#  Maurer-Cartan type cohomology on generalized Reynolds operators and NS-
structures on Lie triple systems

R.Gharbi, S. Mabrouk, A. Makhlouf Rahma Gharbi1,3 111 E-mail:
<EMAIL_ADDRESS>, Sami Mabrouk2 222 E-mail<EMAIL_ADDRESS>and
Abdenacer Makhlouf1 333 E-mail<EMAIL_ADDRESS>$($Corresponding
author$)$
1\. IRIMAS - Département de Mathématiques, 18, rue des frères Lumière, F-68093
Mulhouse, France
2\. University of Gafsa, Faculty of Sciences, 2112 Gafsa, Tunisia
3\. University of Sfax, Faculty of Sciences Sfax, BP 1171, 3038 Sfax, Tunisia

###### Abstract

The purpose of this paper is to introduce and study the notion of generalized
Reynolds operators on Lie triple systems with representations (Abbr. L.t.sRep
pairs) as generalization of weighted Reynolds operators on Lie triple systems.
First, We construct an $L_{\infty}$-algebra whose Maurer-Cartan elements are
generalized Reynolds operators. This allows us to define a Yamaguti cohomology
of a generalized Reynolds operator. This cohomology can be seen as the
Yamaguti cohomology of a certain Lie triple system with coefficients in a
suitable representation. Next, we study deformations of generalized Reynolds
operators from cohomological points of view and we investigate the obstruction
class of an extendable deformation of order $n$. We end this paper by
introducing a new algebraic structure, in connection with generalized Reynolds
operator, called NS-Lie triple system. Moreover, we show that NS-Lie triple
systems can be derived from NS-Lie algebras.

Key words : Lie triple system, generalized Reynolds operator, Maurer-Cartan
element,

$L_{\infty}$-algebra, Lie-Yamaguti cohomology, deformation, NS-Lie triple
system.

Mathematics Subject Classification (2020) : 17B15, 17A40, 17B56, 17B10, 17B38.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Generalized Reynolds operators on Lie triple systems with representations
    1. 3.1 Weighted Reynolds operators on L.t.s
    2. 3.2 Generalized Reynolds operators on L.t.sRep pairs
  4. 4 Yamaguti cohomology of generalized Reynolds operators
    1. 4.1 $L_{\infty}$-algebra and Maurer-Cartan characterization
    2. 4.2 Yamaguti cohomology
  5. 5 Obstruction class of a generalized Reynolds operator
  6. 6 NS-Lie triple systems
    1. 6.1 Definitions and constructions
    2. 6.2 From NS-Lie algebras to NS-Lie triple systems

##  1 Introduction

The concept of Lie triple system was introduced first by Jacobson [15] and the
present formulation is due to Yamguti [29]. Moreover, it appeared in Cartan’s
work on Riemannian Geometry [2] and was strongly developed for Symmetric
spaces and related spaces. Indeed, the tangent space of a symmetric space is a
Lie triple system. It turns out that they have important applications in
physics, in particular, in elementary particle theory and the theory of
quantum mechanics, as well as numerical analysis of differential equations.
They have become an interesting subject in mathematics, their structure have
been studied first by Lister in [21].

The notion of a Rota-Baxter operator on an associative algebra appeared first
in probability theory when G. Baxter studied algebraically the Spitzer
identity. The theory was developed first by Rota in the combinatorics field.
Later relevant application was considered in the Connes-Kreimer’s algebraic
approach to renormalization of quantum field theory [5]. For further details,
see [11, 12]. A Rota-Baxter operator on a Lie algebra is naturally the
operator form of a classical $r$-matrix [25] under certain conditions. To
better understand such connection in general, Kupershmidt introduced the
notion of an $\mathcal{O}$-operator (also called a relative Rota-Baxter
operator [24] or a generalized Rota-Baxter operator [28]) on a Lie algebra in
[18]. Recently a deformation theory and a cohomology theory of relative Rota-
Baxter operators on both Lie and associative algebras are studied in [7, 27].
Reynolds operators were introduced by Reynolds in [23] in the study of
fluctuation theory in fluid dynamics. In [16], the author coined the concept
of the Reynolds operator and regarded the operator as a mathematical subject
in general. In [31], the authors provided examples and properties of the
Reynolds operators and studied the free Reynolds algebras.

NS-algebras were introduced by Leroux in [20]. In [19], the authors studied
the relationship between the category of Nijenhuis algebras and the category
of NS-algebras. Uchino defined the notion of generalized Reynolds operators on
associative algebras and observed that a generalized Reynolds operator induces
an NS-algebra in [28]. In [9], Das defined a cohomology of generalized
Reynolds operators on associative algebras and a cohomology of NS-algebras,
various applications were provided. He also introduced the notions of
generalized Reynolds operators, Reynolds operators on Lie algebras and NS-Lie
algebras, and showed that NS-Lie algebras are the underlying algebraic
structures of generalized Reynolds operators on Lie algebras in [8].

Inspired by these works, we aim in this paper to study Maurer-Cartan type
cohomology on generalized Reynolds operators on Lie triple systems with
representations (Abbr. L.t.sRep pairs). Moreover, we introduce the notion of
NS-Lie triple system as the underlying structure of generalized Reynolds
operators.

This is paper is organized as follow, in Section 2, we summarize some basics
and briefly recall representations and cohomology of Lie triple systems. In
Section 3, we introduce a generalization of weighted Reynolds operators on a
Lie triple system called generalized Reynolds operators on a Lie triple system
and we give some constructions. In Section 4, we construct an
$L_{\infty}$-algebra whose Maurer–Cartan elements are given by generalized
Reynolds operators. This motivates us to define a Yamaguti cohomology of
generalized Reynolds operator on Lie triple system using the underlying Lie
triple systems of the generalized Reynolds operator. In Section 5, we describe
the obstruction class of a deformation of order $n$ to be extended to order
$n+1$. Finally, Section 6 deals a new algebraic structure called NS-Lie triple
system that is related to generalized Reynolds operators. We show that Lie
algebras, NS-Lie algebras, Lie triple systems and NS-Lie triple systems are
closely related.

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

##  2 Preliminaries

In this section, we recall some basic notions as representations and
cohomology theory of Lie triple systems (L.t.s) introduced in [21]. A Lie
triple system is a vector space $L$ endowed with a ternary bracket
$[\cdot,\cdot,\cdot]:\wedge^{2}L\otimes L\to L$ satisfying

$\displaystyle[x,y,z]+[y,z,x]+[z,x,y]=0,$ (2.1)
$\displaystyle[x,y,[z,t,e]]=[[x,y,z],t,e]+[z,[x,y,t],e]+[z,t,[x,y,e]]\quad\forall
x,y,z,t,e\in L.$ (2.2)

A morphism
$\phi:(L,[\cdot,\cdot,\cdot])\rightarrow(L^{{}^{\prime}},[\cdot,\cdot,\cdot]^{{}^{\prime}})$
of L.t.s is a linear map satisfying :

$\phi([x,y,z])=[\phi(x),\phi(y),\phi(z)]^{{}^{\prime}}\quad\forall x,y,z\in
L.$

An isomorphism is a bijective morphism. We set $X=(x,y)\in L\otimes L$ and
denote by $\mathfrak{R}(X),\ \mathfrak{D}(X):L\rightarrow L$ maps defined by
$\mathfrak{R}(X)z=[z,x,y]$ and $\mathfrak{D}(X)z=[x,y,z]$. Then the identities
(2.1) and (2.2) can be rewritten respectively in the form
$\mathfrak{D}(X)=\mathfrak{R}(\tau(X))-\mathfrak{R}(X)$, where
$\tau(X)=y\otimes x$, and

$\mathfrak{D}(X)[z_{1},z_{2},z_{3}]=[\mathfrak{D}(X)z_{1},z_{2},z_{3}]+[z_{1},\mathfrak{D}(X)z_{2},z_{3}]+[z_{1},z_{2},\mathfrak{D}(X)z_{3}].$

This means that $\mathfrak{D}(X)$ is a derivation with respect to the bracket
$[\cdot,\cdot,\cdot]$.

In [29], K. Yamaguti introduced the notion of representation and cohomology
theory of L.t.s. Later, the authors in [13, 14, 17] studied the cohomology
theory of L.t.s from a different point of view. K. Yamaguti’s work can be
described as follows.

A representation of a L.t.s $L$ on a vector space $M$ is a bilinear map
$\theta:\otimes^{2}L\rightarrow End(M)$, such that the following conditions
are satisfied :

$\displaystyle\theta(z,t)\theta(x,y)-\theta(y,t)\theta(x,z)-\theta(x,[y,z,t])+D(y,z)\theta(x,t)=0,$
(2.3)
$\displaystyle\theta(z,t)D(x,y)-D(x,y)\theta(z,t)+\theta([x,y,z],t)+\theta(z,[x,y,t])=0\quad\forall
x,y,z,t\in L,$ (2.4)

where $D(x,y)=\theta(y,x)-\theta(x,y)$. It is denoted by a pair $(M,\theta)$.
We say that we have a L.t.sRep pair and refer to it with the tuple
$(L,[\cdot,\cdot,\cdot],\theta)$.

######  Remark 2.1.

  1. 1.

It can be concluded from (2.4) that

$\displaystyle D(z,t)D(x,y)-D(x,y)D(z,t)+D([x,y,z],t)+D(z,[x,y,t])=0.$ (2.5)

  2. 2.

If $M=L$ then $\mathfrak{R}$ is a representation of $L$ in it self, which is
called the regular representation.

Recall that, a representation of a Lie algebra $(L,[\cdot,\cdot])$ on a vector
space $M$ is a linear map $\rho:L\to End(M)$ satisfying :

$\displaystyle\rho([x,y])=\rho(x)\rho(y)-\rho(y)\rho(x)\quad\forall x,y\in L.$

Note that a LieRep pair $(L,[\cdot,\cdot],\rho)$ is a Lie algebra
$(L,[\cdot,\cdot])$ with a representation $\rho$.

######  Theorem 2.1.

Let $(L,[\cdot,\cdot],\rho)$ be a LieRep pair. Then
$(L,[\cdot,\cdot,\cdot],\theta_{\rho})$ is a L.t.sRep pair, where

$[x,y,z]=[[x,y],z]\quad\text{and}\quad\theta_{\rho}(x,y)=\rho(y)\rho(x)\quad\forall
x,y,z\in L.$ (2.6)

Let $(L,[\cdot,\cdot,\cdot],\theta)$ be a L.t.sRep pair. For each $n\geqslant
0$, we denote by $\mathcal{C}_{L.t.s}^{2n+1}(L,M)$, the vector space of
$(2n+1)$-cochains of $L$ with coefficients in $M$ :
$\psi\in\mathcal{C}_{L.t.s}^{2n+1}(L,M)$ is a multilinear function of
$\psi:\times^{2n+1}L\to M$ satisfying

$\displaystyle\quad\quad\quad\psi(x_{1},x_{2},\cdots,x_{2n-2},x,x,y)=0,$
$\displaystyle\circlearrowleft_{x,y,z}\psi(x_{1},x_{2},\cdots,x_{2n-2},x,y,z)=0,$

where $\circlearrowleft_{x,y,z}$ means that we are taking a cyclic summation.

A Yamaguti coboundary operator
$\delta^{2n-1}:\mathcal{C}_{L.t.s}^{2n-1}(L,M)\rightarrow\mathcal{C}_{L.t.s}^{2n+1}(L,M)$
is defined by:

$\displaystyle\delta^{2n-1}\psi(x_{1},x_{2},\cdots,x_{2n+1})$ $\displaystyle=$
$\displaystyle\theta(x_{2n},x_{2n+1})\psi(x_{1},x_{2},\cdots,x_{2n-1})-\theta(x_{2n-1},x_{2n+1})\psi(x_{1},x_{2},\cdots,x_{2n-2},x_{2n})$
$\displaystyle+\sum_{k=1}^{n}(-1)^{n+k}D(x_{2k-1},x_{2k})\psi(x_{1},x_{2},\cdots,\widehat{x}_{2k-1},\widehat{x}_{2k},\cdots,x_{2n+1})$
$\displaystyle+\sum_{k=1}^{n}\sum_{j=2k+1}^{2n+1}(-1)^{n+k+1}\psi(x_{1},x_{2},\cdots,\widehat{x}_{2k-1},\widehat{x}_{2k},\cdots,[x_{2k-1},x_{2k},x_{j}],\cdots,x_{2n+1}),$
(2.7)

for all $\psi\in\mathcal{C}_{L.t.s}^{2n-1}(L,M),n\geqslant 1$, where
$\;\widehat{}\;$ means that the element is omitted.

######  Proposition 2.2.

The Yamaguti coboundary operator $\delta^{2n-1}$ defined above is a square
zero map, i.e., $\delta^{2n+1}\circ\delta^{2n-1}=0$. Therefore, the set of
Yamaguti cochains forms a complex with this coboundary as follow:

$\mathcal{C}_{L.t.s}^{1}(L,M)\overset{\delta^{1}}{\longrightarrow}\mathcal{C}_{L.t.s}^{3}(L,M)\overset{\delta^{3}}{\longrightarrow}\mathcal{C}_{L.t.s}^{5}(L,M)\longrightarrow\cdots.$

Hence we get the Yamaguti cohomology group
$\mathcal{H}_{L.t.s}^{\bullet}(L,M)=\mathcal{Z}_{L.t.s}^{\bullet}(L,M)/\mathcal{B}_{L.t.s}^{\bullet}(L,M)$,
where $\mathcal{Z}_{L.t.s}^{\bullet}(L,M)$ is the space of cocycles and
$\mathcal{B}_{L.t.s}^{\bullet}(L,M)$ is the space of coboundaries. See [29,
30] for more details. A $1$-cochain $\varphi\in\mathcal{C}_{L.t.s}^{1}(L,M)$
is a $1$-cocycle if

$D(x_{1},x_{2})\varphi(x_{3})-\theta(x_{1},x_{3})\varphi(x_{2})+\theta(x_{2},x_{3})\varphi(x_{1})-\varphi([x_{1},x_{2},x_{3}])=0.$
(2.8)

A $3$-cochain $\mathcal{H}\in\mathcal{C}_{L.t.s}^{3}(L,M)$ is a $3$-coboundary
if there exists a map $\varphi\in\mathcal{C}_{L.t.s}^{1}(L,M)$ such that
$\mathcal{H}=\delta^{1}\varphi$. A $3$-coboundary is called a $3$-cocycle if
for all $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}\in L$ :

$\displaystyle\mathcal{H}(x_{1},x_{1},x_{2})=0,$ (2.9)
$\displaystyle\mathcal{H}(x_{1},x_{2},x_{3})+\mathcal{H}(x_{2},x_{3},x_{1})+\mathcal{H}(x_{3},x_{1},x_{2})=0,$
(2.10)
$\displaystyle\mathcal{H}(x_{1},x_{2},[y_{1},y_{2},y_{3}])+D(x_{1},x_{2})\mathcal{H}(y_{1},y_{2},y_{3})-\mathcal{H}([x_{1},x_{2},y_{1}],y_{2},y_{3})-\mathcal{H}(y_{1},[x_{1},x_{2},y_{2}],y_{3})$
$\displaystyle=$
$\displaystyle\mathcal{H}(y_{1},y_{2},[x_{1},x_{2},y_{3}])+\theta(y_{2},y_{3})\mathcal{H}(x_{1},x_{2},y_{1})-\theta(y_{1},y_{3})\mathcal{H}(x_{1},x_{2},y_{2})+D(y_{1},y_{2})\mathcal{H}(x_{1},x_{2},y_{3}).$
(2.11)

######  Theorem 2.3.

[4] Let $\phi\in\mathcal{Z}^{2}_{Lie}(L,M)$. Then
$\omega(x,y,z)=\phi([x,y],z)-\rho(z)\phi(x,y)$ is a $3$-cocycle of the
L.t.sRep pair $(L,[\cdot,\cdot,\cdot],\theta_{\rho})$ given in Theorem 2.1.

##  3 Generalized Reynolds operators on Lie triple systems with
representations

In this section, we introduce the notion of generalized Reynolds operator on
L.t.sRep pairs which provides a generalization of weighted Reynolds operators
on L.t.s. Also, We give its characterization by a graph and provide some
constructions. This notion is an algebraic analogue of generalized Reynolds
operator on a Lie algebra introduced in [6] and on a $3$-Lie algebras
introduced in [3, 26].

###  3.1 Weighted Reynolds operators on L.t.s

######  Definition 3.1.

Let $(L,[\cdot,\cdot])$ be a Lie algebra and $\lambda$ be a non-null scalar .
A  $\lambda$-weighted Reynolds operator on a Lie algebra is a linear map
$\mathcal{R}:L\to L$ satisfying:

$\displaystyle[\mathcal{R}x,\mathcal{R}y]=\mathcal{R}\Big{(}[\mathcal{R}x,y]+[x,\mathcal{R}y]+\lambda[\mathcal{R}x,\mathcal{R}y]\Big{)}\quad\forall
x,y\in L.$ (3.1)

Note that $(L,[\cdot,\cdot],\mathcal{R})$ is $\lambda$-weighted Reynolds Lie
algebra.

######  Remark 3.1.

If $\lambda=-1$, we recover the definition of Reynolds operator on a Lie
algebra introduced by A. Das in [6] in the study of generalized Reynolds
operators on a Lie algebra.

######  Definition 3.2.

Let $(L,[\cdot,\cdot,\cdot])$ be a L.t.s and $\lambda$ be a scalar. A
$\lambda$-weighted Reynolds operator on a L.t.s is a linear map
$\mathcal{R}:L\to L$ satisfying:

$\displaystyle[\mathcal{R}x,\mathcal{R}y,\mathcal{R}z]=\mathcal{R}\Big{(}[\mathcal{R}x,\mathcal{R}y,z]+[x,\mathcal{R}y,,\mathcal{R}z]+[\mathcal{R}x,y,\mathcal{R}z]+\lambda[\mathcal{R}x,\mathcal{R}y,\mathcal{R}z]\Big{)}\quad\forall
x,y,z\in L.$ (3.2)

Note that $(L,[\cdot,\cdot,\cdot],\mathcal{R})$ is $\lambda$-weighted Reynolds
L.t.s.

