# Twisting of Lie triple systems, $L_{\infty}$-algebras, and (generalized)
matched pairs

Jia Zhao  Jia Zhao, School of Mathematics and Statistics, Nantong University,
Nantng 226019, Jiangsu, China<EMAIL_ADDRESS>and  Haobo Xia  Haobo Xia
(corresponding author), Department of Mathematics, Jilin University, Changchun
130012, Jilin, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we introduce notions of (proto-, quasi-)twilled Lie triple
systems and give their equivalent descriptions using the controlling algebra
and bidegree convention. Then we construct an $L_{\infty}$-algebra via a
twilled Lie triple system. Besides, we establish the twisting theory of Lie
triple systems and then characterize the twisting as a Maurer-Cartan element
in the constructed $L_{\infty}$-algebra. Finally, we clarify the relationship
between twilled Lie triple systems and matched pairs and clarify the
relationship between twilled Lie triple systems and relative Rota-Baxter
operators respectively so that we obtain the relationship between matched
pairs of Lie triple systems and relative Rota-Baxter operators.

###### Key words and phrases:

twilled Lie triple system, $L_{\infty}$-algebra, twisting, matched pair,
realtive Rota-Baxter operator

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries: Lie triple systems and their controlling algebras
  3. 3 (Proto-, quasi-)twilled Lie triple systems
    1. 3.1 Lift and bidegree
    2. 3.2 Proto-twilled and quasi-twilled Lie triple systems
    3. 3.3 Twilled Lie triple systems and $L_{\infty}$-algebras
  4. 4 Twisting of twilled Lie triple systems
  5. 5 Matched pairs and relative Rota-Baxter operators
    1. 5.1 Matched pairs and twilled Lie triple systems
    2. 5.2 Relative Rota-Baxter operators and twilled Lie triple systems

##  1\. Introduction

Lie triple systems can be traced back to Cartan’s study on symmetric spaces
[4]. It was abstracted as an algebraic object and was named as Lie triple
systems by Jacobson [10], and then representation of Lie triple systems was
defined by Hodge and Parshall [8]. Indeed, Leibniz algebras and Nambu algebras
are related with Lie triple systems closely and Lie triple systems play an
important role in Lie theory. The relationship between Lie triple systems and
other algebraic structures is reflected in the following two aspects: on the
one hand, a Lie triple system is a special Nambu algebra; on the other hand,
the space of fundamental objects of Lie triple systems is endowed with a
Leibniz algebraic structure. Due to the importance of Lie triple systems,
Lister established a structure theory of Lie triple systems [16]. Moreover,
applications of Lie triple systems on numerical analysis of differential
equations were studied in [17] and that $T^{*}$-extension of Lie triple
systems is compatible with nilpotency and solvability was examined in [15].

Motivated by studies on (quasi-)Lie bialgebras and (quasi-)Hopf algebras,
Drinfeld introduced an operation called the twisting in [7]. One provides a
method to study Manin triples via the twisting operations. Twisting operations
play an important role in the context of bialgebra theory and Poisson
geometry, see [11, 12, 19, 20] for more details. It is well known that a
graded commutative algebra $\wedge^{\bullet}(V\oplus V^{*})$ is equipped with
a Poisson bracket $\\{\cdot,\cdot\\}$ defined to be
$\\{v,v^{\prime}\\}=\\{\epsilon,\epsilon^{\prime}\\}=0$ and
$\\{v,\epsilon\\}=\langle v,\epsilon\rangle$ for any $v,v^{\prime}\in V$ and
$\epsilon,\epsilon^{\prime}\in V^{*}$. Then a Lie algebra structure on
$V\oplus V^{*}$ is an element $\Theta$ in $\wedge^{3}(V\oplus V^{*})$ such
that $\\{\Theta,\Theta\\}=0$. Moreover, the structure $\Theta$ has a close
connection with Lie bialgebra structures. A Lie bialgebra structure is pair
$(\mu,\nu)$, such that $\Theta:=\mu+\nu$ is a Lie algebra structure on
$V\oplus V^{*}$, where $\mu\in(\wedge^{2}V^{*})\otimes V$ and $\nu\in
V^{*}\otimes(\wedge^{2}V)$. If $(\mu,\nu)$ forms a Lie bialgebra structure on
$V$, then $(V\oplus V^{*},\mu+\nu)$ is called a Drinfeld double. Suppose that
$r\in\wedge^{2}V$, then by definition, the twisting of a structure $\Theta$ by
$r$ is a transformation

$\Theta^{r}:=\exp(X_{r})(\Theta),$

where $X_{r}$ is a Hamiltonian vector field $X_{r}:=\\{\cdot,r\\}$. The
relationship between twisting and the classical Yang-Baxter equation is
reinforced in the sequel. Let $(\mu,\nu=0)$ be a Lie bialgebra on the vector
space $V$, then the Drinfeld double is the space $V\oplus V^{*}$ with the
structure $\Theta:=\mu$. If $r$ is a solution to the classical Yang-Baxter
equation

$[r,r]=0,$

then a pair $(\mu,\\{\mu,r\\})$ is a Lie bialgebra structure and the double
$\mu+\\{\mu,r\\}=\Theta^{r}$, where $[r,r]:=\\{\\{\mu,r\\},r\\}$. Generally,
Lie 2-(co-, bi-)algebra structures are characterized as Maurer-Cartan elements
in a graded Poisson algebra $(S^{\bullet}(V[2]\oplus
V^{*}[1]),\\{\cdot,\cdot\\})$ in [5].

Thanks to our study on Lie-Yamaguti bialgebra theory in [28], we obtain the
bialgebra theory of Lie triple systems naturally. In order to establish a
comprehensive theoretical system for Lie triple system bialgebra, the aim of
this paper is to build the twisting theoty of Lie triple systems. We introduce
the notions of (proto-, quasi-)twilled Lie triple systems and then we
construct an $L_{\infty}$-algebra from a twilled Lie triple system.
Consequently, we characterize the twisting as a Maurer-Cartan element of the
constructed $L_{\infty}$-algebra. Finally, we introduce notions of matched
pairs and generalized matched pairs of Lie triple systems to clarify the
relationship between (generalized) matched pairs and relative Rota-Baxter
operators. Similar to the case of $3$-Lie algebras, the reason why a solution
to the classical Yang-Baxter equation (treated as a special relative Rota-
Baxter operator) can not give rise to a double Lie triple system bialgebra is
that a relative Rota-Baxter operator does not corresponds to a usual matched
pair of Lie triple systems, but corresponds to a generalized matched pair.
Relations among twilled Lie triple systems (twilled LTSs for short in the
diagram), relative Rota-Baxter operators (relative RB-operators for short in
the diagram) and matched pairs of Lie triple systems can be shown in the
following diagram:

mathched pairsTheorem 5.4strict twilled LTS sgeneralized matched pairs Theorem
5.11 twilled LTS sTheorem 5.14 relative RB-operators

The conclusions shown in the above diagram indicate that generalized matched
pairs of Lie triple systems may lead to generalized bialgebra theory for Lie
triple systems and we expect new studies in this direction. Besides, one can
read [9, 21, 25] for more details about twisting theory of associative
algebras, Leibniz algebras and $3$-Lie algebras, and see [2, 3] for bialgebra
theories for left-symmetric algebras and 3-Lie algebras.

In this paper, all the vector spaces are over $\mathbb{K}$, a field of
characteristic $0$.

##  2\. Preliminaries: Lie triple systems and their controlling algebras

In this section, we first recall some basic notions such as Lie triple
systems, representations and their controlling algebras.

######  Definition 2.1.

[10] A Lie triple system is a vector space $\mathfrak{g}$, together with a
trilinear bracket
$\left\llbracket\cdot,\cdot,\cdot\right\rrbracket:\wedge^{2}\mathfrak{g}\otimes\mathfrak{g}\to\mathfrak{g}$
such that the following equations are satisfied for all
$x,y,z,w,t\in\mathfrak{g}$,

(1) $\displaystyle\left\llbracket x,y,z\right\rrbracket+\left\llbracket
y,z,x\right\rrbracket+\left\llbracket z,x,y\right\rrbracket=0,$ (2)
$\displaystyle\left\llbracket x,y,\left\llbracket
z,w,t\right\rrbracket\right\rrbracket=\left\llbracket\left\llbracket
x,y,z\right\rrbracket,w,t\right\rrbracket+\left\llbracket z,\left\llbracket
x,y,w\right\rrbracket,t\right\rrbracket+\left\llbracket z,w,\left\llbracket
x,y,t\right\rrbracket\right\rrbracket.$

Here, Eq. (2) is called the fundamental identity. We denote a Lie triple
system by a pair
$(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$.

######  Remark 2.2.

A Lie triple system structure induces a Leibniz algebra structure on its
tensor space. Let
$(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$ be a Lie
triple system. Define an operation $[\cdot,\cdot]_{\mathsf{F}}$ on
$\otimes^{2}\mathfrak{g}$ to be

$[\mathcal{X},\mathcal{Y}]_{\mathsf{F}}:=\left\llbracket
x_{1},x_{2},y_{1}\right\rrbracket\otimes y_{2}+y_{1}\otimes\left\llbracket
x_{1},x_{2},y_{2}\right\rrbracket,\quad\forall\mathcal{X}=x_{1}\otimes
x_{2},~{}\mathcal{Y}=y_{1}\otimes y_{2}\in\otimes^{2}\mathfrak{g},$

then $(\otimes^{2}\mathfrak{g},[\cdot,\cdot]_{\mathsf{F}})$ is a Leibniz
algebra. Elements in $\otimes^{2}\mathfrak{g}$ are called the fundamental
objects.

######  Example 2.3.

Let $(\mathfrak{g},[\cdot,\cdot])$ be a Lie algebra. We define
$\left\llbracket\cdot,\cdot,\cdot\right\rrbracket:\wedge^{2}\mathfrak{g}\otimes\mathfrak{g}\,\rightarrow\,\mathfrak{g}$
by

$\left\llbracket x,y,z\right\rrbracket:=[[x,y],z],\quad\forall
x,y,z\in\mathfrak{g}.$

Then $(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$ becomes
a Lie triple system naturally.

######  Definition 2.4.

[27] Let $(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$ be
a Lie triple system and $V$ a vector space. A representation of $\mathfrak{g}$
on $V$ consists of a bilinear map
$\rho:\otimes^{2}\mathfrak{g}\to\mathfrak{gl}(V)$ such that for all
$x,y,z,w\in\mathfrak{g}$,

(3) $\displaystyle\rho(z,w)\rho(x,y)-\rho(y,w)\rho(x,z)-\rho(x,\left\llbracket
y,z,w\right\rrbracket)+D_{\rho}(y,z)\rho(x,w)=0,$ (4)
$\displaystyle\rho(\left\llbracket
x,y,z\right\rrbracket,w)+\rho(z,\left\llbracket
x,y,w\right\rrbracket)=[D_{\rho}(x,y),\rho(z,w)],$

where the bilinear map $D_{\rho}:\otimes^{2}\mathfrak{g}\to\mathfrak{gl}(V)$
is given by

(5) $\displaystyle D_{\rho}(x,y):=\rho(y,x)-\rho(x,y),\quad\forall
x,y\in\mathfrak{g}.$

It is obvious that $D_{\rho}$ is skew-symmetric and we write $D$ instead of
$D_{\rho}$ without ambiguities in the sequel. We denote a representation of
$\mathfrak{g}$ on $V$ by $(V;\rho)$.

######  Example 2.5.

Let $(\mathfrak{g},[\cdot,\cdot,\cdot])$ be a Lie triple system. We define
$\mathfrak{R}:\otimes^{2}\mathfrak{g}\to\mathfrak{gl}(\mathfrak{g})$ by
$(x,y)\mapsto\mathfrak{R}_{x,y}$ , where $\mathfrak{R}_{x,y}z=\left\llbracket
z,x,y\right\rrbracket$ for all $z\in\mathfrak{g}$. Then
$(\mathfrak{g};\mathfrak{R})$ forms a representation of $\mathfrak{g}$ on
itself, called the adjoint representation. By (1), $\mathfrak{L}\triangleq
D_{\mathfrak{R}}=\mathfrak{R}_{y,x}-\mathfrak{R}_{x,y}$ is given by for all
$x,y\in\mathfrak{g}$,

$\displaystyle\mathfrak{L}_{x,y}z=\left\llbracket
x,y,z\right\rrbracket,\quad\forall z\in\mathfrak{g}.$

By a direct calculation, we have the following

######  Proposition 2.6.

Let $(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$ be a Lie
triple system and $V$ a vector space. Let
$\rho:\otimes^{2}\mathfrak{g}\to\mathfrak{gl}(V)$ be a linear map. Then
$(V;\rho)$ is a representation of
$(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$ if and only
if there is a Lie triple system structure
$\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\rho}$ on the direct sum
$\mathfrak{g}\oplus V$ which is defined by for all
$x,y,z\in\mathfrak{g},u,v,w\in V$,

$\displaystyle~{}\left\llbracket x+u,y+v,z+w\right\rrbracket_{\rho}$
$\displaystyle=$ $\displaystyle\left\llbracket
x,y,z\right\rrbracket+D(x,y)w+\rho(y,z)u-\rho(x,z)v,$

where $D$ is given by (5). This Lie triple system $(\mathfrak{g}\oplus
V,\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\mu})$ is called the
semidirect product Lie triple system, denoted by
$\mathfrak{g}\ltimes_{\rho}V$, or simply by $\mathfrak{g}\ltimes V$.

