra of the descendent local Lie group $(V\subset
G,\star,{\dagger})$ is the descendent Lie algebra
$(\mathfrak{g},[\cdot,\cdot]_{B})$, where
$B=\mathbf{\mathfrak{D}}(\mathcal{B})$.

  * •

$\mathcal{B}:(V\subset G,\star,{\dagger})\to G$ is a local Lie group
homomorphism. This in particular implies that $\mathrm{Im}(\mathcal{B})$ and
$G_{-}$ are locally isomorphic near the identity, where $G_{-}$ is the Lie
group integrating the Lie algebra $\mathfrak{g}_{-}:=\mathrm{Im}(B)$.

###  6.2. Application in the local factorization problem

As mentioned in Remark 2.3, Rota-Baxter operators $B$ on a Lie algebra
$\mathfrak{g}$ one-to-one correspond to modified $r$-matrices $R$. The study
of integration of such modified $r$-matrices gives arise to the Adler-Kostant-
Symes (AKS) theory [31]. The AKS theory [20] allows one to construct an
explicit integral curve $L(t)$ of certain Hamiltonian function $h$ on
$\mathfrak{g}^{*}$, by

(42) $L(t)=\mathrm{Ad}^{*}_{g_{+}(t)}L_{0}=\mathrm{Ad}^{*}_{g_{-}(t)}L_{0},$

with initial value $L_{0}\in\mathfrak{g}^{*}$, as long as we know the solution
$(g_{-}(t),g_{+}(t))$ of the local factorization problem, which in turn comes
from a split (i.e. an infinitesimal factorization) of $\mathfrak{g}$. However,
usually, the solution $(g_{-}(t),g_{+}(t))$ is not explicitly given. With the
help of the local integration of Rota-Baxter operators, we can give such a
local factorization explicitly.

To state this more precisely, let us first recall some facts adapted to the
Rota-Baxter operators context from [16, 31], and we refer the readers to the
references therein for the original references. Let $B$ be a Rota-Baxter
operator on $\mathfrak{g}$. Then it gives rise to an infinitesimal
factorization (or a split) [31, Proposition 9] of $\mathfrak{g}$ as follows.
Denote by
$\mathfrak{g}_{+}=\mathrm{Im}(B+{\rm{Id}}),~{}\mathfrak{g}_{-}=\mathrm{Im}(B).$
We further denote by $\mathfrak{k}_{+}:=\ker(B)$ and
$\mathfrak{k}_{-}:=\ker(B+{\rm{Id}})$. Since both $B$ and $B+{\rm{Id}}$ are
Lie algebra homomorphisms from the descendent Lie algebra $\mathfrak{g}_{B}$
to $\mathfrak{g}$, $\mathfrak{g}_{+},\mathfrak{g}_{-},\mathfrak{k}_{+}$ and
$\mathfrak{k}_{-}$ are Lie subalgebras. Moreover,
$\mathfrak{k}_{\pm}\subset\mathfrak{g}_{\pm}$ are Lie ideals, and
$\theta:\mathfrak{g}_{+}/\mathfrak{k}_{+}\to\mathfrak{g}_{-}/\mathfrak{k}_{-}$
given by $(B+{\rm{Id}})(u)\mapsto B(u)$ on the representatives of equivalence
classes is well-defined and is a Lie algebra isomorphism. The map $\theta$ is
called the Cayley transform of $B$. Define
$\mathfrak{g}_{\theta}=\mathfrak{g}_{+}\oplus\mathfrak{g}_{-}$ by

$\mathfrak{g}_{\theta}=\\{(x,y)\in\mathfrak{g}_{+}\oplus\mathfrak{g}_{-}~{}\mbox{
such that }~{}\theta(\bar{x})=\bar{y}\\}.$

Then any $x\in\mathfrak{g}$ can be uniquely expressed as $x=x_{+}-x_{-}$ for
$(x_{+},x_{-})\in\mathfrak{g}_{\theta}.$

Let $G$ be a Lie group of $\mathfrak{g}$, and let $G_{\pm}$ and $K_{\pm}$ be
the Lie subgroups of $G$ integrating the above Lie subalgebras
$\mathfrak{g}_{\pm}$ and $\mathfrak{k}_{\pm}$ correspondingly. Then the Cayley
transform $\theta$ integrates also to the group level into a (local) Lie group
homomorphism $\Theta:G_{+}/K_{+}\to G_{-}/K_{-}$, by Lie’s II Theorem. Let us
denote by $\bar{g}$ the equivalence class of $g$. Then the solution
$(g_{+}(t),g_{-}(t))\in G_{+}\times G_{-}$ of the (local) factorization
problem in [31, Theorem 11] is the solution of the following equations,