######  Example 3.1.

For a fixed complex number $q$, there is a Lie triple system structure
analogue to the Lie algebra of Block type given in [1], generated by the basis
$\\{L_{m,i}|m,i\in\mathbb{Z}\\}$ and given by the following bracket

$[L_{m,i},L_{n,j},L_{p,k}]=(n(i+q)-m(j+q))(p(i+j+q)-(m+n)(p+q))L_{m+n+p,i+j+k}.$

Let $B(q)_{\geq 0}=\\{L_{m,i}|m,i\in\mathbb{Z}_{+}\\}$, then the linear map
$\mathcal{R}:B(q)_{\geq 0}\to B(q)_{\geq 0}$ defined by

$\displaystyle\mathcal{R}(L_{m,i})=\frac{-1}{\lambda(m+i+1)}\ L_{m,i}\ \ \ \
\forall m\in B(q)_{\geq 0},\lambda\neq 0$ (3.3)

is a $\lambda$-weighted Reynolds operator on $B(q)_{\geq 0}$.

######  Definition 3.3.

Let $(L,[\cdot,\cdot,\cdot],\mathcal{R})$ and
$(L^{\prime},[\cdot,\cdot,\cdot]^{\prime},\mathcal{R^{\prime}})$ be two
$\lambda$-weighted Reynolds L.t.s. A linear map $\alpha:L\to L^{\prime}$ is
called morphism of $\lambda$-weighted Reynolds L.t.s if $\alpha$ is morphism
of L.t.s satisfying $\alpha\circ\mathcal{R}=\mathcal{R^{\prime}}\circ\alpha.$

######  Theorem 3.2.

Let $(L,[\cdot,\cdot,\cdot],\mathcal{R})$ be $\lambda$-weighted Reynolds
L.t.s. Define $[\cdot,\cdot,\cdot]_{\mathcal{R}}$ on $L$ by

$\displaystyle[x,y,z]_{\mathcal{R}}=[\mathcal{R}x,\mathcal{R}y,z]+[\mathcal{R}x,y,\mathcal{R}z]+[x,\mathcal{R}y,\mathcal{R}z]+\lambda[\mathcal{R}x,\mathcal{R}y,\mathcal{R}z]\quad\forall
x,y,z\in L.$ (3.4)

Then $(L,[\cdot,\cdot,\cdot]_{\mathcal{R}},\mathcal{R})$ is a
$\lambda$-weighted Reynolds L.t.s. Moreover $\mathcal{R}$ is a morphism of
$\lambda$-weighted Reynolds operators L.t.s from
$(L,[\cdot,\cdot,\cdot]_{\mathcal{R}},\mathcal{R})$ to
$(L,[\cdot,\cdot,\cdot],\mathcal{R})$, that is
$\mathcal{R}([x,y,z]_{\mathcal{R}})=[\mathcal{R}(x),\mathcal{R}(y),\mathcal{R}(z)].$

######  Proposition 3.3.

Let $\mathcal{R}:L\to L$ be a $\lambda$-weighted Reynolds operator on a Lie
algebra, then $\mathcal{R}$ is a $2\lambda$-weighted Reynolds operator on the
induced L.t.s.

######  Example 3.4.

Let $W$ be the Witt algebra generated by basis elements
$\\{l_{n}\\}_{n\in\mathbb{Z}}$ and the Lie bracket given by:

$\displaystyle[l_{m},l_{n}]=(m-n)l_{m+n}\ \ \ \ \ \forall m,n\in\mathbb{Z}.$
(3.5)

By Theorem 2.1, we obtain a L.t.s defined by the bracket

$\displaystyle[l_{m},l_{n},l_{p}]=(m-n)(m+n-p))l_{m+n+p}\ \ \ \ \ \forall
m,n,p\in\mathbb{Z}.$

Note that $W_{\geq 0}=span\\{l_{n}|n\geq 0\\}$ is a L.t.s subalgebra of $W$.
The linear map $\mathcal{R}:W_{\geq 0}\to W_{\geq 0}$ defined by

$\displaystyle\mathcal{R}(l_{m})=\frac{-1}{\lambda(m+1)}\ l_{m}\ \ \ \ \forall
m\in W_{\geq 0}\ and\ \lambda\neq 0$ (3.6)

is a $\lambda$-weighted Reynolds operator on the Lie algebra $W_{\geq 0}$.
Then, according to Proposition 3.3 $\mathcal{R}$ is a Reynolds operator of
weight $2\lambda$ on the induced L.t.s. This example will be more clear in
Section 6 when we will introduce NS-Lie triple systems and a functor from the
category of NS-Lie triple systems to the category of generalized Reynolds
operator on L.t.s.

###  3.2 Generalized Reynolds operators on L.t.sRep pairs

Let $(L,[\cdot,\cdot,\cdot],\theta)$ be a L.t.sRep pair. For any $3$-cocycle
$\mathcal{H}\in\mathcal{Z}_{L.t.s}^{3}(L,M)$, there is a Lie triple system
structure on the direct sum of vector spaces $L\oplus M$, defined by:

$\displaystyle[x+u,y+v,z+w]_{\mathcal{H}}=\Big{(}[x,y,z],\
\theta(y,z)u-\theta(x,z)v+D(x,y)w+\mathcal{H}(x,y,z)\Big{)}\quad\forall~{}~{}x,y,z\in
L.~{}~{}u,v,w\in M.$ (3.7)

This Lie triple system is called the twisted semi-direct product Lie triple
system and denoted by $L\ltimes_{\theta}^{\mathcal{H}}M$.

In the sequel, $\mathcal{H}$ will always denote a $3$-cocycle.

######  Definition 3.4.

A linear map $T:M\to L$ is said to be a  generalized Reynolds operator  on a
L.t.sRep pair $(L,[\cdot,\cdot,\cdot],\theta)$ if it satisfies:

$\displaystyle[Tu,Tv,Tw]=T\Big{(}D(Tu,Tv)w+\theta(Tv,Tw)u-\theta(Tu,Tw)v+\mathcal{H}(Tu,Tv,Tw)\Big{)}\quad\forall
u,v,w\in M.$ (3.8)

######  Example 3.5.

Recall that a Rota-Baxter operator of weight zero on a L.t.s $L$ is a linear
map $\mathcal{R}:L\rightarrow L$ satisfying :

$[\mathcal{R}(x),\mathcal{R}(y),\mathcal{R}(z)]=\mathcal{R}\Big{(}[\mathcal{R}(x),\mathcal{R}(y),z]+[\mathcal{R}(x),y,\mathcal{R}(z)]+[x,\mathcal{R}(y),\mathcal{R}(z)]\Big{)}\quad\forall
x,y,z\in L.$

Any Rota–Baxter operator of weight zero on a L.t.s is a generalized Reynolds
operator with respect to the regular representation and $\mathcal{H}=0$.

######  Remark 3.2.

The notion of generalized Reynolds operator is also called twisted
$\mathcal{O}$-operator or twisted Kupershmidt operator. In [4], the authors
introduced the notion of an $\mathcal{O}$-operator on a L.t.sRep pair. It is
straightforward to see that generalized Reynolds operators on L.t.sRep pairs
are generalization of Rota-Baxter operators of weight 0 and
$\mathcal{O}$-operators on L.t.sRep pairs.

######  Example 3.6.

Any $\lambda$-weighted Reynolds operator on a L.t.s. $(L,[\cdot,\cdot,\cdot])$
is generalized Reynolds operator with respect to the regular representation,
where $\mathcal{H}$ is given by:

$\displaystyle\mathcal{H}(x,y,z)=\lambda[x,y,z]\ \ \ \forall x,y,z\in L.$

######  Example 3.7.

Let $(L,[\cdot,\cdot,\cdot],\theta)$ be a L.t.sRep pair. Suppose that
$\varphi\in\mathcal{C}_{L.t.s}^{1}(L,M)$ is an invertible $1$-cochain. Then
$T=\varphi^{-1}:M\to L$ is a generalized Reynolds operator with
$\mathcal{H}=-\delta^{1}\varphi$. Indeed, we have

$\displaystyle\delta^{1}\varphi(Tu,Tv,Tw)=D(Tu,Tv)w-\theta(Tu,Tw)v+\theta(Tv,Tw)u-\varphi([Tu,Tv,Tw])\quad\forall
u,v,w\in M.$

By applying $T$ to both sides we get the result.

######  Definition 3.5.

Let $T$ and $T^{\prime}$ be two generalized Reynolds operators on a L.t.sRep
pair. A morphism of generalized Reynolds operators from $T$ to $T^{\prime}$
consists of a L.t.s morphism $\phi:L\to L$ and an endomorphism $\psi:M\to M$
such that

$\displaystyle\phi\circ T=T^{{}^{\prime}}\circ\psi,$ (3.9)
$\displaystyle\psi(\theta(x,y)u)=\theta(\phi(x),\phi(y))\psi(u),$ (3.10)
$\displaystyle\psi(\mathcal{H}(x,y,z))=\mathcal{H}(\phi(x),\phi(y),\phi(z))\quad\forall\
x,y,z\in L,\ u\in M.$ (3.11)

In particular, if both $\phi$ and $\psi$ are invertible then $(\phi,\psi)$ is
called an isomorphism.

Let $\varphi\in\mathcal{C}_{L.t.s}^{1}(L,M)$ be a $1$-cochain. Define
$\kappa_{\varphi}:L\oplus M\to L\oplus M$ by
$\kappa_{\varphi}=\begin{pmatrix}Id_{L}&0\\\ \varphi&Id_{M}\end{pmatrix}.$

######  Proposition 3.8.

Let $(L,[\cdot,\cdot,\cdot],\theta)$ be a L.t.sRep pair and
$\varphi\in\mathcal{C}_{L.t.s}^{1}(L,M)$ be a $1$-cochain. Then
$\kappa_{\varphi}$ defined above is an isomorphism from the
$\mathcal{H}$-twisted semi-direct product L.t.s
$L\ltimes_{\theta}^{\mathcal{H}}M$ to the ${\mathcal{H}^{\prime}}$-semi-direct
product L.t.s $L\ltimes_{\theta}^{\mathcal{H}^{\prime}}M$, where
${\mathcal{H}^{\prime}}=\mathcal{H}-\delta^{1}\varphi.$

The identity (3.8) can be characterized by the graph of $T$ being a
subalgebra.

######  Proposition 3.9.

A linear map $T:M\rightarrow L$ is a generalized Reynolds operator on a
L.t.sRep pair if and only if its graph $Gr(T)=\\{(Tu,u)|\;u\in M\\}$ is a
subalgebra of the $\mathcal{H}$-twisted semi-direct product
$L\ltimes_{\theta}^{\mathcal{H}}M$.

###### Proof.

Let $T:M\rightarrow L$ be a linear map. For all $u,vw\in M$, we have:

$\displaystyle[Tu+u,Tv+v,Tw+w]_{\mathcal{H}}$ $\displaystyle=$
$\displaystyle\Big{(}[Tu,Tv,Tw],~{}~{}\theta(Tv,Tw)u-\theta(Tu,Tw)v+D(Tu,Tv)w+\mathcal{H}(Tu,Tv,Tw)\Big{)}.$

which implies that the graph $Gr(T)$ is a subalgebra of the
$\mathcal{H}$-twisted semi-direct product $L\ltimes_{\theta}^{\mathcal{H}}M$
if and only if $T$ satisfies

$\displaystyle[Tu,Tv,Tw]=T\Big{(}\theta(Tv,Tw)u-\theta(Tu,Tw)v+D(Tu,Tv)w+\mathcal{H}(Tu,Tv,Tw)\Big{)},$

which means that $T$ is a generalized Reynolds operator. ∎

Since $M$ and $Gr(T)$ are isomorphic as vector spaces, we get the following
corollary.

######  Corollary 3.10.

Let $T:M\rightarrow L$ be a generalized Reynolds operator on a L.t.sRep pair
$(L,[\cdot,\cdot,\cdot],\theta)$. Then there is a L.t.s structure on $M$ given
by:

$\displaystyle[u,v,w]_{T}=\theta(Tv,Tw)u-\theta(Tu,Tw)v+D(Tu,Tv)w+\mathcal{H}(Tu,Tv,Tw)\quad\forall~{}u,v,w\in
M.$ (3.12)

Furthermore, $T$ is a morphism from $(M,[\cdot,\cdot,\cdot]_{T})$ to
$(L,[\cdot,\cdot,\cdot])$.

######  Definition 3.6.

Let $T:M\to L$ be a generalized Reynolds operator on a L.t.sRep pair
$(L,[\cdot,\cdot,\cdot],\theta)$. The map
$\varphi\in\mathcal{Z}_{L.t.s}^{1}(L,M)$ is called a $T$-admissible
$1$-cocycle if the linear map $(Id+\varphi\circ T):M\to M$ is invertible.

######  Proposition 3.11.

Let $\varphi$ be a $T$-admissible $1$-cocycle, then $T\circ(Id+\varphi\circ
T)^{-1}:M\to L$ is a generalized Reynolds operator, which we denote by
$T_{\varphi}$.

###### Proof.

Let $\varphi$ be a $T$-admissible $1$-cocycle. By Propositions 3.8 and 3.9, we
have

$\kappa_{\varphi}(Gr(T))=\\{(Tu,u+\varphi(Tu))|u\in M\\}\subseteq
L\ltimes_{\theta}^{\mathcal{H}^{\prime}}M,$

is a subalgebra of the $\mathcal{H}^{\prime}$-twisted semi-direct product
L.t.s $L\ltimes_{\theta}^{\mathcal{H}^{\prime}}M$, where
${\mathcal{H}^{\prime}}=\mathcal{H}-\delta^{1}\varphi$. Since the linear map
$(Id+\varphi\circ T):M\to M$ is invertible, so $\kappa_{\varphi}(Gr(T))$ is
the graph of $T\circ(Id+\varphi\circ T)^{-1}:M\to L$, which implies that
$T\circ(Id+\varphi\circ T)^{-1}$ is a generalized Reynolds operator. This
completes the proof. ∎

Recall from Corollary 3.10 that a generalized Reynolds operator induces a Lie
triple system on $M$. We have the following proposition.

######  Proposition 3.12.

Let $T$ be a generalized Reynolds operator and $\varphi$ be a $T$-admissible
$1$-cocycle. Then the Lie triple system structures on $M$ induced by $T$ and
$T_{\varphi}$ are isomorphic.

###### Proof.

Consider the linear map $(Id+\varphi\circ T):M\to M$. For all $u,v,w\in M$, we
have

$\displaystyle[(Id+\varphi\circ T)u,(Id+\varphi\circ T)v,(Id+\varphi\circ
T)w]_{T_{\varphi}}$ $\displaystyle=$
$\displaystyle\theta(Tu,Tv)(Id+\varphi\circ T)w-\theta(Tu,Tw)(Id+\varphi\circ
T)v$ $\displaystyle+D(Tv,Tw)(Id+\varphi\circ T)u+\mathcal{H}(Tu,Tv,Tw)$
$\displaystyle=$
$\displaystyle\theta(Tu,Tv)w-\theta(Tu,Tw)v+D(Tv,Tw)u+\mathcal{H}(Tu,Tv,Tw)$
$\displaystyle+\theta(Tu,Tv)(\varphi\circ T)w-\theta(Tu,Tw)(\varphi\circ
T)v+D(Tv,Tw)(\varphi\circ T)u$ $\displaystyle\overset{\eqref{1cocycle}}{=}$
$\displaystyle[u,v,w]_{T}+\varphi([Tu,Tv,Tw])$ $\displaystyle=$
$\displaystyle[u,v,w]_{T}+\varphi\circ T([u,v,w]_{T})$ $\displaystyle=$
$\displaystyle(Id+\varphi\circ T)[u,v,w]_{T}.$

Thus $(Id+\varphi\circ T)$ is an isomorphism of Lie triple systems. ∎

In Ref. [6], the author introduced a generalized Reynolds operator on LieRep
pair $(L,[\cdot,\cdot],\rho)$ with a $2$-cocycle $H$ in the Chevalley-
Eilenberg cohomology, as a linear map $T:M\to L$ satisfying

$\displaystyle[Tu,Tv]=T\Big{(}\rho(Tu)v+\rho(Tv)u+H(Tu,Tv)\Big{)}\quad\forall
u,v\in M.$

######  Proposition 3.13.

Let $T:M\to L$ be a generalized Reynolds operator on a LieRep pair
$(L,[\cdot,\cdot],\rho)$. Then $T$ is also a generalized Reynolds operator on
the induced L.t.sRep pair.

##  4 Yamaguti cohomology of generalized Reynolds operators

In this section, after recalling the notion of $L_{\infty}$-algebra, we
construct one on a given graded vector space whose Maurer-Cartan elements are
generalized Reynolds operators on L.t.sRep pairs. It is the Maurer-Cartan
characterization of generalized Reynolds operator $T$. This characterization
allows us to introduce the Yamaguti cohomology of $T$. Next, we show that the
cohomology of $T$ is equivalently described by the Yamaguti cohomology of $M$
with coefficients in a suitable representation $L$.

###  4.1 $L_{\infty}$-algebra and Maurer-Cartan characterization

A permutation $\sigma\in\mathbb{S}_{n}$ is called an $(i,n-i)$-shuffle if
$\sigma(1)<....<\sigma(i)$ and $\sigma(i+1)<....<\sigma(n)$. If $i=0$ or
$i=n$, we assume $\sigma=Id$. The set of all $(i,n-i)$-shuffles will be
denoted by $\mathbb{S}_{(i,n-i)}$.

