# Pre-Leibniz algebras

and  Apurba Das  Department of Mathematics, Indian Institute of Technology,
Kharagpur 721302, West Bengal, India<EMAIL_ADDRESS>
###### Abstract.

The notion of pre-Leibniz algebras was recently introduced in the study of
Rota-Baxter operators on Leibniz algebras. In this paper, we first construct a
graded Lie algebra whose Maurer-Cartan elements are pre-Leibniz algebras.
Using this characterization, we define the cohomology of a pre-Leibniz algebra
with coefficients in a representation. This cohomology is shown to split the
Loday-Pirashvili cohomology of Leibniz algebras. As applications of our
cohomology, we study formal and finite order deformations of a pre-Leibniz
algebra. Finally, we define homotopy pre-Leibniz algebras and classify some
special types of homotopy pre-Leibniz algebras.

2010 MSC classification: 17A32, 17A99, 17B56, 16S80, 18G55.

Keywords: Pre-Leibniz algebras, Maurer-Cartan elements, Cohomology,
Deformations, Homotopy pre-Leibniz algebras.

###### Contents

  1. 1 Introduction
  2. 2 Leibniz algebras and the Loday-Pirashvili cohomology
  3. 3 Pre-Leibniz algebras and their Maurer-Cartan characterizations
  4. 4 Representations and cohomology of pre-Leibniz algebras
  5. 5 Deformations of pre-Leibniz algebras
  6. 6 Homotopy pre-Leibniz algebras

##  1\. Introduction

The notion of pre-Lie algebras was first introduced by Gerstenhaber in his
work on deformations of algebras [15, 14]. A pre-Lie algebra naturally induces
a Lie algebra structure on the underlying vector space. Pre-Lie algebras have
found important connections with rooted trees, vector fields on affine spaces,
deformation theory and Rota-Baxter operators on Lie algebras [9, 11, 12, 18].
Cohomology and deformation theory for pre-Lie algebras was introduced by
Dzhumadil’daev [13]. Such study generalizes the classical Hochschild
cohomology theory and deformation theory for associative algebras [15, 16].
The notion of homotopy pre-Lie algebras was described by Chapoton and Livernet
in terms of rooted trees [9]. For more on pre-Lie algebras, see [8, 2, 3, 23].

On the other hand, Leibniz algebras were first considered by Bloh [6, 7] and
rediscovered by Loday [19] in the study of periodicity phenomenon in algebraic
$K$-theory. Leibniz algebras are roughly a noncommutative generalization of
Lie algebras. In [20] Loday and Pirashvili introduced a cohomology theory for
Leibniz algebras with coefficients in a representation. Given a vector space
$\mathfrak{g}$, Balavoine [4] constructs a graded Lie algebra (known as
Balavoine’s graded Lie algebra) on the space of multilinear maps on
$\mathfrak{g}$ whose Maurer-Cartan elements correspond to Leibniz algebra
structures on $\mathfrak{g}$. Recently, Rota-Baxter operators and relative
Rota-Baxter operators on Leibniz algebras are introduced and their relation
with Leibniz Yang-Baxter equation and Leibniz bialgebras are discovered in
[22, 24, 25]. Like a Rota-Baxter operator on a Lie algebra induces a pre-Lie
algebra structure, a (relative) Rota-Baxter operator on a Leibniz algebra
gives rise to a pre-Leibniz algebra (already introduced in [24] by the name of
Leibniz-dendriform algebra) structure. A pre-Leibniz algebra
$(\mathfrak{a},\triangleleft,\triangleright)$ is a vector space $\mathfrak{a}$
together with two bilinear operations
$\triangleleft,\triangleright:\mathfrak{a}\otimes\mathfrak{a}\rightarrow\mathfrak{a}$
satisfying a set of three identities (see Definition 3.1). It turns out that
the sum of the two operations is indeed a Leibniz algebra structure on
$\mathfrak{a}$, called the total Leibniz algebra and denoted by
$\mathfrak{a}_{\mathrm{Tot}}$. Thus, pre-Leibniz algebras can be considered as
a splitting of Leibniz algebras. Nothing is known about this newly defined
pre-Leibniz algebra structure. Our aim in this paper is to give an extensive
study of this new structure.

Given a vector space $\mathfrak{a}$, we first construct a graded Lie algebra
whose Maurer-Cartan elements are in one-to-one correspondence with pre-Leibniz
algebra structures on $\mathfrak{a}$. This graded Lie algebra can be
considered as a splitting of the Balavoine’s graded Lie algebra. Next, we
introduce representations of a pre-Leibniz algebra and define the cohomology
of a pre-Leibniz algebra with coefficients in a representation. We show that
there is a morphism from the cohomology of a pre-Leibniz algebra
$(\mathfrak{a},\triangleleft,\triangleright)$ to the Loday-Pirashvili
cohomology of the total Leibniz algebra $\mathfrak{a}_{\mathrm{Tot}}$ (cf.
Theorem 4.5).

Next, we consider formal, infinitesimal and finite order deformations of a
pre-Leibniz algebra $(\mathfrak{a},\triangleleft,\triangleright)$. We show
that there is a one-to-one correspondence between the equivalence classes of
infinitesimal deformations of $(\mathfrak{a},\triangleleft,\triangleright)$
and the second cohomology group with coefficients in the adjoint
representation (cf. Theorem 5.3 and Remark 5.4). A pre-Leibniz algebra is said
to be rigid if any formal deformation is trivial. We show that the vanishing
of the second cohomology group implies that the pre-Leibniz algebra is rigid
(cf. Theorem 5.7, Remark 5.8). Finally, given a finite order deformation of a
pre-Leibniz algebra, we associate a third cohomology class (called the
obstruction class) which is the obstruction to extend the given deformation to
the next order (cf. Theorem 5.12).

The notion of Leibniz∞-algebras (homotopy Leibniz algebras) was first
introduced by Ammar and Poncin [1] as the homotopy analogue of Leibniz
algebras. See [10, 17, 26] for more on Leibniz∞-algebras. In this paper, we
introduce pre-Leibniz∞-algebras as the homotopy version of pre-Leibniz
algebras. We show that a pre-Leibniz∞-algebra induces a Leibniz∞-algebra
structure on the underlying graded vector space (cf. Theorem 6.3). This
generalizes the construction of the total Leibniz algebra associated with a
pre-Leibniz algebra. We introduce Rota-Baxter operators on Leibniz∞-algebras
that gives rise to a pre-Leibniz∞-algebras (cf. Theorem 6.5). Next, we focus
on pre-Leibniz∞-algebras whose underlying graded vector space is concentrated
in degree $-1$ and $0$. Some particular classes of such homotopy algebras are
given by ‘skeletal’ and ‘strict’ pre-Leibniz∞-algebras. We show that skeletal
pre-Leibniz∞-algebras are classified by third cocycles of pre-Leibniz algebras
(cf. Theorem 6.8). Finally, we introduce crossed module of pre-Leibniz
algebras and show that strict pre-Leibniz∞-algebras are in one-to-one
correspondence with crossed module of pre-Leibniz algebras (cf. Theorem 6.10).

The paper is organized as follows. In Section 2, we recall Leibniz algebras,
their cohomology and Balavoine’s graded lie algebra. Pre-Leibniz algebras are
considered and a new graded Lie algebra associated to a vector space
$\mathfrak{a}$ is constructed in Section 3. The Maurer-Cartan elements of this
graded Lie algebra correspond to pre-Leibniz algebra structures on
$\mathfrak{a}$. In Section 4, we introduce representations of pre-Leibniz
algebras and define cohomology with coefficients in a representation.
Deformation theory of pre-Leibniz algebras is extensively studied in Section 5
in terms of cohomology theory. Finally, we introduce pre-Leibniz∞-algebras,
their relation with Leibniz∞-algebras, and classification of skeletal and
strict pre-Leibniz∞-algebras are given in Section 6.

All vector spaces, (multi)linear maps and tensor products are over a field
${\bf k}$ of characteristic $0$.

##  2\. Leibniz algebras and the Loday-Pirashvili cohomology

In this preliminary section, we recall Leibniz algebras, their representations
and the Loday-Pirashvili cohomology of Leibniz algebras. Our main references
are [4, 19, 20].

######  2.1 Definition.

A (left) Leibniz algebra is a pair $(\mathfrak{g},[~{},~{}])$ consists of a
vector space $\mathfrak{g}$ and a bilinear bracket
$[~{},~{}]:\mathfrak{g}\otimes\mathfrak{g}\rightarrow\mathfrak{g}$ satisfying

$\displaystyle[x,[y,z]]=[[x,y],z]+[y,[x,z]],~{}\text{ for
}x,y,z\in\mathfrak{g}.$ (1)

Sometimes we denote a Leibniz algebra simply by $\mathfrak{g}$ when the
bracket is clear from the context.

######  2.2 Remark.

Note that the identity (1) is equivalent to to say that the left
multiplications $[x,-]:\mathfrak{g}\rightarrow\mathfrak{g}$ are derivations
for the bracket. This is the reason the Leibniz algebras of Definition 2.1 are
called left Leibniz algebras. Similarly, one has right Leibniz algebras as a
pair $(\mathfrak{g},[~{},~{}])$ for which the right multiplications
$[-,x]:\mathfrak{g}\rightarrow\mathfrak{g}$ are derivations for the bracket.
It is easy to see that if $(\mathfrak{g},[~{},~{}])$ is a left Leibniz algebra
then $(\mathfrak{g},[~{},~{}]^{op})$ is a right Leibniz algebra and vice
versa, where $[x,y]^{op}=[y,x]$, for $x,y\in\mathfrak{g}$. Therefore, any
results for left Leibniz algebras can be easily generalized to right Leibniz
algebras without much hard work. Throughout the paper, by a Leibniz algebra,
we shall always mean a left Leibniz algebra.