In the following, we recall the controlling algebras of Lie triple systems.
Let $\mathfrak{g}$ be a vector space and
$C^{*}(\mathfrak{g},\mathfrak{g})=\oplus_{p\geqslant
0}C^{p}(\mathfrak{g},\mathfrak{g})$, where
$C^{p}(\mathfrak{g},\mathfrak{g})=\mathrm{Hom}(\otimes^{2p+1}\mathfrak{g},\mathfrak{g})$
and degrees of elements in $C^{p}(\mathfrak{g},\mathfrak{g})$ are assumed to
be $p$. For all $P\in C^{p}(\mathfrak{g},\mathfrak{g})$ and $Q\in
C^{q}(\mathfrak{g},\mathfrak{g})$, define a graded bracket (called the Nambu
bracket) $[\cdot,\cdot]_{\mathsf{N}}$ to be

$[P,Q]_{\mathsf{N}}=P\circ Q-(-1)^{pq}Q\circ P,$

where $P\circ Q\in C^{p+q}(\mathfrak{g},\mathfrak{g})$ is defined by

$\displaystyle P\circ Q(\mathcal{X}_{1},\cdots,\mathcal{X}_{p+q},x)$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{p}(-1)^{(k-1)q}(-1)^{\sigma}\sum_{\sigma\in\mathbb{S}(k-1,q)}P\Big{(}\mathcal{X}_{\sigma(1)},\cdots,\mathcal{X}_{\sigma(k-1)},Q\big{(}\mathcal{X}_{\sigma(k)},\cdots,\mathcal{X}_{\sigma(k+q-1)},x_{k+q}\big{)}\otimes
y_{k+q},\mathcal{X}_{k+q+1},\cdots,\mathcal{X}_{p+q},x\Big{)}$ (6)
$\displaystyle+\sum_{k=1}^{p}(-1)^{(k-1)q}(-1)^{\sigma}\sum_{\sigma\in\mathbb{S}(k-1,q)}P\Big{(}\mathcal{X}_{\sigma(1)},\cdots,\mathcal{X}_{\sigma(k-1)},x_{k+q}\otimes
Q\big{(}\mathcal{X}_{\sigma(k)},\cdots,\mathcal{X}_{\sigma(k+q-1)},y_{k+q}\big{)},\mathcal{X}_{k+q+1},\cdots,\mathcal{X}_{p+q},x\Big{)}$
$\displaystyle+\sum_{\sigma\in\mathbb{S}(p,q)}(-1)^{\sigma}P\Big{(}\mathcal{X}_{\sigma(1)},\cdots,\mathcal{X}_{\sigma(p)},Q\big{(}\mathcal{X}_{\sigma(p+1)},\cdots,\mathcal{X}_{\sigma(p+q)},x\big{)}\Big{)},$

for all $\mathcal{X}_{i}\in\otimes^{2}\mathfrak{g},~{}i=1,2,\cdots,p+q$ and
$x\in\mathfrak{g}$.

Then $(C^{*}(\mathfrak{g},\mathfrak{g}),[\cdot,\cdot]_{\mathsf{N}})$ is a
graded Lie algebra and its Maurer-Cartan elements corresponds to Nambu algebra
structures [18]. Let
$C^{*}_{\mathsf{LTS}}(\mathfrak{g},\mathfrak{g})=\oplus_{p\geqslant
0}C_{\mathsf{LTS}}^{p}(\mathfrak{g},\mathfrak{g})=\oplus_{p\geqslant
0}\mathrm{Hom}_{\mathsf{LTS}}^{p}(\otimes^{2p+1}\mathfrak{g},\mathfrak{g})$ be
a graded subspace of $C^{*}(\mathfrak{g},\mathfrak{g})$ such that for any
$P\in C^{p}_{\mathsf{LTS}}(\mathfrak{g},\mathfrak{g})$, $P$ satisfies

$\displaystyle P(\mathcal{X}_{1},\cdots,\mathcal{X}_{p-1},x,x,y)$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle~{}P(\mathcal{X}_{1},\cdots,\mathcal{X}_{p-1},x,y,z)+P(\mathcal{X}_{1},\cdots,\mathcal{X}_{p-1},y,z,x)+P(\mathcal{X}_{1},\cdots,\mathcal{X}_{p-1},z,x,y)$
$\displaystyle=$ $\displaystyle 0.$

The corresponding Nambu bracket is denoted by $[\cdot,\cdot]_{\mathsf{LTS}}$,
when the graded vector space is restricted to
$C^{*}_{\mathsf{LTS}}(\mathfrak{g},\mathfrak{g})=\oplus_{p\geqslant
0}C_{\mathsf{LTS}}^{p}(\mathfrak{g},\mathfrak{g})$. Then
$(C^{*}_{\mathsf{LTS}}(\mathfrak{g},\mathfrak{g})=\oplus_{p\geqslant
0}C_{\mathsf{LTS}}^{p}(\mathfrak{g},\mathfrak{g}),[\cdot,\cdot]_{\mathsf{LTS}})$
is a graded subalgebra and its Maurer-Cartan elements corresponds to Lie
triple system structures [26]. In the present paper, we always consider the
graded Lie algebra
$(C^{*}_{\mathsf{LTS}}(\mathfrak{g},\mathfrak{g}),[\cdot,\cdot]_{\mathsf{LTS}})$.

##  3\. (Proto-, quasi-)twilled Lie triple systems

In this section, we introduce notions of (proto-, quasi-)twilled Lie triple
systems, and use the graded Lie algebra bracket in the controlling algebra to
give structures of (proto-, quasi-)twilled Lie triple systems. For this
purpose, we have to examine the lift of a given linear map and its bidegree
first.

###  3.1. Lift and bidegree

Let $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ be vector spaces, and elements
in $\mathfrak{g}_{1}$ are denoted by $x,y,z,x_{i}$ and those in
$\mathfrak{g}_{2}$ by $u,v,w,u_{i}$. We denote by $\mathfrak{g}^{l,k}$ the
direct sum of all $(l+k)$-tensor power of $\mathfrak{g}_{1}$ and
$\mathfrak{g}_{2}$: $\otimes^{n}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$,
where $n=l+k$, and $l$ (resp. $k$) means the quantities of $\mathfrak{g}_{1}$
(resp. $\mathfrak{g}_{2}$). For example,
$\mathfrak{g}^{2,1}\subset\otimes^{3}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
can be written as

$\mathfrak{g}^{2,1}=(\mathfrak{g}_{1}\otimes\mathfrak{g}_{1}\otimes\mathfrak{g}_{2})\oplus(\mathfrak{g}_{1}\otimes\mathfrak{g}_{2}\otimes\mathfrak{g}_{1})\oplus(\mathfrak{g}_{2}\otimes\mathfrak{g}_{1}\otimes\mathfrak{g}_{1}).$

Then
$\displaystyle{\otimes^{n}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})=\oplus_{l+k=n}\mathfrak{g}^{l,k}}$.
For example,

$\otimes^{3}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})=\mathfrak{g}^{3,0}\oplus\mathfrak{g}^{2,1}\oplus\mathfrak{g}^{1,2}\oplus\mathfrak{g}^{0,3}.$

By $\mathrm{Hom}$-functor, we have

(7) $\displaystyle
C^{p}_{\mathsf{LTS}}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})=\bigoplus_{l+k=2p}C^{p}_{\mathsf{LTS}}(\mathfrak{g}^{l,k},\mathfrak{g}_{1})\oplus\bigoplus_{l+k=2p}C^{p}_{\mathsf{LTS}}(\mathfrak{g}^{l,k},\mathfrak{g}_{2}).$

For a linear map $f\in
C^{p}_{\mathsf{LTS}}(\mathfrak{g}^{l,k},\mathfrak{g}_{1})$ (resp. $f\in
C^{p}_{\mathsf{LTS}}(\mathfrak{g}^{l,k},\mathfrak{g}_{2})$), $f$ naturally
induces a linear map $\hat{f}\in
C^{p}_{\mathsf{LTS}}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
defined to be

$\displaystyle\hat{f}:=\begin{cases}f,\quad\text{on
}~{}~{}\mathfrak{g}^{l,k},\\\ 0,\quad\text{all other cases.}\end{cases}$

The linear map $\hat{f}$ is called a lift of $f$. For example, the lifts of
linear maps
$\alpha:\otimes^{3}\mathfrak{g}_{1}\longrightarrow\mathfrak{g}_{1}$,
$\beta:\otimes^{2}\mathfrak{g}_{1}\otimes\mathfrak{g}_{2}\longrightarrow\mathfrak{g}_{2}$
and
$\gamma:\mathfrak{g}_{2}\otimes(\otimes^{2}\mathfrak{g}_{1})\longrightarrow\mathfrak{g}_{2}$
are defined to be

$\displaystyle\hat{\alpha}((x,u),(y,v),(z,w))$ $\displaystyle=$
$\displaystyle(\alpha(x,y,z),0),$
$\displaystyle\hat{\beta}((x,u),(y,v),(z,w))$ $\displaystyle=$
$\displaystyle(0,\beta(x,y,w)),$
$\displaystyle\hat{\gamma}((x,u),(y,v),(z,w))$ $\displaystyle=$
$\displaystyle(0,\gamma(u,y,z))$

respectively. A linear map $H:\mathfrak{g}_{2}\longrightarrow\mathfrak{g}_{1}$
induces its lift given by $\hat{H}(x,u)=(H(u),0)$. It is straightforward to
see that $\hat{H}\circ\hat{H}=0$.

Now we give the notion of bidegree of linear maps in
$C^{p}_{\mathsf{LTS}}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
in the sequel.

######  Definition 3.1.

Let $f\in
C^{p}_{\mathsf{LTS}}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
be a linear map and $l,k\in\mathbb{Z}$. Suppose that $l$ and $k$ satisfies the
following conditions:

  * (i)

$l+k=2p$;

  * (ii)

If $X\in\mathfrak{g}^{l+1,k}$, then $f(X)\in\mathfrak{g}_{1}$;

  * (iii)

If $X\in\mathfrak{g}^{l,k+1}$, then $f(X)\in\mathfrak{g}_{2}$;

  * (iv)

All the other cases, $f(X)=0$,

then we say that the bidegree of $f$ is $l|k$, denoted by $||f||=l|k$. A
linear map $f$ is said to be homogeneous if $f$ has a bidegree.

Notice that $l+k\geqslant 0,~{}k,l\geqslant-1$ since $p\geqslant 0$ and
$l+1,k+1\geqslant 0$. For example, $||\hat{H}||=-1|1$, where
$H:\mathfrak{g}_{2}\longrightarrow\mathfrak{g}_{1}$ is a linear map;
$||\hat{\alpha}||=||\hat{\beta}||=||\hat{\gamma}||=2|0$. Consequently, we
obtain a homogeneous element
$\hat{\mu}:=\hat{\alpha}+\hat{\beta}+\hat{\gamma}$ whose bidegree is $2|0$:

$\hat{\mu}((x,u),(y,v),(z,w))=(\alpha(x,y,z),\beta(x,y,w)+\gamma(u,y,z)-\gamma(v,x,z)).$

However, there does not exist any linear map whose lift is $\hat{\mu}$, but we
focus on $\hat{\mu}$ since it is a multiplication of the semidirect product
type.

We give some lemmas in the following.

######  Lemma 3.2.

Let $f_{1},\cdots,f_{n}\in
C^{p}_{\mathsf{LTS}}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
be homogeneous linear maps and the bidegrees of $f_{i}$ are different. Then
$\sum_{i=1}^{n}f_{i}=0$ if and only if $f_{i}=0,~{}i=1,2,\cdots,n$.

######  Lemma 3.3.

If $||f||=-1|l$ (resp. $l|-1$) and $||g||=-1|k$ (resp. $||g||=k|-1$), then
$[f,g]_{\mathsf{LTS}}=0$.

###### Proof.

Suppose that $||f||=-1|l$ and $||g||=-1|k$. Then both $f$ and $g$ are lifts of
some linear maps in $C^{*}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})$.
Hence, we have $f\circ g=g\circ f=0$, which leads to $[f,g]_{\mathsf{LTS}}=0$.
∎

######  Lemma 3.4.

Let $f\in
C^{p}_{\mathsf{LTS}}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
and $g\in
C^{q}_{\mathsf{LTS}}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
be homogeneous linear maps whose bidegrees are $l_{f}|l_{f}$ and $l_{g}|k_{g}$
respectively. Then $f\circ g\in
C^{p+q}_{\mathsf{LTS}}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
is a homogeneous linear map of bidegree $l_{f}+l_{g}|k_{f}+k_{g}$.

######  Lemma 3.5.

Suppose that $||f||=l_{f}|k_{f}$ and $||g||=l_{g}|k_{g}$, then
$||[f,g]_{\mathsf{LTS}}||=l_{f}+l_{g}|k_{f}+k_{g}$.

###### Proof.

Since $[P,Q]_{\mathsf{LTS}}=P\circ Q-(-1)^{pq}Q\circ P,$ thus by Lemma 3.4, we
obtain that $||[f,g]_{\mathsf{LTS}}||=l_{f}+l_{g}|k_{f}+k_{g}$. ∎

###  3.2. Proto-twilled and quasi-twilled Lie triple systems

######  Definition 3.6.

Let
$(\mathcal{G},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\mathcal{G}})$
be a Lie triple system with a decomposition of two subspaces:
$\mathcal{G}=\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$. Then we call the triple
$(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$ a proto-twilled Lie triple
system.

It is not necessary that $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ be
subalgebras of $\mathcal{G}$ in a proto-twilled Lie triple system
$(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$.

######  Lemma 3.7.

Any $1$-cochain $\Theta\in C^{1}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$ can
be decomposed into five linear maps whose bidegrees are
$3|-1,~{}2|0,~{}1|1,~{}0|2$ and $-1|3$:

$\Theta=\hat{\phi}_{1}+\hat{\mu}_{1}+\hat{\psi}+\hat{\mu}_{2}+\hat{\phi}_{2}.$

###### Proof.

Notice that
$C^{1}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})=\mathrm{Hom}_{\mathsf{LTS}}(\otimes^{3}\mathcal{G},\mathcal{G})$,
and by (7), we have

$C^{1}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})=(3|-1)\oplus(2|0)\oplus(1|1)\oplus(0|2)\oplus(-1|3),$

where $(l|k)$ denotes the space of linear maps of bidegree $l|k$ such that
$l+k=2$. By Lemma 3.2, $\Theta$ can be decomposed into five homogeneous linear
maps of bidegrees $3|-1,~{}2|0,~{}1|1,~{}0|2$ and $-1|3$. The proof is
finished. ∎

The operation $\left\llbracket(x,u),(y,v),(z,w)\right\rrbracket_{\mathcal{G}}$
of $\mathcal{G}$ is uniquely decomposed into 12 multiplications by the
canonical projections $\mathcal{G}\longrightarrow\mathfrak{g}_{1}$ and
$\mathcal{G}\longrightarrow\mathfrak{g}_{2}$:

$\displaystyle\left\llbracket x,y,z\right\rrbracket_{\mathcal{G}}$
$\displaystyle=$ $\displaystyle(\left\llbracket
x,y,z\right\rrbracket_{1},\left\llbracket
x,y,z\right\rrbracket_{2}),\quad\left\llbracket
x,y,w\right\rrbracket_{\mathcal{G}}=(\left\llbracket
x,y,w\right\rrbracket_{1},\left\llbracket x,y,w\right\rrbracket_{2}),$
$\displaystyle~{}\left\llbracket x,v,z\right\rrbracket_{\mathcal{G}}$
$\displaystyle=$ $\displaystyle(\left\llbracket
x,v,z\right\rrbracket_{1},\left\llbracket
x,v,z\right\rrbracket_{2}),\quad\left\llbracket
x,v,w\right\rrbracket_{\mathcal{G}}=(\left\llbracket
x,v,w\right\rrbracket_{1},\left\llbracket x,v,w\right\rrbracket_{2}),$
$\displaystyle~{}\left\llbracket u,v,z\right\rrbracket_{\mathcal{G}}$
$\displaystyle=$ $\displaystyle(\left\llbracket
u,v,z\right\rrbracket_{1},\left\llbracket
u,v,z\right\rrbracket_{2}),\quad\left\llbracket
u,v,w\right\rrbracket_{\mathcal{G}}=(\left\llbracket
u,v,w\right\rrbracket_{1},\left\llbracket u,v,z\right\rrbracket_{2}).$

Here the operation $\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{1}$
(resp. $\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{2}$) denotes the
projection of $\mathcal{G}$ onto $\mathfrak{g}_{1}$ (resp.
$\mathfrak{g}_{2}$).