######  Definition 4.1.

An $L_{\infty}$-algebra is a $\mathbb{Z}$-graded vector space
$\mathfrak{(}\mathfrak{g}=\oplus_{k\in\mathbb{Z}}\mathfrak{g}^{k})$ equipped
with a collection $(k\geq 1)$ of linear maps
$l_{k}:\otimes^{k}\mathfrak{g}\to\mathfrak{g}$ of degree $1$ with the property
that, for any homogeneous elements $x_{1},\cdots,x_{n}\in\mathfrak{g}$, we
have

  * (i)

graded symmetry:

$\displaystyle
l_{n}(x_{\sigma(1)},\cdots,x_{\sigma(n-1)},x_{\sigma(n)})=\varepsilon(\sigma)l_{n}(x_{1},\cdots,x_{n-1},x_{n})\quad\forall\sigma\in\mathbb{S}_{n},$

  * (ii)

generalized Jacobi identity:

$\displaystyle\sum_{i=1}^{n}\sum_{\sigma\in\mathbb{S}_{(i,n-i)}}\varepsilon(\sigma)l_{n-i+1}(l_{i}(x_{\sigma(1)},\cdots,x_{\sigma(i)}),x_{\sigma(i+1)},\cdots,x_{\sigma(n)})=0\quad\forall
n\geq 1.$

######  Definition 4.2.

A Maurer-Cartan element of an $L_{\infty}$-algebra
$(\mathfrak{g}=\oplus_{k\in\mathbb{Z}}\mathfrak{g}^{k},\\{l_{i}\\}_{i=1}^{+\infty})$
is an element $\pi\in\mathfrak{g}^{0}$ satisfying the Maurer-Cartan equation

$\displaystyle\sum_{n=1}^{+\infty}\frac{1}{n!}l_{n}(\pi,\cdots,\pi)=0.$ (4.1)

Let $\pi$ be a Maurer-Cartan element of an $L_{\infty}$-algebra
$(\mathfrak{g},\\{l_{i}\\}_{i=1}^{+\infty})$. For all $k\geq 1$ and
$x_{1},\cdots,x_{n}\in\mathfrak{g},$ define a series of linear maps
$l_{k}^{\pi}:\otimes^{k}\mathfrak{g}\to\mathfrak{g}$ of degree $1$ by

$\displaystyle
l^{\pi}_{k}(x_{1},\cdots,x_{k})=\sum^{+\infty}_{n=0}\frac{1}{n!}l_{n+k}\\{\underbrace{\pi,\cdots,\pi}_{n},x_{1},\cdots,x_{k}\\}.$
(4.2)

######  Theorem 4.1.

[10] With the above notations,
$(\mathfrak{g},\\{l^{\pi}_{i}\\}_{i=1}^{+\infty})$ is an $L_{\infty}$-algebra,
obtained from the $L_{\infty}$-algebra
$(\mathfrak{g},\\{l_{i}\\}_{i=1}^{+\infty})$ by twisting with the Maurer-
Cartan element $\pi$. Moreover, $\pi+\pi^{\prime}$ is a Maurer-Cartan element
of $(\mathfrak{g},\\{l_{i}\\}_{i=1}^{+\infty})$ if and only if $\pi^{\prime}$
is a Maurer-Cartan element of the twisted $L_{\infty}$-algebra
$(\mathfrak{g},\\{l^{\pi}_{i}\\}_{i=1}^{+\infty})$ .

Let $L$ be a vector space. Consider the graded vector space
$C^{\ast}(L,L)=\oplus_{n\geq 0}C^{n}(L,L),$ where $C^{n}(L,L)$ is the set of
linear maps
$P\in\text{Hom}(\underbrace{(\wedge^{2}L)\otimes\cdots\otimes(\wedge^{2}L)}_{n\geq
0}\otimes L,L),$ satisfying

$\displaystyle
P(\mathfrak{X}_{1},\mathfrak{X}_{2},\cdots,\mathfrak{X}_{n-1},x,x,y)=$
$\displaystyle 0,$ (4.3)
$\displaystyle\circlearrowleft_{x,y,z}P(\mathfrak{X}_{1},\mathfrak{X}_{2},\cdots,\mathfrak{X}_{n-1},x,y,z)=$
$\displaystyle 0.$ (4.4)

The degree of an element in $C^{n}(L,L)$ is defined to be $n$. Define

$\displaystyle[P,Q]_{L.t.s}=P{\circ}Q-(-1)^{pq}Q{\circ}P\quad\forall~{}P\in\mathfrak{C}^{p}(L,L),Q\in\mathfrak{C}^{q}(L,L),$
(4.5)

with $P{\circ}Q\in C^{p+q}(L,L)$ and defined by

$\displaystyle(P{\circ}Q)(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{p+q},x)$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{p}(-1)^{(k-1)q}\sum_{\sigma\in\mathcal{S}(k-1,q)}(-1)^{\sigma}P(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(k-1)},Q(\mathfrak{X}_{\sigma(k)},\cdots,\mathfrak{X}_{\sigma(k+q-1)},x_{k+q})\wedge
y_{k+q},\mathfrak{X}_{k+q+1},\cdots,\mathfrak{X}_{p+q},x)$
$\displaystyle+\sum_{k=1}^{p}(-1)^{(k-1)q}\sum_{\sigma\in\mathcal{S}(k-1,q)}(-1)^{\sigma}P(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(k-1)},x_{k+q}\wedge
Q(\mathfrak{X}_{\sigma(k)},\cdots,\mathfrak{X}_{\sigma(k+q-1)},y_{k+q}),\mathfrak{X}_{k+q+1},\cdots,\mathfrak{X}_{p+q},x)$
$\displaystyle+\sum_{\sigma\in\mathcal{S}(p,q)}(-1)^{pq}(-1)^{\sigma}P(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(p)},Q(\mathfrak{X}_{\sigma(p+1)},\cdots,\mathfrak{X}_{\sigma(p+q-1)},\mathfrak{X}_{\sigma(p+q)},x)),$

where $\sigma$ is a permutation in $(k-1,q)$-shuffle.

######  Proposition 4.2.

[4] The graded vector space $C^{*}(L,L)$ equipped with the graded commutator
bracket defined by (4.5) is a graded Lie algebra.

######  Proposition 4.3.

Let $L$ be a vector space. Then $\pi\in C^{1}(L,L)$ defines a Lie triple
system structure on $L$ if and only if $[\pi,\pi]_{L.t.s}=0$, i.e., $\pi$ is a
Maurer-Cartan element of the graded Lie algebra
$(C^{*}(L,L),[\cdot,\cdot]_{L.t.s})$. Moreover,
$(C^{*}(L,L),[\cdot,\cdot]_{L.t.s},{\rm d}_{\pi})$ is a differential graded
Lie algebra, where ${\rm d}_{\pi}$ is defined by

$\displaystyle{\rm d}_{\pi}:=(-1)^{n-1}[\pi,\cdot]_{L.t.s}.$ (4.6)

Let $(L,[\cdot,\cdot,\cdot],\theta)$ be a L.t.sRep pair. For convenience, we
use $\pi:\wedge^{2}L\otimes L\to L$ to indicate the Lie triple system
structure $[\cdot,\cdot,\cdot]$. Then $\pi+\theta+\mathcal{H}$ corresponds to
the semidirect product Lie triple system structure on $L\oplus M$ given by

$\displaystyle[x+u,y+v,z+w]_{\mathcal{H}}=\Big{(}[x,y,z],\
\theta(y,z)u-\theta(x,z)v+D(x,y)w+\mathcal{H}(x,y,z)\Big{)}.$ (4.7)

Therefore, we have

$[\pi+\theta+\mathcal{H},\ \pi+\theta+\mathcal{H}]_{L.t.s}=0.$

Consider the graded vector space $C^{\ast}(M,L)=\oplus_{n\geq 0}C^{n}(M,L)$,
where $C^{n}(M,L)$ is the set of linear maps $f\in
Hom(\underbrace{(\wedge^{2}M)\otimes\cdots\otimes(\wedge^{2}M)}_{n\geq
0}\otimes M,L),$ satisfying

$\displaystyle
f(\mathfrak{U}_{1},\mathfrak{U}_{2},\cdots,\mathfrak{U}_{n-1},u,u,v)=$
$\displaystyle 0,$ (4.8)
$\displaystyle\circlearrowleft_{u,v,w}f(\mathfrak{U}_{1},\mathfrak{U}_{2},\cdots,\mathfrak{U}_{n-1},u,v,w)=$
$\displaystyle 0,\ \ \forall\mathfrak{U}_{i}\in\otimes^{2}M,\;1\leq i\leq
n-1.$ (4.9)

Define

$\displaystyle
l_{3}:\mathfrak{C}^{p}(M,L)\times\mathfrak{C}^{q}(M,L)\times\mathfrak{C}^{r}(M,L)\to\mathfrak{C}^{p+q+r+1}(M,L),$
$\displaystyle
l_{4}:\mathfrak{C}^{p}(M,L)\times\mathfrak{C}^{q}(M,L)\times\mathfrak{C}^{r}(M,L)\times\mathfrak{C}^{s}(M,L)\to\mathfrak{C}^{p+q+r+s+1}(M,L),$

by

$\displaystyle l_{3}(P,Q,R)=$
$\displaystyle[[[\pi+\theta,P]_{L.t.s},Q]_{L.t.s},R]_{L.t.s},$ $\displaystyle
l_{4}(P,Q,R,S)=$
$\displaystyle[[[[\mathcal{H},P]_{L.t.s},Q]_{L.t.s},R]_{L.t.s},S]_{L.t.s}.$

Now we give the Maurer-Cartan characterization of a generalized Reynolds
operator on a LieRep pair $(L,[\cdot,\cdot,\cdot],\theta)$.

######  Proposition 4.4.

The graded vector space $C^{\ast}(M,L)$ is an $L_{\infty}$-algebra with
$\\{l_{1}=l_{2}=0,l_{3}(\cdot,\cdot,\cdot),l_{4}(\cdot,\cdot,\cdot,\cdot)\\}$
and higher brackets are trivial.

######  Theorem 4.5.

A linear map $T:M\to L$ is a generalized Reynolds operator if and only if $T$
is a solution of the Maurer-Cartan equation of the $L_{\infty}$-algebra
$(C^{\ast}(M,L),l_{3},l_{4})$, i.e.

$\frac{1}{3!}l_{3}(T,T,T)+\frac{1}{4!}l_{4}(T,T,T,T)=0.$

###### Proof.

Using the above discussion, the first part follows. For the second part, we
have that for any $T\in Hom(M,L)$,

$l_{4}(T,T,T,T)(u,v,w)=-24T(\mathcal{H}(Tu,Tv,Tw)).$ (4.10)

Next, according to the Proof of Theorem 4.4 in [4] we have

$l_{3}(T,T,T)(u,v,w)=6\Big{(}[Tu,Tv,Tw]-T\big{(}\theta(Tv,Tw)u-\theta(Tu,Tw)v+D(Tu,Tv)w\big{)}\Big{)}.$
(4.11)

Hence we obtain

$\displaystyle\Big{(}\frac{1}{3!}l_{3}(T,T,T)+\frac{1}{4!}l_{4}(T,T,T,T)\Big{)}(u,v,w)$
$\displaystyle=$
$\displaystyle[Tu,Tv,Tw]-T\big{(}\theta(Tv,Tw)u-\theta(Tu,Tw)v+D(Tu,Tv)w+\mathcal{H}(Tu,Tv,Tw)).$

Thus, a linear map $T\in Hom(M,L)$ is a generalized Reynolds operator on a
L.t.sRep pair $(L,[\cdot,\cdot,\cdot],\theta)$ if and only if $T$ is a Maurer-
Cartan element of the $L_{\infty}$-algebra $(C^{\ast}(M,L),l_{3},l_{4})$. ∎

######  Proposition 4.6.

Let $T$ be a generalized Reynolds operator on a L.t.sRep pair
$(L,[\cdot,\cdot,\cdot],\theta)$. Then $C^{\ast}(M,L)$ carries a twisted
$L_{\infty}$-algebra structure given by

$\displaystyle
l_{1}^{T}(P)=\frac{1}{2}l_{3}(T,T,P)+\frac{1}{6}l_{4}(T,T,T,P),$ (4.12)
$\displaystyle l_{2}^{T}(P,Q)=l_{3}(T,P,Q)+\frac{1}{2}l_{4}(T,T,P,Q),$ (4.13)
$\displaystyle l_{3}^{T}(P,Q,R)=l_{3}(P,Q,R)+l_{4}(T,P,Q,R),$ (4.14)
$\displaystyle l_{4}^{T}(P,Q,R,S)=l_{4}(P,Q,R,S),$ (4.15) $\displaystyle
l_{k}^{T}=0,\quad k\geq 5,$ (4.16)

where $P\in C^{p}(M,L),Q\in C^{q}(M,L),R\in C^{r}(M,L)$ and S $\in
C^{s}(M,L)$. Moreover, for any linear map $T^{\prime}:M\to L$, the sum
$T+T^{\prime}$ is a generalized Reynolds operator if and only if $T^{\prime}$
is a Maurer-Cartan element in the twisted $L_{\infty}$-algebra
$(C^{\ast}(M,L),l_{1}^{T},l_{2}^{T},l_{3}^{T},l_{4}^{T})$, i.e satisfies

$l_{1}^{T}(T^{\prime})+\frac{1}{2!}l_{2}^{T}(T^{\prime},T^{\prime})+\frac{1}{3!}l_{3}^{T}(T^{\prime},T^{\prime},T^{\prime})+\frac{1}{4!}l_{4}^{T}(T^{\prime},T^{\prime},T^{\prime},T^{\prime})=0.$

###### Proof.

For the first part, since $T$ is a Maurer-Cartan element of the
$L_{\infty}$-algebra $(C^{\ast}(M,L),l_{3},l_{4})$, by Theorem 4.1, we have
that $C^{\ast}(M,L)$ carries a twisted $L_{\infty}$-algebra structure. For the
second part, by Theorem 4.5, $T+T^{\prime}$ is a generalized Reynolds operator
if and only if

$\frac{1}{3!}l_{3}(T+T^{\prime},T+T^{\prime},T+T^{\prime})+\frac{1}{4!}l_{4}(T+T^{\prime},T+T^{\prime},T+T^{\prime},T+T^{\prime})=0.$
(4.17)

Applying $\frac{1}{3!}l_{3}(T,T,T)+\frac{1}{4!}l_{4}(T,T,T,T)=0$, the above
condition is equivalent to

$\displaystyle\frac{1}{3!}\Big{(}3l_{3}(T,T,T^{\prime})+3l_{3}(T,T^{\prime},T^{\prime})+l_{3}(T^{\prime},T^{\prime},T^{\prime})\Big{)}$
$\displaystyle+\frac{1}{4!}\Big{(}4l_{4}(T,T,T,T^{\prime})+6l_{4}(T,T,T^{\prime},T^{\prime})+4l_{4}(T,T^{\prime},T^{\prime},T^{\prime})+l_{4}(T^{\prime},T^{\prime},T^{\prime},T^{\prime})\Big{)}=0,$

that is,
$l_{1}^{T}(T^{\prime})+\frac{1}{2!}l_{2}^{T}(T^{\prime},T^{\prime})+\frac{1}{3!}l_{3}^{T}(T^{\prime},T^{\prime},T^{\prime})+\frac{1}{4!}l_{4}^{T}(T^{\prime},T^{\prime},T^{\prime},T^{\prime})=0$,
which implies that $T^{\prime}$ is a Maurer-Cartan element of the twisted
$L_{\infty}$-algebra
$(C^{\ast}(M,L),l_{1}^{T},l_{2}^{T},l_{3}^{T},l_{4}^{T})$. ∎

The above characterization of a generalized Reynolds operator $T$ allows us to
define a cohomology associated to $T$. More precisely, we define
$C^{n}_{T}(M,L)=Hom(\underbrace{\otimes^{2}M\otimes\cdots\otimes(\otimes^{2}M)}_{n\geq
0}\wedge M,L)$, for $n\geq 0$ and the differential operator
$d_{T}:C^{n}_{T}(M,L)\to C^{n+1}_{T}(M,L)$ by

$\displaystyle
d_{T}(f)=l_{1}^{T}(f)=\frac{1}{2}l_{3}(T,T,f)+\frac{1}{6}l_{4}(T,T,T,f).$
(4.18)

The corresponding cohomology groups are

$H^{n}_{T}(M,L)=\frac{Z^{n}_{T}(M,L)}{B^{n}_{T}(M,L)}=\frac{\\{f\in
C^{n}_{T}(M,L)|d_{T}(f)=0\\}}{\\{d_{T}(g)|g\in C^{n-1}_{T}(M,L)\\}}.$

###  4.2 Yamaguti cohomology

Let $T:M\to L$ be a generalized Reynolds operator on a L.t.sRep pair
$(L,[\cdot,\cdot,\cdot],\theta)$. Once a L.t.s structure on the representation
space $M$ is given, we construct a representation of the representation space
(viewed as a L.t.s ) on the L.t.sRep pair (viewed as a vector space) as
follow: Define the linear map $\theta_{T}:\otimes^{2}M\to End(L)$ by

$\theta_{T}(u,v)x=[x,Tu,Tv]-T\Big{(}D(x,Tu)v-\theta(x,Tv)u+\mathcal{H}(x,Tu,Tv)\Big{)}\quad\forall
u,v\in M~{}and~{}x\in L.$ (4.19)

######  Proposition 4.7.

With the above notations, $(L,\theta_{T})$ is a representation of the L.t.s
$(M,[\cdot,\cdot,\cdot]_{T}).$

###### Proof.

Note that

$\displaystyle D_{T}(u,v)x=$ $\displaystyle\theta_{T}(v,u)x-\theta_{T}(u,v)x$
$\displaystyle=$
$\displaystyle[Tu,Tv,x]-T(\theta(Tv,x))u-\theta(Tu,x))v+\mathcal{H}(Tu,Tv,x))\quad\forall
u,v\in M~{}and~{}x\in L.$

According to Eqs. (2.1)-(2.5), (2.10)- (2.11) and (3.8)-(3.12), for all
$u_{i}\in M,1\leq i\leq 4$ and $x\in L$, we get