######  2.3 Definition.

Let $(\mathfrak{g},[~{},~{}])$ be a Leibniz algebra. A representation of it
consists of a triple $(M,\rho^{L},\rho^{R})$ in which $M$ is a vector space
and $\rho^{L}:\mathfrak{g}\otimes M\rightarrow M$,
$\rho^{R}:M\otimes\mathfrak{g}\rightarrow M$ are bilinear maps (called left
and right $\mathfrak{g}$-actions) satisfying for $x,y\in\mathfrak{g}$, $u\in
M$,

$\displaystyle\rho^{L}(x,\rho^{L}(y,u))=~{}$
$\displaystyle\rho^{L}([x,y],u)+\rho^{L}(y,\rho^{L}(x,u)),$ (2)
$\displaystyle\rho^{L}(x,\rho^{R}(u,y))=~{}$
$\displaystyle\rho^{R}(\rho^{L}(x,u),y)+\rho^{R}(u,[x,y]),$ (3)
$\displaystyle\rho^{R}(u,[x,y])=~{}$
$\displaystyle\rho^{R}(\rho^{R}(u,x),y)+\rho^{L}(x,\rho^{R}(u,y)).$ (4)

We may also denote a representation $(M,\rho^{L},\rho^{R})$ simply by $M$ when
the left and right $\mathfrak{g}$-actions are clear. Let
$(\mathfrak{g},[~{},~{}])$ be a Leibniz algebra. Then $\mathfrak{g}$ is a
representation of itself with both left and right $\mathfrak{g}$-actions are
given by the Leibniz bracket, i.e. $\rho^{L}(x,y)=\rho^{R}(x,y)=[x,y]$, for
$x,y\in\mathfrak{g}$. This is called the adjoint representation.

Let $(\mathfrak{g},[~{},~{}])$ be a Leibniz algebra and
$(M,\rho^{L},\rho^{R})$ be a representation of it. The Loday-Pirashvili
cochain complex
$\\{C^{\ast}_{\mathsf{LP}}(\mathfrak{g},M),\delta_{\mathsf{LP}}\\}$ is given
by the following: the $n$-th cochain group
$C^{n}_{\mathsf{LP}}(\mathfrak{g},M)=\mathrm{Hom}(\mathfrak{g}^{\otimes n},M)$
and the coboundary map
$\delta_{\mathsf{LP}}:C^{n}_{\mathsf{LP}}(\mathfrak{g},M)\rightarrow
C^{n+1}_{\mathsf{LP}}(\mathfrak{g},M)$ given by

$\displaystyle(\delta_{\mathsf{LP}}f)(x_{1},\ldots,x_{n+1})=~{}$
$\displaystyle\sum_{i=1}^{n}(-1)^{i+1}~{}\rho^{L}\big{(}x_{i},f(x_{1},\ldots,\widehat{x_{i}},\ldots,x_{n+1})\big{)}+(-1)^{n+1}~{}\rho^{R}\big{(}f(x_{1},\ldots,x_{n}),x_{n+1}\big{)}$
(5) $\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i}~{}f(x_{1},\ldots,\widehat{x_{i}},\ldots,x_{j-1},[x_{i},x_{j}],x_{j+1},\ldots,x_{n+1}),$

for $x_{1},\ldots,x_{n+1}\in\mathfrak{g}$. The cohomology groups of the
cochain complex
$\\{C^{\ast}_{\mathsf{LP}}(\mathfrak{g},M),\delta_{\mathsf{LP}}\\}$ are called
the Loday-Pirashvili cohomology of the Leibniz algebra $\mathfrak{g}$ with
coefficients in the representation $M$, and they are denoted by
$H^{\ast}_{\mathsf{LP}}(\mathfrak{g},M)$.

Next we recall the graded Lie algebra which characterize Leibniz algebras as
Maurer-Cartan elements [4]. Let $\mathfrak{g}$ be a vector space, not
necessarily a Leibniz algebra. Then the graded space
$C^{\ast}_{\mathsf{LP}}(\mathfrak{g},\mathfrak{g})=\oplus_{n\geq
1}C^{n}_{\mathsf{LP}}(\mathfrak{g},\mathfrak{g})=\oplus_{n\geq
1}\mathrm{Hom}(\mathfrak{g}^{\otimes n},\mathfrak{g})$ of multilinear maps on
$\mathfrak{g}$ carries a degree $-1$ graded Lie bracket (called the Balavoine
bracket)

$\displaystyle\llbracket
f,g\rrbracket_{\mathsf{B}}:=\sum_{i=1}^{m}(-1)^{(i-1)(n-1)}~{}f\circ_{i}g~{}-~{}(-1)^{(m-1)(n-1)}\sum_{i=1}^{n}(-1)^{(i-1)(m-1)}~{}g\circ_{i}f,$
(6)

for $f\in C^{m}_{\mathsf{LP}}(\mathfrak{g},\mathfrak{g})$ and $g\in
C^{n}_{\mathsf{LP}}(\mathfrak{g},\mathfrak{g})$, where

$\displaystyle(f\circ_{i}g)$ $\displaystyle(x_{1},\ldots,x_{m+n-1})$ (7)
$\displaystyle=\sum_{\sigma\in
Sh(i-1,n-1)}(-1)^{\sigma}~{}f(x_{\sigma(1)},\ldots,x_{\sigma(i-1)},g(x_{\sigma(i)},\ldots,x_{\sigma(i+n-2)},x_{i+n-1}),\ldots,x_{m+n-1}).$

In other words,
$(C^{\ast+1}_{\mathsf{LP}}(\mathfrak{g},\mathfrak{g}),\llbracket~{},~{}\rrbracket_{\mathsf{B}})$
is a graded Lie algebra, called the Balavoine’s graded Lie algebra.

Let $(\mathfrak{g},[~{},~{}])$ be a Leibniz algebra. Let $\mu\in
C^{2}_{\mathsf{LP}}(\mathfrak{g},\mathfrak{g})$ denotes the element
corresponding to the bracket $[~{},~{}]$, i.e., $\mu(x,y)=[x,y]$, for
$x,y\in\mathfrak{g}$. Then the Leibniz identity (1) is equivalent to
$\llbracket\mu,\mu\rrbracket_{\mathsf{B}}=0$. In fact, we have the following.

######  2.4 Proposition.

Let $\mathfrak{g}$ be a vector space. Then there is a one-to-one
correspondence between Leibniz algebra structures on $\mathfrak{g}$ and
Maurer-Cartan elements in the graded Lie algebra
$(C^{\ast+1}_{\mathsf{LP}}(\mathfrak{g},\mathfrak{g}),\llbracket~{},~{}\rrbracket_{\mathsf{B}})$.

With the above notation, the coboundary map (5) of the Leibniz algebra
$\mathfrak{g}$ with coefficients in itself is given by
$\delta_{\mathsf{LP}}f=(-1)^{n-1}\llbracket\mu,f\rrbracket_{\mathsf{B}},~{}\text{
for }f\in C^{n}_{\mathsf{LP}}(\mathfrak{g},\mathfrak{g}).$

##  3\. Pre-Leibniz algebras and their Maurer-Cartan characterizations

The notion of a pre-Leibniz algebra first appeared in the recent article [24]
where it was named as Leibniz-dendriform algebra. In this section, we first
recall pre-Leibniz algebras and their relation with relative Rota-Baxter
operators. Next, we construct a new graded Lie algebra associated with a
vector space, whose Maurer-Cartan elements are precisely pre-Leibniz algebra
structures on that vector space.

######  3.1 Definition.

[24] A (left) pre-Leibniz algebra is a triple
$(\mathfrak{a},\triangleleft,\triangleright)$ consisting of a vector space
$\mathfrak{a}$ together with two bilinear operations
$\triangleleft,\triangleright:\mathfrak{a}\otimes\mathfrak{a}\rightarrow\mathfrak{a}$
satisfying the following identities

$\displaystyle x\triangleleft(y\triangleleft z+y\triangleright z)=~{}$
$\displaystyle(x\triangleleft y)\triangleleft z+y\triangleright(x\triangleleft
z),$ (8) $\displaystyle x\triangleright(y\triangleleft z)=~{}$
$\displaystyle(x\triangleright y)\triangleleft z+y\triangleleft(x\triangleleft
z+x\triangleright z),$ (9) $\displaystyle x\triangleright(y\triangleright
z)=~{}$ $\displaystyle(x\triangleleft y+x\triangleright y)\triangleright
z+y\triangleright(x\triangleright z),~{}\text{ for }x,y,z\in\mathfrak{a}.$
(10)

######  3.2 Remark.

Let $(\mathfrak{a},\triangleleft,\triangleright)$ be a pre-Leibniz algebra in
which $x\triangleleft y=-y\triangleright x$, for $x,y\in\mathfrak{a}$ (we call
such pre-Leibniz algebras ‘skew-symmetric’). In this case, all the three
identities (8)-(10) are equivalent to

$\displaystyle(x\triangleleft y)\triangleleft z-x\triangleleft(y\triangleleft
z)=(x\triangleleft z)\triangleleft y-x\triangleleft(z\triangleleft
y),~{}\text{ for }x,y,z\in\mathfrak{a}.$

In other words, $(\mathfrak{a},\triangleleft)$ is a pre-Lie algebra [14].
Therefore, skew-symmetric pre-Leibniz algebras are equivalent to pre-Lie
algebras. In this regard, one can think pre-Leibniz algebras as the
noncommutative analogue of pre-Lie algebras.

######  3.3 Definition.

Let $(\mathfrak{a},\triangleleft,\triangleright)$ and
$(\mathfrak{a}^{\prime},\triangleleft^{\prime},\triangleright^{\prime})$ be
two pre-Leibniz algebras. A morphism of pre-Leibniz algebras from
$(\mathfrak{a},\triangleleft,\triangleright)$ to
$(\mathfrak{a}^{\prime},\triangleleft^{\prime},\triangleright^{\prime})$ is a
linear map $\phi:\mathfrak{a}\rightarrow\mathfrak{a}^{\prime}$ satisfying
$\phi(x\triangleleft y)=\phi(x)\triangleleft^{\prime}\phi(y)$ and
$\phi(x\triangleright y)=\phi(x)\triangleright^{\prime}\phi(y)$, for
$x,y\in\mathfrak{a}.$

Pre-Leibniz algebras are splitting of Leibniz algebras in the following sense
[24].

######  3.4 Proposition.

Let $(\mathfrak{a},\triangleleft,\triangleright)$ be a pre-Leibniz algebra.
Then $(\mathfrak{a},[~{},~{}]_{\triangleleft,\triangleright})$ is a Leibniz
algebra, where

$\displaystyle[x,y]_{\triangleleft,\triangleright}:=x\triangleleft
y+x\triangleright y,\text{ for }x,y\in\mathfrak{a}.$

We call this the total Leibniz algebra and denoted by
$\mathfrak{a}_{\mathrm{Tot}}$.

A class of pre-Leibniz algebras arise from (relative) Rota-Baxter operators in
the Leibniz algebra context.