In the following, we use the notation $\Theta$ to denote the operation
$\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\mathcal{G}}$ on
$\mathcal{G}$, i.e.,

$\Theta((x,u),(y,v),(z,w))=\left\llbracket(x,u),(y,v),(z,w)\right\rrbracket_{\mathcal{G}}.$

Write
$\Theta=\hat{\phi}_{1}+\hat{\mu}_{1}+\hat{\psi}+\hat{\mu}_{2}+\hat{\phi}_{2}$
as in Lemma 3.7, then we obtain that

(13)
$\displaystyle\left\\{\begin{array}[]{rcl}~{}~{}\hat{\phi}_{1}((x,u),(y,v),(z,w))&=&(0,\left\llbracket
x,y,z\right\rrbracket_{2}),\\\
~{}~{}\hat{\mu}_{1}((x,u),(y,v),(z,w))&=&(\left\llbracket
x,y,z\right\rrbracket_{1},\left\llbracket
x,y,w\right\rrbracket_{2}+\left\llbracket
u,y,z\right\rrbracket_{2}-\left\llbracket v,x,z\right\rrbracket_{2}),\\\
~{}~{}\hat{\psi}((x,u),(y,v),(z,w))&=&(\left\llbracket
x,y,w\right\rrbracket_{1}+\left\llbracket
u,y,z\right\rrbracket_{1}-\left\llbracket
v,x,z\right\rrbracket_{1},\left\llbracket
u,v,z\right\rrbracket_{2}+\left\llbracket
x,v,w\right\rrbracket_{2}-\left\llbracket y,u,w\right\rrbracket_{2}),\\\
~{}~{}\hat{\mu}_{2}((x,u),(y,v),(z,w))&=&(\left\llbracket
u,v,z\right\rrbracket_{1}+\left\llbracket
x,v,w\right\rrbracket_{1}-\left\llbracket
y,u,w\right\rrbracket_{1},\left\llbracket u,v,w\right\rrbracket_{2}),\\\
~{}~{}\hat{\phi}_{2}((x,u),(y,v),(z,w))&=&(\left\llbracket
u,v,w\right\rrbracket_{1},0).\end{array}\right.$

It is easy to see that $\hat{\phi}_{1}$ and $\hat{\phi}_{2}$ are lifts of
linear maps $\phi_{1}\in
C^{1}_{\mathsf{LTS}}(\mathfrak{g}_{1},\mathfrak{g}_{2})$ and $\phi_{2}\in
C^{1}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})$ respectively, where
$\phi_{1}(x,y,z):=\left\llbracket x,y,z\right\rrbracket_{2}$ and
$\phi_{2}(u,v,w):=\left\llbracket u,v,w\right\rrbracket_{1}$.

######  Proposition 3.8.

The Maurer-Cartan equation $[\Theta,\Theta]_{\mathsf{LTS}}=0$ is equivalent to
the following conditions:

$\displaystyle\left\\{\begin{array}[]{rcl}[\hat{\phi}_{1},\hat{\mu}_{1}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\psi},\hat{\phi}_{1}]_{\mathsf{LTS}}+\frac{1}{2}[\hat{\mu}_{1},\hat{\mu}_{1}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\phi}_{1},\hat{\mu}_{2}]_{\mathsf{LTS}}+[\hat{\psi},\hat{\mu}_{1}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\psi},\hat{\phi}_{2}]_{\mathsf{LTS}}+[\hat{\mu}_{1},\hat{\mu}_{2}]_{\mathsf{LTS}}+\frac{1}{2}[\hat{\psi},\hat{\psi}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\mu}_{1},\hat{\phi}_{2}]_{\mathsf{LTS}}+[\hat{\psi},\hat{\mu}_{2}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\psi},\hat{\phi}_{2}]_{\mathsf{LTS}}+\frac{1}{2}[\hat{\mu}_{2},\hat{\mu}_{2}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\mu}_{2},\hat{\phi}_{2}]_{\mathsf{LTS}}&=&0.\end{array}\right.$

###### Proof.

Since
$\Theta=\hat{\phi}_{1}+\hat{\mu}_{1}+\hat{\psi}+\hat{\mu}_{2}+\hat{\phi}_{2}\in
C^{1}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$, then expanding the expression
$[\Theta,\Theta]_{\mathsf{LTS}}\in
C^{2}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$ yields the following nonzero
terms by bidegree classification:

$\displaystyle\left\\{\begin{array}[]{rcl}[\hat{\phi}_{1},\hat{\mu}_{1}]_{\mathsf{LTS}}&\in&(5|-1),\\\
~{}2[\hat{\psi},\hat{\phi}_{1}]_{\mathsf{LTS}}+[\hat{\mu}_{1},\hat{\mu}_{1}]_{\mathsf{LTS}}&\in&(4|0),\\\
~{}2[\hat{\phi}_{1},\hat{\mu}_{2}]_{\mathsf{LTS}}+2[\hat{\psi},\hat{\mu}_{1}]_{\mathsf{LTS}}&\in&(3|1),\\\
~{}2[\hat{\psi},\hat{\phi}_{2}]_{\mathsf{LTS}}+2[\hat{\mu}_{1},\hat{\mu}_{2}]_{\mathsf{LTS}}+[\hat{\psi},\hat{\psi}]_{\mathsf{LTS}}&\in&(2|2),\\\
~{}2[\hat{\mu}_{1},\hat{\phi}_{2}]_{\mathsf{LTS}}+2[\hat{\psi},\hat{\mu}_{2}]_{\mathsf{LTS}}&\in&(1|3),\\\
~{}2[\hat{\psi},\hat{\phi}_{2}]_{\mathsf{LTS}}+[\hat{\mu}_{2},\hat{\mu}_{2}]_{\mathsf{LTS}}&\in&(0|4)\\\
~{}[\hat{\mu}_{2},\hat{\phi}_{2}]_{\mathsf{LTS}}&\in&(-1|5).\end{array}\right.$

Thus the Maurer-Cartan equation $[\Theta,\Theta]_{\mathsf{LTS}}=0$ holds if
and only if all of its expanding parts vanish. ∎

######  Definition 3.9.

Let $(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$ be a proto-twilled Lie
triple system and
$\Theta=\hat{\phi}_{1}+\hat{\mu}_{1}+\hat{\psi}+\hat{\mu}_{2}+\hat{\phi}_{2}\in
C^{1}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$ its structure. If $\phi_{2}=0$,
or equivalently, $\mathfrak{g}_{2}$ is a subalgebra, then the proto-twilled
Lie triple system $(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$ is called
a quasi-twilled Lie triple system.

######  Remark 3.10.

Since
$\mathcal{G}=\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}=\mathfrak{g}_{2}\oplus\mathfrak{g}_{1}$,
thus the condition in Definition 3.9 is adapted in the case that $\phi_{1}=0$.
Hence the condition making the proto-twilled Lie triple system
$(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$ into a quasi-twilled Lie
triple system is that either $\mathfrak{g}_{1}$ or $\mathfrak{g}_{2}$ is a
sublagebra. In the following, we always assume that $\mathfrak{g}_{2}$ is a
subalgebra when quasi-twilled Lie triple system is referred.

It is direct to obtain the following proposition.

######  Proposition 3.11.

The triple $(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$ is a quasi-
twilled Lie triple system if and only if the following conditions are
satisfied:

$\displaystyle\left\\{\begin{array}[]{rcl}[\hat{\phi}_{1},\hat{\mu}_{1}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\psi},\hat{\phi}_{1}]_{\mathsf{LTS}}+\frac{1}{2}[\hat{\mu}_{1},\hat{\mu}_{1}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\phi}_{1},\hat{\mu}_{2}]_{\mathsf{LTS}}+[\hat{\psi},\hat{\mu}_{1}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\mu}_{1},\hat{\mu}_{2}]_{\mathsf{LTS}}+\frac{1}{2}[\hat{\psi},\hat{\psi}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\psi},\hat{\mu}_{2}]_{\mathsf{LTS}}&=&0,\\\
~{}\frac{1}{2}[\hat{\mu}_{2},\hat{\mu}_{2}]_{\mathsf{LTS}}&=&0.\end{array}\right.$

###  3.3. Twilled Lie triple systems and $L_{\infty}$-algebras

In this subsection, we introduce the notion of twilled Lie triple systems,
which is our main object in the present paper, and then we construct an
$L_{\infty}$-algebra from a twilled Lie triple system via the higher derived
brackets.

######  Definition 3.12.

Let $(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$ be a proto-twilled Lie
triple system and
$\Theta=\hat{\phi}_{1}+\hat{\mu}_{1}+\hat{\psi}+\hat{\mu}_{2}+\hat{\phi}_{2}\in
C^{1}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$ its structure. If both
$\phi_{1}$ and $\phi_{2}$ all vanish, or equivalently, both $\mathfrak{g}_{1}$
and $\mathfrak{g}_{2}$ are subalgebras of $\mathcal{G}$, then the proto-
twilled Lie triple system $(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$ is
called a twilled Lie triple system.

######  Remark 3.13.

In [22], the first author et al. explored product structures on Lie-Yamaguti
algebras, and we obtain the corresponding product structure on Lie triple
system if we restrict the binary operation on Lie-Yamaguti algebra to zero. By
Definition 3.12, we see that $\mathcal{G}$ is a twilled Lie triple system if
and only if there exists a product structure on $\mathcal{G}$.

It is easy to deduce the following proposition.

######  Proposition 3.14.

The triple $(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$ is a twilled Lie
triple system if and only if the following conditions are satisfied:

$\displaystyle\left\\{\begin{array}[]{rcl}~{}\frac{1}{2}[\hat{\mu}_{1},\hat{\mu}_{1}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\psi},\hat{\mu}_{1}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\mu}_{1},\hat{\mu}_{2}]_{\mathsf{LTS}}+\frac{1}{2}[\hat{\psi},\hat{\psi}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\psi},\hat{\mu}_{2}]_{\mathsf{LTS}}&=&0,\\\
~{}\frac{1}{2}[\hat{\mu}_{2},\hat{\mu}_{2}]_{\mathsf{LTS}}&=&0.\end{array}\right.$

######  Remark 3.15.

From Proposition 3.14, we obtain that $\hat{\mu}_{1}\in
C^{1}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$ is a Lie triple system
structure on $\mathcal{G}=\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$. Moreover,
by (13), we see that
$\hat{\mu}_{1}|_{\mathfrak{g}_{1}\otimes\mathfrak{g}_{1}}$ is a Lie triple
system structure on $\mathfrak{g}_{1}$, and we denote this Lie triple system
by
$(\mathfrak{g}_{1},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\mathfrak{g}_{1}})$.
Set $\rho_{1}(x,y)u:=\hat{\mu}_{1}(u,x,y)$, then
$D_{1}(x,y)u=D_{\rho_{1}}(x,y)u=\hat{\mu}_{1}(x,y,u)$ since $\mathfrak{g}_{2}$
is a Lie triple system. Consequently, $(\mathfrak{g}_{2};\rho_{1})$ is a
representation of $\mathfrak{g}_{1}$. Similarly, we have
$\hat{\mu}_{2}|_{\mathfrak{g}_{1}\otimes\mathfrak{g}_{2}}$ is a Lie triple
system structure on $\mathfrak{g}_{2}$, and $(\mathfrak{g}_{1};\rho_{2})$ is a
representation of $\mathfrak{g}_{2}$, where
$\rho_{2}(u,v)x:=\hat{\mu}_{2}(x,u,v)$ and
$D_{2}(u,v)x=D_{\rho_{2}}(u,v)x=\hat{\mu}_{2}(u,v,x)$.

In the sequel, we construct an $L_{\infty}$-algebra via a given twilled Lie
triple system. Before this, let us recall the method of higher derived
brackets of a differential graded Lie algebra. Let
$(\mathfrak{g},[\cdot,\cdot],d)$ be a differential graded Lie algebra and
$d_{t}:=\sum_{i=0}^{\infty}d_{i}t^{i}$ a formal deformation of $d$ with
$d_{0}=d$. Then $d_{t}$ is a differential on $\mathfrak{g}[[t]]$, which is a
Lie triple system of formal series with coefficients in $\mathfrak{g}$. The
square zero condition $d_{t}\circ d_{t}=0$ is equivalent to

$\sum_{i+j=n,\atop i,j\geqslant 0}d_{i}\circ d_{j}=0,\quad
n\in\mathbb{Z}_{\geqslant 0}.$

Then set $l_{k}:\otimes^{k}\mathfrak{g}\longrightarrow\mathfrak{g}$ to be

(18) $\displaystyle
l_{k}(x_{1},\cdots,x_{k})=[\cdots[[d_{k-1}(x_{1}),x_{2}],x_{3}],\cdots,x_{k}]$

for all homogeneous elements $x_{1},\cdots,x_{k}\in\mathfrak{g}$. The
collection of graded linear maps $\\{l_{k}\\}_{k=1}^{\infty}$ is called the
higher derived brackets.

A suitable $L_{\infty}$-algebra can be obtained from a differential graded Lie
algebra via the method of higher derived brackets. The notion of
$L_{\infty}$-algebras was introduced in [23]. See [13, 14] for more details.

######  Proposition 3.16.

([24]) Let $(\mathfrak{g},[\cdot,\cdot],d)$ be a differential graded Lie
algebra, and $\mathfrak{h}\subset\mathfrak{g}$ an abelian subalgebra, i.e.,
$[\mathfrak{h},\mathfrak{h}]=0$. If $l_{k}$ is closed on $\mathfrak{h}$, where
$l_{k}$ is given by (18), then $(\mathfrak{h},\\{l_{k}\\}_{i=1}^{\infty})$ is
an $L_{\infty}$-algebra.