$\displaystyle\theta_{T}(u_{3},u_{4})\theta_{T}(u_{1},u_{2})-\theta_{T}(u_{2},u_{4})\theta_{T}(u_{1},u_{3})-\theta_{T}(u_{1},[u_{2},u_{3},u_{4}]_{T})+D_{T}(u_{2},u_{3})\theta_{T}(u_{1},u_{4})x$
$\displaystyle=$
$\displaystyle[[x,Tu_{1},Tu_{2}],Tu_{3},Tu_{4}]+[Tu_{2},[x,Tu_{1},Tu_{3}],Tu_{4}]+[Tu_{2},Tu_{3},[x,Tu_{1},Tu_{4}]]$
$\displaystyle-[x,Tu_{1},[Tu_{2},Tu_{3},Tu_{4}]]+T\Big{(}\theta(Tu_{3},Tu_{4})\theta(x,Tu_{2})u_{1}-\theta(Tu_{3},Tu_{4})\mathcal{H}(x,Tu_{1},Tu_{2})$
$\displaystyle-\theta(Tu_{3},Tu_{4})D(x,Tu_{1})u_{2}+\theta(Tu_{2},Tu_{4})\mathcal{H}(x,Tu_{1},Tu_{3})-\theta(Tu_{2},Tu_{4})\theta(x,Tu_{3})u_{1}$
$\displaystyle+\theta(Tu_{2},Tu_{4})D(x,Tu_{1})u_{3}+D(x,Tu_{1})\theta(Tu_{3},Tu_{4})u_{2}-D(x,Tu_{1})\theta(Tu_{2},Tu_{4})u_{3}$
$\displaystyle+D(x,Tu_{1})D(Tu_{2},Tu_{3})u_{4}+D(x,Tu_{1})\mathcal{H}(Tu_{2},Tu_{3},Tu_{4})+\mathcal{H}(x,Tu_{1},[Tu_{2},Tu_{3},Tu_{4}])$
$\displaystyle-\theta(x,[Tu_{2},Tu_{3},Tu_{4}])u_{1}+D(Tu_{3},Tu_{2})\mathcal{H}(x,Tu_{1},Tu_{4})+D(Tu_{2},Tu_{3})\theta(x,Tu_{4})u_{1}$
$\displaystyle+D(Tu_{3},Tu_{2})D(x,Tu_{1})u_{2}-D([x,Tu_{1},Tu_{2}],Tu_{3})u_{4}+D([x,Tu_{1},Tu_{2}],Tu_{4})u_{3}$
$\displaystyle+\theta([x,Tu_{1},Tu_{2}],Tu_{4})u_{3}-\theta([x,Tu_{1},Tu_{3}],Tu_{4})u_{2}-D([x,Tu_{1},Tu_{4}],Tu_{3})u_{2}$
$\displaystyle+\theta([x,Tu_{1},Tu_{4}],Tu_{2})u_{3}+D([x,Tu_{1},Tu_{4}],Tu_{2})u_{3}-\theta([x,Tu_{1},Tu_{4}],Tu_{3})u_{2}\Big{)}$
$\displaystyle=$ $\displaystyle
T\Big{(}\mathcal{H}(x,Tu_{1},[Tu_{2},Tu_{3},Tu_{4}])$
$\displaystyle+D(x,Tu_{1})\mathcal{H}(Tu_{2},Tu_{3},Tu_{4})-\mathcal{H}([x,Tu_{1},Tu_{2}],Tu_{3},Tu_{4})$
$\displaystyle-\mathcal{H}(Tu_{2},[x,Tu_{1},Tu_{3}],Tu_{4})-\mathcal{H}(Tu_{2},Tu_{3},[x,Tu_{1},Tu_{4}])-\theta(Tu_{3},Tu_{4})\mathcal{H}(x,Tu_{1},Tu_{4})$
$\displaystyle+\theta(Tu_{2},Tu_{4})\mathcal{H}(x,Tu_{1},Tu_{3})-D(Tu_{2},Tu_{3})\mathcal{H}(x,Tu_{1},Tu_{4})\Big{)}$
$\displaystyle=$ $\displaystyle
T(\delta^{3}\mathcal{H}(x,Tu_{1},Tu_{2},Tu_{3},Tu_{4}))=0.$

Similarly, we have

$\displaystyle\theta_{T}(u_{3},u_{4})D_{T}(u_{1},u_{2})-D_{T}(u_{1},u_{2})\theta_{T}(u_{3},u_{4})+\theta_{T}([u_{1},u_{2},u_{3}]_{T},u_{4})+\theta_{T}(u_{3},[u_{1},u_{2},u_{4}]_{T})x$
$\displaystyle=$
$\displaystyle[[Tu_{1},Tu_{2},x],Tu_{3},Tu_{4}]+[x,[Tu_{1},Tu_{2},Tu_{3}],Tu_{4}]+[x,Tu_{3},[Tu_{1},Tu_{2},Tu_{4}]]$
$\displaystyle-[Tu_{1},Tu_{2},[x,Tu_{3},Tu_{4}]]+T\Big{(}\theta(Tu_{3},Tu_{4})\theta(Tu_{1},x)u_{2}-\theta(Tu_{3},Tu_{4})\theta(Tu_{2},x)u_{1}$
$\displaystyle-\theta(Tu_{3},Tu_{4})\mathcal{H}(Tu_{1},Tu_{2},x)-\mathcal{H}([Tu_{1},Tu_{2},x],Tu_{3},Tu_{4})+D(Tu_{1},Tu_{2})\mathcal{H}(x,Tu_{3},Tu_{4})$
$\displaystyle-D(Tu_{1},Tu_{2})\theta(x,Tu_{4})u_{3}+D(Tu_{1},Tu_{2})D(x,Tu_{3})u_{4}+\mathcal{H}(Tu_{1},Tu_{2},[x,Tu_{3},Tu_{4}])$
$\displaystyle+\theta(x,Tu_{4})\theta(Tu_{2},Tu_{3})u_{1}-\theta(x,Tu_{4})\theta(Tu_{1},Tu_{3})u_{2}+\theta(x,Tu_{4})D(Tu_{1},Tu_{2})u_{3}$
$\displaystyle+\theta(x,Tu_{4})\mathcal{H}(Tu_{1},Tu_{2},Tu_{3})-D(x,[Tu_{1},Tu_{2},Tu_{3}])u_{4}-\mathcal{H}(x,[Tu_{1},Tu_{2},Tu_{3}],Tu_{4})$
$\displaystyle+\theta(x,[Tu_{1},Tu_{2},Tu_{4}])u_{3}-D(x,Tu_{3})\theta(Tu_{2},Tu_{4})u_{1}+D(x,Tu_{3})\theta(Tu_{1},Tu_{4})u_{2}$
$\displaystyle-D(x,Tu_{3})D(Tu_{1},Tu_{2})u_{4}-D(x,Tu_{3})\mathcal{H}(Tu_{1},Tu_{2},Tu_{1})-\mathcal{H}(x,Tu_{3},[Tu_{1},Tu_{2},Tu_{4}])$
$\displaystyle-D([Tu_{1},Tu_{2},x],Tu_{3})u_{4}+\theta([Tu_{1},Tu_{2},x],Tu_{4})u_{3}-\theta(Tu_{2},[x,Tu_{3},Tu_{4}])u_{1}$
$\displaystyle-\theta(Tu_{1},[x,Tu_{3},Tu_{4}])u_{2}\Big{)}$ $\displaystyle=$
$\displaystyle
T\Big{(}\mathcal{H}(Tu_{1},Tu_{2},[x,Tu_{3},Tu_{4}])-\mathcal{H}([Tu_{1},Tu_{2},x],Tu_{3},Tu_{4})+D(Tu_{1},Tu_{2})\mathcal{H}(x,Tu_{3},Tu_{4})$
$\displaystyle-\mathcal{H}(x,[Tu_{1},Tu_{2},Tu_{3}],Tu_{4})-\mathcal{H}(x,Tu_{3},[Tu_{1},Tu_{2},Tu_{4}])-\theta(Tu_{3},Tu_{4})\mathcal{H}(Tu_{1},Tu_{2},x)$
$\displaystyle+\theta(x,Tu_{4})\mathcal{H}(Tu_{1},Tu_{2},Tu_{3})-D(x,Tu_{3})\mathcal{H}(Tu_{1},Tu_{2},Tu_{4})\Big{)}$
$\displaystyle=$ $\displaystyle
T(\delta^{3}\mathcal{H}(Tu_{1},Tu_{2},x,Tu_{3},Tu_{4}))=0.$

Therefore, we deduce that $\theta_{T}$ is a representation of the L.t.s
$(M,[\cdot,\cdot,\cdot]_{T})$ on $L$. ∎

It follows from the above Proposition that we may consider The Yamaguti
cohomology of L.t.s $(M,[\cdot,\cdot,\cdot]_{T})$ with coefficients in the
representation $(L,\theta_{T})$. More precisely, for each $n\geqslant 0$, we
denote by $\mathfrak{C}^{2n+1}(M,L)$ the Yamaguti $(2n+1)$-cochains of $M$
with coefficients in $L$, that a $(2n+1)$-cochain
$\varphi\in\mathfrak{C}^{2n+1}(M,L)$ is a linear map of $\otimes^{2n+1}M$ into
$L$ satisfying

$\displaystyle\varphi(v_{1},v_{2},\cdots,v_{2n-2},v,v,u)=0,$
$\displaystyle\circlearrowleft_{u,v,w}\varphi(v_{1},v_{2},\cdots,v_{2n-2},u,v,w)=0.$

Define the corresponding coboundary operator
$\delta^{2n-1}_{T}:\mathfrak{C}^{2n-1}(M,L)\to\mathfrak{C}^{2n+1}(M,L)$ by

$\displaystyle\delta^{2n-1}_{T}\varphi(v_{1},v_{2},\cdots,v_{2n+1})$
$\displaystyle=$
$\displaystyle\theta_{T}(v_{2n},v_{2n+1})\varphi(v_{1},v_{2},\cdots,v_{2n-1})-\theta_{T}(v_{2n-1},v_{2n+1})\varphi(v_{1},v_{2},\cdots,v_{2n-2},v_{2n})$
$\displaystyle+\sum_{k=1}^{n}(-1)^{n+k}D_{T}(v_{2k-1},v_{2k})\varphi(v_{1},v_{2},\cdots,\widehat{v}_{2k-1},\widehat{v}_{2k},\cdots,v_{2n+1})$
$\displaystyle+\sum_{k=1}^{n}\sum_{j=2k+1}^{2n+1}(-1)^{n+k+1}\varphi(v_{1},v_{2},\cdots,\widehat{v}_{2k-1},\widehat{v}_{2k},\cdots,[v_{2k-1},v_{2k},v_{j}]_{T},\cdots,v_{2n+1})$
$\displaystyle=$
$\displaystyle[\varphi(v_{1},v_{2},\cdots,v_{2n-1}),Tv_{2n},Tv_{2n+1}]-TD(\varphi(v_{1},v_{2},\cdots,v_{2n-1}),Tv_{2n})v_{2n+1}$
$\displaystyle+T\theta(\varphi(v_{1},v_{2},\cdots,v_{2n-1}),Tv_{2n+1})v_{2n}-T\mathcal{H}(\varphi(v_{1},v_{2},\cdots,v_{2n-1}),Tv_{2n},Tv_{2n+1})$
$\displaystyle-[\varphi(v_{1},v_{2},\cdots,v_{2n-2},v_{2n}),Tv_{2n-1},Tv_{2n+1}]+TD(\varphi(v_{1},\cdots,v_{2n-2},v_{2n}),Tv_{2n-1})v_{2n+1}$
$\displaystyle-T\theta(\varphi(v_{1},\cdots,v_{2n-2},v_{2n}),Tv_{2n+1})v_{2n-1}+T\mathcal{H}(\varphi(v_{1},\cdots,v_{2n-2},v_{2n}),Tv_{2n-1},Tv_{2n+1})$
$\displaystyle+\sum_{k=1}^{n}(-1)^{n+k}\Big{(}[T(v_{2k-1},Tv_{2k},\varphi(v_{1},\cdots,\widehat{v}_{2k-1},\widehat{v}_{2k},\cdots,v_{2n+1})]$
$\displaystyle\quad-T\theta(Tv_{2k},\varphi(v_{1},\cdots,\widehat{v}_{2k-1},\widehat{v}_{2k},\cdots,v_{2n+1})v_{2k-1}+T\theta(Tv_{2k-1},\varphi(v_{1},\cdots,\widehat{v}_{2k-1},\widehat{v}_{2k},\cdots,v_{2n+1})v_{2k}$
$\displaystyle\quad-T\mathcal{H}(Tv_{2k-1},Tv_{2k},\varphi(v_{1},\cdots,\widehat{v}_{2k-1},\widehat{v}_{2k},\cdots,v_{2n+1}))\Big{)}$
$\displaystyle+\sum_{k=1}^{n}\sum_{j=2k+1}^{2n+1}(-1)^{n+k+1}\Big{(}\varphi(v_{1},\cdots,\widehat{v}_{2k-1},\widehat{v}_{2k},\cdots,\theta(Tv_{2k},Tv_{j})v_{2k-1}-\theta(Tv_{2k-1},Tv_{j})v_{2k},\cdots,v_{2n+1})$
$\displaystyle\quad+\varphi(v_{1},v_{2},\cdots,\widehat{v}_{2k-1},\widehat{v}_{2k},\cdots,D(Tv_{2k-1},Tv_{2k})v_{j}+\mathcal{H}(Tv_{2k-1},Tv_{2k},Tv_{j}),\cdots,v_{2n+1})\Big{)}.$

With this coboundary operator the Yamaguti cochain forms a complex

$\mathfrak{C}^{1}(M,L)\overset{\delta_{T}^{1}}{\longrightarrow}\mathfrak{C}^{3}(M,L)\overset{\delta_{T}^{3}}{\longrightarrow}\mathfrak{C}^{5}(M,L)\longrightarrow\cdots,$

such that $\delta_{T}^{2n+1}\circ\delta_{T}^{2n-1}=0$ for all $n\geq 1$. In
particular, a $1$-cochain $\varphi\in\mathfrak{C}^{1}(M,L)$ is $1$-cocycle if

$\displaystyle T\mathcal{H}(\varphi(v_{1}),Tv_{2},Tv_{3})$ $\displaystyle=$
$\displaystyle[\varphi(v_{1}),Tv_{2},Tv_{3}]+[Tv_{1},\varphi(v_{2}),Tv_{3}]+[Tv_{1},Tv_{2},\varphi(v_{3})]+T\theta(\varphi(v_{1}),Tv_{3})v_{2}$
$\displaystyle-
TD(\varphi(v_{1}),Tv_{2})v_{3}+TD(\varphi(v_{2}),Tv_{1})v_{3}-T\theta(\varphi(v_{2}),Tv_{3})v_{1}+T\mathcal{H}(\varphi(v_{2}),Tv_{1},Tv_{3})$
$\displaystyle-
TD(\varphi(v_{1}),Tv_{2})v_{3}-T\theta(Tv_{2},\varphi(v_{3}))v_{1}+T\theta(Tv_{1},\varphi(v_{3}))v_{2}-T\mathcal{H}(Tv_{1},Tv_{2},\varphi(v_{3}))$
$\displaystyle-\varphi(\theta(Tv_{2},Tv_{3})v_{1})-\varphi(\theta(Tv_{1},Tv_{3})v_{2})+\varphi(D(Tv_{1},Tv_{2})v_{3})+\varphi(\mathcal{H}(Tv_{1},Tv_{2},Tv_{3}).$
(4.20)

For all $\chi=(x_{1},x_{2})\in L\otimes L$, we define
$\partial_{T}(\chi):M\rightarrow L$ by

$\partial_{T}(\chi)u=T\Big{(}D(\chi)u+\mathcal{H}(\chi,Tu)\Big{)}-[\chi,Tu]\quad\forall
u\in M.$ (4.21)

######  Proposition 4.8.

Let $T$ be a generalized Reynolds operator on a L.t.sRep pair
$(L,[\cdot,\cdot,\cdot],\theta)$. Then $\partial_{T}(\chi)$ is a $1$-cocycle
on the L.t.s $(M,[\cdot,\cdot,\cdot]_{T})$ with coefficients in
$(L,\theta_{T})$.

###### Proof.

For any $u_{1},u_{2},u_{3}\in M$, we have:

$\displaystyle\delta^{1}_{T}\circ\partial_{T}(\chi)(u_{1},u_{2},u_{3})$
$\displaystyle=$
$\displaystyle[TD(\chi)u_{1},Tu_{2},Tu_{3}]+[Tu_{1},TD(\chi)u_{2},Tu_{3}]+[Tu_{1},Tu_{2},TD(\chi)u_{3}]+[T\mathcal{H}(\chi,Tu_{1}),Tu_{2},Tu_{3}]$
$\displaystyle+[Tu_{1},T\mathcal{H}(\chi,Tu_{2}),Tu_{3}]+[Tu_{1},Tu_{2},T\mathcal{H}(\chi,Tu_{3})]-[[\chi,Tu_{1}],Tu_{2},Tu_{3}]$
$\displaystyle-[Tu_{1},[\chi,Tu_{2}],Tu_{3}]-[Tu_{1},Tu_{2},[\chi,Tu_{3}]]+[\chi,[Tu_{1},Tu_{2},Tu_{3}]]$
$\displaystyle-T\Big{(}D(TD(\chi)u_{1},Tu_{2})u_{3}+D(T\mathcal{H}(\chi)u_{1},Tu_{2})u_{3}-D([\chi,Tu_{1}],Tu_{2})u_{3}$
$\displaystyle-\theta(TD(\chi)u_{1},Tu_{3})u_{2}-\theta(T\mathcal{H}(\chi)u_{1},Tu_{3})u_{2}+\theta([\chi,Tu_{1}],Tu_{3})u_{2}$
$\displaystyle+\mathcal{H}(TD(\chi)u_{1},Tu_{2},Tu_{3})+\mathcal{H}(T\mathcal{H}(\chi,Tu_{1}),Tu_{2},Tu_{3})-\mathcal{H}([\chi,Tu_{1}],Tu_{2},Tu_{3})$
$\displaystyle-D(TD(\chi)u_{2},Tu_{1})u_{3}-D(T\mathcal{H}(\chi,Tu_{2}),Tu_{1})u_{3}+D([\chi,Tu_{2}],Tu_{1})u_{3}$
$\displaystyle+\theta(TD(\chi)u_{2},Tu_{3})u_{1}+\theta(T\mathcal{H}(\chi,Tu_{2}),Tu_{3})u_{1}-\theta([\chi,Tu_{2}],Tu_{3})u_{1}$
$\displaystyle+\mathcal{H}(Tu_{1},TD(\chi)u_{2},Tu_{3})-\mathcal{H}(Tu_{1},T\mathcal{H}(\chi,Tu_{2}),Tu_{1})+\mathcal{H}(Tu_{1},[\chi,Tu_{2}],Tu_{3})$
$\displaystyle+\theta(Tu_{2},TD(\chi)u_{3})u_{1}+\theta(Tu_{2},T\mathcal{H}(\chi,Tu_{3}))u_{1}-\theta(Tu_{2},[\chi,Tu_{3}])u_{1}$
$\displaystyle-\theta(Tu_{1},TD(\chi)u_{3})u_{2}-\theta(Tu_{1},T\mathcal{H}(\chi,Tu_{3}))u_{2}+\theta(Tu_{1},[\chi,Tu_{3}])u_{2}$
$\displaystyle+\mathcal{H}(Tu_{1},Tu_{2},TD(\chi)u_{3})+\mathcal{H}(Tu_{1},Tu_{2},T\mathcal{H}(\chi,Tu_{3}))-\mathcal{H}(Tu_{1},Tu_{2},[\chi,Tu_{3}])\Big{)}$
$\displaystyle-
TD(\chi)(\theta(Tu_{2},Tu_{3})u_{1}-\theta(Tu_{1},Tu_{3})u_{2}+D(Tu_{1},Tu_{2})u_{3}+\mathcal{H}(Tu_{1},Tu_{2},Tu_{3}))$
$\displaystyle-\mathcal{H}(\chi,[Tu_{1},Tu_{2},Tu_{3}])+[\chi,[Tu_{1},Tu_{2},Tu_{3}]]$
$\displaystyle\overset{\eqref{lts 2}\eqref{O op on lts}}{=}$ $\displaystyle
T\Big{(}\theta(Tu_{2},Tu_{3})D(\chi)u_{1}-\theta(Tu_{1},Tu_{3})D(\chi)u_{2}+D(Tu_{1},Tu_{2})D(\chi)u_{3}$
$\displaystyle-\theta(Tu_{1},Tu_{3})\mathcal{H}(\chi,Tu_{2})+D(Tu_{1},Tu_{2})\mathcal{H}(\chi,\mathcal{T}u_{3})+D([\chi,Tu_{1}],Tu_{2})u_{3}-\theta([\chi,Tu_{1}],Tu_{3})u_{2}$
$\displaystyle+\mathcal{H}([\chi,Tu_{1}],Tu_{2},Tu_{3})-D([\chi,Tu_{2}],Tu_{1})u_{3}+\theta([\chi,Tu_{2}],Tu_{2})u_{1}+\mathcal{H}(Tu_{1},[\chi,Tu_{2}],Tu_{3})$
$\displaystyle+\theta(Tu_{2},[\chi,Tu_{3}],)u_{1}-\theta(Tu_{1},[\chi,Tu_{3}],)u_{2}+\mathcal{H}(Tu_{1},Tu_{2},[\chi,Tu_{3}])-\mathcal{H}(\chi,[Tu_{1},Tu_{2},Tu_{3}])$
$\displaystyle-D(\chi)\theta(Tu_{2},Tu_{3})u_{1}+D(\chi)\theta(Tu_{1},Tu_{3})u_{2}-D(\chi)D(Tu_{1},Tu_{2})u_{3}-D(\chi)\mathcal{H}(Tu_{1},Tu_{2},Tu_{3}))\Big{)}$
$\displaystyle\overset{\eqref{2.12}}{=}$ $\displaystyle
T\Big{(}(\theta(Tu_{2},Tu_{3})D(\chi)+\theta(Tu_{2},[\chi,Tu_{3}])+\theta([\chi,Tu_{2}],Tu_{2})-D(\chi)\theta(Tu_{2},Tu_{3}))u_{1}$
$\displaystyle-(\theta(Tu_{1},Tu_{3})D(\chi)-\theta([\chi,Tu_{1}],Tu_{3})-\theta(Tu_{1},[\chi,Tu_{3}])+D(\chi)\theta(Tu_{1},Tu_{3}))u_{2}$
$\displaystyle+(D(Tu_{1},Tu_{2})D(\chi)+D([\chi,Tu_{1}],Tu_{2})-D([\chi,Tu_{2}],Tu_{1})-D(\chi)D(Tu_{1},Tu_{2}))u_{3}\Big{)}$
$\displaystyle\overset{\eqref{rep lts 2}\eqref{rep lts 3}}{=}$ $\displaystyle
0.$