######  3.5 Definition.

Let $(\mathfrak{g},[~{},~{}])$ be a Leibniz algebra and
$(M,\rho^{L},\rho^{R})$ be a representation of it. A relative Rota-Baxter
operator on $M$ over the Leibniz algebra $\mathfrak{g}$ is a linear map
$T:M\rightarrow\mathfrak{g}$ satisfying

$\displaystyle[Tu,Tv]=T(\rho^{R}(u,Tv)+\rho^{L}(Tu,v)),~{}\text{ for }u,v\in
M.$

A Rota-Baxter operator on a Leibniz algebra $\mathfrak{g}$ is a relative Rota-
Baxter operator on $\mathfrak{g}$ (considered as the adjoint representation
space) over the Leibniz algebra $\mathfrak{g}$.

######  3.6 Proposition.

Let $T:M\rightarrow\mathfrak{g}$ be a relative Rota-Baxter operator on $M$
over the Leibniz algebra $\mathfrak{g}$. Then
$(M,\triangleleft,\triangleright)$ is a pre-Leibniz algebra, where

$\displaystyle u\triangleleft v:=\rho^{R}(u,Tv)~{}~{}~{}\text{ and
}~{}~{}~{}u\triangleright v:=\rho^{L}(Tu,v),~{}\text{ for }u,v\in M.$

Therefore, $(M,[~{},~{}]_{T})$ is a Leibniz algebra, where
$[u,v]_{T}:=\rho^{R}(u,Tv)+\rho^{L}(Tu,v)$, for $u,v\in M$.

In the previous proposition, we show that a relative Rota-Baxter operator on
the representation space over a Leibniz algebra induces a pre-Leibniz algebra
structure. We will now consider the converse. Let
$(\mathfrak{a},\triangleleft,\triangleright)$ be a pre-Leibniz algebra. Take
the total Leibniz algebra
$\mathfrak{a}_{\mathrm{Tot}}=(\mathfrak{a},[~{},~{}]_{\triangleleft,\triangleright})$
given in Proposition 3.4. It can be checked that the total Leibniz algebra
$\mathfrak{a}_{\mathrm{Tot}}$ has a representation on the vector space
$\mathfrak{a}$ with left and right $\mathfrak{a}_{\mathrm{Tot}}$-actions

$\displaystyle\varrho^{L}(u,v)=u\triangleright v~{}~{}~{}\text{ and
}~{}~{}~{}\varrho^{R}(v,u)=v\triangleleft u,~{}\text{for
}u\in\mathfrak{a}_{\mathrm{Tot}},v\in\mathfrak{a}.$

With this notation, the identity map
$\mathrm{id}:\mathfrak{a}\rightarrow\mathfrak{a}_{\mathrm{Tot}}$ is a relative
Rota-Baxter operator on $\mathfrak{a}$ over the total Leibniz algebra
$\mathfrak{a}_{\mathrm{Tot}}$. Moreover, the induced pre-Leibniz algebra
structure on $\mathfrak{a}$ coincides with the given one.

A new graded Lie algebra. Given a vector space, we construct a graded Lie
algebra whose Maurer-Cartan elements correspond to pre-Leibniz structures on
the vector space.

Let $C_{n}$ be the set of first $n$ natural numbers. For some convenience, we
will denote the elements of $C_{n}$ as certain symbols, and write
$C_{n}=\\{[1],[2],\ldots,[n]\\}$, for $n\geq 1$. Let $m,n\geq 1$ and $1\leq
i\leq m$. For any $\sigma\in Sh(i-1,n-1)$, we put the elements of $C_{m+n-1}$
into $m$ many boxes as follows

$\displaystyle\framebox{$[\sigma(1)]$}\quad\framebox{$[\sigma(2)]$}\quad\cdots\quad\framebox{$[\sigma(i-1)]$}\quad\framebox{$[\sigma(i)]$\quad$\cdots$
$[\sigma(i+n-2)]$ ~{}
[i+n-1]}\quad\framebox{[i+n]}\quad\cdots\quad\framebox{[m+n-1]}~{}.$

Note that, we put exactly one element in each of the first $(i-1)$ boxes, put
$n$ elements in the $i$-th box, and exactly one element in each of the last
$(m-i)$ many boxes. With the above notations, we will now define maps

$\displaystyle R^{\sigma}_{m;i,n}:C_{m+n-1}\rightarrow C_{m}~{}~{}~{}~{}\text{
and }~{}~{}~{}~{}S^{\sigma}_{m;i,n}:C_{m+n-1}\rightarrow{\bf k}[C_{n}]$ (11)

as follows:

$\displaystyle R^{\sigma}_{m;i,n}([r])=~{}$ $\displaystyle[k]\in
C_{m}\qquad\text{ if }[r]\text{ appears in the }k\text{-th box},$
$\displaystyle S^{\sigma}_{m;i,n}([r])=~{}$
$\displaystyle\begin{cases}[1]+\cdots+[n]\qquad\text{ if }[r]\text{ does not
appears in the }i\text{-th box},\\\ [j]\qquad\text{ if }[r]\text{ appears in
the }i\text{-th box and in the }j\text{-th position}.\end{cases}$

Let $\mathfrak{a}$ be a vector space. For any $n\geq 1$, we consider the space

$\displaystyle
C^{n}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}):=\mathrm{Hom}(\mathbf{k}[C_{n}]\otimes\mathfrak{a}^{\otimes
n},\mathfrak{a}).$

For any $f\in C^{m}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$, $g\in
C^{n}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ and $1\leq i\leq m$, we define
an element $f\diamond_{i}g\in
C^{m+n-1}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ by

$\displaystyle(f\diamond_{i}g)([r];x_{1},\ldots,x_{m+n-1}):=$
$\displaystyle\sum_{\sigma\in
Sh(i-1,n-1)}(-1)^{\sigma}~{}f(R^{\sigma}_{m;i,n}[r];x_{\sigma(1)},\ldots,x_{\sigma(i-1)},g(S^{\sigma}_{m;i,n}[r];x_{\sigma(i)},\ldots,x_{\sigma(i+n-2)},x_{i+n-1}),\ldots,x_{m+n-1}),$

for $[r]\in C_{m+n-1}$ and $x_{1},\ldots,x_{m+n-1}\in\mathfrak{a}$. We define
a bracket

$\displaystyle\llbracket~{},~{}\rrbracket_{\mathsf{pL}}:C^{m}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})\otimes
C^{n}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})\rightarrow
C^{m+n-1}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}),\text{ for }m,n\geq 1$
$\displaystyle\llbracket
f,g\rrbracket_{\mathsf{pL}}:=\sum_{i=1}^{m}(-1)^{(i-1)(n-1)}~{}f\diamond_{i}g~{}-(-1)^{(m-1)(n-1)}\sum_{i=1}^{n}(-1)^{(i-1)(m-1)}~{}g\diamond_{i}f.$
(12)

######  3.7 Theorem.

With the above notations, we have the followings:

  * (i)

$(C^{\ast}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}),\llbracket~{},~{}\rrbracket)$
is a degree $-1$ graded Lie algebra. In other words,
$(C^{\ast+1}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}),\llbracket~{},~{}\rrbracket)$
is a graded Lie algebra.

  * (ii)

There is a one-to-one correspondence between pre-Leibniz algebra structures on
$\mathfrak{a}$ and Maurer-Cartan elements in the graded Lie algebra
$(C^{\ast+1}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}),\llbracket~{},~{}\rrbracket)$.

###### Proof.

(i) Consider the double vector space
$D(\mathfrak{a})=\mathfrak{a}\oplus\mathfrak{a}$. Then any $f\in
C^{m}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ gives rise to an element
$D(f)\in C^{m}_{\mathsf{LP}}(D(\mathfrak{a}),D(\mathfrak{a}))$ given by

$\displaystyle D(f)\big{(}(x_{1},0),\ldots,(x_{m},0)\big{)}=~{}$
$\displaystyle\sum_{r=1}^{m}\big{(}f([r];x_{1},\ldots,x_{m}),0\big{)},$
$\displaystyle
D(f)\big{(}(x_{1},0),\ldots,(0,x_{r}),\ldots,(x_{m},0)\big{)}=~{}$
$\displaystyle\big{(}0,f([r];x_{1},\ldots,x_{m})\big{)},$ $\displaystyle
D(f)\big{(}\ldots,(0,x_{r}),\ldots,(0,x_{s}),\ldots\big{)}=~{}$ $\displaystyle
0.$

(It is easy to see that $D(f)=0$ implies that $f=0$). If $f\in
C^{m}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$, $g\in
C^{n}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ and $1\leq i\leq m$, then it
is straightforward to verify that $D(f\diamond_{i}g)=D(f)\circ_{i}D(g)$. As a
consequence, we get that ${D\llbracket f,g\rrbracket_{\mathsf{pL}}=\llbracket
D(f),D(g)\rrbracket_{\mathsf{B}}}.$ This shows that
$(C^{\ast}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}),\llbracket~{},~{}\rrbracket_{\mathsf{pL}})$
is a degree $-1$ graded Lie algebra as
$(C^{\ast}_{\mathsf{LP}}(D(\mathfrak{a}),D(\mathfrak{a})),\llbracket~{},~{}\rrbracket_{\mathsf{LP}})$
is so.