Let $(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$ be a twilled Lie triple
system, and
$C^{p}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})=\mathrm{Hom}_{\mathsf{LTS}}(\otimes^{2p+1}\mathfrak{g}_{2},\mathfrak{g}_{1})$
a subspace of
$(C^{*}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G}),[\cdot,\cdot]_{\mathsf{LTS}})$.
For any $f\in C^{p}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1}),~{}g\in
C^{q}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1}),~{}h\in
C^{r}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})$, define

$\displaystyle
l_{1}:C^{p}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})\longrightarrow
C^{p+1}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1}),$ $\displaystyle
l_{2}:C^{p}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})\times
C^{q}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})\longrightarrow
C^{p+q+1}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1}),$ $\displaystyle
l_{3}:C^{p}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})\times
C^{q}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})\times
C^{r}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})\longrightarrow
C^{p+q+r+1}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})$

to be

$\displaystyle l_{1}(f)$ $\displaystyle=$
$\displaystyle[\hat{\mu}_{2},\hat{f}]_{\mathsf{LTS}},$
$\displaystyle~{}l_{2}(f,g)$ $\displaystyle=$
$\displaystyle[[\hat{\psi},\hat{f}]_{\mathsf{LTS}},\hat{g}]_{\mathsf{LTS}},$
$\displaystyle~{}l_{3}(f,g,h)$ $\displaystyle=$
$\displaystyle[[[\hat{\mu}_{1},\hat{f}]_{\mathsf{LTS}},\hat{g}]_{\mathsf{LTS}},\hat{h}]_{\mathsf{LTS}}$

respectively. Here $\hat{f},~{}\hat{g}$ and $\hat{h}$ are lifts of linear maps
$f,~{}g$ and $h$ respectively.

######  Theorem 3.17.

Let $(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$ be a twilled Lie triple
system. Then with the above notations,
$(C^{*}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1}),l_{1},l_{2},l_{3})$
is an $L_{\infty}$-algebra.

###### Proof.

We define derivations $d_{i}$ to be

$d_{0}:=[\hat{\mu}_{2},\cdot]_{\mathsf{LTS}},\quad
d_{1}:=[\hat{\psi},\cdot]_{\mathsf{LTS}},\quad
d_{2}:=[\hat{\mu}_{1},\cdot]_{\mathsf{LTS}},\quad d_{k}=0,\quad\forall
k\geqslant 3.$

For all $f\in C^{*}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$, since
$2[[\hat{\mu}_{2},\hat{\mu}_{2}]_{\mathsf{LTS}},f]_{\mathsf{LTS}}=[[\hat{\mu}_{2},\hat{\mu}_{2}]_{\mathsf{LTS}},f]_{\mathsf{LTS}}$,
we deduce that

$d_{0}\circ d_{0}=0,$

which implies that
$(C^{*}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G}),[\cdot,\cdot]_{\mathsf{LTS}},d_{0})$
is a differential graded Lie algebra. Moreover, for all $f\in
C^{*}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$, by Proposition 3.14, we have

$\displaystyle(d_{0}\circ d_{1}+d_{1}\circ d_{0})(f)$ $\displaystyle=$
$\displaystyle[\hat{\mu}_{2},[\hat{\psi},f]_{\mathsf{LTS}}]_{\mathsf{LTS}}+[\hat{\psi},[\hat{\mu}_{2},f]_{\mathsf{LTS}}]_{\mathsf{LTS}}$
$\displaystyle=$
$\displaystyle[[\hat{\mu}_{2},\hat{\psi}]_{\mathsf{LTS}},f]_{\mathsf{LTS}}=0,$
$\displaystyle~{}(d_{0}\circ d_{2}+d_{1}\circ d_{1}+d_{2}\circ d_{0})(f)$
$\displaystyle=$
$\displaystyle[\hat{\mu}_{2},[\hat{\mu}_{1},f]_{\mathsf{LTS}}]_{\mathsf{LTS}}+[\hat{\psi},[\hat{\psi},f]_{\mathsf{LTS}}]_{\mathsf{LTS}}+[\hat{\mu}_{1},[\hat{\mu}_{2},f]_{\mathsf{LTS}}]_{\mathsf{LTS}}$
$\displaystyle=$
$\displaystyle[[\hat{\mu}_{1},\hat{\mu}_{2}]_{\mathsf{LTS}},f]_{\mathsf{LTS}}+\frac{1}{2}[[\hat{\psi},\hat{\psi}]_{\mathsf{LTS}},f]_{\mathsf{LTS}}=0,$
$\displaystyle~{}(d_{1}\circ d_{2}+d_{2}\circ d_{1})(f)$ $\displaystyle=$
$\displaystyle[\hat{\psi},[\hat{\mu}_{1},f]_{\mathsf{LTS}}]_{\mathsf{LTS}}+[\hat{\mu}_{1},[\hat{\psi},f]_{\mathsf{LTS}}]_{\mathsf{LTS}}$
$\displaystyle=$
$\displaystyle[[\hat{\psi},\hat{\mu}_{1}]_{\mathsf{LTS}},f]_{\mathsf{LTS}}=0,$
$\displaystyle~{}(d_{2}\circ d_{2})(f)$ $\displaystyle=$
$\displaystyle[\hat{\mu}_{1},[\hat{\mu}_{1},f]_{\mathsf{LTS}}]_{\mathsf{LTS}}=\frac{1}{2}[[\hat{\mu}_{1},\hat{\mu}_{1}]_{\mathsf{LTS}},f]_{\mathsf{LTS}}=0.$

Thus we obtain that $\displaystyle{\sum_{i+j=n,\atop n\geqslant 0}d_{i}\circ
d_{j}=0}$ and we can construct the higher derived brackets on
$C^{*}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$ as follows:

$\displaystyle l_{1}(f)$ $\displaystyle=$
$\displaystyle[\hat{\mu}_{2},f]_{\mathsf{LTS}},$ $\displaystyle~{}l_{2}(f,g)$
$\displaystyle=$
$\displaystyle[[\hat{\psi},f]_{\mathsf{LTS}},g]_{\mathsf{LTS}},$
$\displaystyle~{}l_{3}(f,g,h)$ $\displaystyle=$
$\displaystyle[[[\hat{\mu}_{1},f]_{\mathsf{LTS}},g]_{\mathsf{LTS}},h]_{\mathsf{LTS}},$
$\displaystyle~{}l_{k}$ $\displaystyle=$ $\displaystyle 0,\quad k\geqslant 4,$

for all $f\in C^{p}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G}),~{}g\in
C^{q}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G}),~{}h\in
C^{r}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$. It is direct to see that
$C^{*}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})$ is an abelian
subalgebra of
$(C^{*}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G}),[\cdot,\cdot]_{\mathsf{LTS}})$,
and we show that $l_{1},l_{2},l_{3}$ are closed on
$C^{*}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})$ in the sequel. For
all $f\in C^{p}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})$, we have
$||\hat{f}||=-1|2p+1$. Then by Lemma 3.5, we deduce that
$||l_{1}(f)||=||[\hat{\mu}_{2},\hat{f}]_{\mathsf{LTS}}||=-1|2p+3$, which
implies that $l_{1}(f)\in
C^{*}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})$. Similarly, for all
$g\in C^{q}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})$ and $h\in
C^{r}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})$, we have

$||l_{2}(f,g)||=-1|2p+2q+3,\quad||l_{3}(f,g,h)||=-1|2p+2q+2r+3,$

and thus $l_{2}(f,g),l_{3}(f,g,h)\in
C^{*}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})$, which implies that
$l_{1},l_{2}$ and $l_{3}$ are closed on
$C^{*}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1})$. Consequently, by
Proposition 3.16, we obtain that
$(C^{*}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1}),l_{1},l_{2},l_{3})$
is an $L_{\infty}$-algebra. ∎

##  4\. Twisting of twilled Lie triple systems

Let $f\in C^{0}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$ be a $0$-cochain. Set

$\exp(X_{f})(\cdot):=\sum_{k=0}^{\infty}\frac{1}{k!}X_{f}^{k},$

where
$X_{f}^{k}:=\underbrace{[\cdots[[}_{k}\cdot,f]_{\mathsf{LTS}},f]_{\mathsf{LTS}},\cdots,f]_{\mathsf{LTS}}$.
Note that $\exp(X_{f})(\cdot)$ is not well defined in general.

Let $(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$ be a proto-twilled Lie
triple system with the structure
$\Theta:=\hat{\phi}_{1}+\hat{\mu}_{1}+\hat{\psi}+\hat{\mu}_{2}+\hat{\phi}_{2}$
and $\hat{H}\in C^{0}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$ be the lift of
a linear map $H:\mathfrak{g}_{2}\longrightarrow\mathfrak{g}_{1}$. Then
$\exp(X_{\hat{H}})(\cdot)$ is a well-defined operator since
$\hat{H}\circ\hat{H}=0$.

######  Definition 4.1.

With the above notations, the transformation $\exp(X_{\hat{H}})(\Theta)$ is
called a twisting of $\Theta$ by $H$, which is denoted by $\Theta^{H}$.

We need to show that the twisting of the structure is again a Lie triple
system structure. Before this, we show a property of the twisting.

######  Lemma 4.2.

We have the following equation:

$\Theta^{H}=\exp(-\hat{H})\circ\Theta\circ\Big{(}\exp(\hat{H})\otimes\exp(\hat{H})\otimes\exp(\hat{H})\Big{)}.$

###### Proof.

For all $(x_{1},u_{1}),(x_{2},u_{2}),(x_{3},u_{3})\in\mathcal{G}$, we have

$\displaystyle[\Theta,\hat{H}]_{\mathsf{LTS}}\Big{(}(x_{1},u_{1}),(x_{2},u_{2}),(x_{3},u_{3})\Big{)}$
$\displaystyle=$
$\displaystyle(\Theta\circ\hat{H}-\hat{H}\circ\Theta)\Big{(}(x_{1},u_{1}),(x_{2},u_{2}),(x_{3},u_{3})\Big{)}$
$\displaystyle=$
$\displaystyle\Theta\bigg{(}H(x_{1},u_{1}),(x_{2},u_{2}),(x_{3},u_{3})\bigg{)}+\Theta\bigg{(}(x_{1},u_{1}),H(x_{2},u_{2}),(x_{3},u_{3})\bigg{)}+\Theta\bigg{(}(x_{1},u_{1}),(x_{2},u_{2}),H(x_{3},u_{3})\bigg{)}$
$\displaystyle-\hat{H}\bigg{(}\Theta\Big{(}(x_{1},u_{1}),(x_{2},u_{2}),(x_{3},u_{3})\Big{)}\bigg{)}$
$\displaystyle=$
$\displaystyle\Theta\bigg{(}(H(u_{1}),0),(x_{2},u_{2}),(x_{3},u_{3})\bigg{)}+\Theta\bigg{(}(x_{1},u_{1}),(H(u_{2}),0),(x_{3},u_{3})\bigg{)}+\Theta\bigg{(}(x_{1},u_{1}),(x_{2},u_{2}),(H(u_{3}),0)\bigg{)}$
$\displaystyle-\hat{H}\bigg{(}\Theta\Big{(}(x_{1},u_{1}),(x_{2},u_{2}),(x_{3},u_{3})\Big{)}\bigg{)}.$

Thus we obtain that

(19) $\displaystyle
X_{\hat{H}}(\Theta)=\Theta\circ(\hat{H}\otimes{\rm{Id}}\otimes{\rm{Id}})+\Theta\circ({\rm{Id}}\otimes\hat{H}\otimes{\rm{Id}})+\Theta\circ({\rm{Id}}\otimes{\rm{Id}}\otimes\hat{H})-\hat{H}\circ\Theta.$

By Eq. (19) and the fact that $\hat{H}\circ\hat{H}=0$, we compute that

$\displaystyle[[\Theta,\hat{H}]_{\mathsf{LTS}},\hat{H}]_{\mathsf{LTS}}\Big{(}(x_{1},u_{1}),(x_{2},u_{2}),(x_{3},u_{3})\Big{)}$
$\displaystyle=$
$\displaystyle[\Theta,\hat{H}]_{\mathsf{LTS}}\Big{(}(H(u_{1}),0),(x_{2},u_{2}),(x_{3},u_{3})\Big{)}+[\Theta,\hat{H}]_{\mathsf{LTS}}\Big{(}(x_{1},u_{1}),(H(u_{2}),0),(x_{3},u_{3})\Big{)}$
$\displaystyle+[\Theta,\hat{H}]_{\mathsf{LTS}}\Big{(}(x_{1},u_{1}),(x_{2},u_{2}),(H(u_{3}),0)\Big{)}-\hat{H}\circ[\Theta,\hat{H}]_{\mathsf{LTS}}\Big{(}(x_{1},u_{1}),(x_{2},u_{2}),(x_{3},u_{3})\Big{)}$
$\displaystyle=$ $\displaystyle
2\Theta\Big{(}(H(u_{1}),0),(H(u_{2}),0),(x_{3},u_{3})\Big{)}+2\Theta\Big{(}(H(u_{1}),0),(x_{2},u_{2}),(H(u_{3}),0)\Big{)}$
$\displaystyle+2\Theta\Big{(}(x_{1},u_{1}),(H(u_{2}),0),(H(u_{3}),0)\Big{)}-2\hat{H}\bigg{(}\Theta\Big{(}(H(u_{1}),0),(x_{2},u_{2}),(x_{3},u_{3})\Big{)}\bigg{)}$
$\displaystyle-2\hat{H}\bigg{(}\Theta\Big{(}(x_{1},u_{1}),(H(u_{2}),0),(x_{3},u_{3})\Big{)}\bigg{)}-2\hat{H}\bigg{(}\Theta\Big{(}(x_{1},u_{1}),(x_{2},u_{2}),(H(u_{3}),0)\Big{)}\bigg{)}.$

Hence we obtain that

$\displaystyle X_{\hat{H}}^{2}(\Theta)$ $\displaystyle=$ $\displaystyle
2\Theta\circ(\hat{H}\otimes\hat{H}\otimes{\rm{Id}})+2\Theta\circ(\hat{H}\otimes{\rm{Id}}\otimes\hat{H})+2\Theta\circ({\rm{Id}}\otimes\hat{H}\otimes\hat{H})$
(20)
$\displaystyle-2\hat{H}\circ\Theta\circ(\hat{H}\otimes{\rm{Id}}\otimes{\rm{Id}})-2\hat{H}\circ\Theta\circ({\rm{Id}}\otimes\hat{H}\otimes{\rm{Id}})-2\hat{H}\circ\Theta\circ({\rm{Id}}\otimes{\rm{Id}}\otimes\hat{H}).$

Similarly, we obtain that

(21) $\displaystyle X_{\hat{H}}^{3}(\Theta)$ $\displaystyle=$ $\displaystyle
6\Theta\circ(\hat{H}\otimes\hat{H}\otimes\hat{H})-6\hat{H}\circ\Theta\circ(\hat{H}\otimes\hat{H}\otimes{\rm{Id}})$
$\displaystyle-6\hat{H}\circ\Theta\circ(\hat{H}\otimes{\rm{Id}}\otimes\hat{H})-6\hat{H}\circ\Theta\circ({\rm{Id}}\otimes{\rm{Id}}\otimes\hat{H}),$
(22) $\displaystyle~{}X_{\hat{H}}^{4}(\Theta)$ $\displaystyle=$
$\displaystyle-24\hat{H}\circ\Theta\circ(\hat{H}\otimes\hat{H}\otimes\hat{H}),$
(23) $\displaystyle~{}X_{\hat{H}}^{k}(\Theta)$ $\displaystyle=$ $\displaystyle
0,\quad\forall k\geqslant 5.$