Thus, we deduce that $\partial_{T}(\chi)$ is a $1$-cocycle. ∎

Define the set of $(2n-1)$-cochains by

$\mathcal{C}_{T}^{2n-1}(M,L)=\begin{cases}L\otimes L\quad\quad\quad\;\;\;\ if\
n=0,\\\ \mathfrak{C}^{2n-1}(M,L)\quad if\ n\geq 1.\end{cases}$ (4.22)

Define $\Lambda_{T}:\mathcal{C}_{T}^{2n-1}(M,L)\to\mathcal{C}_{T}^{2n+1}(M,L)$
by

$\Lambda_{T}=\begin{cases}\partial_{T}\;\;\;\ \ if\ n=0,\\\ \delta_{T}\quad\
if\ n\geq 1.\end{cases}$ (4.23)

Now we give the cohomology of generalized Reynolds operator on a L.t.sRep pair
$(L,[\cdot,\cdot,\cdot],\theta)$.

######  Definition 4.3.

Let $T$ be a generalized Reynolds operator on a L.t.sRep pair
$(L,[\cdot,\cdot,\cdot],\theta)$. Denote the set of cocycles by
${\mathcal{Z}}^{\ast}(M,L)$, the set of coboundaries by
${\mathcal{B}}^{\ast}(M,L)$ and the cohomology group by

${\mathbf{H}}_{T}^{\ast}(M,L)={\mathcal{Z}}^{\ast}(M,L)/{\mathcal{B}}^{\ast}(M,L).$

The cohomology groups correspond to the cohomology groups for the generalized
Reynolds operator $T$.

The coboundary operator $\Lambda_{T}$ coincides with the differential $d_{T}$
defined by (4.18) using the Maurer-Cartan element $T$ of the
$L_{\infty}$-algebra $(C^{\ast}(M,L),l_{3},l_{4})$.

######  Theorem 4.9.

Let $T$ be a generalized Reynolds operator on a L.t.sRep pair
$(L,[\cdot,\cdot,\cdot],\theta)$. Then we have

$d_{T}(f)=(-1)^{n-1}\Lambda_{T}(f)\quad\forall f\in
Hom(\underbrace{M\otimes\cdots\otimes M}_{2n-1},L),\;n\geq 1.$ (4.24)

###### Proof.

In [4], the authors showed that

$\displaystyle\frac{1}{2}l_{3}(T,T,f)(\mathfrak{U}_{1},\cdots,\mathfrak{U}_{n},u_{n+1})$
$\displaystyle=$
$\displaystyle[[[\pi+\theta,T]_{L.t.s},T]_{L.t.s},f]_{L.t.s}(\mathfrak{U}_{1},\cdots,\mathfrak{U}_{n},u_{n+1})$
$\displaystyle=$
$\displaystyle(-1)^{n-1}[[\pi+\theta,T]_{L.t.s},T]_{L.t.s}\Big{(}f(\mathfrak{U}_{1},\cdots,\mathfrak{U}_{n-1},u_{n}),v_{n},u_{n+1}\Big{)}$
$\displaystyle+(-1)^{n-1}[[\pi+\theta,T]_{L.t.s},T]_{L.t.s}\Big{(}u_{n},f(\mathfrak{U}_{1},\cdots,\mathfrak{U}_{n-1},v_{n}),u_{n+1}\Big{)}$
$\displaystyle+(-1)^{n-1}\sum_{i=1}^{n}(-1)^{n-1}(-1)^{i-1}[[\pi+\theta,T]_{L.t.s},T]_{L.t.s}\Big{(}\mathfrak{U}_{i},f(\mathfrak{U}_{1}\cdots,\widehat{\mathfrak{U}_{i}},\cdots,\mathfrak{U}_{n},u_{n+1})\Big{)}$
$\displaystyle-\sum_{k=1}^{n-1}\sum_{i=1}^{k}(-1)^{i+1}f\Big{(}\mathfrak{U}_{1}\cdots,\widehat{\mathfrak{U}_{i}},\cdots,\mathfrak{U}_{k},[[\pi+\theta,T]_{L.t.s},T]_{L.t.s}(\mathfrak{U}_{i},u_{k+1}),v_{k+1},\mathfrak{U}_{k+2},\cdots,\mathfrak{U}_{n},u_{n+1}\Big{)}$
$\displaystyle-\sum_{k=1}^{n-1}\sum_{i=1}^{k}(-1)^{i+1}f\Big{(}\mathfrak{U}_{1}\cdots,\widehat{\mathfrak{U}_{i}},\cdots,\mathfrak{U}_{k},u_{k+1},[[\pi+\theta,T]_{L.t.s},T]_{L.t.s}(\mathfrak{U}_{i},v_{k+1}),\mathfrak{U}_{k+2},\cdots,\mathfrak{U}_{n},u_{n+1}\Big{)}$
$\displaystyle-\sum_{i=1}^{n}f\Big{(}\mathfrak{U}_{1}\cdots,\widehat{\mathfrak{U}_{i}},\cdots,\mathfrak{U}_{n},[[\pi+\theta,T]_{L.t.s},T]_{L.t.s}(\mathfrak{U}_{i},u_{n+1})\Big{)}.$

Now, we add one argument and show that

$\displaystyle
l_{4}(T,T,T,f)(\mathfrak{U}_{1},\cdots,\mathfrak{U}_{n},u_{n+1})$
$\displaystyle=$
$\displaystyle[\mathcal{H},T]_{L.t.s},T]_{L.t.s},T]_{L.t.s},f]_{L.t.s}(\mathfrak{U}_{1},\cdots,\mathfrak{U}_{n},u_{n+1})$
$\displaystyle=$
$\displaystyle[\mathcal{H},T]_{L.t.s},T]_{L.t.s},T]_{L.t.s}(f(\mathfrak{U}_{1},\cdots,\mathfrak{U}_{n-1},u_{n}),v_{n},u_{n+1})$
$\displaystyle+[\mathcal{H},T]_{L.t.s},T]_{L.t.s},T]_{L.t.s}(u_{n},f(\mathfrak{U}_{1},\cdots,\mathfrak{U}_{n-1},v_{n}),u_{n+1})$
$\displaystyle+\sum_{i=1}^{n}(-1)^{n-1}(-1)^{i-1}[\mathcal{H},T]_{L.t.s},T]_{L.t.s},T]_{L.t.s}\Big{(}\mathfrak{U}_{i},f(\mathfrak{U}_{1},\cdots,\widehat{\mathfrak{U}_{i}},\cdots,\mathfrak{U}_{n},u_{n+1})\Big{)}$
$\displaystyle-(-1)^{n-1}\sum_{k=1}^{n-1}\sum_{i=1}^{k}(-1)^{i+1}f\Big{(}\mathfrak{U}_{1},\cdots,\widehat{\mathfrak{U_{i}}},\cdots,\mathfrak{U}_{k},[\mathcal{H},T]_{L.t.s},T]_{L.t.s},T]_{L.t.s}(\mathfrak{U}_{i},u_{k+1}),v_{k+1},\mathfrak{U}_{k+2},\cdots,\mathfrak{U}_{n},u_{n+1}\Big{)}$
$\displaystyle-(-1)^{n-1}\sum_{k=1}^{n-1}\sum_{i=1}^{k}(-1)^{i+1}f\Big{(}\mathfrak{U}_{1},\cdots,\widehat{\mathfrak{U_{i}}},\cdots,\mathfrak{U}_{k},u_{k+1},[\mathcal{H},T]_{L.t.s},T]_{L.t.s},T]_{L.t.s}(\mathfrak{U}_{i},v_{k+1}),\mathfrak{U}_{k+2},\cdots,\mathfrak{U}_{n},u_{n+1}\Big{)}$
$\displaystyle-(-1)^{n-1}\sum_{i=1}^{n}(-1)^{i+1}f\Big{(}\mathfrak{U}_{1},\cdots,\widehat{\mathfrak{U_{i}}},\cdots,\mathfrak{U}_{n},[\mathcal{H},T]_{L.t.s},T]_{L.t.s},T]_{L.t.s}(\mathfrak{U}_{i},u_{n+1})\Big{)}$
$\displaystyle=$
$\displaystyle(-1)^{n-1}6\Bigg{\\{}(-1)^{n+1}\Big{(}-T\mathcal{H}(f(\mathfrak{U}_{1},\cdots,\mathfrak{U}_{n-1},u_{n}),Tv_{n},Tu_{n+1})-T\mathcal{H}(f(\mathfrak{U}_{1},\cdots,\mathfrak{U}_{n-1},v_{n}),Tu_{n+1},Tu_{n})\Big{)}$
$\displaystyle-\sum_{j=1}^{n}(-1)^{j+1}T\mathcal{H}(f(\mathfrak{U}_{1},\cdots,\widehat{\mathfrak{U}_{j}},\cdots,\mathfrak{U}_{n},u_{n+1}),Tu_{j},T_{j})$
$\displaystyle+\sum_{j=1}^{n}(-1)^{j}f\Big{(}\mathfrak{U}_{1},\cdots,\widehat{\mathfrak{U}_{j}},\cdots,\mathfrak{U}_{n},\mathcal{H}(Tu_{j},Tv_{j},Tu_{n+1})\Big{)}$
$\displaystyle+\sum_{1\leq j<k\leq
n}(-1)^{j}f\Big{(}\mathfrak{U}_{1},\cdots,\widehat{\mathfrak{U}_{j}},\cdots,\mathfrak{U}_{k-1},\mathcal{H}(Tu_{j},Tv_{j},Tu_{k}),v_{k}+u_{k},\mathcal{H}(Tu_{j},Tv_{j},Tv_{k}),\mathfrak{U}_{k+1},\cdots,\mathfrak{U}_{n},u_{n+1}\Big{)}\Bigg{\\}}.$

Hence
$d_{T}(f)=\frac{1}{2}l_{3}(T,T,f)+\frac{1}{6}l_{4}(T,T,T,f)=(-1)^{n-1}\partial_{T}(f)$.
The proof is finished. ∎

######  Proposition 4.10.

Let $T$ and $T^{{}^{\prime}}$ be two generalized Reynolds operators on a
L.t.sRep pair $(L,[\cdot,\cdot,\cdot],\theta)$ and $(\phi,\psi)$ be a morphism
from $T$ to $T^{{}^{\prime}}$. Then

  * •

$\psi$ is a L.t.s morphism from $(M,[\cdot,\cdot,\cdot]_{T})$ to
$(M,[\cdot,\cdot,\cdot]_{T^{{}^{\prime}}})$.

  * •

The following diagram commute for all $u,v\in M$,

$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{T}(u,v)}$$\scriptstyle{\phi}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{T^{{}^{\prime}}}(\psi(u),\psi(v))}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{L}$

where $(L,\theta_{T})$ is the induced representation of the L.t.s
$(M,[\cdot,\cdot,\cdot]_{T})$ and $(L,\theta_{T^{{}^{\prime}}})$ is the
induced representation of the L.t.s $(M,[\cdot,\cdot,\cdot]_{T}^{\prime})$.

Motivated by Proposition 3.8, we show the relevance of our cohomology theory.
Let $T$ and $T^{{}^{\prime}}$ be two generalized Reynolds operators on $L$
with respect to a representation $(M,\theta)$ and $(\phi,\psi)$ be a morphism
from $T$ to $T^{{}^{\prime}}$ such that $\psi$ is invertible. Denote by
$C^{2n-1}_{T}(M,L)$ the space of $(2n-1)$-cochains of a L.t.s
$(M,[\cdot,\cdot,\cdot]_{T})$ with coefficients in a representation
$(L,\theta_{T})$. Define

$\begin{cases}\Theta:L\otimes L\rightarrow L\otimes
L\quad\quad\quad\quad\quad\quad\quad\quad\;\;\;\ if\ n=0,\\\
\Theta:C^{2n-1}_{T}(M,L)\rightarrow
C^{2n-1}_{T^{{}^{\prime}}}(M,L)\quad\quad\quad if\ n\geq 1.\end{cases}$

by

$\begin{cases}\Theta(\chi)=\phi(\chi)=(\phi(x_{1}),\phi(x_{2}))\quad\quad\quad\quad\quad\forall\
\chi=(x_{1},x_{2})\in L\otimes L\quad\;\;\;\ \ if\ n=0,\\\
\Theta(\varphi)(u_{1},\cdots,u_{2n-1})=\phi(\varphi(\psi^{-1}(u_{1}),\cdots,\psi^{-1}(u_{2n-1})))\;\
\forall u_{i}\in M\quad\quad\ if\ n\geq 1.\end{cases}$

######  Theorem 4.11.

With the above notations, $\Theta$ is a cochain map from the cochain complex
$(C^{\star}_{T}(M,L),\Lambda_{T})$ to the cochain complex
$(C^{\star}_{T^{{}^{\prime}}}(M,L),\Lambda_{T^{{}^{\prime}}})$. Consequently,
it induces a morphism $\overline{\Theta}$ from the cohomology group
${\mathbf{H}}_{T}^{\star}(M,L)$ to ${\mathbf{H}}_{T^{\prime}}^{\star}(M,L)$.

###### Proof.

For $n=0$, let $\chi\in L\otimes L$ and $(\phi,\psi)$ be a morphism from $T$
to $T^{{}^{\prime}}$. Applying Eqs (3.9)-(3.11), we obtain

$\displaystyle\Theta\circ\partial_{T}(\chi)u=$
$\displaystyle\Phi(\partial_{T}(\chi)(\psi^{-1}(u))$ $\displaystyle=$
$\displaystyle\phi\circ
T\Big{(}D(\chi)(\psi^{-1}(u))+\mathcal{H}(\chi,T(\psi^{-1}(u)))\Big{)}-\phi[\chi,T(\psi^{-1}(u))]$
$\displaystyle=$ $\displaystyle
T^{{}^{\prime}}\circ\psi\Big{(}D(\chi)(\psi^{-1}(u))+\mathcal{H}(\chi,T(\psi^{-1}(u)))\Big{)}-[\phi(\chi),\phi\circ
T(\psi^{-1}(u))]$ $\displaystyle=$ $\displaystyle
T^{{}^{\prime}}\Big{(}D(\phi(\chi))\psi(\psi^{-1}(u))+\mathcal{H}(\phi(\chi),\phi\circ
T(\psi^{-1}(u)))\Big{)}-[\phi(\chi),T^{{}^{\prime}}\circ\psi(\psi^{-1}(u))]$
$\displaystyle=$ $\displaystyle
T^{{}^{\prime}}\Big{(}D(\phi(\chi))u+\mathcal{H}(\phi(\chi),T^{{}^{\prime}}u))\Big{)}-[\phi(\chi),T^{{}^{\prime}}u]$
$\displaystyle=$ $\displaystyle
T^{{}^{\prime}}\Big{(}D(\Theta(\chi))u+\mathcal{H}(\Theta(\chi),T^{{}^{\prime}}u))\Big{)}-[\Theta(\chi),T^{{}^{\prime}}u]$
$\displaystyle=$ $\displaystyle\partial_{T}^{\prime}(\Theta(\chi))u.$