$\exp 2tX_{0}=g_{+}(t)\cdot
g_{-}(t)^{-1},\quad\Theta(\bar{g}_{+}(t))=\bar{g}_{-}(t),\quad\text{where}\>X_{0}=dh|_{L_{0}}\in\mathfrak{g},$

with the initial value $g_{\pm}(0)=e\in G$, for sufficiently small $t$.

######  Theorem 6.4.

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},B)$ be a Rota-Baxter Lie
algebra. Then the above solution $(g_{+},g_{-})$ of the (local) factorization
problem has an explicit expression,

(43) $g_{-}(t)=\mathcal{B}(\exp 2tX_{0}),\quad g_{+}(t)=\exp
2tX_{0}\cdot\mathcal{B}(\exp 2tX_{0}),$

for sufficiently small $t$, with $\mathcal{B}$ given explicitly by (34)
locally.

###### Proof.

Let $(G,\cdot,\mathcal{B})$ be the integrated local Rota-Baxter Lie group of
$(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},B)$ given in Theorem 5.3. By
Remark 6.3, $\mathrm{Im}(\mathcal{B})$ and $G_{-}$ are locally isomorphic.
Define $\mathcal{B}_{+}:U\to G$ by

$\mathcal{B}_{+}(g)=g\cdot\mathcal{B}(g).$

By the same discussion in [16, Proposition 3.1], $\mathcal{B}_{+}$ is a local
Lie group homomorphism from the descendent local Lie group to $G$. Since
$\mathrm{Im}(\mathcal{B}_{+})$ and $G_{+}$ have the same Lie algebra
$\mathfrak{g}_{+}$, so they are also locally isomorphic. Obviously, locally we
have

$g=g\cdot\mathcal{B}(g)\cdot(\mathcal{B}(g))^{-1},\quad
g\cdot\mathcal{B}(g)\in\mathrm{Im}(\mathcal{B}_{+}),~{}\mathcal{B}(g)\in\mathrm{Im}(\mathcal{B}).$

Therefore, we have $\exp 2tX_{0}=g_{+}(t)\cdot g_{-}(t)^{-1}$, where
$g_{+}(t)$ and $g_{-}(t)$ are given by (43). ∎

###  6.3. Application in the integration of matched pairs

A matched pair of Lie algebras consists of a pair of Lie algebras
$(\mathfrak{g},\mathfrak{h})$, a representation
$\rho:\mathfrak{g}\to\mathfrak{gl}(\mathfrak{h})$ of $\mathfrak{g}$ on
$\mathfrak{h}$ and a representation
$\mu:\mathfrak{h}\to\mathfrak{gl}(\mathfrak{g})$ of $\mathfrak{h}$ on
$\mathfrak{g}$ such that

(44) $\displaystyle\rho(x)[\xi,\eta]_{\mathfrak{h}}$ $\displaystyle=$
$\displaystyle[\rho(x)\xi,\eta]_{\mathfrak{h}}+[\xi,\rho(x)\eta]_{\mathfrak{h}}+\rho\big{(}(\mu(\eta)x\big{)}\xi-\rho\big{(}\mu(\xi)x\big{)}\eta,$
(45) $\displaystyle\mu(\xi)[x,y]_{\mathfrak{g}}$ $\displaystyle=$
$\displaystyle[\mu(\xi)x,y]_{\mathfrak{g}}+[x,\mu(\xi)y]_{\mathfrak{g}}+\mu\big{(}\rho(y)\xi\big{)}x-\mu\big{(}\rho(x)\xi\big{)}y,$

for all $x,y\in\mathfrak{g}$ and $\xi,\eta\in\mathfrak{h}$.

It is well known that if there is a Lie algebra $\mathfrak{k}$ such that both
$\mathfrak{g}$ and $\mathfrak{h}$ are Lie subalgebras, and $\mathfrak{k}$ is
isomorphic to $\mathfrak{g}\oplus\mathfrak{h}$ as vector spaces, then
$(\mathfrak{g},\mathfrak{h})$ is a matched pair of Lie algebras.