(ii) Suppose there are bilinear maps
$\triangleleft,\triangleright:\mathfrak{a}\otimes\mathfrak{a}\rightarrow\mathfrak{a}$.
They corresponds to an element $\pi\in
C^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ given by

$\displaystyle\pi([1];x,y)=x\triangleleft y~{}~{}~{}~{}\text{ and
}~{}~{}~{}~{}\pi([2];x,y)=x\triangleright y,~{}\text{ for
}x,y\in\mathfrak{a}.$ (13)

By a direct calculation using the definition, we get that

$\displaystyle\llbracket\pi,\pi\rrbracket_{\mathsf{pL}}([1];x,y,z)=~{}$
$\displaystyle 2\big{(}(x\triangleleft y)\triangleleft
z-x\triangleleft(y\triangleleft z+y\triangleright
z)+y\triangleright(x\triangleleft z)\big{)},$
$\displaystyle\llbracket\pi,\pi\rrbracket_{\mathsf{pL}}([2];x,y,z)=~{}$
$\displaystyle 2\big{(}(x\triangleright y)\triangleleft
z-x\triangleright(x\triangleleft z)+y\triangleleft(x\triangleleft
z+x\triangleright z)\big{)},$
$\displaystyle\llbracket\pi,\pi\rrbracket_{\mathsf{pL}}([3];x,y,z)=~{}$
$\displaystyle 2\big{(}(x\triangleleft y+x\triangleright y)\triangleright
z-x\triangleright(y\triangleright z)+y\triangleright(x\triangleright
z)\big{)},$

for $x,y,z\in\mathfrak{a}$. This shows that
$(\mathfrak{a},\triangleleft,\triangleright)$ is a pre-Leibniz algebra if and
only if $\llbracket\pi,\pi\rrbracket_{\mathsf{pL}}=0$. Hence the result
follows. ∎

Let $(\mathfrak{a},\triangleleft,\triangleright)$ be a pre-Leibniz algebra
with the corresponding Maurer-Cartan element $\pi\in
C^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$. Then there is a differential

$\displaystyle
d_{\pi}:=\llbracket\pi,-\rrbracket_{\mathsf{pL}}:C^{n}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})\rightarrow
C^{n+1}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$

which makes the triple
$(C^{\ast+1}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}),\llbracket~{},~{}\rrbracket_{\mathsf{pL}},d_{\pi})$
into a differential graded Lie algebra. This differential graded Lie algebra
is useful to study Maurer-Cartan deformations of a pre-Leibniz algebra.

######  3.8 Proposition.

Let $(\mathfrak{a},\triangleleft,\triangleright)$ be a pre-Leibniz algebra.
Then for any bilinear operations
$\triangleleft^{\prime},\triangleright^{\prime}:\mathfrak{a}\otimes\mathfrak{a}\rightarrow\mathfrak{a}$,
the triple
$(\mathfrak{a},\triangleleft+\triangleleft^{\prime},\triangleright+\triangleright^{\prime})$
is a pre-Leibniz algebra if and only if the element $\pi^{\prime}\in
C^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ defined by
$\pi^{\prime}([1];x,y)=x\triangleleft^{\prime}y$ and
$\pi^{\prime}([2];x,y)=x\triangleright^{\prime}y$ for $x,y\in\mathfrak{a}$, is
a Maurer-Cartan element in the differential graded Lie algebra
$(C^{\ast+1}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}),\llbracket~{},~{}\rrbracket_{\mathsf{pL}},d_{\pi})$.

###### Proof.

Observe that

$\displaystyle\llbracket\pi+\pi^{\prime},\pi+\pi^{\prime}\rrbracket_{\mathsf{pL}}=~{}$
$\displaystyle\llbracket\pi,\pi\rrbracket_{\mathsf{pL}}+\llbracket\pi,\pi^{\prime}\rrbracket_{\mathsf{pL}}+\llbracket\pi^{\prime},\pi\rrbracket_{\mathsf{pL}}+\llbracket\pi^{\prime},\pi^{\prime}\rrbracket_{\mathsf{pL}}$
$\displaystyle=~{}$ $\displaystyle
2\big{(}\llbracket\pi,\pi^{\prime}\rrbracket_{\mathsf{pL}}+\frac{1}{2}\llbracket\pi^{\prime},\pi^{\prime}\rrbracket_{\mathsf{pL}}\big{)}=2\big{(}d_{\pi}(\pi^{\prime})+\frac{1}{2}\llbracket\pi^{\prime},\pi^{\prime}\rrbracket_{\mathsf{pL}}\big{)}.$

This shows that $\pi+\pi^{\prime}$ (equivalently, the pair
$(\triangleleft+\triangleleft^{\prime},\triangleright+\triangleright^{\prime})$
) defines a pre-Leibniz algebra structure on $\mathfrak{a}$ if and only if
$\pi^{\prime}$ is a Maurer-Cartan element in the differential graded Lie
algebra
$(C^{\ast+1}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}),\llbracket~{},~{}\rrbracket_{\mathsf{pL}},d_{\pi})$.
∎

##  4\. Representations and cohomology of pre-Leibniz algebras

In this section, we first introduce representations of a pre-Leibniz algebra
and define the cohomology of a pre-Leibniz algebra with coefficients in a
representation. This cohomology in a certain sense splits the Loday-Pirashvili
cohomology of Leibniz algebra. We also find the relation between the
cohomology of a relative Rota-Baxter operator introduced in [24, 25] and the
cohomology of the induced pre-Leibniz algebra.

Representations of pre-Leibniz algebras. Here we introduce representations of
a pre-Leibniz algebra and construct the corresponding semidirect product.

######  4.1 Definition.

Let $(\mathfrak{a},\triangleleft,\triangleright)$ be a pre-Leibniz algebra. A
representation of it is consists of a vector space $M$ together with four
bilinear operations

$\displaystyle\triangleleft^{L}:\mathfrak{a}\otimes M\rightarrow
M,\qquad\triangleright^{L}:\mathfrak{a}\otimes M\rightarrow
M,\qquad\triangleleft^{R}:M\otimes\mathfrak{a}\rightarrow M~{}~{}~{}~{}\text{
and }~{}~{}~{}~{}\triangleright^{R}:M\otimes\mathfrak{a}\rightarrow M$ (14)

satisfying the $9$ identities where each set of $3$ identities correspond to
the identities (8)-(10) with exactly one of $x,y,z$ is replaced by an element
of $M$ and the corresponding bilinear operation is replaced by the respective
operation from (14). A representation as above may be denoted by
$(M,\triangleleft^{L},\triangleright^{L},\triangleleft^{R},\triangleright^{R})$.

Any pre-Leibniz algebra $(\mathfrak{a},\triangleleft,\triangleright)$ can be
regarded as a representation of itself with
$\triangleleft^{L}=\triangleleft^{R}=\triangleleft$ and
$\triangleright^{L}=\triangleright^{R}=\triangleright$. This is called the
adjoint representation.

######  4.2 Proposition.

Let $(\mathfrak{a},\triangleleft,\triangleright)$ be a pre-Leibniz algebra and
$(M,\triangleleft^{L},\triangleleft^{R},\triangleright^{L},\triangleright^{R})$
be a representation of it. Then the direct sum $\mathfrak{a}\oplus M$ carries
a pre-Leibniz algebra structure given by

$\displaystyle(x,u)~{}\overline{\triangleleft}~{}(y,v):=~{}$
$\displaystyle(x\triangleleft y,~{}x\triangleleft^{L}v+u\triangleleft^{R}y),$
$\displaystyle(x,u)~{}\overline{\triangleright}~{}(y,v):=~{}$
$\displaystyle(x\triangleright
y,~{}x\triangleright^{L}v+u\triangleright^{R}y).$

This is called the semidirect product pre-Leibniz algebra, denoted by
$\mathfrak{a}\ltimes M$.

Cohomology theory. Let $(\mathfrak{a},\triangleleft,\triangleright)$ be a pre-
Leibniz algebra and
$(M,\triangleleft^{L},\triangleright^{L},\triangleleft^{R},\triangleright^{R})$
be a representation of it. We define elements $\pi^{L}\in\mathrm{Hom}({\bf
k}[C_{2}]\otimes\mathfrak{a}\otimes M,M)$ and $\pi^{R}\in\mathrm{Hom}({\bf
k}[C_{2}]\otimes M\otimes\mathfrak{a},M)$ by

$\displaystyle\begin{cases}\pi^{L}([1];x,u)=x\triangleleft^{L}u\\\
\pi^{L}([2];x,u)=x\triangleright^{L}u,\end{cases}~{}~{}~{}~{}\text{ and
~{}~{}}~{}~{}~{}~{}\begin{cases}\pi^{R}([1];u,x)=u\triangleleft^{R}x\\\
\pi^{R}([2];u,x)=u\triangleright^{R}x,\end{cases}$

for $x\in\mathfrak{a}$ and $u\in M$. With these notations, we are now in a
position to define the cohomology of the pre-Leibniz algebra
$(\mathfrak{a},\triangleleft,\triangleright)$ with coefficients in the
representation
$(M,\triangleleft^{L},\triangleright^{L},\triangleleft^{R},\triangleright^{R})$.

For each $n\geq 1$, we define the $n$-th cochain group
$C^{n}_{\mathsf{pL}}(\mathfrak{a},M)$ as

$\displaystyle
C^{n}_{\mathsf{pL}}(\mathfrak{a},M):=\mathrm{Hom}(\mathbf{k}[C_{n}]\otimes\mathfrak{a}^{\otimes
n},M),$

and a map $\delta_{\mathsf{pL}}:C^{n}_{\mathsf{pL}}(\mathfrak{a},M)\rightarrow
C^{n+1}_{\mathsf{pL}}(\mathfrak{a},M)$ given by

$\displaystyle(\delta_{\mathsf{pL}}f)([r];x_{1},\ldots,x_{n+1})$ (15)
$\displaystyle=\sum_{i=1}^{n}(-1)^{i+1}~{}\pi^{L}\big{(}R^{\sigma_{i}}_{2;1,n}([r]);x_{i},f(S^{\sigma_{i}}_{2;1,n}([r]);x_{1},\ldots,\widehat{x_{i}},\ldots,x_{n+1})\big{)}$
$\displaystyle+(-1)^{n+1}~{}\pi^{R}\big{(}R^{id}_{2;n,1}([r]);f(S^{id}_{2;n,1}([r]);x_{1},\ldots,x_{n}),x_{n+1}\big{)}$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i}~{}f\big{(}R^{\sigma_{i,j}}_{n;j-1,2}([r]);x_{1},\ldots,\widehat{x_{i}},\ldots,x_{j-1},\pi\big{(}S^{\sigma_{i,j}}_{n;j-1,2}([r]);x_{i},x_{j}\big{)},x_{j+1},\ldots,x_{n+1}\big{)},$

for $[r]\in C_{n+1}$ and $x_{1},\ldots,x_{n+1}\in\mathfrak{a}$. Note that, in
the first summation, we denote by $\sigma_{i}\in Sh(1,n-1)$ the unique
permutation on the set $\\{1,\ldots,n\\}$ such that $\sigma_{i}(1)=i$.
Similarly, on the third summation, we denote $\sigma_{i,j}\in Sh(j-2,1)$ the
unique permutation on the set $\\{1,\ldots,j-1\\}$ such that

$\displaystyle\sigma_{i,j}(1)=1,~{}\ldots,~{}\sigma_{i,j}(i-1)=i-1,~{}\sigma_{i,j}(i)=i+1,~{}\ldots,~{}\sigma_{i,j}(j-2)=j-1,~{}\sigma_{i,j}(j-1)=i.$

Then we have the following.