For $n\geq 1$, let $\varphi\in\mathcal{C}_{T}^{2n-1}(M,L)$, we have

$\displaystyle\delta_{T}^{\prime}\Theta(\varphi)(v_{1},v_{2},\cdots,v_{2n+1})$
$\displaystyle=$
$\displaystyle\theta_{T}^{\prime}(v_{2n},v_{2n+1})\Theta(\varphi)(v_{1},v_{2},\cdots,v_{2n-1})-\theta_{T}^{\prime}(v_{2n-1},v_{2n+1})\Theta(\varphi)(v_{1},v_{2},\cdots,v_{2n-2},v_{2n})$
$\displaystyle+\sum_{k=1}^{n}(-1)^{n+k}D_{T}^{\prime}(v_{2k-1},v_{2k})\Theta(\varphi)(v_{1},v_{2},\cdots,\widehat{v}_{2k-1},\widehat{v}_{2k},\cdots,v_{2n+1})$
$\displaystyle+\sum_{k=1}^{n}\sum_{j=2k+1}^{2n+1}(-1)^{n+k+1}\Theta(\varphi)(v_{1},v_{2},\cdots,\widehat{v}_{2k-1},\widehat{v}_{2k},\cdots,[v_{2k-1},v_{2k},v_{j}]_{T}^{\prime},\cdots,v_{2n+1})$
$\displaystyle=$
$\displaystyle\theta_{T}^{\prime}(v_{2n},v_{2n+1})\phi(\varphi(\psi^{-1}(v_{1}),\cdots,\psi^{-1}(v_{2n-1}))-\theta_{T}^{\prime}(v_{2n-1},v_{2n+1})\phi(\varphi(\psi^{-1}(v_{1}),\cdots,\psi^{-1}(v_{2n}))$
$\displaystyle+\sum_{k=1}^{n}(-1)^{n+k}D_{T}^{\prime}(v_{2k-1},v_{2k})\phi(\varphi(\psi^{-1}(v_{1}),\cdots,\widehat{\psi^{-1}(v_{2k-1})},\widehat{\psi^{-1}(v_{2k})},\cdots,\psi^{-1}(v_{2n+1}))$
$\displaystyle+\sum_{k=1}^{n}\sum_{j=2k+1}^{2n+1}(-1)^{n+k+1}\phi(\varphi(\psi^{-1}(v_{1}),\cdots,[\psi^{-1}(v_{2k-1}),\psi^{-1}(v_{2k}),\psi^{-1}(v_{j})]_{T}^{\prime},\cdots,\psi^{-1}(v_{2n+1}))$
$\displaystyle=$
$\displaystyle\theta_{T}^{\prime}(\psi\circ\psi^{-1}(v_{2n}),\psi\circ\psi^{-1}(v_{2n+1}))\phi(\varphi(\psi^{-1}(v_{1}),\cdots,\psi^{-1}(v_{2n-1}))$
$\displaystyle-\theta_{T}^{\prime}(\psi\circ\psi^{-1}(v_{2n-1}),\psi\circ\psi^{-1}(v_{2n+1}))\phi(\varphi(\psi^{-1}(v_{1}),\cdots,\psi^{-1}(v_{2n}))$
$\displaystyle+\sum_{k=1}^{n}(-1)^{n+k}D_{T}^{\prime}(\psi\circ\psi^{-1}(v_{2k-1}),\psi\circ\psi^{-1}(v_{2k}))\phi(\varphi(\psi^{-1}(v_{1}),\cdots,\psi^{-1}(v_{2n+1}))$
$\displaystyle+\sum_{k=1}^{n}\sum_{j=2k+1}^{2n+1}(-1)^{n+k+1}\phi(\varphi(\psi^{-1}(v_{1}),\cdots,[\psi^{-1}(v_{2k-1}),\psi^{-1}(v_{2k}),\psi^{-1}(v_{j})]_{T},\cdots,\psi^{-1}(v_{2n+1}))$
$\displaystyle=$
$\displaystyle\phi\Big{(}\theta_{T}(\psi^{-1}(v_{2n}),\psi^{-1}(v_{2n+1}))\varphi(\psi^{-1}(v_{1}),\cdots,\psi^{-1}(v_{2n-1}))$
$\displaystyle-\theta_{T}(\psi^{-1}(v_{2n-1}),\Psi^{-1}(v_{2n+1}))\varphi(\psi^{-1}(v_{1}),\cdots,\psi^{-1}(v_{2n}))$
$\displaystyle+\sum_{k=1}^{n}(-1)^{n+k}D_{T}(\psi^{-1}(v_{2k-1}),\psi^{-1}(v_{2k}))\varphi(\psi^{-1}(v_{1}),\cdots,\psi^{-1}(v_{2n+1}))$
$\displaystyle+\sum_{k=1}^{n}\sum_{j=2k+1}^{2n+1}(-1)^{n+k+1}\varphi(\psi^{-1}(v_{1}),\cdots,[\psi^{-1}(v_{2k-1}),\psi^{-1}(v_{2k}),\psi^{-1}(v_{j})]_{T},\cdots,\psi^{-1}(v_{2n+1}))\Big{)}$
$\displaystyle=$
$\displaystyle\phi\Big{(}\delta_{T}\varphi(\psi^{-1}(v_{1}),\psi^{-1}(v_{2}),\cdots,\psi^{-1}(v_{2n+1}))\Big{)}$
$\displaystyle=$
$\displaystyle\Theta\Big{(}\delta_{T}\varphi\Big{)}(v_{1},v_{2},\cdots,v_{2n+1}).$

The proof is finished. ∎

##  5 Obstruction class of a generalized Reynolds operator

In this section, we will use the Yamaguti cohomology theory constructed in the
previous section to investigate formal deformations of generalized Reynolds
operator on a L.t.sRep pair by introducing a special cohomology class
associated to an order $n$ deformation. We show that a deformation of order
$n$ is extendable if and only if this cohomology class in the third cohomology
group is trivial. Thus we call this cohomology class the obstruction class of
a deformation of order $n$ being extendable.

Let $\mathbb{K}[[\lambda]]$ be the ring of power series in one variable
$\lambda$. For any $\mathbb{K}$-linear space $M$, we denote by $M[[\lambda]]$
the vector space of formal power series in $\lambda$ with coefficients in $M$.
If in addition, we have a structure of L.t.s $(L,[\cdot,\cdot,\cdot])$ over
$\mathbb{K}$, then there is a L.t.s structure over the ring
$\mathbb{K}[[\lambda]]$ on $L[[\lambda]]$ given by

$\Big{[}\sum_{i=0}^{+\infty}\lambda^{i}x_{i},\sum_{j=0}^{+\infty}\lambda^{j}y_{j},\sum_{k=0}^{+\infty}\lambda^{k}z_{k}\Big{]}=\sum_{s=0}^{+\infty}\sum_{i+j+k=s}\lambda^{s}[x_{i},y_{j},z_{k}],\;\forall
x_{i},y_{j},z_{k}\in L.$ (5.1)

For any representation $(M,\theta)$ of $(L,[\cdot,\cdot,\cdot])$, there is a
natural representation of the L.t.s $L[[\lambda]]$ on the
$\mathbb{K}[[\lambda]]$-module $M[[\lambda]]$, which is given by

$\theta\Big{(}\sum_{i=0}^{+\infty}\lambda^{i}x_{i},\sum_{j=0}^{+\infty}\lambda^{j}y_{j}\Big{)}\Big{(}\sum_{k=0}^{+\infty}\lambda^{k}m_{k}\Big{)}=\sum_{s=0}^{+\infty}\sum_{i+j+k=s}\lambda^{s}\theta(x_{i},y_{j})m_{k},\;\forall
x_{i},y_{j}\in L,\;m_{k}\in M.$ (5.2)

Similarly, the $3$-cocycle $\mathcal{H}$ can be extended to a $3$-cocycle on
the L.t.s $L[[\lambda]]$ with coefficients in $M[\lambda]]$, denoted by the
same notation $\mathcal{H}$. Consider a power series

$T_{\lambda}=\sum_{i=0}^{+\infty}\lambda^{i}T_{i},\quad\;T_{i}\in\mathcal{C}_{T}^{1}(M,L),$
(5.3)

that is $T_{\lambda}\in\mathcal{C}_{T}^{1}(M,L[[\lambda]])$. Extend it to be a
$\mathbb{K}[[\lambda]]$-module map from $M[[\lambda]]$ to $L[[\lambda]]$ which
is still denoted by $T_{\lambda}$.

######  Definition 5.1.

If the power series
$T_{\lambda}=\displaystyle\sum_{i=0}^{+\infty}T_{i}\lambda^{i}$ with $T_{0}=T$
satisfies:

$\displaystyle[T_{\lambda}u,T_{\lambda}v,T_{\lambda}w]=T_{\lambda}\Big{(}D(T_{\lambda}u,T_{\lambda}v)w+\theta(T_{\lambda}v,T_{\lambda}w)u-\theta(T_{\lambda}u,T_{\lambda}w)v+\mathcal{H}(T_{\lambda}u,T_{\lambda}v,T_{\lambda}w)\Big{)},$

we say that it is a formal deformation of the generalized Reynolds operator
$T$.

######  Remark 5.1.

If $T_{\lambda}=T+\lambda T_{1}$ is a generalized Reynolds operator on a
L.t.sRep pair $(L,[\cdot,\cdot,\cdot],\theta)$, we say that $T_{1}$ generates
a one-parameter infinitesimal deformation of $T$.

Based on the relationship between generalized Reynolds operator and L.t.s
structure, we have:

######  Proposition 5.1.

Let $T_{\lambda}=\displaystyle\sum_{i=0}^{+\infty}\lambda^{i}T_{i}$ be a
formal deformation of a generalized Reynolds operator $T$ on a L.t.sRep pair
$(L,[\cdot,\cdot,\cdot],\theta)$. Then $[\cdot,\cdot,\cdot]_{T_{\lambda}}$
defined by

$[u,v,w]_{T_{\lambda}}=\displaystyle\sum_{k=0}^{+\infty}\sum_{i+j=k}\lambda^{k}\Big{(}D(T_{i}u,T_{j}v)w+\theta(T_{i}v,T_{j}w)u-\theta(T_{i}u,T_{j}w)v+\sum_{i+j+s=k}\mathcal{H}(T_{i}u,T_{j}v,T_{s}v)\Big{)}\quad\forall
u,v,w\in M,$

is a formal deformation of the associated L.t.s $(M,[\cdot,\cdot,\cdot]_{T})$
given in Corollary 3.10.

######  Definition 5.2.

Let $T_{\lambda}=\displaystyle\sum_{i=0}^{+\infty}\lambda^{i}T_{i}$ and
$T^{{}^{\prime}}_{\lambda}=\displaystyle\sum_{i=0}^{+\infty}\lambda^{i}T{{}^{\prime}}_{i}$
be two formal deformations of a generalized Reynolds operator on a L.t.sRep
pair $(L,[\cdot,\cdot,\cdot],\theta)$. They are said to be equivalent if there
exists an element $\chi\in L\wedge L$, $\phi_{i}\in gl(L)$ and $\psi_{i}\in
gl(M)$ $(i\geq 2)$, such that the pair

$\Big{(}\phi_{\lambda}=Id_{L}+\lambda[\chi,-]+\sum_{i=2}^{+\infty}\lambda^{i}\phi_{i},\quad\;\psi_{\lambda}=Id_{V}+\lambda
D(\chi)(-)+\lambda\mathcal{H}(\chi,-)+\sum_{i=2}^{+\infty}\lambda^{i}\psi_{i}\Big{)}$
(5.4)

is a morphism of generalized Reynolds operators from $T_{\lambda}$ to
$T^{{}^{\prime}}_{\lambda}$. In particular, a formal deformation $T_{\lambda}$
of a generalized Reynolds operator $T$ is said to be trivial if there exists
an element $\chi\in L\wedge L$, $\phi_{i}\in gl(L)$ and $\psi_{i}\in gl(M)$ (
$i\geq 2$) such that $(\phi_{\lambda},\psi_{\lambda})$ defined by Eq. (5.4)
gives an equivalence between $T_{\lambda}$ and $T$, with the latter regarded
as a deformation of itself.

######  Theorem 5.2.

If two formal deformations of a generalized Reynolds operator $T$ on a
L.t.sRep pair $(L,[\cdot,\cdot,\cdot],\theta)$ are equivalent, then their
infinitesimals are in the same cohomology class in
${\mathbf{H}}_{T}^{1}(M,L)$.

######  Definition 5.3.

Let $T$ be a generalized Reynolds operator on a L.t.sRep pair
$(L,[\cdot,\cdot,\cdot],\theta)$. If
$T_{\lambda}=\displaystyle\sum_{i=0}^{n}\lambda^{i}T_{i}$ defines a
$\mathbb{K}[[\lambda]]/\lambda^{n+1}$-module from $M[[\lambda]]/\lambda^{n+1}$
to the L.t.s $L[[\lambda]]/\lambda^{n+1}$ satisfying

$\displaystyle[T_{\lambda}u,T_{\lambda}v,T_{\lambda}w]=T_{\lambda}\Big{(}D(T_{\lambda}u,T_{\lambda}v)w+\theta(T_{\lambda}v,T_{\lambda}w)u-\theta(T_{\lambda}u,T_{\lambda}w)v+\mathcal{H}(T_{\lambda}u,T_{\lambda}v,T_{\lambda}w)\Big{)},$

we say that $T_{\lambda}$ is an order $n$ deformation of the generalized
Reynolds operator $T$.

######  Definition 5.4.

Let $T_{\lambda}=\displaystyle\sum_{i=0}^{n}\lambda^{i}T_{i}$ be an order $n$
deformation of a generalized Reynolds operator $T$ on a L.t.sRep pair
$(L,[\cdot,\cdot,\cdot],\theta)$. If there exists a $1$-cochain
$T_{n+1}\in\mathcal{C}_{T}^{1}(M,L)$ such that
$\widetilde{T}_{\lambda}=T_{\lambda}+\lambda^{n+1}T_{n+1}$ is an order $(n+1)$
deformation of $T$ then we say that $T_{\lambda}$ is extendable.

######  Theorem 5.3.

Let $T_{\lambda}=\displaystyle\sum_{i=0}^{n}\lambda^{i}T_{i}$ be an order $n$
deformation of $T$. Then $T_{\lambda}$ is extendable if and only if the
cohomology class $[Obs^{T}]\in{\mathbf{H}}_{T}^{3}(M,L)$ is trivial, where
$Obs^{T}$ is a $3$-cochain in $\mathcal{C}_{T}^{3}(M,L)$ defined by :

$\displaystyle Obs^{T}(u,v,w)$ (5.5) $\displaystyle=$
$\displaystyle\sum_{0\leq i,j,k\leq n+1\atop
i+j+k>n+1}[T_{i}u,T_{j}v,T_{k}w]-T_{i}\big{(}D(T_{j}u,T_{k}v)w+\theta(T_{j}v,T_{k}w)u-\theta(T_{j}u,T_{k}w)v+\mathcal{H}(T_{i}u,T_{j}v,T_{k}w)\Big{)}.$

###### Proof.

Let $\widetilde{T}_{\lambda}=\displaystyle\sum_{i=0}^{n+1}\lambda^{i}T_{i}$ be
the extension of $T_{\lambda}$, then for all $u,v,w\in M$

$\displaystyle[\mathcal{\widetilde{T}}_{\lambda}u,\mathcal{\widetilde{T}}_{\lambda}v,\mathcal{\widetilde{T}}_{\lambda}w]=\mathcal{\widetilde{T}}_{\lambda}\Big{(}D(\widetilde{T}_{\lambda}u,\widetilde{T}_{\lambda}v)w+\theta(\widetilde{T}_{\lambda}v,\widetilde{T}_{\lambda}w)u-\theta(\widetilde{T}_{\lambda}u,\widetilde{T}_{\lambda}w)v+\mathcal{H}(\widetilde{T}_{\lambda}u,\widetilde{T}_{\lambda}v,\widetilde{T}_{\lambda}w)\Big{)}.$

Expanding the equation and comparing the coefficients of $\lambda^{n}$ yields
that:

$\displaystyle[T_{n+1}u,Tv,Tw]+[Tu,T_{n+1}v,Tw]+[Tu,Tv,T_{n+1}w]-T\big{(}D(T_{n+1}u,Tv)w-\theta(T_{n+1}u,Tw)v$
$\displaystyle+\mathcal{H}(T_{n+1}u,Tv,Tw)+D(Tu,T_{n+1}v)w+\theta(T_{n+1}v,Tw)v+\mathcal{H}(Tu,T_{n+1}v,Tw)$
$\displaystyle+\theta(Tv,T_{n+1}w)u-\theta(Tu,T_{n+1}w)v+\mathcal{H}(Tu,Tv,T_{n+1}w)\big{)}$
$\displaystyle+\sum_{0\leq i,j,k\leq n+1\atop
i+j+k>n+1}[T_{i}u,T_{j}v,T_{k}w]-T_{i}\big{(}D(T_{j}u,T_{k}v)w+\theta(T_{j}v,T_{k}w)u-\theta(T_{j}u,T_{k}w)v+\mathcal{H}(T_{i}u,T_{j}v,T_{k}w)\Big{)}$
$\displaystyle=$ $\displaystyle 0,$

which is equivalent to

$\displaystyle Obs^{T}=\delta^{1}_{T}T_{n+1}.$ (5.6)

Thus, the cohomology class $[Obs^{T}]$ is trivial. Conversely, suppose that
the cohomology class $Obs^{T}$ is trivial, then there exists a $1$-cochain
$T_{n+1}\in\mathcal{C}_{T}^{1}(M,L)$ such that
$Obs^{T}=\delta^{1}_{T}(T_{n+1})$. Set
$\widetilde{T}_{\lambda}=T_{\lambda}+\lambda^{n+1}T_{n+1}$. Then
$\widetilde{T}_{\lambda}$ satisfies

$\displaystyle\sum_{i+j+k=s}[T_{i}u,T_{j}v,T_{k}w]-T_{i}\big{(}D(T_{j}u,T_{k}v)w+\theta(T_{j}v,T_{k}w)u-\theta(T_{j}u,T_{k}w)v+\mathcal{H}(T_{i}u,T_{j}v,T_{k}w)\Big{)}\
\forall 0\leq s\leq n+1,$

which implies that $\widetilde{T}_{\lambda}$ is an order $(n+1)$ deformation
of $T$. Hence it is an extension of $T_{\lambda}$. ∎

######  Definition 5.5.

Let $T_{\lambda}=\displaystyle\sum_{i=0}^{n}\lambda^{i}T_{i}$ be an order $n$
deformation of $T$. Then the cohomology class
$[Obs^{T}]\in{\mathbf{H}}_{T}^{3}(M,L)$ defined in Theorem 5.3 is called the
obstruction class of $T_{\lambda}$ being extendable.

######  Corollary 5.4.

Let $T$ be a generalized Reynolds operator on a L.t.sRep pair
$(L,[\cdot,\cdot,\cdot],\theta)$. If ${\mathbf{H}}_{T}^{3}(M,L)=0$, then every
$1$-cocycle in $\mathcal{Z}^{1}(M,L)$ is the infinitesimal of some formal
deformation of $T$.

##  6 NS-Lie triple systems

The aim of this section is to introduce the notion of a NS-Lie triple system.
A NS-Lie triple system gives rise to a L.t.s. and a representation on itself.
We show that a generalized Reynolds operator induces a NS-Lie triple system.
Thus, NS-Lie triple systems can be viewed as the underlying algebraic
structures of generalized Reynolds operators on L.t.sRep pairs. Also, we show
that Lie algebras, NS-Lie algebras, Lie triple systems and NS-Lie triple
systems are closely related.