######  Lemma 6.5.

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},B)$ be a Rota-Baxter Lie
algebra. Then $(\mathfrak{g}_{B},\mathfrak{g}_{diag})$ is a matched pair of
Lie algebras, where $\mathfrak{g}_{diag}=\\{(x,x)|\forall x\in\mathfrak{g}\\}$
and $\mathfrak{g}_{B}=\\{(B(x),x+B(x))|\forall x\in\mathfrak{g}\\}$.

###### Proof.

Consider the direct sum Lie algebra $\mathfrak{g}\oplus\mathfrak{g}$, in which
the Lie bracket is given by

$[(x,y),(x^{\prime},y^{\prime})]=([x,x^{\prime}]_{\mathfrak{g}},[y,y^{\prime}]_{\mathfrak{g}}).$

It is obvious that $\mathfrak{g}_{diag}$ is a Lie subalgebra of
$\mathfrak{g}\oplus\mathfrak{g}$. Since
$(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},B)$ is a Rota-Baxter Lie algebra,
we have

$\displaystyle[(B(x),x+B(x)),(B(y),y+B(y))]$ $\displaystyle=$
$\displaystyle([B(x),B(y)]_{\mathfrak{g}},[x,y]_{\mathfrak{g}}+[x,B(y)]_{\mathfrak{g}}+[B(x),y]_{\mathfrak{g}}+[B(x),B(y)]_{\mathfrak{g}})$
$\displaystyle=$
$\displaystyle\Big{(}B([x,y]_{\mathfrak{g}}+[x,B(y)]_{\mathfrak{g}}+[B(x),y]_{\mathfrak{g}}),$
$\displaystyle[x,y]_{\mathfrak{g}}+[x,B(y)]_{\mathfrak{g}}+[B(x),y]_{\mathfrak{g}}+B([x,y]_{\mathfrak{g}}+[x,B(y)]_{\mathfrak{g}}+[B(x),y]_{\mathfrak{g}})\Big{)}$
$\displaystyle\in$ $\displaystyle\mathfrak{g}_{B},$

which implies that $\mathfrak{g}_{B}$ is a Lie subalgebra of
$\mathfrak{g}\oplus\mathfrak{g}$. Moreover, it is obvious that
$\mathfrak{g}_{diag}\oplus\mathfrak{g}_{B}$ is isomorphic to
$\mathfrak{g}\oplus\mathfrak{g}$ as vector spaces. Therefore,
$(\mathfrak{g}_{B},\mathfrak{g}_{diag})$ is a matched pair of Lie algebras. ∎

A pair of Lie groups $(P,Q)$ is called a matched pair of Lie groups ([23]) if
there is a left action of $P$ on $Q$ and a right action of $Q$ on $P$:

$P\times Q\to Q,\qquad(p,q)\mapsto p\triangleright q;\qquad P\times Q\to
P\qquad(p,q)\mapsto p\triangleleft q,$

such that

(46) $\displaystyle p\triangleright(q_{1}q_{2})$ $\displaystyle=$
$\displaystyle(p\triangleright q_{1})\big{(}(p\triangleleft
q_{1})\triangleright q_{2}\big{)};$ (47)
$\displaystyle(p_{1}p_{2})\triangleleft q$ $\displaystyle=$
$\displaystyle\big{(}p_{1}\triangleleft(p_{2}\triangleright
q)\big{)}(p_{2}\triangleleft q).$

Similar to the case of Lie algebras, the following equivalent characterization
of matched pairs of Lie groups is well known.

######  Proposition 6.6.

A pair of Lie groups $(P,Q)$ is a matched pair if there exists a Lie group
$K$, and injective Lie group homomorphism $i_{P}:P\to K$ and $i_{Q}:Q\to K$,
such that the map $P\times Q\,\rightarrow\,K$ defined by $(p,q)\to
i_{P}(p)i_{Q}(q)$ is a diffeomorphism.

It is not hard to see that a matched pair of Lie groups differentiates to a
matched pair of Lie algebras. However, not every matched pair of Lie algebras
integrates to a matched pair of Lie groups (e.g. [12, Example 2.7] gives rise
to a counter example). Some partial result is available for this integration
problem. For example, it is proved in [23] that a matched pair of Lie algebras
$(\mathfrak{g},\mathfrak{h})$ integrates to a matched pair of Lie groups
$(G,H)$, if the simply connected Lie groups $G$ and $H$ integrating
$\mathfrak{g}$ and $\mathfrak{h}$ respectively are compact. In the following,
we give another integration result for matched pairs coming from Rota-Baxter
operators using the integration of Rota-Baxter operators.