By Eqs. (19)-(23) and by using the fact that $\hat{H}\circ\hat{H}=0$ again, we
have

$\displaystyle\exp(-\hat{H})\circ\Theta\circ\Big{(}\exp(\hat{H})\otimes\exp(\hat{H})\otimes\exp(\hat{H})\Big{)}$
$\displaystyle=$
$\displaystyle\Big{(}{\rm{Id}}-\hat{H}\Big{)}\circ\Theta\circ\Big{(}({\rm{Id}}+\hat{H})\otimes({\rm{Id}}+\hat{H})\otimes({\rm{Id}}+\hat{H})\Big{)}$
$\displaystyle=$
$\displaystyle\Theta\circ({\rm{Id}}\otimes{\rm{Id}}\otimes{\rm{Id}})+\Theta\circ({\rm{Id}}\otimes\hat{H}\otimes{\rm{Id}})+\Theta\circ(\hat{H}\otimes{\rm{Id}}\otimes{\rm{Id}})+\Theta\circ(\hat{H}\otimes\hat{H}\otimes{\rm{Id}})$
$\displaystyle+\Theta\circ({\rm{Id}}\otimes{\rm{Id}}\otimes\hat{H})+\Theta\circ({\rm{Id}}\otimes\hat{H}\otimes\hat{H})+\Theta\circ(\hat{H}\otimes{\rm{Id}}\otimes\hat{H})+\Theta\circ(\hat{H}\otimes\hat{H}\otimes\hat{H})$
$\displaystyle-\hat{H}\circ\Theta\circ({\rm{Id}}\otimes{\rm{Id}}\otimes{\rm{Id}})-\hat{H}\circ\Theta\circ({\rm{Id}}\otimes\hat{H}\otimes{\rm{Id}})-\hat{H}\circ\Theta\circ(\hat{H}\otimes{\rm{Id}}\otimes{\rm{Id}})-\hat{H}\circ\Theta\circ(\hat{H}\otimes\hat{H}\otimes{\rm{Id}})$
$\displaystyle-\hat{H}\circ\Theta\circ({\rm{Id}}\otimes{\rm{Id}}\otimes\hat{H})-\hat{H}\circ\Theta\circ({\rm{Id}}\otimes\hat{H}\otimes\hat{H})-\hat{H}\circ\Theta\circ(\hat{H}\otimes{\rm{Id}}\otimes\hat{H})-\hat{H}\circ\Theta\circ(\hat{H}\otimes\hat{H}\otimes\hat{H})$
$\displaystyle=$
$\displaystyle\Theta+X_{\hat{H}}(\Theta)+\frac{1}{2}X_{\hat{H}}^{2}(\Theta)+\frac{1}{3!}X_{\hat{H}}^{3}(\Theta)+\frac{1}{4!}X_{\hat{H}}^{4}(\Theta)$
$\displaystyle=$ $\displaystyle\Theta^{H}.$

This completes the proof. ∎

######  Proposition 4.3.

The twisting $\Theta^{H}$ is a Lie triple system structure on $\mathcal{G}$.

###### Proof.

For all $(x,u),(y,v),(z,w)\in\mathcal{G}$, one has

$\displaystyle\Theta^{H}((x,u),(x,u),(y,v))$ $\displaystyle=$
$\displaystyle\exp(-\hat{H})\circ\Theta\circ\Big{(}\exp(\hat{H})\otimes\exp(\hat{H})\otimes\exp(\hat{H})\Big{)}((x,u),(x,u),(y,v))$
$\displaystyle=$
$\displaystyle\Big{(}{\rm{Id}}-\hat{H}\Big{)}\bigg{(}\Theta\Big{(}(x+H(u),u),(x+H(u),u),(y+H(v),v)\Big{)}\bigg{)}=0,$

and

$\displaystyle\Theta^{H}((x,u),(y,v),(z,w))+c.p.$ $\displaystyle=$
$\displaystyle\exp(-\hat{H})\bigg{(}\Theta\Big{(}(x+H(u),u),(y+H(v),v),(z+H(w),w)\Big{)}+c.p.\bigg{)}$
$\displaystyle=$ $\displaystyle 0.$

Here, the last equalities hold since $\Theta$ is a Lie triple system
structure. Moreover, one needs to show that $\Theta^{\hat{H}}$ satisfies the
fundamental identity, or equivalently, $\Theta^{\hat{H}}$ satisfies the
following Maurer-Cartan equation:

$[\Theta^{H},\Theta^{H}]_{\mathsf{LTS}}=0.$

In fact, by Lemma 4.2, we have

$\displaystyle~{}[\Theta^{H},\Theta^{H}]_{\mathsf{LTS}}$ $\displaystyle=$
$\displaystyle 2\Theta^{H}\circ\Theta^{H}$ $\displaystyle=$ $\displaystyle
2\exp(-\hat{H})\circ(\Theta\circ\Theta)\circ\Big{(}\exp(\hat{H})\otimes\exp(\hat{H})\otimes\exp(\hat{H})\Big{)}$
$\displaystyle=$
$\displaystyle\exp(-\hat{H})\circ[\Theta,\Theta]_{\mathsf{LTS}}\circ\Big{(}\exp(\hat{H})\otimes\exp(\hat{H})\otimes\exp(\hat{H})\Big{)}=0$

This completes the proof. ∎

######  Corollary 4.4.

The linear map

$\exp(\hat{H}):(\mathcal{G},\Theta^{H})\longrightarrow(\mathcal{G},\Theta)$

is an isomorphism between Lie triple systems.

Let $(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$ be a proto-twilled Lie
triple system with its structure $\Theta$. It is easy to see that
$(\mathcal{G},\Theta^{H})$ is also a proto-twilled Lie triple system, and thus
$\Theta^{H}$ can also be decomposed into 5 terms:
$\Theta^{H}=\hat{\phi}_{1}^{H}+\hat{\mu}_{1}^{H}+\hat{\psi}^{H}+\hat{\mu}_{2}^{H}+\hat{\phi}_{2}^{H}$,
where the bidegrees of
$\hat{\phi}_{1}^{H},\hat{\mu}_{1}^{H},\hat{\psi}^{H},\hat{\mu}_{2}^{H}$ and
$\hat{\phi}_{2}^{H}$ are $3|-1,2|0,1|1,0|2$ and $-1|3$ respectively. Then the
decomposed terms of twisting operations are determined by the following
result.

######  Theorem 4.5.

With the above notations, we have

$\displaystyle\left\\{\begin{array}[]{rcl}\hat{\phi}_{1}^{H}&=&\hat{\phi}_{1},\\\
\hat{\mu}_{1}^{H}&=&\hat{\mu}_{1}+X_{\hat{H}}(\hat{\phi}_{1}),\\\
\hat{\psi}^{H}&=&\hat{\psi}+X_{\hat{H}}(\hat{\mu}_{1})+\frac{1}{2}X_{\hat{H}}^{2}(\hat{\phi}_{1}),\\\
\hat{\mu}_{2}^{H}&=&\hat{\mu}_{2}+X_{\hat{H}}(\hat{\psi})+\frac{1}{2}X_{\hat{H}}^{2}(\hat{\mu}_{1})+\frac{1}{6}X_{\hat{H}}^{3}(\hat{\phi}_{1}),\\\
\hat{\phi}_{2}^{H}&=&\hat{\phi}_{2}+X_{\hat{H}}(\hat{\mu}_{2})+\frac{1}{2}X_{\hat{H}}^{2}(\hat{\psi})+\frac{1}{6}X_{\hat{H}}^{3}(\hat{\mu}_{1})+\frac{1}{24}X_{\hat{H}}^{4}(\hat{\phi}_{1}).\end{array}\right.$

###### Proof.

By the proof of Lemma 4.2, we have

(25)
$\displaystyle\Theta^{H}=\Theta+X_{\hat{H}}(\Theta)+\frac{1}{2}X_{\hat{H}}^{2}(\Theta)+\frac{1}{3!}X_{\hat{H}}^{3}(\Theta)+\frac{1}{4!}X_{\hat{H}}^{4}(\Theta).$

Thus the first term is
$\Theta=\hat{\phi}_{1}+\hat{\mu}_{1}+\hat{\psi}+\hat{\mu}_{2}+\hat{\phi}_{2}$.
By linearity of $X_{\hat{H}}(\cdot)$ and by Lemma 3.3 which leads to
$X_{\hat{H}}(\phi_{2})=0$, we have

$X_{\hat{H}}(\Theta)=X_{\hat{H}}(\phi_{1})+X_{\hat{H}}(\mu_{1})+X_{\hat{H}}(\psi)+X_{\hat{H}}(\mu_{2}).$

Here the bidegrees of each terms is
$||X_{\hat{H}}(\phi_{1})||=2|0,||X_{\hat{H}}(\mu_{1})||=1|1,||X_{\hat{H}}(\psi)||=0|2$
and $||X_{\hat{H}}(\mu_{2})||=-1|3$ respectively. Similarly, the remaining
terms are

$X_{\hat{H}}^{2}(\Theta)=X_{\hat{H}}^{2}(\phi_{1})+X_{\hat{H}}^{2}(\mu_{1})+X_{\hat{H}}^{2}(\psi),$

where
$||X_{\hat{H}}^{2}(\phi_{1})||=1|1,||X_{\hat{H}}^{2}(\mu_{1})||=0|2,||X_{\hat{H}}^{2}(\psi)||=-1|3$;

$X_{\hat{H}}^{3}(\Theta)=X_{\hat{H}}^{3}(\phi_{1})+X_{\hat{H}}^{2}(\mu_{1}),$

where $||X_{\hat{H}}^{3}(\phi_{1})||=0|2$ and
$||X_{\hat{H}}^{2}(\mu_{1})||=-1|3$; and

$X_{\hat{H}}^{4}(\Theta)=X_{\hat{H}}^{4}(\hat{\phi}_{1})$

whose bidegree is $||X_{\hat{H}}^{4}(\hat{\phi}_{1})||=-1|3$. Thus by Eq. (25)
and the bidegree convention, we obtain that the $3|-1$-component is
$\hat{\phi}_{1}$, which gives $\hat{\phi}_{1}^{H}$; the sum of all
$2|0$-components is $\hat{\mu}_{1}+X_{\hat{H}}(\hat{\phi}_{1})$, which gives
$\hat{\mu}_{1}^{H}$; the sum of all $1|1$-components is
$\hat{\psi}+X_{\hat{H}}(\hat{\mu}_{1})+\frac{1}{2}X_{\hat{H}}^{2}(\hat{\phi}_{1})$,
which gives $\hat{\psi}^{H}$; the sum of all $0|2$-components is
$\hat{\mu}_{2}+X_{\hat{H}}(\hat{\psi})+\frac{1}{2}X_{\hat{H}}(\hat{\mu}_{1})+\frac{1}{6}X_{\hat{H}}^{3}(\hat{\phi}_{1})$,
which gives $\hat{\mu}_{2}^{H}$; and the sum of all $-1|3$-components is
$\hat{\phi}_{2}+X_{\hat{H}}(\hat{\mu}_{2})+\frac{1}{2}X_{\hat{H}}(\hat{\psi})+\frac{1}{6}X_{\hat{H}}^{3}(\hat{\mu}_{1})+\frac{1}{24}X_{\hat{H}}^{4}(\hat{\phi}_{1})$,
which gives $\hat{\phi}_{2}^{H}$. This finishes the proof. ∎

Now we are ready to give the Maurer-Cartan characterizations of twisting of
twilled Lie triple systems.

######  Theorem 4.6.

Let $((\mathcal{G},\Theta),\mathfrak{g}_{1},\mathfrak{g}_{2})$ be a twilled
Lie triple system, and $H\in C^{0}(\mathfrak{g}_{2},\mathfrak{g}_{1})$ a
linear map. Then the twisting
$((\mathcal{G},\Theta^{H}),\mathfrak{g}_{1},\mathfrak{g}_{2})$ is also a
twilled Lie triple system if and only if $H$ is a Maurer-Cartan element of the
$L_{\infty}$-algebra
$(C^{*}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1}),l_{1},l_{2},l_{3})$
constructed in Theorem 3.17, i.e., $H$ satisfies the following Maurer-Cartan
equation

$l_{1}(H)+\frac{1}{2}l_{2}(H,H)+\frac{1}{3!}l_{3}(H,H,H)=0.$

###### Proof.

Write
$\Theta=\hat{\phi}_{1}+\hat{\mu}_{1}+\hat{\psi}+\hat{\mu}_{2}+\hat{\phi}_{2}$,
since $(\mathcal{G},\Theta)$ is a twilled Lie triple system, then we have
$\phi_{1}=\phi_{2}=0$. By Theorem 4.5, we have

$\displaystyle\left\\{\begin{array}[]{rcl}\hat{\mu}_{1}^{H}&=&\hat{\mu}_{1},\\\
\hat{\psi}^{H}&=&\hat{\psi}+X_{\hat{H}}(\hat{\mu}_{1}),\\\
\hat{\mu}_{2}^{H}&=&\hat{\mu}_{2}+X_{\hat{H}}(\hat{\psi})+\frac{1}{2}X_{\hat{H}}^{2}(\hat{\mu}_{1}),\\\
\hat{\phi}_{2}^{H}&=&X_{\hat{H}}(\hat{\mu}_{2})+\frac{1}{2}X_{\hat{H}}^{2}(\hat{\psi})+\frac{1}{6}X_{\hat{H}}^{3}(\hat{\mu}_{1}).\end{array}\right.$

Thus the twisting
$((\mathcal{G},\Theta^{H}),\mathfrak{g}_{1},\mathfrak{g}_{2})$ is also a
twilled Lie triple system if and only if $\hat{\phi}_{2}^{H}=0$, which implies
that $H$ is a Maurer-Cartan element of the $L_{\infty}$-algebra
$(C^{*}_{\mathsf{LTS}}(\mathfrak{g}_{2},\mathfrak{g}_{1}),l_{1},l_{2},l_{3})$.
∎

The twisting of twilled Lie triple systems gives rise to a Lie triple system
structure on its decomposed subspace.

######  Proposition 4.7.

Let $((\mathcal{G},\Theta),\mathfrak{g}_{1},\mathfrak{g}_{2})$ and its
twisting $((\mathcal{G},\Theta^{H}),\mathfrak{g}_{1},\mathfrak{g}_{2})$ both
be twilled Lie triple systems. Then for all $u,v,w\in\mathfrak{g}_{2},$

(27) $\displaystyle\left\llbracket u,v,w\right\rrbracket_{H}$
$\displaystyle:=$ $\displaystyle\left\llbracket
u,v,w\right\rrbracket_{2}+\left\llbracket
H(u),v,w\right\rrbracket_{2}+\left\llbracket
u,H(v),w\right\rrbracket_{2}+\left\llbracket u,v,H(w)\right\rrbracket_{2}$
$\displaystyle+\left\llbracket H(u),H(v),w\right\rrbracket_{2}+\left\llbracket
u,H(v),H(w)\right\rrbracket_{2}+\left\llbracket
H(u),v,H(w)\right\rrbracket_{2},$

defines a Lie triple system structure on $\mathfrak{g}_{2}$.