######  4.3 Proposition.

The map $\delta_{\mathsf{pL}}$ satisfies
$\delta_{\mathsf{pL}}\circ\delta_{\mathsf{pL}}=0$.

###### Proof.

Since $(\mathfrak{a},\triangleleft,\triangleright)$ is a pre-Leibniz algebra
and
$(M,\triangleleft^{L},\triangleright^{L},\triangleleft^{R},\triangleright^{R})$
is a representation, it follows that $(\mathfrak{a}\oplus
M,\overline{\triangleleft},\overline{\triangleright})$ is a pre-Leibniz
algebra (Proposition 4.2). Define an element
$\overline{\pi}\in\mathrm{Hom}({\bf k}[C_{2}]\otimes(\mathfrak{a}\oplus
M)^{\otimes 2},\mathfrak{a}\oplus M)$ by

$\displaystyle\overline{\pi}([1];(x,u),(y,v))=(x,u)~{}\overline{\triangleleft}~{}(y,v)~{}~{}~{}~{}\text{
and
}~{}~{}~{}~{}\overline{\pi}([2];(x,u),(y,v))=(x,u)~{}\overline{\triangleright}~{}(y,v),$

for $(x,u),(y,v)\in\mathfrak{a}\oplus M$. Then by Theorem 3.7,
$\overline{\pi}$ is a Maurer-Cartan element in the graded Lie algebra
$(C^{\ast+1}_{\mathsf{pL}}(\mathfrak{a}\oplus M,\mathfrak{a}\oplus
M),\llbracket~{},~{}\rrbracket_{\mathsf{pL}})$. Hence $\overline{\pi}$ induces
a cochain complex $\\{C^{\ast}_{\mathsf{pL}}(\mathfrak{a}\oplus
M,\mathfrak{a}\oplus M),\overline{\delta_{\mathsf{pL}}}\\}$, where

$\displaystyle C^{n}_{\mathsf{pL}}(\mathfrak{a}\oplus M,\mathfrak{a}\oplus
M)=~{}$ $\displaystyle\mathrm{Hom}({\bf k}[C_{n}]\otimes(\mathfrak{a}\oplus
M)^{\otimes n},\mathfrak{a}\oplus M),\text{ for }n\geq 1~{}~{}~{}~{}~{}\text{
and }$ $\displaystyle~{}~{}~{}\overline{\delta_{\mathsf{pL}}}(F)=~{}$
$\displaystyle(-1)^{n-1}\llbracket\overline{\pi},F\rrbracket_{\mathsf{pL}},\text{
for }F\in C^{n}_{\mathsf{pL}}(\mathfrak{a}\oplus M,\mathfrak{a}\oplus M).$

Then it can be easily checked that the inclusions

$\displaystyle C^{n}_{\mathsf{pL}}(\mathfrak{a},M)=\mathrm{Hom}({\bf
k}[C_{n}]\otimes\mathfrak{a}^{\otimes n},M)\hookrightarrow\mathrm{Hom}({\bf
k}[C_{n}]\otimes(\mathfrak{a}\oplus M)^{\otimes n},\mathfrak{a}\oplus
M)=C^{n}_{\mathsf{pL}}(\mathfrak{a}\oplus M,\mathfrak{a}\oplus M)$

makes $\\{C^{\ast}_{\mathsf{pL}}(\mathfrak{a},M),\delta_{\mathsf{pL}}\\}$ into
a subcomplex of $\\{C^{\ast}_{\mathsf{pL}}(\mathfrak{a}\oplus
M,\mathfrak{a}\oplus M),\overline{\delta_{\mathsf{pL}}}\\}$. This implies that
$\delta_{\mathsf{pL}}\circ\delta_{\mathsf{pL}}=0$. ∎

It follows from the above proposition that
$\\{C^{\ast}_{\mathsf{pL}}(\mathfrak{a},M),\delta_{\mathsf{pL}}\\}$ is a
cochain complex. Let $Z^{n}_{\mathsf{pL}}(\mathfrak{a},M)\subset
C^{n}_{\mathsf{pL}}(\mathfrak{a},M)$ denote the space of $n$-cocycles and
$B^{n}_{\mathsf{pL}}(\mathfrak{a},M)\subset
C^{n}_{\mathsf{pL}}(\mathfrak{a},M)$ denote the space of $n$-coboundaries.
Then we have $B^{n}_{\mathsf{pL}}(\mathfrak{a},M)\subset
Z^{n}_{\mathsf{pL}}(\mathfrak{a},M)$. The corresponding quotients

$\displaystyle
H^{n}_{\mathsf{pL}}(\mathfrak{a},M):=\frac{Z^{n}_{\mathsf{pL}}(\mathfrak{a},M)}{B^{n}_{\mathsf{pL}}(\mathfrak{a},M)},\text{
for }n\geq 1$

are called the cohomology of the pre-Leibniz algebra $\mathfrak{a}$ with
coefficients in the representation $M$.

Let $(\mathfrak{a},\triangleleft,\triangleright)$ be a pre-Leibniz algebra.
Consider the corresponding Maurer-Cartan element $\pi\in
C^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ in the graded Lie algebra
$(C^{\ast+1}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}),\llbracket~{},~{}\rrbracket_{\mathsf{pL}})$.
It follows from the expressions of (12) and (15) that the differential
$\delta_{\mathsf{pL}}:C^{n}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})\rightarrow
C^{n+1}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ of the pre-Leibniz algebra
$\mathfrak{a}$ with coefficients in the adjoint representation is given by

$\displaystyle\delta_{\mathsf{pL}}f=(-1)^{n-1}\llbracket\pi,f\rrbracket_{\mathsf{pL}},\text{
for }f\in C^{n}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}).$ (16)

The corresponding cohomology groups are denoted by
$H^{\ast}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}).$ It follows that the
cohomology of the pre-Leibniz algebra $\mathfrak{a}$ with coefficients in the
adjoint representation is induced by the Maurer-Cartan element $\pi\in
C^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ in the graded Lie algebra
$(C^{\ast+1}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}),\llbracket~{},~{}\rrbracket_{\mathsf{pL}}).$
As a consequence, we get the following.

######  4.4 Theorem.

Let $(\mathfrak{a},\triangleleft,\triangleright)$ be a pre-Leibniz algebra.
Then the shifted graded space of cohomology
$H^{\ast+1}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ carries a graded Lie
algebra structure.

Relation with the Loday-Pirashvili cohomology. Let
$(\mathfrak{a},\triangleleft,\triangleright)$ be a pre-Leibniz algebra and let
$(M,\triangleleft^{L},\triangleright^{L},\triangleleft^{R},\triangleright^{R})$
be a representation of it. Then we know from Proposition 3.4 that
$\mathfrak{a}_{\mathrm{Tot}}=(\mathfrak{a},[~{},~{}]_{\triangleleft,\triangleright})$
is a Leibniz algebra. Moreover, it is easy to see that $(M,\rho^{L},\rho^{R})$
is a representation of the Leibniz algebra $\mathfrak{a}_{\mathrm{Tot}}$,
where

$\displaystyle\rho^{L}=\triangleleft^{L}+\triangleright^{L}\qquad\text{ and
}\qquad\rho^{R}=\triangleleft^{R}+\triangleright^{R}.$

We denote this representation by $M_{\mathrm{Tot}}.$ Therefore, we may
consider the Loday-Pirashvili cohomology of the Leibniz algebra
$\mathfrak{a}_{\mathrm{Tot}}$ with coefficients in the representation
$M_{\mathrm{Tot}}$. The following result finds the connection between the pre-
Leibniz cohomology and the Loday-Pirashvili cohomology.

######  4.5 Theorem.

The collection $\\{\Phi_{n}\\}_{n\geq 1}$ of maps

$\displaystyle\Phi_{n}:$
$\displaystyle~{}C^{n}_{\mathsf{pL}}(\mathfrak{a},M)\rightarrow
C^{n}_{\mathsf{LP}}(\mathfrak{a}_{\mathrm{Tot}},M_{\mathrm{Tot}}),~{}f\mapsto
f_{\mathrm{Tot}}$ $\displaystyle\text{ where
}~{}~{}f_{\mathrm{Tot}}(x_{1},\ldots,x_{n})=\sum_{r=1}^{n}f([r];x_{1},\ldots,x_{n})$

give rise to a morphism from the cochain complex
$\\{C^{\ast}_{\mathsf{pL}}(\mathfrak{a},M),\delta_{\mathsf{pL}}\\}$ defining
the pre-Leibniz cohomology to the Loday-Pirashvili cochain complex
$\\{C^{\ast}_{\mathsf{LP}}(\mathfrak{a}_{\mathrm{Tot}},M_{\mathrm{Tot}}),\delta_{\mathsf{LP}}\\}$
defining the Leibniz algebra cohomology. As a consequence, there is a morphism
$H^{\ast}_{\mathsf{pL}}(\mathfrak{a},M)\rightarrow
H^{\ast}_{\mathsf{LP}}(\mathfrak{a}_{\mathrm{Tot}},M_{\mathrm{Tot}})$ of
cohomologies.