######  Theorem 6.7.

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},B)$ be a Rota-Baxter Lie
algebra. Assume that $B$ is integrable and $(G,\cdot,\mathcal{B})$ is the
integrated Rota-Baxter Lie group of
$(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},B)$. Then
$(G_{\mathcal{B}},G_{diag})$ is a matched pair of Lie groups, where
$G_{diag}=\\{(g,g)|\forall g\in G\\}$ and
$G_{\mathcal{B}}=\\{(\mathcal{B}(g),g\cdot\mathcal{B}(g))|\forall g\in G\\}$.
Furthermore, the differentiation of $(G_{\mathcal{B}},G_{diag})$ is the
matched pair $(\mathfrak{g}_{B},\mathfrak{g}_{diag})$.

###### Proof.

Consider the direct product Lie group $G\times G$, in which the group
multiplication is given by

$(g,h)(g^{\prime},h^{\prime})=(g\cdot g^{\prime},h\cdot h^{\prime}).$

It is obvious that $G_{diag}$ is a Lie subgroup. Since $(G,\cdot,\mathcal{B})$
is a Rota-Baxter Lie group, for any $g,h\in G$, we have

$\displaystyle\Big{(}\mathcal{B}(g),g\cdot\mathcal{B}(g)\Big{)}\Big{(}\mathcal{B}(h),h\cdot\mathcal{B}(h)\Big{)}$
$\displaystyle=$
$\displaystyle\Big{(}\mathcal{B}(g)\cdot\mathcal{B}(h),g\cdot\mathcal{B}(g)\cdot
h\cdot\mathcal{B}(h)\Big{)}$ $\displaystyle=$
$\displaystyle\Big{(}\mathcal{B}(g\cdot\mathcal{B}(g)\cdot
h\cdot(\mathcal{B}(g))^{-1}),g\cdot\mathcal{B}(g)\cdot
h\cdot(\mathcal{B}(g))^{-1}\cdot\mathcal{B}(g\cdot\mathcal{B}(g)\cdot
h\cdot(\mathcal{B}(g))^{-1})\Big{)}$ $\displaystyle\in$ $\displaystyle
G_{\mathcal{B}}$

and

$\displaystyle(\mathcal{B}(g),g\cdot\mathcal{B}(g))^{-1}$ $\displaystyle=$
$\displaystyle\Big{(}\mathcal{B}((\mathcal{B}(g))^{-1}\cdot
g^{-1}\cdot\mathcal{B}(g)),(\mathcal{B}(g))^{-1}\cdot
g^{-1}\cdot\mathcal{B}(g)\cdot\mathcal{B}((\mathcal{B}(g))^{-1}\cdot
g^{-1}\cdot\mathcal{B}(g))\Big{)}$ $\displaystyle\in$ $\displaystyle
G_{\mathcal{B}},$

which implies that $G_{\mathcal{B}}$ is a subgroup. Moreover, since
$\mathcal{B}$ is a smooth map, it follows that $G_{\mathcal{B}}$ is a closed
set in $G\times G$. Thus $G_{\mathcal{B}}$ is a Lie subgroup. For any
$(a,b)\in G\times G$, there is

$(a,b)=\Big{(}\mathcal{B}(b\cdot a^{-1}),b\cdot a^{-1}\cdot\mathcal{B}(b\cdot
a^{-1})\Big{)}\Big{(}(\mathcal{B}(b\cdot a^{-1}))^{-1}\cdot
a,(\mathcal{B}(b\cdot a^{-1}))^{-1}\cdot a\Big{)}.$

Thus the map $G_{\mathcal{B}}\times G_{diag}\,\rightarrow\,G\times G$ defined
by
$\Big{(}(\mathcal{B}(g),g\cdot\mathcal{B}(g)),(h,h)\Big{)}\,\rightarrow\,\Big{(}\mathcal{B}(g)\cdot
h,g\cdot\mathcal{B}(g)\cdot h\Big{)}$ is a diffeomorphism. Therefore, by
Proposition 6.6, $(G_{\mathcal{B}},G_{diag})$ is a matched pair of Lie groups.
The other conclusion is straightforward. ∎

######  Remark 6.8.

Despite there are explicit counter examples of nonintegrable matched pairs, we
do not know yet an explicit example of nonintegrable Rota-Baxter operators.

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