###### Proof.

Since $((\mathcal{G},\Theta^{H}),\mathfrak{g}_{1},\mathfrak{g}_{2})$ is a
twilled Lie triple system, by Proposition 3.14, we deduce that
$\hat{\mu}_{2}^{H}$ is a Lie triple system structure on $\mathcal{G}$.
Moreover, for all $u,v,w\in\mathfrak{g}_{2}$, by Eqs. (6) and (13), we compute
that

$\displaystyle\hat{\mu}_{2}(u,v,w)$ $\displaystyle=$
$\displaystyle\left\llbracket u,v,w\right\rrbracket_{2},$
$\displaystyle~{}X_{\hat{H}}(\hat{\psi})(u,v,w)$ $\displaystyle=$
$\displaystyle[\hat{\psi},\hat{H}]_{\mathsf{LTS}}(u,v,w)=\Big{(}\hat{\psi}\circ\hat{H}-\hat{H}\circ\hat{\psi}\Big{)}(u,v,w)$
$\displaystyle=$
$\displaystyle\hat{\psi}(H(u),v,w)+\hat{\psi}(u,H(v),w)+\hat{\psi}(u,v,H(w))$
$\displaystyle=$ $\displaystyle\left\llbracket
H(u),v,w\right\rrbracket_{2}+\left\llbracket
u,H(v),w\right\rrbracket_{2}+\left\llbracket u,v,H(w)\right\rrbracket_{2},$
$\displaystyle~{}\frac{1}{2}X_{\hat{H}}^{2}(\hat{\mu}_{1})(u,v,w)$
$\displaystyle=$
$\displaystyle\frac{1}{2}[[\hat{\mu}_{1},\hat{H}]_{\mathsf{LTS}},\hat{H}]_{\mathsf{LTS}}(u,v,w)$
$\displaystyle=$ $\displaystyle\left\llbracket
H(u),H(v),w\right\rrbracket_{2}+\left\llbracket
H(u),v,H(w)\right\rrbracket_{2}+\left\llbracket
u,H(v),H(w)\right\rrbracket_{2}.$

Adding these terms yields Eq. (27). This completes the proof. ∎

##  5\. Matched pairs and relative Rota-Baxter operators

In this section, we examine the relationship between twilled Lie triple
systems and matched pairs and relationship between twilled Lie triple systems
and relative Rota-Baxter operators respectively in order to obtain the
relationship between matched pairs of Lie triple systems and relative Rota-
Baxter operators. This work help us achieve bialgebra theory for Lie triple
systems.

###  5.1. Matched pairs and twilled Lie triple systems

The notion of matched pairs of Lie-Yamaguti algebras was introduced in [28],
and here we restrict the Lie-Yamaguti algebras to the context of Lie triple
systems to give the definition of matched pairs of Lie triple systems.

######  Definition 5.1.

A matched pair of Lie triple systems consists of two Lie triple systems
$(\mathfrak{g}_{1},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\mathfrak{g}_{1}})$
and
$(\mathfrak{g}_{2},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\mathfrak{g}_{2}})$,
and two linear maps
$\rho_{1}:\otimes^{2}\mathfrak{g}_{1}\longrightarrow\mathfrak{gl}(\mathfrak{g}_{2})$
and
$\rho_{2}:\otimes^{2}\mathfrak{g}_{2}\longrightarrow\mathfrak{gl}(\mathfrak{g}_{1})$,
such that the following conditions are satisfied:

  * (i)

$(\mathfrak{g}_{2};\rho_{1})$ is a representation of $\mathfrak{g}_{1}$;

  * (ii)

$(\mathfrak{g}_{1};\rho_{2})$ is a representation of $\mathfrak{g}_{2}$;

  * (iii)

For all $x,y,z\in\mathfrak{g}_{1}$ and $u,v,w\in\mathfrak{g}_{2}$, the
following equalities hold:

$\displaystyle~{}\rho_{2}(u,v)\left\llbracket
x,y,z\right\rrbracket_{\mathfrak{g}_{1}}$ $\displaystyle=$
$\displaystyle\left\llbracket
x,y,\rho_{2}(u,v)z\right\rrbracket_{\mathfrak{g}_{1}}-\rho_{2}(D_{1}(x,y)u,v)z-\rho_{2}(u,D_{1}(x,y)v)z,$
$\displaystyle~{}\left\llbracket
x,y,\rho_{2}(u,v)z\right\rrbracket_{\mathfrak{g}_{1}}$ $\displaystyle=$
$\displaystyle\rho_{2}(u,D_{1}(x,y)v)z+\rho_{2}(\rho_{1}(z,y)u,v)x-\rho_{2}(\rho_{1}(z,x)u,v)y,$
$\displaystyle~{}\left\llbracket\rho_{2}(u,v)x,y,z\right\rrbracket_{\mathfrak{g}_{1}}$
$\displaystyle=$
$\displaystyle\rho_{2}(u,\rho_{1}(y,z)v)x+D_{2}(v,\rho_{1}(x,y)u)z-\rho_{2}(v,\rho_{1}(x,z)u)y,$
$\displaystyle~{}\rho_{1}(x,y)\left\llbracket
u,v,w\right\rrbracket_{\mathfrak{g}_{2}}$ $\displaystyle=$
$\displaystyle\left\llbracket
u,v,\rho_{1}(x,y)w\right\rrbracket_{\mathfrak{g}_{2}}-\rho_{1}(D_{2}(u,v)x,y)w-\rho_{1}(x,D_{2}(u,v)y)w$
$\displaystyle~{}\left\llbracket
u,v,\rho_{1}(x,y)w\right\rrbracket_{\mathfrak{g}_{2}}$ $\displaystyle=$
$\displaystyle\rho_{1}(x,D_{2}(u,v)y)w+\rho_{1}(\rho_{2}(w,v)x,y)u-\rho_{1}(\rho_{2}(w,u)x,y)v,$
$\displaystyle~{}\left\llbracket\rho_{1}(x,y)u,v,w\right\rrbracket_{\mathfrak{g}_{2}}$
$\displaystyle=$
$\displaystyle\rho_{1}(x,\rho_{2}(v,w)y)u+D_{1}(y,\rho_{2}(u,v)x)y-\rho_{1}(y,\rho_{2}(u,w)x)v,$

where $D_{1}(x,y)=D_{\rho_{1}}(x,y)=\rho_{1}(y,x)-\rho_{1}(x,y)$ and
$D_{2}(u,v)=D_{\rho_{2}}(u,v)=\rho_{2}(v,u)-\rho_{2}(u,v)$. We denote a
matched pair of Lie triple systems by a quadruple
$(\mathfrak{g}_{1},\mathfrak{g}_{2};\rho_{1},\rho_{2})$.

The following proposition demonstrates that a matched pair of Lie triple
systems gives rise to a Lie triple systems structure on its direct sum.

######  Proposition 5.2.

Let $(\mathfrak{g}_{1},\mathfrak{g}_{2};\rho_{1},\rho_{2})$ be a matched pair
of Lie triple systems, where
$\rho_{1}:\otimes^{2}\mathfrak{g}_{1}\longrightarrow\mathfrak{gl}(\mathfrak{g}_{2})$
and
$\rho_{2}:\otimes^{2}\mathfrak{g}_{2}\longrightarrow\mathfrak{gl}(\mathfrak{g}_{1})$.
Define an operation
$\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\bowtie}$ on
$\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$ to be

(28) $\displaystyle\left\llbracket x+u,y+v,z+w\right\rrbracket_{\bowtie}$
$\displaystyle=$ $\displaystyle\left\llbracket
x,y,z\right\rrbracket_{\mathfrak{g}_{1}}+D_{2}(u,v)z+\rho_{2}(v,w)x-\rho_{2}(u,w)y$
$\displaystyle+\left\llbracket
u,v,w\right\rrbracket_{\mathfrak{g}_{2}}+D_{1}(x,y)w+\rho_{1}(y,z)u-\rho_{1}(x,z)v.$

Then
$(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\bowtie})$
is a Lie triple system, which is called the double of $\mathfrak{g}_{1}$ and
$\mathfrak{g}_{2}$.

###### Proof.

Let the quadruple $(\mathfrak{g}_{1},\mathfrak{g}_{2};\rho_{1},\rho_{2})$ be a
matched pair of Lie triple systems, then we have the following equations:

$\displaystyle\left\\{\begin{array}[]{rcl}[\hat{\pi}_{1},\hat{\rho}_{1}]_{\mathsf{LTS}}+\frac{1}{2}[\hat{\rho}_{1},\hat{\rho}_{1}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\pi}_{2},\hat{\rho}_{2}]_{\mathsf{LTS}}+\frac{1}{2}[\hat{\rho}_{2},\hat{\rho}_{2}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\pi}_{2},\hat{\rho}_{1}]_{\mathsf{LTS}}+[\hat{\rho}_{2},\hat{\pi}_{1}+\hat{\rho}_{1}]_{\mathsf{LTS}}&=&0.\end{array}\right.$

Here, $\hat{\pi}_{1},~{}\hat{\rho}_{1},~{}\hat{\pi}_{2},~{}\hat{\rho}_{2}\in
C^{1}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$ are given by

$\displaystyle\hat{\pi}_{1}((x,u),(y,v),(z,w))$ $\displaystyle=$
$\displaystyle\left\llbracket x,y,z\right\rrbracket_{\mathfrak{g}_{1}},$
$\displaystyle\hat{\rho}_{1}((x,u),(y,v),(z,w))$ $\displaystyle=$
$\displaystyle D_{1}(x,y)w+\rho_{1}(y,z)u-\rho_{1}(x,z)v,$
$\displaystyle~{}\hat{\pi}_{2}((x,u),(y,v),(z,w))$ $\displaystyle=$
$\displaystyle\left\llbracket u,v,w\right\rrbracket_{\mathfrak{g}_{2}},$
$\displaystyle\hat{\rho}_{2}((x,u),(y,v),(z,w))$ $\displaystyle=$
$\displaystyle D_{2}(u,v)z+\rho_{2}(v,w)x-\rho_{2}(u,w)y,\quad\forall
x,y,z\in\mathfrak{g}_{1},~{}u,v,w\in\mathfrak{g}_{2},$

respectively. It is straightforward to see that the equation

$[\hat{\pi}_{1},\hat{\rho}_{1}]_{\mathsf{LTS}}+\frac{1}{2}[\hat{\rho}_{1},\hat{\rho}_{1}]_{\mathsf{LTS}}=0~{}~{}(\text{resp.}~{}~{}[\hat{\pi}_{2},\hat{\rho}_{2}]_{\mathsf{LTS}}+\frac{1}{2}[\hat{\rho}_{2},\hat{\rho}_{2}]_{\mathsf{LTS}}=0)$

is equivalent to $(\mathfrak{g}_{2};\rho_{1})$ (resp.
$(\mathfrak{g}_{1};\rho_{2})$) is a representation of $\mathfrak{g}_{1}$
(resp. $\mathfrak{g}_{2}$), and that the equation

$[\hat{\pi}_{1},\hat{\rho}_{2}]_{\mathsf{LTS}}+[\hat{\rho}_{1},\hat{\pi}_{2}+\hat{\rho}_{2}]_{\mathsf{LTS}}=0$

is equivalent to Condition (iii). Since $\mathfrak{g}_{1}$ and
$\mathfrak{g}_{2}$ are Lie triple systems, i.e.,

$[\pi_{1},\pi_{1}]_{\mathsf{LTS}}=[\pi_{2},\pi_{2}]_{\mathsf{LTS}}=0,$

then we have

$[\hat{\pi}_{1}+\hat{\rho}_{1}+\hat{\pi}_{2}+\hat{\rho}_{2},\hat{\pi}_{1}+\hat{\rho}_{1}+\hat{\pi}_{2}+\hat{\rho}_{2}]_{\mathsf{LTS}}=0,$

which implies that
$(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\bowtie})$
is a Lie triple system. This finishes the proof. ∎

To demonstrate the relationship between matched pairs and twilled Lie triple
systems, we introduce the notion of strict twilled Lie tripe systems as
follows.

######  Definition 5.3.

Let $(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$ be a twilled Lie triple
system equipped with the structure
$\Theta=\hat{\mu}_{1}+\hat{\psi}+\hat{\mu}_{2}$. Then
$(\mathcal{G},\mathfrak{g}_{1},\mathfrak{g}_{2})$ is called strict if
$\psi=0$, or equivalently, the following conditions are satisfied:

$\displaystyle\left\\{\begin{array}[]{rcl}~{}\frac{1}{2}[\hat{\mu}_{1},\hat{\mu}_{1}]_{\mathsf{LTS}}&=&0,\\\
~{}[\hat{\mu}_{1},\hat{\mu}_{2}]_{\mathsf{LTS}}&=&0,\\\
~{}\frac{1}{2}[\hat{\mu}_{2},\hat{\mu}_{2}]_{\mathsf{LTS}}&=&0.\end{array}\right.$

######  Theorem 5.4.

There is a one-to-one correspondence between matched pairs and strict twilled
Lie triple systems.

###### Proof.

Let the quadruple $(\mathfrak{g}_{1},\mathfrak{g}_{2};\rho_{1},\rho_{2})$ be a
matched pair of Lie triple systems, then by Proposition 5.2, the double
operation $\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\bowtie}$ is a
Lie triple system structure on the direct sum of vector space
$\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$. The letter is equivalent to that
$\hat{\mu}_{1}+\hat{\mu}_{2}\in C^{1}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$
is a Maurer-Cartan element of the graded Lie algebra
$(C^{*}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G}),[\cdot,\cdot]_{\mathsf{LTS}})$,
i.e.,

$[\hat{\mu}_{1}+\hat{\mu}_{2},\hat{\mu}_{1}+\hat{\mu}_{2}]_{\mathsf{LTS}}=0,$

where $\hat{\mu}_{1}$ and $\hat{\mu}_{2}$ are given by

$\displaystyle\hat{\mu}_{1}(x+u,y+v,z+w)$ $\displaystyle=$
$\displaystyle\Big{(}\left\llbracket
x,y,z\right\rrbracket_{\mathfrak{g}_{1}},D_{1}(x,y)w+\rho_{1}(y,z)u-\rho_{1}(x,z)v\Big{)},$
$\displaystyle\hat{\mu}_{2}(x+u,y+v,z+w)$ $\displaystyle=$
$\displaystyle\Big{(}D_{2}(u,v)z+\rho_{2}(v,w)x-\rho_{2}(u,w)y,\left\llbracket
u,v,w\right\rrbracket_{\mathfrak{g}_{2}}\Big{)},$

for all $x,y,z\in\mathfrak{g}_{1},~{}u,v,w\in\mathfrak{g}_{2}$. By the
convention in (13), define

$\rho_{1}(x,y)u:=\left\llbracket
u,x,y\right\rrbracket_{2},\quad\rho_{2}(u,v)x:=\left\llbracket
x,u,v\right\rrbracket_{1}.$

Consequently we obtain

$D_{1}(x,y)u=\rho_{1}(y,x)u-\rho_{1}(x,y)u=\left\llbracket
x,y,u\right\rrbracket_{2},$

and similarly

$D_{2}(u,v)x=\rho_{2}(v,u)x-\rho_{2}(u,v)x=\left\llbracket
u,v,x\right\rrbracket_{1}.$

Then the structure $\Theta=\hat{\mu}_{1}+\hat{\mu}_{2}$, which demonstrates
that $\mathcal{G}=\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$ is a strict twilled
Lie triple system. This finishes the proof. ∎

It is natural to ask what object a usual twilled Lie triple system corresponds
to. To answer this question, we shall introduce the notion of generalized
matched pairs of Lie triple systems. Before this, we give the definition of
generalized representation of Lie triple systems first.