###### Proof.

Case I. (adjoint representation) Let
$(M,\triangleleft^{L},\triangleright^{L},\triangleleft^{R},\triangleright^{R})$
be the adjoint representation of the pre-Leibniz algebra
$(\mathfrak{a},\triangleleft,\triangleright)$. That is, $M=\mathfrak{a}$ and
$\triangleleft^{L}=\triangleleft^{R}=\triangleleft$,
$\triangleright^{L}=\triangleright^{R}=\triangleright$. We have seen in (16)
that the differential $\delta_{\mathsf{pL}}$ in the cochain complex
$\\{C^{\ast}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}),\delta_{\mathsf{pL}}\\}$
is given by
$\delta_{\mathsf{pL}}(f)=(-1)^{n-1}\llbracket\pi,f\rrbracket_{\mathsf{pL}},$
for $f\in C^{n}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}).$ On the other hand,
we know that the differential $\delta_{\mathsf{LP}}$ in the cochain complex
$\\{C^{\ast}_{\mathsf{LP}}(\mathfrak{a}_{\mathrm{Tot}},\mathfrak{a}_{\mathrm{Tot}}),\delta_{\mathsf{LP}}\\}$
is given by

$\displaystyle\delta_{\mathsf{LP}}(F)=(-1)^{n-1}\llbracket\pi_{\mathrm{Tot}},F\rrbracket_{\mathsf{B}},~{}\text{
for }F\in
C^{n}_{\mathsf{LP}}(\mathfrak{a}_{\mathrm{Tot}},\mathfrak{a}_{\mathrm{Tot}}).$

Here $\pi_{\mathrm{Tot}}\in
C^{2}_{\mathsf{LP}}(\mathfrak{a}_{\mathrm{Tot}},\mathfrak{a}_{\mathrm{Tot}})$
is the element defined by $\pi_{\mathrm{Tot}}(x,y)=x\triangleleft
y+x\triangleright y$, for $x,y\in\mathfrak{a}_{\mathrm{Tot}}$. For any $f\in
C^{m}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$, $g\in
C^{n}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ and $1\leq i\leq m$, it is
easy to see that
$\Phi_{m+n-1}(f\diamond_{i}g)=\Phi_{m}(f)\circ_{i}\Phi_{n}(g)$. As a
consequence, we get that $\Phi_{m+n-1}(\llbracket
f,g\rrbracket_{\mathsf{pL}})=\llbracket\Phi_{m}(f),\Phi_{n}(g)\rrbracket_{\mathsf{B}}$.
Moreover, since $\Phi_{2}(\pi)=\pi_{\mathrm{Tot}}$, we have

$\displaystyle(\delta_{\mathsf{LP}}\circ\Phi_{n})(f)=(-1)^{n-1}\llbracket\pi_{\mathrm{Tot}},\Phi_{n}(f)\rrbracket_{\mathsf{B}}=~{}$
$\displaystyle(-1)^{n-1}\llbracket\Phi_{2}(\pi),\Phi_{n}(f)\rrbracket_{B}$
$\displaystyle=~{}$
$\displaystyle(-1)^{n-1}\Phi_{n+1}(\llbracket\pi,f\rrbracket_{\mathsf{pL}})=(\Phi_{n+1}\circ\delta_{\mathsf{pL}})(f),$

for $f\in C^{n}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$. This shows that the
collection $\\{\Phi_{n}\\}_{n\geq 1}$ defines a morphism of cochain complexes
from
$\\{C^{\ast}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}),\delta_{\mathsf{pL}}\\}$
to
$\\{C^{\ast}_{\mathsf{LP}}(\mathfrak{a}_{\mathrm{Tot}},\mathfrak{a}_{\mathrm{Tot}}),\delta_{\mathsf{LP}}\\}.$

Case II. (arbitrary representation) Let
$(M,\triangleleft^{L},\triangleright^{L},\triangleleft^{R},\triangleright^{R})$
be an arbitrary representation of the pre-Leibniz algebra
$(\mathfrak{a},\triangleleft,\triangleright)$. Consider the semidirect product
pre-Leibniz algebra $(\mathfrak{a}\oplus
M,\overline{\triangleleft},\overline{\triangleright})$ given in Proposition
4.2. Let $(\mathfrak{a}\oplus M)_{\mathrm{Tot}}=(\mathfrak{a}\oplus
M,[~{},~{}]_{\overline{\triangleleft},\overline{\triangleright}})$ be the
corresponding total Leibniz algebra. By case I, the collection
$\\{\Psi_{n}\\}_{n\geq 1}$ of maps

$\displaystyle\Psi_{n}:C^{n}_{\mathsf{pL}}(\mathfrak{a}\oplus
M,\mathfrak{a}\oplus M)\rightarrow C^{n}_{\mathsf{LP}}((\mathfrak{a}\oplus
M)_{\mathrm{Tot}},(\mathfrak{a}\oplus M)_{\mathrm{Tot}}),~{}f\mapsto
f_{\mathrm{Tot}}$

forms a morphism of cochain complexes from
$\\{C^{\ast}_{\mathsf{pL}}(\mathfrak{a}\oplus M,\mathfrak{a}\oplus
M),\delta_{\mathsf{pL}}\\}$ to $\\{C^{\ast}_{\mathsf{LP}}((\mathfrak{a}\oplus
M)_{\mathrm{Tot}},(\mathfrak{a}\oplus
M)_{\mathrm{Tot}}),\delta_{\mathsf{LP}}\\}$. We observe that

$\displaystyle\\{C^{\ast}_{\mathsf{pL}}(\mathfrak{a},M),\delta_{\mathsf{pL}}\\}$
$\displaystyle\text{ is a subcomplex of
}\\{C^{\ast}_{\mathsf{pL}}(\mathfrak{a}\oplus M,\mathfrak{a}\oplus
M),\delta_{\mathsf{pL}}\\},\text{ and }$
$\displaystyle\\{C^{\ast}_{\mathsf{LP}}(\mathfrak{a}_{\mathrm{Tot}},M_{\mathrm{Tot}}),\delta_{\mathsf{LP}}\\}$
$\displaystyle\text{ is a subcomplex of
}\\{C^{\ast}_{\mathsf{LP}}((\mathfrak{a}\oplus
M)_{\mathrm{Tot}},(\mathfrak{a}\oplus
M)_{\mathrm{Tot}}),\delta_{\mathsf{LP}}\\}.$

Finally, the result follows as the restriction of $\Psi_{\ast}$ to the
subcomplex $\\{C^{\ast}_{\mathsf{pL}}(\mathfrak{a},M),\delta_{\mathsf{pL}}\\}$
is given by $\Phi_{\ast}$. Hence the result follows. ∎

##  5\. Deformations of pre-Leibniz algebras

In this section, we study formal and finite order deformations of pre-Leibniz
algebras following the classical theory of Gerstenhaber [15].

Formal deformations. Let $(\mathfrak{a},\triangleleft,\triangleright)$ be a
pre-Leibniz algebra. Consider the space $\mathfrak{a}[[t]]$ of formal power
series in $t$ with coefficients from $\mathfrak{a}$. Then $\mathfrak{a}[[t]]$
is a $\mathbf{k}[[t]]$-module.

######  5.1 Definition.

A formal deformation of the pre-Leibniz algebra
$(\mathfrak{a},\triangleleft,\triangleright)$ consists of a pair
$(\triangleleft_{t},\triangleright_{t})$ of two formal power series in $t$
with coefficients from bilinear maps on $\mathfrak{a}$,

$\displaystyle\triangleleft_{t}=\triangleleft_{0}+t\triangleleft_{1}+~{}t^{2}\triangleleft_{2}+\cdots~{}~{}\text{
and
}~{}~{}\triangleright_{t}=\triangleright_{0}+t\triangleright_{1}+~{}t^{2}\triangleright_{2}+\cdots$

with $\triangleleft_{0}=\triangleleft$ and
$\triangleright_{0}=\triangleright$, such that
$(\mathfrak{a}[[t]],\triangleleft_{t},\triangleright_{t})$ is a pre-Leibniz
algebra over $\mathbf{k}[[t]].$

For each $i\geq 0$, we define an element $\pi_{i}\in
C^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})=\mathrm{Hom}(\mathbf{k}[C_{2}]\otimes\mathfrak{a}^{\otimes
2},\mathfrak{a})$ by

$\displaystyle\pi_{i}([1];x,y)=x\triangleleft_{i}y\quad\text{ and
}\quad\pi_{i}([2];x,y)=x\triangleright_{i}y,~{}\text{ for
}x,y\in\mathfrak{a}.$

With these notations, we have $\pi_{0}=\pi$. Moreover,
$(\triangleleft_{t},\triangleright_{t})$ defines a deformation of the pre-
Leibniz algebra $\mathfrak{a}$ if and only if
$\llbracket\pi_{0}+t\pi_{1}+t^{2}\pi_{2}+\cdots,\pi_{0}+t\pi_{1}+t^{2}\pi_{2}+\cdots\rrbracket_{\mathsf{pL}}=0.$
This is equivalent to

$\displaystyle\sum_{i+j=n}\llbracket\pi_{i},\pi_{j}\rrbracket_{\mathsf{pL}}=0,~{}\text{
for }n=0,1,2,\ldots.$ (17)

Note that (17) holds automatically for $n=0$ as
$\llbracket\pi,\pi\rrbracket_{\mathsf{pL}}=0$. However, for $n=1$, we get
$\llbracket\pi,\pi_{1}\rrbracket_{\mathsf{pL}}=0$, which implies that the
linear term $\pi_{1}\in C^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ is a
$2$-cocycle in the cohomology of the pre-Leibniz algebra $\mathfrak{a}$ with
coefficients in itself.

######  5.2 Definition.

Two formal deformations $(\triangleleft_{t},\triangleright_{t})$ and
$(\triangleleft_{t}^{\prime},\triangleright_{t}^{\prime})$ of a pre-Leibniz
algebra $(\mathfrak{a},\triangleleft,\triangleright)$ are said to be
equivalent (we write
$(\triangleleft_{t},\triangleright_{t})\sim(\triangleleft_{t}^{\prime},\triangleright_{t}^{\prime})$)
if there is a formal power series

$\displaystyle\phi_{t}=\phi_{0}+t\phi_{1}+t^{2}\phi_{2}+\cdots,\text{ with
}\phi_{i}\in\mathrm{Hom}(\mathfrak{a},\mathfrak{a})\text{ and
}\phi_{0}=\mathrm{id}_{\mathfrak{a}}$

such that the $\mathbf{k}[[t]]$-linear map
$\phi_{t}:\mathfrak{a}[[t]]\rightarrow\mathfrak{a}[[t]]$ is a morphism of pre-
Leibniz algebras from
$(\mathfrak{a}[[t]],\triangleleft_{t},\triangleright_{t})$ to
$(\mathfrak{a}[[t]],\triangleleft_{t}^{\prime},\triangleright_{t}^{\prime})$.

The following result is straightforward, hence we omit the details.