###  6.1 Definitions and constructions

######  Definition 6.1.

Let $L$ be a vector space together with two trilinear maps
$\\{\cdot,\cdot,\cdot\\}$, $[\cdot,\cdot,\cdot]:\otimes^{3}L\to L.$ The triple
$(L,\\{\cdot,\cdot,\cdot\\},[\cdot,\cdot,\cdot])$ is called NS-Lie triple
system if the following identities hold for all $x_{1},x_{2}$ and
$y_{1},y_{2},y_{3}\in L$

$\displaystyle[y_{1},y_{2},y_{3}]=-[y_{2},y_{1},y_{3}],\quad\quad\circlearrowleft_{y_{1},y_{2},y_{3}}[y_{1},y_{2},y_{3}]=0,$
(6.1)
$\displaystyle\\{x_{1},x_{2},[\\![y_{1},y_{2},y_{3}]\\!]\\}=\\{\\{x_{1},x_{2},y_{1}\\},y_{2},y_{3}\\}-\\{\\{x_{1},x_{2},y_{2}\\},y_{1},y_{3}\\}+\\{y_{1},y_{2},\\{x_{1},x_{2},y_{3}\\}\\}^{\ast},$
(6.2)
$\displaystyle\\{x_{1},x_{2},\\{y_{1},y_{2},y_{3}\\}\\}^{\ast}=\\{\\{x_{1},x_{2},y_{1}\\}^{\ast},y_{2},y_{3}\\}+\\{y_{1},[\\![x_{1},x_{2},y_{2}]\\!],y_{3}\\}+\\{y_{1},y_{2},[\\![x_{1},x_{2},y_{3}]\\!]\\},$
(6.3)
$\displaystyle[x_{1},x_{2},[\\![y_{1},y_{2},y_{3}]\\!]]=[[\\![x_{1},x_{2},y_{1}]\\!],y_{2},y_{3}]+[y_{1},[\\![x_{1},x_{2},y_{2}]\\!],y_{3}]+[y_{1},y_{2},[\\![x_{1},x_{2},y_{3}]\\!]]$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad+\\{[x_{1},x_{2},y_{1}],y_{2},y_{3}\\}-\\{[x_{1},x_{2},y_{2}],y_{1},y_{3}\\}+\\{y_{1},y_{2},[x_{1},x_{2},y_{3}]\\}^{\ast}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad-\\{x_{1},x_{2},[y_{1},y_{2},y_{3}]\\}^{\ast},$
(6.4)

where $\\{\cdot,\cdot,\cdot\\}^{\ast}$ and $[\\![\cdot,\cdot,\cdot]\\!]$ are
defined by

$\displaystyle\\{y_{1},y_{2},y_{3}\\}^{\ast}=\\{y_{3},y_{2},y_{1}\\}-\\{y_{3},y_{1},y_{2}\\},$
$\displaystyle[\\![y_{1},y_{2},y_{3}]\\!]=\\{y_{1},y_{2},y_{3}\\}^{\ast}+\\{y_{1},y_{2},y_{3}\\}-\\{y_{2},y_{1},y_{3}\\}+[y_{1},y_{2},y_{3}].$
(6.5)

######  Remark 6.1.

Let $(L,\\{\cdot,\cdot,\cdot\\},[\cdot,\cdot,\cdot])$ be a NS-Lie triple
system. In the one hand, if $\\{\cdot,\cdot,\cdot\\}$ is trivial, we get that
$(A,[\cdot,\cdot,\cdot])$ is a L.t.s. In the other hand, if
$[\cdot,\cdot,\cdot]$ is trivial, then $(A,\\{\cdot,\cdot,\cdot\\})$ is a
pre-L.t.s introduced in [22]. Thus, NS-Lie triple systems are generalizations
of both L.t.s and pre-L.t.s.
In the following, we show that NS-Lie triple systems split L.t.sRep pairs.

######  Theorem 6.1.

Let $(L,\\{\cdot,\cdot,\cdot\\},[\cdot,\cdot,\cdot])$ be a NS-Lie triple
system. Then $(L,[\\![\cdot,\cdot,\cdot]\\!])$ is a L.t.s which is called the
sub-adjacent L.t.s of $(L,\\{\cdot,\cdot,\cdot\\},[\cdot,\cdot,\cdot])$ and
denoted by $\mathfrak{L}$. Moreover, $(L,\varrho)$ is a representation of
$\mathfrak{L}$, where the linear map $\varrho:L\otimes L\to End(L)$ is defined
by :

$\displaystyle\varrho(x,y)z=\\{z,x,y\\},\quad\quad\forall x,y,z\in L.$

###### Proof.

It obvious that $[\\![\cdot,\cdot,\cdot]\\!]$ satisfies (2.1).
By Eqs. (6.2)-(6.4), for all $x_{1},x_{2}$ and $y_{1},y_{2},y_{3}\in L$, we
have

$\displaystyle[\\![x_{1},x_{2},[\\![y_{1},y_{2},y_{3}]\\!]]\\!]-[\\![[\\![x_{1},x_{2},y_{1}]\\!],y_{2},y_{3}]\\!]-[\\![y_{1},[\\![x_{1},x_{2},y_{2}]\\!],y_{3}]\\!]-[\\![y_{1},y_{2},[\\![x_{1},x_{2},y_{3}]\\!]]\\!]$
$\displaystyle=$
$\displaystyle\\{x_{1},x_{2},\\{y_{1},y_{2},y_{3}\\}\\}^{\ast}+\\{x_{1},x_{2},\\{y_{3},y_{2},y_{1}\\}\\}^{\ast}-\\{x_{1},x_{2},\\{y_{3},y_{1},y_{2}\\}\\}^{\ast}-\\{x_{1},x_{2},\\{y_{2},y_{1},y_{3}\\}\\}^{\ast}$
$\displaystyle-\\{x_{1},x_{2},[y_{1},y_{2},y_{3}]\\}^{\ast}-\\{x_{1},x_{2},[\\![y_{1},y_{2},y_{3}]\\!]\\}+\\{x_{2},x_{1},[\\![y_{1},y_{2},y_{3}]\\!]\\}-[x_{1},x_{2},[\\![y_{1},y_{2},y_{3}]\\!]]$
$\displaystyle-\\{y_{3},y_{2},[\\![x_{1},x_{2},y_{1}]\\!]\\}+\\{y_{3},[\\![x_{1},x_{2},y_{1}]\\!],y_{2}\\}-\\{\\{x_{1},x_{2},y_{1}\\}^{\ast},y_{2},y_{3}\\}-\\{\\{x_{1},x_{2},y_{1}\\},y_{2},y_{3}\\}$
$\displaystyle+\\{\\{x_{2},x_{1},y_{1}\\}^{\ast},y_{2},y_{3}\\}-\\{[x_{1},x_{2},y_{1}],y_{2},y_{3}\\}+\\{y_{2},[\\![x_{1},x_{2},y_{1}]\\!],y_{3}\\}-[[\\![x_{1},x_{2},y_{1}]\\!],y_{2},y_{3}]$
$\displaystyle-\\{y_{3},[\\![x_{1},x_{2},y_{2}]\\!],y_{1}\\}+\\{y_{3},y_{1},[\\![x_{1},x_{2},y_{2}]\\!]\\}-\\{y_{1},[\\![x_{1},x_{2},y_{2}]\\!],y_{3}\\}+\\{\\{x_{1},x_{2},y_{2}\\}^{\ast},y_{1},y_{3}\\}$
$\displaystyle+\\{\\{x_{1},x_{2},y_{2}\\},y_{1},y_{3}\\}-\\{\\{x_{2},x_{1},y_{2}\\},y_{1},y_{3}\\}+\\{[x_{1},x_{2},y_{2}],y_{1},y_{3}\\}-[y_{1},[\\![x_{1},x_{2},y_{2}]\\!],y_{3}]$
$\displaystyle-\\{\\{x_{1},x_{2},y_{3}\\}^{\ast},y_{2},y_{1}\\}+\\{\\{x_{1},x_{2},y_{3}\\}^{\ast},y_{1},y_{2}\\}-\\{y_{1},y_{2},\\{x_{1},x_{2},y_{3}\\}\\}^{\ast}+\\{y_{1},y_{2},\\{x_{2},x_{1},y_{3}\\}\\}^{\ast}$
$\displaystyle-\\{y_{1},y_{2},[x_{1},x_{2},y_{3}]\\}^{\ast}+\\{y_{1},y_{2},[\\![x_{1},x_{2},y_{3}]\\!]\\}-[y_{1},y_{2},[\\![x_{1},x_{2},y_{3}]\\!]]$
$\displaystyle=$ $\displaystyle 0,$

which implies that $\mathfrak{L}$ is a L.t.s. By Eq.(6.2), Eq.(6.3), we have

$\displaystyle\varrho(x_{1},x_{2})\varrho(y_{1},y_{2})-\varrho(y_{2},x_{2})\varrho(y_{1},x_{1})-\varrho(y_{1},[\\![y_{2},x_{1},x_{2}]\\!])+D_{\varrho}(y_{2},x_{1})\varrho(y_{1},x_{2})=0,$
$\displaystyle\varrho(x_{1},x_{2})D_{\varrho}(y_{1},y_{2})-D^{\ast}(y_{1},y_{2})\varrho(x_{1},x_{2})+\varrho([\\![y_{1},y_{2},x_{1}]\\!],x_{2})+\varrho(x_{1},[\\![y_{1},y_{2},x_{2}]\\!])=0,$

where $D_{\varrho}(x_{1},x_{2})=\varrho(x_{2},x_{1})-\varrho(x_{1},x_{2})$.
Therefore, $\varrho$ is a representation of $\mathfrak{L}$ on $L$. Then
$(L,[\\![\cdot,\cdot,\cdot]\\!],\varrho)$ is a L.t.sRep pair. ∎

The following results illustrate that NS-Lie triple systems can be viewed as
the underlying algebraic structures of generalized Reynolds operators on
L.t.sRep pairs.

######  Theorem 6.2.

Let $T:M\to L$ be a generalized Reynolds operator on a L.t.sRep pair
$(L,[\cdot,\cdot,\cdot],\theta)$. Then

$\displaystyle\\{u,v,w\\}=\theta(Tv,Tw)u\quad\text{and}\quad[u,v,w]=\mathcal{H}(Tu,Tv,Tw)$
(6.6)

defines a NS-Lie triple system structure on $M$. It is called the induced NS-
Lie triple system.

###### Proof.

For any $u,v,w\in L$, it is obvious that

$\displaystyle[u,v,w]=-[v,u,w]\quad
and\quad\circlearrowleft_{u,v,w}[u,v,w]=0.$

Furthermore, for all $u_{1},u_{2},v_{1},v_{2},v_{3}\in M$, by Eqs. (2.3)-(2.4)
and Eq. (6.5), we have

$\displaystyle\\{u_{1},u_{2},[\\![v_{1},v_{2},v_{3}]\\!]\\}-\\{\\{u_{1},u_{2},v_{1}\\},v_{2},v_{3}\\}+\\{\\{u_{1},u_{2},v_{2}\\},v_{1},v_{3}\\}-\\{v_{1},v_{2},\\{u_{1},u_{2},v_{3}\\}\\}$
$\displaystyle=$
$\displaystyle\\{u_{1},u_{2},\\{v_{1},v_{2},v_{3}\\}^{\ast}\\}+\\{u_{1},u_{2},\\{v_{1},v_{2},v_{3}\\}\\}-\\{u_{1},u_{2},\\{v_{2},v_{1},v_{3}\\}\\}+\\{u_{1},u_{2},[v_{1},v_{2},v_{3}]\\}$
$\displaystyle-\\{\\{u_{1},u_{2},,v_{1}\\},v_{2},v_{3}\\}+\\{\\{u_{1},u_{2},v_{2}\\},v_{1},v_{3}\\}-\\{v_{1},v_{2},\\{u_{1},u_{2},v_{3}\\}\\}^{\ast}$
$\displaystyle=$
$\displaystyle\\{u_{1},u_{2},\\{v_{3},v_{2},v_{1}\\}\\}-\\{u_{1},u_{2},\\{v_{3},v_{1},v_{2}\\}\\}+\\{u_{1},u_{2},\\{v_{1},v_{2},v_{3}\\}\\}-\\{u_{1},u_{2},\\{v_{2},v_{1},v_{3}\\}\\}$
$\displaystyle+\\{u_{1},u_{2},[v_{1},v_{2},v_{3}]\\}-\\{\\{u_{1},u_{2},v_{1}\\},v_{2},v_{3}\\}+\\{\\{u_{1},u_{2},v_{2}\\},v_{1},v_{3}\\}-\\{\\{u_{1},u_{2},v_{3}\\},v_{2},v_{1}\\}$
$\displaystyle+\\{\\{u_{1},u_{2},v_{3}\\},v_{1},v_{2}\\}$ $\displaystyle=$
$\displaystyle\theta(Tu_{2},[Tv_{1},Tv_{2},Tv_{3}])u_{1}-\theta(Tv_{2},Tv_{3})\theta(Tu_{2},Tv_{1})u_{1}+\theta(Tv_{1},Tv_{3})\theta(Tu_{2},Tv_{2})u_{1}$
$\displaystyle-\theta(Tv_{2},Tv_{1})\theta(Tu_{2},Tv_{3})u_{1}+\theta(Tv_{1},Tv_{2})\theta(Tu_{2},Tv_{3})u_{1}=0.$

This implies that Eq.(6.2) in Definition 6.1 holds. Similarly for Eq.(6.3).
Moreover, since $\mathcal{H}$ is a $3$-cocycle of $L$ with coefficients in
$M$, then Eq.(6.4) holds. Therfore,
$(L,\\{\cdot,\cdot,\cdot\\},[\cdot,\cdot,\cdot])$ is a NS-Lie triple system. ∎

######  Corollary 6.3.

The subadjacent L.t.s $(M,[\\![\cdot,\cdot,\cdot]\\!])$ of the above NS-L.t.s
$(M,\\{\cdot,\cdot,\cdot\\},[\cdot,\cdot,\cdot])$ is exactly the L.t.s given
in Corollary 3.10.

######  Definition 6.2.

A morphism from a NS-L.t.s $(L,\\{\cdot,\cdot,\cdot\\},[\cdot,\cdot,\cdot])$
to
$(L^{\prime},\\{\cdot,\cdot,\cdot\\}^{\prime},[\cdot,\cdot,\cdot]^{\prime})$
is a linear map $\psi:L\to L^{\prime}$ satisfying

$\displaystyle\psi\\{x,y,z\\}=\\{\psi(x),\psi(y),\psi(z)\\}^{\prime}\quad
and\quad\psi[x,y,z]=[\psi(x),\psi(y),\psi(z)]^{\prime}\quad\forall x,y,z\in
L.$

######  Proposition 6.4.

Let $T$ and $T^{\prime}$ be two generalized Reynolds operators on a L.t.sRep
pair $(L,[\cdot,\cdot,\cdot],\theta)$, and
$(L,\\{\cdot,\cdot,\cdot\\},[\cdot,\cdot,\cdot])$,
$(L^{\prime},\\{\cdot,\cdot,\cdot\\}^{\prime},[\cdot,\cdot,\cdot]^{\prime})$
be the induced NS-L.t.s. Let $(\phi,\psi)$ be a morphism from $T$ to
$T^{\prime}$, then $\psi$ is a morphism from the NS-L.t.s
$(L,\\{\cdot,\cdot,\cdot\\},[\cdot,\cdot,\cdot])$ to
$(L^{\prime},\\{\cdot,\cdot,\cdot\\}^{\prime},[\cdot,\cdot,\cdot]^{\prime})$.

###### Proof.

It is straightforward by Eq.(3.9)-(3.11) and Eq.(6.6). ∎

Since $\lambda$-weighted Reynolds operators L.t.s is a particular generalized
Reynolds operator on a L.t.sRep pair, then the following corollary is obvious.

######  Corollary 6.5.

Let $(L,[\cdot,\cdot,\cdot],\mathcal{R})$ be a $\lambda$-weighted Reynolds
operator L.t.s. Then
$(L,\\{\cdot,\cdot,\cdot\\},\lambda[\cdot,\cdot,\cdot]\circ(\mathcal{R}\otimes\mathcal{R}\otimes\mathcal{R}))$
is a NS-L.t.s, where

$\displaystyle\\{x,y,z\\}=[x,\mathcal{R}y,\mathcal{R}z]\ \forall x,y,z\in L.$

######  Example 6.6.

Example 3.4 can be enhanced to NS-Lie triple system. More precisely, we make a
$\lambda$-weighted Reynolds operator on a Lie algebra $\mathcal{R}$ to
construct a generalized Reynolds operator on the induced L.t.s
$(L,[[\cdot,\cdot],\cdot])$. Moreover by Corollary 6.5, we obtain a NS-L.t.s
on $W_{\geq 0}$ defined by the structures

$\displaystyle\\{l_{m},l_{n},l_{p}\\}$
$\displaystyle=[l_{m},\mathcal{R}(l_{n}),\mathcal{R}(l_{p})]=\frac{1}{\lambda^{2}(n+1)(p+1)}[l_{m},l_{n},l_{p}]$
$\displaystyle=\frac{(m-n)(m+n-p))}{\lambda^{2}(n+1)(p+1)}l_{m+n+p},$
$\displaystyle
2\lambda[\mathcal{R}(l_{m}),\mathcal{R}(l_{n}),\mathcal{R}(l_{p})]$
$\displaystyle=2\frac{(m-n)(m+n-p))}{\lambda^{2}(m+1)(n+1)(p+1)}l_{m+n+p}\quad\forall
m,n,p\in\mathbb{Z}_{+}.$

So $\mathcal{R}$ is a generalized Reynolds operator on the induced L.t.s
$(L,[\cdot,\cdot,\cdot])$.

######  Example 6.7.

More generally, consider the $\lambda$-weighted Reynolds operator
$\mathcal{R}:B(q)_{\geq 0}\to B(q)_{\geq 0}$ on the L.t.s $(B(q)_{\geq
0},[\cdot,\cdot,\cdot])$ defined in Example 3.1. Its follows from Corollary
6.5, that $B(q)_{\geq 0}$ carries a NS-L.t.s structure $(B(q)_{\geq
0},\\{\cdot,\cdot,\cdot\\},\lambda[\cdot,\cdot,\cdot])$ given by

$\displaystyle\\{L_{m,i},L_{n,j},L_{p,k}\\}=\frac{(n(i+q)-m(j+q))(p(i+j+q)-(m+n)(p+q))}{\lambda^{2}(n+j+1)(p+k+1)}L_{m+n+p,i+j+k}.$

###  6.2 From NS-Lie algebras to NS-Lie triple systems

Now, we recall the definition of NS-Lie algebra introduced by A. Das in [6],
so that we can construct a NS-Lie triple system starting from a NS-Lie
algebra.