For a linear map
$\tau:\mathfrak{g}\longrightarrow\mathrm{Hom}(\otimes^{2}V,V)$, there induces
a linear map
$\mathcal{D}_{\tau}:\mathfrak{g}\longrightarrow\mathrm{Hom}(\wedge^{2}V,V)$
defined to be

(31)
$\displaystyle\mathcal{D}_{\tau}(x)(u,v)=\tau(x)(v,u)-\tau(x)(u,v),\quad\forall
x\in\mathfrak{g},~{}u,v\in V.$

We denote $\mathcal{D}_{\tau}$ by $\mathcal{D}$ without ambiguities.

######  Definition 5.5.

Let $(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$ be a Lie
triple system and $V$ a vector space. A generalized representation of
$\mathfrak{g}$ on $V$ consists of linear maps
$\rho:\otimes^{2}\mathfrak{g}\longrightarrow\mathfrak{gl}(V)$ and
$\tau:\mathfrak{g}\longrightarrow\mathrm{Hom}(\otimes^{2}V,V)$, such that
$\hat{\pi}+\hat{\rho}+\hat{\tau}\in C^{1}_{\mathsf{LTS}}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V)$ is a Maurer-Cartan element of
$(C^{*}_{\mathsf{LTS}}(\mathfrak{g}\oplus V,\mathfrak{g}\oplus
V),[\cdot,\cdot]_{\mathsf{LTS}})$, i.e.,

$[\hat{\pi}+\hat{\rho}+\hat{\tau},\hat{\pi}+\hat{\rho}+\hat{\tau}]_{\mathsf{LTS}}=0.$

Here $\hat{\pi},~{}\hat{\rho},~{}\hat{\tau}$ are given by

$\displaystyle\hat{\pi}(x+u,y+v,z+w)$ $\displaystyle=$
$\displaystyle\left\llbracket x,y,z\right\rrbracket,$
$\displaystyle\hat{\rho}(x+u,y+v,z+w)$ $\displaystyle=$ $\displaystyle
D(x,y)w+\rho(y,z)u-\rho(x,z)v,$ $\displaystyle\hat{\tau}(x+u,y+v,z+w)$
$\displaystyle=$
$\displaystyle\mathcal{D}(z)(u,v)+\tau(x)(v,w)-\tau(y)(u,w),\quad\forall
x,y,z\in\mathfrak{g},~{}~{}u,v,w\in V,$

respectively, where $\mathcal{D}$ is defined by (31). We denote a generalized
representation of a Lie triple system $\mathfrak{g}$ by a pair
$(V;(\rho,\tau))$.

######  Example 5.6.

Let $(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$ be a Lie
triple system. Define
$\mathcal{R}:\mathfrak{g}\longrightarrow\mathrm{Hom}(\otimes^{2}\mathfrak{g},\mathfrak{g})$
to be

$\mathcal{R}(x)(y,z):=\left\llbracket x,y,z\right\rrbracket,\forall
x,y,z\in\mathfrak{g}.$

Then $(\mathfrak{g};(\mathfrak{R},\mathcal{R}))$ is a generalized
representation of $\mathfrak{g}$ on itself, where
$(\mathfrak{g};\mathfrak{R})$ is the adjoint representation of $\mathfrak{g}$.
Obviously,
$\mathcal{D}_{\mathcal{R}}=\mathcal{L}:\mathfrak{g}\longrightarrow\mathrm{Hom}(\wedge^{2}\mathfrak{g},\mathfrak{g})$
is given by

$\mathcal{L}(x)(y,z)=\left\llbracket y,z,x\right\rrbracket,\quad\forall
x,y,z\in\mathfrak{g}.$

Generalized representation can be characterized by the notion of generalized
semidirect product Lie triple systems.

######  Proposition 5.7.

Let $(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$ be a Lie
triple system, $V$ a vector space, and
$\rho:\otimes^{2}\mathfrak{g}\longrightarrow\mathfrak{gl}(V)$ and
$\tau:\mathfrak{g}\longrightarrow\mathrm{Hom}(\otimes^{2}V,V)$ be linear maps.
Then $(V;(\rho,\tau))$ is a generalized representation of $\mathfrak{g}$ if
and only if there exists a Lie triple system structure
$\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{(\rho,\tau)}$ on the direct
sum $\mathfrak{g}\oplus V$ defined to be for all $x,y,z\in\mathfrak{g}$ and
$u,v,w\in V$,

$\displaystyle\left\llbracket x+u,y+v,z+w\right\rrbracket_{(\rho,\tau)}$
$\displaystyle=$ $\displaystyle\left\llbracket
x,y,z\right\rrbracket+D(x,y)w+\rho(y,z)u-\rho(x,z)v$
$\displaystyle+\mathcal{D}(z)(u,v)+\tau(x)(v,w)-\tau(y)(u,w).$

The Lie triple system
$(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{(\rho,\tau)})$
is called the generalized product Lie triple system.

###### Proof.

It follows from

$\left\llbracket
x+u,y+v,z+w\right\rrbracket_{(\rho,\tau)}=(\hat{\pi}+\hat{\rho}+\hat{\tau})(x+u,y+v,z+w)$

and involves a direct computation. ∎

######  Remark 5.8.

The notion of generalized representation of Lie triple systems given, it is
natural to establish a new cohomology for Lie triple systems and then
deformations and extensions can be explored consequently. We will examine this
problem in the future and we are also looking forward to new studies in this
direction.

Next, we introduce the notion of generalized matched pairs of Lie triple
systems in the sequel.

######  Definition 5.9.

A generalized matched pair of Lie triple systems consists of two Lie triple
systems
$(\mathfrak{g}_{1},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\mathfrak{g}_{1}})$
and
$(\mathfrak{g}_{2},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\mathfrak{g}_{2}})$,
and two pairs of linear maps
$\rho_{1}:\otimes^{2}\mathfrak{g}_{1}\longrightarrow\mathfrak{gl}(\mathfrak{g}_{2}),~{}\tau_{1}:\mathfrak{g}_{1}\longrightarrow\mathrm{Hom}(\otimes^{2}\mathfrak{g}_{2},\mathfrak{g}_{2})$
and
$\rho_{2}:\otimes^{2}\mathfrak{g}_{2}\longrightarrow\mathfrak{gl}(\mathfrak{g}_{1}),~{}\tau_{2}:\mathfrak{g}_{2}\longrightarrow\mathrm{Hom}(\otimes^{2}\mathfrak{g}_{1},\mathfrak{g}_{1})$,
such that the following equalities hold:

(32)
$\displaystyle[\hat{\pi}_{1},\hat{\rho}_{1}+\hat{\tau}_{1}]_{\mathsf{LTS}}+\frac{1}{2}[\hat{\rho}_{1}+\hat{\tau}_{1},\hat{\rho}_{1}+\hat{\tau}_{1}]_{\mathsf{LTS}}$
$\displaystyle=$ $\displaystyle 0,$ (33)
$\displaystyle~{}[\hat{\pi}_{2},\hat{\rho}_{2}+\hat{\tau}_{2}]_{\mathsf{LTS}}+\frac{1}{2}[\hat{\rho}_{2}+\hat{\tau}_{2},\hat{\rho}_{2}+\hat{\tau}_{2}]_{\mathsf{LTS}}$
$\displaystyle=$ $\displaystyle 0,$ (34)
$\displaystyle~{}[\hat{\pi}_{1},\hat{\rho}_{2}+\hat{\tau}_{2}]_{\mathsf{LTS}}+[\hat{\rho}_{1}+\hat{\tau}_{1},\hat{\pi}_{2}]_{\mathsf{LTS}}+[\hat{\rho}_{1}+\hat{\tau}_{1},\hat{\rho}_{2}+\hat{\tau}_{2}]_{\mathsf{LTS}}$
$\displaystyle=$ $\displaystyle 0,$

where

$\displaystyle\hat{\pi}_{1}((x,u),(y,v),(z,w))$ $\displaystyle=$
$\displaystyle\left\llbracket
x,y,z\right\rrbracket_{\mathfrak{g}_{1}},\quad\hat{\rho}_{1}((x,u),(y,v),(z,w))=D_{1}(x,y)w+\rho_{1}(y,z)u-\rho_{1}(x,z)v,$
$\displaystyle\hat{\tau}_{1}((x,u),(y,v),(z,w))$ $\displaystyle=$
$\displaystyle\mathcal{D}_{1}(z)(u,v)+\tau_{1}(x)(v,w)-\tau_{1}(y)(u,w),$
$\displaystyle\hat{\pi}_{2}((x,u),(y,v),(z,w))$ $\displaystyle=$
$\displaystyle\left\llbracket
u,v,w\right\rrbracket_{\mathfrak{g}_{2}},\quad\hat{\rho}_{2}((x,u),(y,v),(z,w))=D_{2}(u,v)z+\rho_{2}(v,w)x-\rho_{2}(u,w)y,$
$\displaystyle\hat{\tau}_{2}((x,u),(y,v),(z,w))$ $\displaystyle=$
$\displaystyle\mathcal{D}_{2}(w)(x,y)+\tau_{2}(u)(y,z)-\tau(v)(x,z),\quad\forall
x,y,z\in\mathfrak{g}_{1},~{}u,v,w\in\mathfrak{g}_{2}.$

We denote a generalized matched pair of Lie triple systems by a quadruple
$\Big{(}\mathfrak{g}_{1},\mathfrak{g}_{2};(\rho_{1},\tau_{1}),(\rho_{2},\tau_{2})\Big{)}$
.

If the quadruple
$\Big{(}\mathfrak{g}_{1},\mathfrak{g}_{2};(\rho_{1},\tau_{1}),(\rho_{2},\tau_{2})\Big{)}$
is a generalized matched pair of Lie triple systems, then it is
straightforward to see that Eq. (32) (resp. Eq. (33)) means that
$(\mathfrak{g}_{2};(\rho_{1},\tau_{1}))$ (resp.
$(\mathfrak{g}_{1};(\rho_{2},\tau_{2}))$) is a generalized representation of
$\mathfrak{g}_{2}$ (resp. $\mathfrak{g}_{1}$), and Eq. (34) means certain
compatibility conditions.

######  Proposition 5.10.

Suppose that the quadruple
$\Big{(}\mathfrak{g}_{1},\mathfrak{g}_{2};(\rho_{1},\tau_{1}),(\rho_{2},\tau_{2})\Big{)}$
is a generalized matched pair of Lie triple systems. For all
$x,y,z\in\mathfrak{g}_{1}$ and $u,v,w\in\mathfrak{g}_{2}$, define a new
operation
$\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}}$
on the direct sum $\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$ to be

$\displaystyle\left\llbracket
x+u,y+v,z+w\right\rrbracket_{\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}}$
$\displaystyle=$ $\displaystyle\left\llbracket
x,y,z\right\rrbracket_{\mathfrak{g}_{1}}+D_{2}(u,v)z+\rho_{2}(v,w)x-\rho_{2}(u,w)y+\mathcal{D}_{2}(w)(x,y)+\tau_{2}(u)(y,z)-\tau_{2}(v)(x,z)$
(35) $\displaystyle+\left\llbracket
u,v,w\right\rrbracket_{\mathfrak{g}_{2}}+D_{1}(x,y)w+\rho_{1}(y,z)u-\rho_{1}(x,z)v+\mathcal{D}_{1}(z)(u,v)+\tau_{1}(x)(v,w)-\tau_{1}(y)(u,w),$

then
$(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}})$
is a Lie triple system.

###### Proof.

It follows from

$\left\llbracket
x+u,y+v,z+w\right\rrbracket_{\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}}=[\hat{\pi}_{1}+\hat{\rho}_{1}+\hat{\tau}_{1}+\hat{\pi}_{2}+\hat{\rho}_{2}+\hat{\tau}_{2},\hat{\pi}_{1}+\hat{\rho}_{1}+\hat{\tau}_{1}+\hat{\pi}_{2}+\hat{\rho}_{2}+\hat{\tau}_{2}]_{\mathsf{LYS}}(x+u,y+v,z+w).$

This completes the proof. ∎

Now, we are ready to give our another key conclusion in this section.

######  Theorem 5.11.

There is a one-to-one correspondence between generalized matched pairs and
twilled Lie triple systems.