######  5.3 Theorem.

The linear terms corresponding to equivalent deformations of $\mathfrak{a}$
are cohomologous. In other words, there is a well-defined map

$\displaystyle(\text{space of formal deformations of
}\mathfrak{a})/\sim~{}\rightarrow
H^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}).$ (18)

######  5.4 Remark.

Note that the map (18) is not a bijection. However, one may consider a notion
of infinitesimal deformations of $\mathfrak{a}$, and the set of equivalence
classes of infinitesimal deformations are in one-to-one correspondence with
$H^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$. An infinitesimal deformation
is a formal deformation over the ring $\mathbf{k}[[t]]/(t^{2})$. They are also
called order $1$ deformations. More precisely, an infinitesimal deformation of
a pre-Leibniz algebra $(\mathfrak{a},\triangleleft,\triangleright)$ is a pair
$(\triangleleft_{t}=\triangleleft+t\triangleleft_{1},\triangleright_{t}=\triangleright+t\triangleright_{1})$
such that $(\mathfrak{a}[[t]]/(t^{2}),\triangleleft_{t},\triangleright_{t})$
is a pre-Leibniz algebra over $\mathbf{k}[[t]]/(t^{2})$. One may also define
equivalences between infinitesimal deformations. The map

$\displaystyle
H^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})\rightarrow(\text{space of
infinitesimal deformations of }\mathfrak{a})/\sim$

is simply given by
$[(\triangleleft_{1},\triangleright_{1})]\mapsto[(\triangleleft_{t},\triangleright_{t})]$,
where
$(\triangleleft_{t}=\triangleleft+t\triangleleft_{1},~{}\triangleright_{t}=\triangleright+t\triangleright_{1})$.

######  5.5 Definition.

A formal deformation $(\triangleleft_{t},\triangleright_{t})$ of a pre-Leibniz
algebra $(\mathfrak{a},\triangleleft,\triangleright)$ is said to be trivial if
$(\triangleleft_{t},\triangleright_{t})$ is equivalent to the underformed one
$(\triangleleft_{t}^{\prime}=\triangleleft,~{}\triangleright_{t}^{\prime}=\triangleright)$.

The following result is standard, hence we omit the proof. See [15,
Proposition 1] in the case of associative algebras.

######  5.6 Proposition.

Let $(\mathfrak{a},\triangleleft,\triangleright)$ be a pre-Leibniz algebra. If
$H^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})=0$ then any formal deformation
$(\triangleleft_{t},\triangleright_{t})$ is equivalent to a deformation
$(\triangleleft_{t}^{\prime},\triangleright_{t}^{\prime})$ of the form

$\displaystyle(\triangleleft_{t}^{\prime}=\triangleleft+t^{p}\triangleleft_{p}^{\prime}+t^{p+1}\triangleleft_{p+1}^{\prime}+\cdots,~{}\triangleright_{t}^{\prime}=\triangleright+t^{p}\triangleright_{p}^{\prime}+t^{p+1}\triangleright_{p+1}^{\prime}+\cdots),$

where the first non-vanishing term
$\pi_{p}^{\prime}=(\triangleleft_{p}^{\prime},\triangleright_{p}^{\prime})$ is
a $2$-cocycle in the cohomology of $\mathfrak{a}$ but not a $2$-coboundary.

As a consequence, we get the following.

######  5.7 Theorem.

If $H^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})=0$ then any formal
deformation of the pre-Leibniz algebra $\mathfrak{a}$ is trivial.

######  5.8 Remark.

A pre-Leibniz algebra $\mathfrak{a}$ is said to be rigid if any formal
deformation of $\mathfrak{a}$ is trivial. Thus, it follows from the above
theorem that the vanishing of the second cohomology group
$H^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ is a sufficient condition for
the rigidity of the pre-Leibniz algebra $\mathfrak{a}$.

Finite order deformations. Here we consider finite order deformations of a
pre-Leibniz algebra $\mathfrak{a}$. Given a finite order deformation of
$\mathfrak{a}$, we associate a third cohomology class (called the obstruction
class) in the cohomology of $\mathfrak{a}$ with coefficients in itself. When
the obstruction class vanishes, the given deformation extends to a deformation
of the next order.

Let $(\mathfrak{a},\triangleleft,\triangleright)$ be a pre-Leibniz algebra.
Let $N\in\mathbb{N}$ be a fixed natural number. Consider the space
$\mathfrak{a}[[t]]/(t^{N+1})$ of degree $N$ polynomials in $t$ with
coefficients from $\mathfrak{a}$. Then $\mathfrak{a}[[t]]/(t^{N+1})$ is a
$\mathbf{k}[[t]]/(t^{N+1})$-module.

######  5.9 Definition.

An order $N$ deformation of the pre-Leibniz algebra
$(\mathfrak{a},\triangleleft,\triangleright)$ consists of a pair
$(\triangleleft_{t}^{N},\triangleright_{t}^{N})$ of two degree $N$ polynomials
of the form

$\displaystyle(\triangleleft_{t}^{N}=\triangleleft_{0}+t\triangleleft_{1}+\cdots+t^{N}\triangleleft_{N},~{}\triangleright_{t}^{N}=\triangleright_{0}+t\triangleright_{1}+\cdots+t^{N}\triangleright_{N})$

with $\triangleleft_{0}=\triangleleft$ and
$\triangleright_{0}=\triangleright$, such that
$(\mathfrak{a}[[t]]/(t^{N+1}),\triangleleft_{t}^{N},\triangleright_{t}^{N})$
is a pre-Leibniz algebra over $\mathbf{k}[[t]]/(t^{N+1})$.

Using previous notations, we see that the pair
$(\triangleleft_{t}^{N},\triangleright_{t}^{N})$ is a deformation of order $N$
if and only if
$\sum_{i+j=n}\llbracket\pi_{i},\pi_{j}\rrbracket_{\mathsf{pL}}=0$, for
$n=0,1,\ldots,N$. These equations can be equivalently written as

$\displaystyle\delta_{\mathsf{pL}}(\pi_{n}):=-\llbracket\pi,\pi_{n}\rrbracket_{\mathsf{pL}}=\frac{1}{2}\sum_{i+j=n;i,j\geq
1}\llbracket\pi_{i},\pi_{j}\rrbracket_{\mathsf{pL}},~{}\text{ for
}n=0,1,\ldots,N.$ (19)

Motivated by the identities in (19), we define an element
${Ob}_{(\triangleleft_{t}^{N},\triangleright_{t}^{N})}\in
C^{3}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ by

$\displaystyle{Ob}_{(\triangleleft_{t}^{N},\triangleright_{t}^{N})}:=\frac{1}{2}\sum_{i+j=N+1;i,j\geq
1}\llbracket\pi_{i},\pi_{j}\rrbracket_{\mathsf{pL}}.$

######  5.10 Proposition.

The element ${Ob}_{(\triangleleft_{t}^{N},\triangleright_{t}^{N})}$ is a
$3$-cocycle, i.e., ${Ob}_{(\triangleleft_{t}^{N},\triangleright_{t}^{N})}\in
Z^{3}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a}).$

###### Proof.

We have

$\displaystyle\delta_{\mathsf{pL}}\big{(}\frac{1}{2}\sum_{i+j=N+1;i,j\geq
1}\llbracket\pi_{i},\pi_{j}\rrbracket_{\mathsf{pL}}\big{)}$
$\displaystyle=\frac{1}{2}\sum_{i+j=N+1;i,j\geq
1}\llbracket\pi,\llbracket\pi_{i},\pi_{j}\rrbracket_{\mathsf{pL}}\rrbracket_{\mathsf{pL}}$
$\displaystyle=\frac{1}{2}\sum_{i+j=N+1;i,j\geq
1}\big{(}\llbracket\llbracket\pi,\pi_{i}\rrbracket_{\mathsf{pL}},\pi_{j}\rrbracket_{\mathsf{pL}}-\llbracket\pi_{i},\llbracket\pi,\pi_{j}\rrbracket_{\mathsf{pL}}\rrbracket_{\mathsf{pL}}\big{)}$
$\displaystyle=-\frac{1}{4}\sum_{i_{1}+i_{2}+j=N+1;i_{1},i_{2},j\geq
1}\llbracket\llbracket\pi_{i_{1}},\pi_{i_{2}}\rrbracket_{\mathsf{pL}},\pi_{j}\rrbracket_{\mathsf{pL}}+\frac{1}{4}\sum_{i+j_{1}+j_{2}=N+1;i,j_{1},j_{2}\geq
1}\llbracket\pi_{i},\llbracket\pi_{j_{1}},\pi_{j_{2}}\rrbracket_{\mathsf{pL}}\rrbracket_{\mathsf{pL}}$
$\displaystyle=-\frac{1}{2}\sum_{i+j+k=N+1;i,j,k\geq
1}\llbracket\llbracket\pi_{i},\pi_{j}\rrbracket_{\mathsf{pL}},\pi_{k}\rrbracket_{\mathsf{pL}}=0.$

Hence the result follows. ∎

It follows from the above proposition that an order $N$ deformation
$(\triangleleft_{t}^{N},\triangleright_{t}^{N})$ of the pre-Leibniz algebra
$\mathfrak{a}$ associates a cohomology class
$[{Ob}_{(\triangleleft_{t}^{N},\triangleright_{t}^{N})}]\in
H^{3}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$, called the obstruction class.

######  5.11 Definition.

An order $N$ deformation $(\triangleleft_{t}^{N},\triangleright_{t}^{N})$ of
the pre-Leibniz algebra $\mathfrak{a}$ is said to be extensible if there
exists an element $\pi_{N+1}=(\triangleleft_{N+1},\triangleright_{N+1})\in
C^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ such that

$\displaystyle(\triangleleft_{t}^{N+1}:=\triangleleft_{t}^{N}+t^{N+1}~{}\triangleleft_{N+1},~{}\triangleright_{t}^{N+1}:=\triangleright_{t}^{N}+t^{N+1}~{}\triangleright_{N+1})~{}\text{
is a deformation of order }N+1.$

The following theorem gives a necessary and sufficient condition for the
extensibility of a finite order deformation.

######  5.12 Theorem.

An order $N$ deformation $(\triangleleft_{t}^{N},\triangleright_{t}^{N})$ of a
pre-Leibniz algebra $\mathfrak{a}$ is extensible if and only if the
corresponding obstruction class
$[{Ob}_{(\triangleleft_{t}^{N},\triangleright_{t}^{N})}]\in
H^{3}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ is trivial.

###### Proof.

Suppose the order $N$ deformation
$(\triangleleft^{N}_{t},\triangleright^{N}_{t})$ is extensible. Let
$\pi_{N+1}=(\triangleleft_{N+1},\triangleright_{N+1})$ be an element which
makes $(\triangleleft^{N+1}_{t},\triangleright^{N+1}_{t})$ into a deformation
of order $N+1$. Then it follows that

$\displaystyle\delta_{\mathsf{pL}}(\pi_{N+1})=\frac{1}{2}\sum_{i+j=N+1;i,j\geq
1}\llbracket\pi_{i},\pi_{j}\rrbracket_{\mathsf{pL}}:={Ob}_{(\triangleleft_{t}^{N},\triangleright_{t}^{N})}.$

This shows that ${Ob}_{(\triangleleft_{t}^{N},\triangleright_{t}^{N})}$ is a
coboundary. Hence the cohomology class
$[{Ob}_{(\triangleleft_{t}^{N},\triangleright_{t}^{N})}]$ is trivial.