######  Definition 6.3.

A NS-Lie algebra is a vector space L together with bilinear operations
$\circ,\curlyvee:L\to L$ in which $\curlyvee$ is skew-symmetric and satisfying
the following two identities:

$\displaystyle(x\ast y)\circ z-x\circ(y\circ z)+y\circ(x\circ z)=0,$ (6.7)
$\displaystyle x\curlyvee(y\ast z)+y\curlyvee(z\ast x)+z\curlyvee(x\ast
z)+x\circ(y\curlyvee z)+y\circ(z\curlyvee x)+z\circ(x\curlyvee
y)=0\quad\forall x,y,z\in L,$ (6.8)

where $x\ast y=x\circ y-y\circ x+x\curlyvee y.$

######  Proposition 6.8.

Let $(L,\circ,\curlyvee)$ be a NS-Lie algebra. Then $(L,\ast)$ is a Lie
algebra called the adjacent Lie algebra.

Let $(L,[\cdot,\cdot],\rho)$ be a LieRep pair. A skew-symmetric bilinear map
$H:L\times L\to M$ is a $2$-cocycle if it satisfies

$\displaystyle\circlearrowleft_{x,y,z}\big{(}\rho(x)H(y,z)+H(x,[y,z])\big{)}=0\
\ \ \ \forall x,y,z\in L.$ (6.9)

A linear map $T:M\to L$ is said to be a generalized Reynolds operator if it
satisfies :

$\displaystyle[T(u),T(v)]=T\Big{(}\rho(Tu)v-\rho(Tv)u+H(Tu,Tv)\Big{)}\ \ \ \ \
\forall u,v\in M.$ (6.10)

Define two multiplications:

$u\circ v=\rho(Tu)v\quad\text{and}\quad u\curlyvee v=H(Tu,Tv)\ \ \ \ \ \forall
u,v\in M.$ (6.11)

######  Proposition 6.9.

Under the above notations, $(M,\circ,\curlyvee)$ is a NS-Lie algebra.

######  Theorem 6.10.

Let $(L,\circ,\curlyvee)$ be a NS-Lie algebra. Then
$(L,\\{\cdot,\cdot,\cdot\\},[\cdot,\cdot,\cdot])$ is a NS-Lie triple system,
where

$\displaystyle\\{x,y,z\\}=z\circ(y\circ x),$ (6.12)
$\displaystyle[x,y,z]=(x\ast y)\curlyvee z-z\circ(x\curlyvee y)\ \ \forall
x,y,z\in L.$ (6.13)

Note that $[\\![x,y,z]\\!]=(x\ast y)\ast z$.

###### Proof.

Let $x_{1},x_{2}$ and $y_{1},y_{2},y_{3}\in L$. It is obvious that the bracket
$[\cdot,\cdot,\cdot]$ is skew-symmetric and
$\circlearrowleft_{y_{1},y_{2},y_{3}}[y_{1},y_{2},y_{3}]=0$. According to
(6.7), we have

$\displaystyle\\{\\{x_{1},x_{2},y_{1}\\},y_{2},y_{3}\\}-\\{\\{x_{1},x_{2},y_{2}\\},y_{1},y_{3}\\}+\\{y_{1},y_{2},\\{x_{1},x_{2},y_{3}\\}\\}^{\ast}-\\{x_{1},x_{2},[\\![y_{1},y_{2},y_{3}]\\!]\\}$
$\displaystyle=$
$\displaystyle\underline{\underline{{y_{3}\circ\Big{(}y_{2}\circ(y_{1}\circ(x_{2}\circ
x_{1}))\Big{)}-y_{3}\circ\Big{(}y_{1}\circ(y_{2}\circ(x_{2}\circ
x_{1}))\Big{)}}}}+\underline{y_{1}\circ\Big{(}y_{2}\circ(y_{3}\circ(x_{2}\circ
x_{1}))}\Big{)}$
$\displaystyle\underline{-y_{2}\circ\Big{(}y_{1}\circ(y_{3}\circ(x_{2}\circ
x_{1}))\Big{)}}-\Big{(}(y_{1}\ast y_{2})\ast y_{3}\Big{)}\circ(x_{2}\circ
x_{1})$ $\displaystyle\overset{\eqref{NS-Lie-1}}{=}$ $\displaystyle(y_{1}\ast
y_{2})\circ\Big{(}y_{3}\circ(x_{2}\circ
x_{1})\Big{)}-y_{3}\circ\Big{(}(y_{1}\ast y_{2})\circ(x_{2}\circ
x_{1}))\Big{)}-\Big{(}(y_{1}\ast y_{2})\ast y_{3}\Big{)}\circ(x_{2}\circ
x_{1})$ $\displaystyle=$ $\displaystyle 0.$

Then Eq.(6.2) holds. Similarly

$\displaystyle\\{\\{y_{1},y_{2},y_{3}\\},x_{1},x_{2}\\}-\\{\\{y_{1},y_{2},y_{3}\\},x_{2},x_{1}\\}+\\{\\{y_{1},x_{2},x_{1}\\},y_{2},y_{3}\\}-\\{\\{y_{1},x_{1},x_{2}\\},y_{2},y_{3}\\}$
$\displaystyle+\\{y_{1},[x_{1},x_{2},y_{2}],y_{3}\\}+\\{y_{1},y_{2},[x_{1},x_{2},y_{3}]\\}$
$\displaystyle=$
$\displaystyle\underline{\underline{x_{2}\circ\Big{(}x_{1}\circ(y_{3}\circ(y_{2}\circ
y_{1}))\Big{)}-x_{1}\circ\Big{(}x_{2}\circ(y_{3}\circ(y_{2}\circ
y_{1}))\Big{)}}}+\underline{y_{3}\circ\Big{(}y_{2}\circ(x_{1}\circ(x_{2},y_{1}))}\Big{)}$
$\displaystyle\underline{-y_{3}\circ\Big{(}y_{2}\circ(x_{2}\circ(x_{1},y_{1}))\Big{)}}+y_{3}\circ\Big{(}((x_{1}\ast
x_{2})\ast y_{2})\circ y_{1}\Big{)}+\Big{(}(x_{1}\ast x_{2})\ast
y_{3}\Big{)}\circ(y_{2}\circ y_{1})$ $\displaystyle\overset{\eqref{NS-
Lie-1}}{=}$ $\displaystyle y_{3}\circ(y_{2}\circ\Big{(}(x_{1}\ast x_{2})\circ
y_{1}\Big{)}-(x_{1}\ast x_{2})\circ\Big{(}y_{3}\circ(y_{2}\circ
y_{1})\Big{)}+y_{3}\circ\Big{(}((x_{1}\ast x_{2})\ast y_{2})\circ
y_{1}\Big{)}$ $\displaystyle+\Big{(}(x_{1}\ast x_{2})\ast
y_{3}\Big{)}\circ(y_{2}\circ y_{1})$ $\displaystyle=$ $\displaystyle
y_{3}\circ\Big{(}(x_{1}\ast x_{2})\circ(y_{2}\circ
y_{1})\Big{)}-y_{3}\circ\Big{(}(x_{1}\ast x_{2})\circ(y_{2}\circ
y_{1})\Big{)}=0.$

Therefore Eq.(6.3) holds. Also, we have

$\displaystyle[[\\![x_{1},x_{2},y_{1}]\\!],y_{2},y_{3}]+[y_{1},[\\![x_{1},x_{2},y_{2}]\\!],y_{3}]+[y_{1},y_{2},[\\![x_{1},x_{2},y_{3}]\\!]]+\\{[x_{1},x_{2},y_{1}],y_{2},y_{3}\\}$
$\displaystyle-\\{[x_{1},x_{2},y_{2}],y_{1},y_{3}\\}+\\{y_{1},y_{2},[x_{1},x_{2},y_{3}]\\}^{\ast}-\\{x_{1},x_{2},[y_{1},y_{2},y_{3}]\\}^{\ast}-[x_{1},x_{2},[\\![y_{1},y_{2},y_{3}]\\!]]$
$\displaystyle=$ $\displaystyle\Big{(}((x_{1}\ast x_{2})\ast y_{1})\ast
y_{2}\Big{)}\curlyvee y_{3}+\cancel{\Big{(}(y_{1}\ast y_{2})\ast
y_{3}\Big{)}\circ(x_{1}\curlyvee x_{2})}-y_{3}\circ\Big{(}((x_{1}\ast
x_{2})\ast y_{1})\curlyvee y_{2}\Big{)}$
$\displaystyle+\Big{(}y_{1}\ast((x_{1}\ast x_{2})\ast y_{2})\Big{)}\curlyvee
y_{3}-y_{3}\circ\Big{(}y_{1}\curlyvee((x_{1}\ast x_{2})\ast
y_{2})\Big{)}+(y_{1}\ast y_{2})\curlyvee\Big{(}(x_{1}\ast x_{2})\ast
y_{3}\Big{)}$ $\displaystyle-\Big{(}(x_{1}\ast x_{2})\ast
y_{3}\Big{)}\circ(y_{1}\curlyvee y_{2})+y_{3}\circ\Big{(}y_{2}\circ((x_{1}\ast
x_{2})\curlyvee
y_{1})-\cancel{y_{3}\circ\Big{(}y_{2}\circ(y_{1}\circ(x_{1}\curlyvee
x_{2}))\Big{)}}$ $\displaystyle-y_{3}\circ\Big{(}y_{1}\circ((x_{1}\ast
x_{2})\curlyvee y_{2})+y_{1}\circ\Big{(}y_{2}\circ((x_{1}\ast x_{2})\curlyvee
y_{3})\Big{)}+\cancel{y_{3}\circ\Big{(}y_{1}\circ(y_{2}\circ(x_{1}\curlyvee
x_{2}))\Big{)}}$ $\displaystyle-y_{2}\circ\Big{(}y_{1}\circ((x_{1}\ast
x_{2})\curlyvee
y_{3})\Big{)}-\cancel{y_{1}\circ\Big{(}y_{2}\circ(y_{3}\circ(x_{1}\curlyvee
x_{2})\Big{)}}+\cancel{y_{2}\circ\Big{(}y_{1}\circ(y_{3}\circ(x_{1}\curlyvee
x_{2})\Big{)}}$ $\displaystyle-x_{1}\circ\Big{(}x_{2}\circ(y_{1}\ast
y_{2})\curlyvee
y_{3}\Big{)}+x_{1}\circ\Big{(}x_{2}\circ(y_{3}\circ(y_{1}\curlyvee
y_{2}))\Big{)}+x_{2}\circ\Big{(}x_{1}\circ(y_{1}\ast y_{2})\curlyvee
y_{3}\Big{)}$ $\displaystyle-
x_{2}\circ\Big{(}x_{1}\circ(y_{3}\circ(y_{1}\curlyvee
y_{2}))\Big{)}-(x_{1}\ast x_{2})\curlyvee\Big{(}(y_{1}\ast y_{2})\ast
y_{3}\Big{)}$ $\displaystyle\overset{\eqref{NS-Lie-1}}{=}$
$\displaystyle\underline{\Big{(}((x_{1}\ast x_{2})\ast y_{1})\ast
y_{2}\Big{)}\curlyvee y_{3}}-y_{3}\circ\Big{(}((x_{1}\ast x_{2})\ast
y_{1})\curlyvee y_{2}\Big{)}+\underline{\Big{(}y_{1}\ast((x_{1}\ast x_{2})\ast
y_{2})\Big{)}\curlyvee y_{3}}$ $\displaystyle-
y_{3}\circ\Big{(}y_{1}\curlyvee((x_{1}\ast x_{2})\ast y_{2})\Big{)}+(y_{1}\ast
y_{2})\curlyvee\Big{(}(x_{1}\ast x_{2})\ast
y_{3}\Big{)}+y_{3}\circ\Big{(}y_{2}\circ((x_{1}\ast x_{2})\curlyvee y_{1})$
$\displaystyle-y_{3}\circ\Big{(}y_{1}\circ((x_{1}\ast x_{2})\curlyvee
y_{2})+y_{1}\circ\Big{(}y_{2}\circ((x_{1}\ast x_{2})\curlyvee
y_{3})\Big{)}-y_{2}\circ\Big{(}y_{1}\circ((x_{1}\ast x_{2})\curlyvee
y_{3})\Big{)}$ $\displaystyle-x_{1}\circ\Big{(}x_{2}\circ(y_{1}\ast
y_{2})\curlyvee y_{3}\Big{)}+x_{2}\circ\Big{(}x_{1}\circ(y_{1}\ast
y_{2})\curlyvee
y_{3}\Big{)}\underline{\underline{+x_{1}\circ\Big{(}x_{2}\circ(y_{3}\circ(y_{1}\curlyvee
y_{2}))\Big{)}}}$
$\displaystyle\underline{\underline{-x_{2}\circ\Big{(}x_{1}\circ(y_{3}\circ(y_{1}\curlyvee
y_{2}))\Big{)}}}-(x_{1}\ast x_{2})\curlyvee\Big{(}(y_{1}\ast y_{2})\ast
y_{3}\Big{)}\underline{\underline{-\Big{(}(x_{1}\ast x_{2})\ast
y_{3}\Big{)}\circ(y_{1}\curlyvee y_{2})}}$ $\displaystyle\overset{\eqref{NS-
Lie-1}}{=}$ $\displaystyle\Big{(}(x_{1}\ast x_{2})\ast(y_{1}\ast
y_{2})\Big{)}\curlyvee y_{3}+y_{3}\circ\Big{(}((x_{1}\ast
x_{2})\circ(y_{1}\curlyvee y_{2}\Big{)}-y_{3}\circ\Big{(}(x_{1}\ast x_{2})\ast
y_{1})\curlyvee y_{2}\Big{)}$ $\displaystyle-
y_{3}\circ\Big{(}y_{1}\curlyvee((x_{1}\ast x_{2})\ast y_{2})\Big{)}+(y_{1}\ast
y_{2})\curlyvee\Big{(}(x_{1}\ast x_{2})\ast
y_{3}\Big{)}+y_{3}\circ\Big{(}y_{2}\circ((x_{1}\ast x_{2})\curlyvee
y_{1})\Big{)}$ $\displaystyle-y_{3}\circ\Big{(}y_{1}\circ((x_{1}\ast
x_{2})\curlyvee
y_{2})\Big{)}\underline{\underline{+y_{1}\circ\Big{(}y_{2}\circ((x_{1}\ast
x_{2})\curlyvee y_{3})\Big{)}-y_{2}\circ\Big{(}y_{1}\circ((x_{1}\ast
x_{2})\curlyvee y_{3})\Big{)}}}$
$\displaystyle-\underline{x_{1}\circ\Big{(}x_{2}\circ(y_{1}\ast
y_{2})\curlyvee y_{3}\Big{)}+x_{2}\circ\Big{(}x_{1}\circ(y_{1}\ast
y_{2})\curlyvee y_{3}\Big{)}}-(x_{1}\ast x_{2})\curlyvee\Big{(}(y_{1}\ast
y_{2})\ast y_{3}\Big{)}$ $\displaystyle\overset{\eqref{NS-Lie-2}}{=}$
$\displaystyle\Big{(}(x_{1}\ast x_{2})\ast(y_{1}\ast y_{2})\Big{)}\curlyvee
y_{3}+\underline{y_{3}\circ\Big{(}((x_{1}\ast x_{2})\circ(y_{1}\curlyvee
y_{2}\Big{)}-y_{3}\circ\Big{(}(x_{1}\ast x_{2})\ast y_{1})\curlyvee
y_{2}\Big{)}}$
$\displaystyle-\underline{y_{3}\circ\Big{(}y_{1}\curlyvee((x_{1}\ast
x_{2})\ast y_{2})\Big{)}}+(y_{1}\ast y_{2})\curlyvee\Big{(}(x_{1}\ast
x_{2})\ast y_{3}\Big{)}+\underline{y_{3}\circ\Big{(}y_{2}\circ((x_{1}\ast
x_{2})\curlyvee y_{1})\Big{)}}$
$\displaystyle-\underline{y_{3}\circ\Big{(}y_{1}\circ((x_{1}\ast
x_{2})\curlyvee y_{2})\Big{)}}-(x_{1}\ast x_{2})\circ\Big{(}(y_{1}\ast
y_{2})\curlyvee y_{3}\Big{)}+(y_{1}\ast y_{2})\circ\Big{(}(x_{1}\ast
x_{2})\curlyvee y_{3}\Big{)}$ $\displaystyle-(x_{1}\ast
x_{2})\curlyvee\Big{(}(y_{1}\ast y_{2})\ast y_{3}\Big{)}$
$\displaystyle\overset{\eqref{NS-Lie-2}}{=}$ $\displaystyle-\Big{(}(x_{1}\ast
x_{2})\curlyvee\Big{(}(y_{1}\ast y_{2})\ast y_{3}\Big{)}+(y_{1}\ast
y_{2})\curlyvee\Big{(}y_{3}\ast(x_{1}\ast
x_{2})\Big{)}+y_{3}\curlyvee\Big{(}(x_{1}\ast x_{2})\ast(y_{1}\ast
y_{2})\Big{)}$ $\displaystyle+(x_{1}\ast x_{2})\circ\Big{(}(y_{1}\ast
y_{2})\curlyvee y_{3}\Big{)}+(y_{1}\ast
y_{2})\circ\Big{(}y_{3}\curlyvee(x_{1}\ast
x_{2})\Big{)}+y_{3}\circ\Big{(}(x_{1}\ast x_{2})\curlyvee(y_{1}\ast
y_{2})\Big{)}\Big{)}$ $\displaystyle=0.$

Then (6.4) holds and $(L,\\{\cdot,\cdot,\cdot\\},[\cdot,\cdot,\cdot])$ is a
NS-Lie triple system. ∎

We have shown that Lie algebras, NS-Lie algebras, Lie triple systems and NS-
Lie triple systems are closely related in the sense of commutative diagram of
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3.0pt\raise 0.0pt\hbox{$\textstyle{\mbox{\bf L.t.sRep pair
$(L,[[\cdot,\cdot],\cdot],\theta_{\rho}).$}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
159.792pt\raise-19.75pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{\mbox{}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 162.792pt\raise-5.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces}\end{split}$
(6.14)

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