###### Proof.

The quadruple
$(\mathfrak{g}_{1},\mathfrak{g}_{2};(\rho_{1},\tau_{1}),(\rho_{2},\tau_{2}))$
is a generalized matched pair of Lie triple systems, then the operation
$\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}}$
is a Lie triple system on the direct sum of vector space
$\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$. The letter is equivalent to that
$\hat{\mu}_{1}+\hat{\psi}+\hat{\mu}_{2}\in
C^{1}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G})$ is a Maurer-Cartan element of
the graded Lie algebra
$(C^{*}_{\mathsf{LTS}}(\mathcal{G},\mathcal{G}),[\cdot,\cdot]_{\mathsf{LTS}})$,
i.e.,

$[\hat{\mu}_{1}+\hat{\psi}+\hat{\mu}_{2},\hat{\mu}_{1}+\hat{\psi}+\hat{\mu}_{2}]_{\mathsf{LTS}}=0,$

where $\hat{\mu}_{1},\hat{\psi},\hat{\mu}_{2}$ are given by

$\displaystyle\hat{\mu}_{1}(x+u,y+v,z+w)$ $\displaystyle=$
$\displaystyle\Big{(}\left\llbracket
x,y,z\right\rrbracket_{\mathfrak{g}_{1}},D_{1}(x,y)w+\rho_{1}(y,z)u-\rho_{1}(x,z)v\Big{)},$
$\displaystyle\hat{\psi}(x+u,y+v,z+w)$ $\displaystyle=$
$\displaystyle\Big{(}\mathcal{D}_{2}(w)(x,y)+\tau_{2}(u)(x,y)-\tau_{2}(v)(x,z),\mathcal{D}_{1}(z)(u,v)+\tau_{1}(x)(v,w)-\tau_{1}(y)(u,w)\Big{)},$
$\displaystyle\hat{\mu}_{2}(x+u,y+v,z+w)$ $\displaystyle=$
$\displaystyle\Big{(}D_{2}(u,v)z+\rho_{2}(v,w)x-\rho_{2}(u,w)y,\left\llbracket
u,v,w\right\rrbracket_{\mathfrak{g}_{2}}\Big{)},$

for all $x,y,z\in\mathfrak{g}_{1},~{}u,v,w\in\mathfrak{g}_{2}$. By the
convention in (13), define

$\displaystyle\rho_{1}(x,y)u:=\left\llbracket
u,x,y\right\rrbracket_{2},\quad\rho_{2}(u,v)x:=\left\llbracket
x,u,v\right\rrbracket_{1},$ $\displaystyle\tau_{1}(x)(u,v):=\left\llbracket
x,u,v\right\rrbracket_{2},\quad\tau_{2}(u)(x,y):=\left\llbracket
u,x,y\right\rrbracket_{1}.$

Consequently we obtain

$\mathcal{D}_{1}(x)(u,v)=\left\llbracket
u,v,x\right\rrbracket_{2},\quad\mathcal{D}_{2}(u)(x,y)=\left\llbracket
x,y,u\right\rrbracket_{1}.$

Then the structure $\Theta=\hat{\mu}_{1}+\hat{\psi}+\hat{\mu}_{2}$, which
demonstrates that $\mathcal{G}=\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$ is a
twilled Lie triple system. This finishes the proof. ∎

######  Remark 5.12.

If we constructed the dual representation of a generalized representation,
then we could establish the generalized bialgebra of Lie triple systems and
even, the generalized Lie-Yamaguti bialgebra theory. We will examine this
projection in the future.

###  5.2. Relative Rota-Baxter operators and twilled Lie triple systems

In this subsection, we construct a twilled Lie triple system via a relative
Rota-Baxter operator and give some examples to end up with this section.

######  Definition 5.13.

Let $(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$ be a Lie
triple system and $(V;\rho)$ a representation of $\mathfrak{g}$. Then a linear
map $T:V\longrightarrow\mathfrak{g}$ is called a relative Rota-Baxter operator
on $\mathfrak{g}$ with respect to $(V;\rho)$ if $T$ satisfies

$\displaystyle\left\llbracket
Tu,Tv,Tw\right\rrbracket=T\Big{(}D(Tu,Tv)w+\rho(Tv,Tw)u-\rho(Tu,Tw)v\Big{)},\quad\forall
u,v,w\in V.$

Besides, a relative Rota-Baxter operator
$T:\mathfrak{g}\longrightarrow\mathfrak{g}$ with respect to the adjoint
representation $(\mathfrak{g};\mathfrak{R})$ is called a Rota-Baxter operator
on $\mathfrak{g}$, i.e., $T$ satisfies

$\left\llbracket Tx,Ty,Tz\right\rrbracket=T\Big{(}\left\llbracket
Tx,Ty,z\right\rrbracket+\left\llbracket
x,Ty,Tz\right\rrbracket+\left\llbracket
Tx,y,Tz\right\rrbracket\Big{)},\quad\forall x,y,z\in\mathfrak{g}.$

In [6], relative Rota-Baxter operators are realized as Maurer-Cartan elements
in a suitable $L_{\infty}$-algebra $(C^{*}(V,\mathfrak{g}),l_{3})$ (This
$L_{\infty}$-algebra was called the Lie 3-algebra in the literature). When the
(strict) twilled Lie triple system is just the semidirect product
$((\mathfrak{g}\ltimes V,\Theta=\hat{\mu}_{1}),\mathfrak{g},V)$, the
$L_{\infty}$-algebra constructed in Theorem 3.17 is just the controlling
algebra of relative Rota-Baxter operators. Thus based on Theorem 4.6, we have
the following result to relate relative Rota-Baxter operators with twilled Lie
triple systems.

######  Proposition 5.14.

Let $(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$ be a Lie
triple system, $(V;\rho)$ a representation of $\mathfrak{g}$, and
$T:V\longrightarrow\mathfrak{g}$ a linear map. Then $((\mathfrak{g}\oplus
V,\Theta^{T}),\mathfrak{g},V)$ is a twilled Lie triple system if and only if
$T$ is a relative Rota-Baxter operator.

###### Proof.

Since the semidirect product $((\mathfrak{g}\ltimes V,\Theta),\mathfrak{g},V)$
is a strict Lie triple system, by Theorem 4.5, the twisting $\Theta^{T}$ has
the structures as follows:

$\displaystyle\left\\{\begin{array}[]{rcl}\hat{\mu}_{1}^{T}&=&\hat{\mu}_{1},\\\
\hat{\psi}^{T}&=&X_{\hat{T}}(\hat{\mu}_{1}),\\\
\hat{\mu}_{2}^{T}&=&\frac{1}{2}X_{\hat{T}}^{2}(\hat{\mu}_{1}),\\\
\hat{\phi}_{2}^{T}&=&\frac{1}{6}X_{\hat{T}}^{3}(\hat{\mu}_{1}).\end{array}\right.$

Thus $((\mathfrak{g}\oplus V,\Theta^{T}),\mathfrak{g},V)$ is a twilled Lie
triple system if and only if $\hat{\phi}_{2}^{T}=0$, which is equivalent to
that $T$ is a relative Rota-Baxter operator. ∎

######  Corollary 5.15.

Let $T:V\longrightarrow\mathfrak{g}$ be a relative Rota-Baxter operator on a
Lie triple system
$(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$ with respect
to a representation $(V;\rho)$. Then $\hat{\mu}_{2}^{T}$ defines a Lie triple
system structure $\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{T}$ on $V$
and a representation
$\varrho:\otimes^{2}V\longrightarrow\mathfrak{gl}(\mathfrak{g})$ as follows:

$\displaystyle\left\llbracket u,v,w\right\rrbracket_{T}$ $\displaystyle=$
$\displaystyle D(Tu,Tv)w+\rho(Tv,Tw)u-\rho(Tu,Tw)v,$
$\displaystyle\varrho(u,v)x$ $\displaystyle=$ $\displaystyle\left\llbracket
x,Tu,Tv\right\rrbracket-T\Big{(}D(x,Tu)v-\rho(x,Tv)u\Big{)},\quad\forall
x\in\mathfrak{g},~{}u,v,w\in V.$

###### Proof.

By Proposition 5.14, $((\mathfrak{g}\oplus V,\Theta^{T}),\mathfrak{g},V)$ is a
twilled Lie triple system, which implies that $\hat{\mu}_{2}^{T}$ defines a
Lie triple system structure on $V$. Moreover, for all $x\in\mathfrak{g}$ and
$u,v,w\in V$, define

$\left\llbracket
u,v,w\right\rrbracket_{T}:=\hat{\mu}_{2}^{T}(u,v,w),\quad\varrho(u,v)x:=\hat{\mu}_{2}^{T}(u,v,x),$

which give a Lie triple system structure on $V$ and a representation of
$(V,\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{T})$ on $\mathfrak{g}$.
∎

At the end of this paper, we need a proposition to give some examples. Before
this, we introduce some notations. Let
$(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$ be a Lie
triple system, $(V;\rho)$ a representation and
$T:V\longrightarrow\mathfrak{g}$ a relative Rota-Baxter operator with respect
to $(V;\rho)$. Let $\Theta$ denote the operation of twilled Lie triple system
$(\mathfrak{g}\ltimes V,\mathfrak{g},V)$. For all $x,y\in\mathfrak{g}$ and
$u,v\in V$, define
$\tau:\mathfrak{g}\longrightarrow\mathrm{Hom}(\otimes^{2}V,V)$ and
$\sigma:V\longrightarrow\mathrm{Hom}(\otimes^{2}\mathfrak{g},\mathfrak{g})$ to
be

$\displaystyle\tau(x)(u,v)$ $\displaystyle=$
$\displaystyle\rho(Tv,x)u+\rho(x,Tu)v,$ $\displaystyle\sigma(u)(x,y)$
$\displaystyle=$ $\displaystyle\left\llbracket
Tu,x,y\right\rrbracket-T\Big{(}\rho(x,y)u\Big{)}$

respectively. Then we have the explicit formula of linear maps
$\mathcal{D}_{\tau}:\mathfrak{g}\longrightarrow\mathrm{Hom}(\wedge^{2}V,V)$
and
$\mathcal{D}_{\sigma}:V\longrightarrow\mathrm{Hom}(\wedge^{2}\mathfrak{g},\mathfrak{g})$
as follows:

$\displaystyle\mathcal{D}_{\tau}(x)(u,v)$ $\displaystyle=$ $\displaystyle
D(x,Tu)v+D(Tv,x)u,$ $\displaystyle\mathcal{D}_{\sigma}(u)(x,y)$
$\displaystyle=$ $\displaystyle\left\llbracket
x,y,Tu\right\rrbracket-T\Big{(}D(x,y)u\Big{)},\quad\forall
x,y\in\mathfrak{g},~{}u,v\in V.$

######  Proposition 5.16.

With the above notations, the twisting of $\Theta$ is given by

(37) $\displaystyle\Theta^{T}\Big{(}(x,u),(y,v),(z,w)\Big{)}$ $\displaystyle=$
$\displaystyle\left\llbracket
x,y,z\right\rrbracket+D_{\varrho}(u,v)z+\varrho(v,w)x-\varrho(u,w)y+\mathcal{D}_{\sigma}(w)(x,y)+\sigma(u)(y,z)-\sigma(v)(x,z)$
$\displaystyle+\left\llbracket
u,v,w\right\rrbracket_{T}+D_{\rho}(x,y)w+\rho(y,z)u-\rho(x,z)v+\mathcal{D}_{\tau}(z)(u,v)+\tau(x)(v,w)-\tau(y)(u,w).$

Note that although $((\mathfrak{g}\oplus V,\Theta^{T}),\mathfrak{g},V)$ is a
twilled Lie triple system, however nether is $(V;(\rho,\tau))$ a generalized
representation of $\mathfrak{g}$, nor $(\mathfrak{g};(\varrho,\sigma))$ is a
generalized representation of
$(V,\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{T})$ in general.
Consequently, the quadruple $(\mathfrak{g},V;(\rho,\tau),(\varrho,\sigma))$
does not form a generalized matched pair any more.

######  Example 5.17.

Let $(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$ be a
2-dimension Lie triple system with a basis $\\{e_{1},e_{2}\\}$. Define a
nonzero operation with respect to the basis $\\{e_{1},e_{2}\\}$ to be

$\left\llbracket e_{1},e_{2},e_{2}\right\rrbracket=e_{1}.$

Then

$\displaystyle T=\begin{pmatrix}0&a\\\ 0&b\end{pmatrix}$

is a Rota-Baxter operator on $\mathfrak{g}$. Then
$((\mathfrak{g}\oplus\mathfrak{g}_{T},\Theta^{T}),\mathfrak{g},\mathfrak{g}_{T})$
is a 4-dimensional Lie triple system with a basis
$\\{\bm{e}_{1},\bm{e}_{2},\bm{e}_{3},\bm{e}_{4}\\}$, where
$\bm{e}_{1}=(e_{1},0),~{}\bm{e}_{2}=(e_{2},0),\bm{e}_{3}=(0,e_{1}),~{}\bm{e}_{4}=(0,e_{2})$.
By Eq. (37), the twisting $\Theta^{T}$ is given by

$\displaystyle\Theta^{T}({\bm{e}}_{1},{\bm{e}}_{3},{\bm{e}}_{2})=-{\bm{e}}_{3},\quad\Theta^{T}(\bm{e}_{2},\bm{e}_{3},\bm{e}_{4})=b\bm{e}_{3},\quad\Theta^{T}(\bm{e}_{2},\bm{e}_{4},\bm{e}_{3})=b\bm{e}_{3},$
$\displaystyle\Theta^{T}(\bm{e}_{1},\bm{e}_{2},\bm{e}_{4})=b\bm{e}_{1}+\bm{e}_{3},\qquad\Theta^{T}(\bm{e}_{1},\bm{e}_{4},\bm{e}_{2})=b\bm{e}_{1}+\bm{e}_{3}.$

######  Example 5.18.

Let $(\mathfrak{g},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$ be a
4-dimension Lie triple system with a basis $\\{e_{1},e_{2},e_{3},e_{4}\\}$.
Define a nonzero operation with respect to the basis
$\\{e_{1},e_{2},e_{3},e_{4}\\}$ to be

$\left\llbracket e_{1},e_{2},e_{1}\right\rrbracket=e_{4}.$

Take

$\displaystyle T=\begin{pmatrix}0&1&0&0\\\ 0&0&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{pmatrix},$

then $T:\mathfrak{g}\longrightarrow\mathfrak{g}$ is a Rota-Baxter operator.
Consequently, the twisting of Lie triple system
$(\mathfrak{g}\oplus\mathfrak{g}_{T},\mathfrak{g},\mathfrak{g}_{T})$ has a
basis $\\{\bm{e}_{1},\bm{e}_{2}\cdots,\bm{e}_{8}\\}$, where
$\bm{e}_{i}=(e_{i},0),1\leqslant i\leqslant 4$ and
$\bm{e}_{j}=(0,e_{j-4}),5\leqslant j\leqslant 8$. Then the twisting
$\Theta^{T}$ is given by

$\displaystyle\Theta^{T}(\bm{e}_{1},\bm{e}_{2},\bm{e}_{1})$ $\displaystyle=$
$\displaystyle\bm{e}_{4},\quad\Theta^{T}(\bm{e}_{1},\bm{e}_{2},\bm{e}_{6})=\bm{e}_{4},\quad\Theta^{T}(\bm{e}_{2},\bm{e}_{6},\bm{e}_{1})=\bm{e}_{4},\quad\Theta^{T}(\bm{e}_{1},\bm{e}_{2},\bm{e}_{5})=\bm{e}_{6},$
$\displaystyle\Theta^{T}(\bm{e}_{2},\bm{e}_{5},\bm{e}_{1})$ $\displaystyle=$
$\displaystyle-\bm{e}_{6},\quad\Theta^{T}(\bm{e}_{2},\bm{e}_{6},\bm{e}_{5})=\bm{e}_{6},\quad\Theta^{T}(\bm{e}_{5},\bm{e}_{6},\bm{e}_{2})=\bm{e}_{6},\quad\Theta^{T}(\bm{e}_{1},\bm{e}_{6},\bm{e}_{1})=\bm{e}_{4}+\bm{e}_{6}.$

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