Conversely, let $(\triangleleft_{t}^{N},\triangleright_{t}^{N})$ be an order
$N$ deformation such that the cohomology class
$[{Ob}_{(\triangleleft_{t}^{N},\triangleright_{t}^{N})}]$ is trivial.
Therefore, ${Ob}_{(\triangleleft_{t}^{N},\triangleright_{t}^{N})}$ is a
coboundary, say
${Ob}_{(\triangleleft_{t}^{N},\triangleright_{t}^{N})}=\delta_{\mathsf{pL}}(\pi_{N+1})$,
where $\pi_{N+1}=(\triangleleft_{N+1},\triangleright_{N+1})$. Then it is easy
to see that
$(\triangleleft_{t}^{N+1}:=\triangleleft_{t}^{N}+t^{N+1}~{}\triangleleft_{N+1},~{}\triangleright_{t}^{N+1}:=\triangleright_{t}^{N}+t^{N+1}~{}\triangleright_{N+1})$
is a deformation of order $N+1.$ In other words,
$(\triangleleft^{N}_{t},\triangleright^{N}_{t})$ is extensible. ∎

######  5.13 Corollary.

Let $\mathfrak{a}$ be a pre-Leibniz algebra with
$H^{3}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})=0$. Then

  * (i)

any finite order deformation is extensible;

  * (ii)

any $2$-cocycle in $Z^{2}_{\mathsf{pL}}(\mathfrak{a},\mathfrak{a})$ is the
linear term of some formal deformation of $\mathfrak{a}$.

##  6\. Homotopy pre-Leibniz algebras

In this section, we first recall the notion of Leibniz∞-algebras from [1].
Then we define pre-Leibniz∞-algebras and show that they give rise to
Leibniz∞-algebras. We also define Rota-Baxter operators on Leibniz∞-algebras
that induce pre-Leibniz∞-algebras. Finally, we consider skeletal and strict
pre-Leibniz∞-algebras and their classification results.

######  6.1 Definition.

[1] A Leibniz∞-algebra is a graded vector space
$\mathfrak{g}=\oplus_{n\in\mathbb{Z}}\mathfrak{g}_{n}$ together with a
collection $\\{\mu_{k}:\mathfrak{g}^{\otimes
k}\rightarrow\mathfrak{g}~{}|~{}\mathrm{deg}(\mu_{k})=2-k\\}_{k\geq 1}$ of
multilinear maps on $\mathfrak{g}$ satisfying the following identities

$\displaystyle\sum_{i+j=n+1}\sum_{\lambda=1}^{i}$
$\displaystyle\sum_{\sigma\in
Sh(\lambda-1,j-1)}\chi(\sigma)(-1)^{\lambda(j-1)}(-1)^{j(|x_{\sigma(1)}|+\cdots+|x_{\sigma(\lambda-1)}|)}$
(20)
$\displaystyle\mu_{i}\big{(}x_{\sigma(1)},\ldots,x_{\sigma(\lambda-1)},\mu_{j}(x_{\sigma(\lambda)},\ldots,x_{\sigma(\lambda+j-2)},x_{\lambda+j-1}),x_{\lambda+j},\ldots,x_{n}\big{)}=0,$

for $n=1,2,\ldots.$ A Leibniz∞-algebra as above may be denoted by
$(\mathfrak{g},\\{\mu_{k}\\}_{k\geq 1})$.

We will now define pre-Leibniz∞-algebras using the combinatorial maps defined
in (11). The definition will be justified by the subsequent results that are
analogous to other homotopy type algebras (see for instance [5]).

######  6.2 Definition.

A pre-Leibniz∞-algebra is a graded vector space
$\mathfrak{a}=\oplus_{n\in\mathbb{Z}}\mathfrak{a}_{n}$ with a collection
$\\{\pi_{k}:\mathbf{k}[C_{k}]\otimes\mathfrak{a}^{\otimes
k}\rightarrow\mathfrak{a}~{}|~{}\mathrm{deg}(\pi_{k})=2-k\\}_{k\geq 1}$ of
maps satisfying

$\displaystyle\sum_{i+j=n+1}$
$\displaystyle\sum_{\lambda=1}^{i}\sum_{\sigma\in
Sh(\lambda-1,j-1)}\chi(\sigma)(-1)^{\lambda(j-1)}(-1)^{j(|x_{\sigma(1)}|+\cdots+|x_{\sigma(\lambda-1)}|)}$
(21)
$\displaystyle\pi_{i}\big{(}R^{\sigma}_{i;\lambda,j}[r];x_{\sigma(1)},\ldots,x_{\sigma(\lambda-1)},\pi_{j}(S^{\sigma}_{i;\lambda,j}[r];x_{\sigma(\lambda)},\ldots,x_{\sigma(\lambda+j-2)},x_{\lambda+j-1}),\ldots,x_{n}\big{)}=0,$

for $[r]\in C_{n}$ and $n=1,2,\ldots$. A pre-Leibniz∞-algebra may be denoted
by $(\mathfrak{a},\\{\pi_{k}\\}_{k\geq 1})$.

In Proposition 3.4, we have seen that a pre-Leibniz algebra induces a Leibniz
algebra. The following theorem generalizes this in the homotopy context.

######  6.3 Theorem.

Let $(\mathfrak{a},\\{\pi_{k}\\}_{k\geq 1})$ be a pre-Leibniz∞-algebra. Then
$(\mathfrak{a},\\{\mu_{k}\\}_{k\geq 1})$ is a Leibniz∞-algebra, where

$\displaystyle\mu_{k}:\mathfrak{a}^{\otimes
k}\rightarrow\mathfrak{a},~{}\mu_{k}(x_{1},\ldots,x_{k}):=\sum_{r=1}^{k}\pi_{k}([r];x_{1},\ldots,x_{k}),~{}\text{
for }k\geq 1.$

###### Proof.

Since $(\mathfrak{a},\\{\pi_{k}\\}_{k\geq 1})$ is a pre-Leibniz∞-algebra, it
follows that (21) holds for any $n\geq 1$ and $[r]\in C_{n}$. For a fixed
$n\geq 1$, by adding these identities for all $[r]\in C_{n}$, we get

$\displaystyle\sum_{i+j=n+1}$
$\displaystyle\sum_{\lambda=1}^{i}\sum_{\sigma\in
Sh(\lambda-1,j-1)}\chi(\sigma)(-1)^{\lambda(j-1)}(-1)^{j(|x_{\sigma(1)}|+\cdots+|x_{\sigma(\lambda-1)}|)}$
$\displaystyle\bigg{(}\sum_{r=1}^{n}\pi_{i}\big{(}R^{\sigma}_{i;\lambda,j}[r];x_{\sigma(1)},\ldots,x_{\sigma(\lambda-1)},\pi_{j}(S^{\sigma}_{i;\lambda,j}[r];x_{\sigma(\lambda)},\ldots,x_{\sigma(\lambda+j-2)},x_{\lambda+j-1}),\ldots,x_{n}\big{)}\bigg{)}=0.$
(22)

For any fixed $i,j,\lambda$ and $\sigma\in Sh(\lambda-1,j-1)$, we can write

$\displaystyle\sum_{r=1}^{n}=\sum_{r\in\\{\sigma(1),\ldots,\sigma(\lambda-1)\\}}+\sum_{r\in\\{\sigma(\lambda),\ldots,\sigma(\lambda+j-2),\lambda+j-1\\}}+\sum_{\lambda+j}^{n}.$

We observe that

$\displaystyle\sum_{r\in\\{\sigma(1),\ldots,\sigma(\lambda-1)\\}}\pi_{i}\big{(}R^{\sigma}_{i;\lambda,j}[r];x_{\sigma(1)},\ldots,x_{\sigma(\lambda-1)},\pi_{j}(S^{\sigma}_{i;\lambda,j}[r];x_{\sigma(\lambda)},\ldots,x_{\sigma(\lambda+j-2)},x_{\lambda+j-1}),\ldots,x_{n}\big{)}$
$\displaystyle=\sum_{r=1}^{\lambda-1}\pi_{i}\big{(}[r];x_{\sigma(1)},\ldots,x_{\sigma(\lambda-1)},\mu_{j}(x_{\sigma(\lambda)},\ldots,x_{\sigma(\lambda+j-2)},x_{\lambda+j-1}),x_{\lambda+j},\ldots,x_{n}\big{)}.$
(23)

Similarly,

$\displaystyle\sum_{r\in\\{\sigma(\lambda),\ldots,\sigma(\lambda+j-2),\lambda+j-1\\}}\pi_{i}\big{(}R^{\sigma}_{i;\lambda,j}[r];x_{\sigma(1)},\ldots,x_{\sigma(\lambda-1)},\pi_{j}(S^{\sigma}_{i;\lambda,j}[r];x_{\sigma(\lambda)},\ldots,x_{\sigma(\lambda+j-2)},x_{\lambda+j-1}),\ldots,x_{n}\big{)}$
$\displaystyle=\pi_{i}\big{(}[\lambda];x_{\sigma(1)},\ldots,x_{\sigma(\lambda-1)},\mu_{j}(x_{\sigma(\lambda)},\ldots,x_{\sigma(\lambda+j-2)},x_{\lambda+j-1}),x_{\lambda+j},\ldots,x_{n}\big{)},$
(24)

and

$\displaystyle\sum_{\lambda+j}^{n}\pi_{i}\big{(}R^{\sigma}_{i;\lambda,j}[r];x_{\sigma(1)},\ldots,x_{\sigma(\lambda-1)},\pi_{j}(S^{\sigma}_{i;\lambda,j}[r];x_{\sigma(\lambda)},\ldots,x_{\sigma(\lambda+j-2)},x_{\lambda+j-1}),\ldots,x_{n}\big{)}$
$\displaystyle=\sum_{r=\lambda+1}^{i}\pi_{i}\big{(}[r];x_{\sigma(1)},\ldots,x_{\sigma(\lambda-1)},\mu_{j}(x_{\sigma(\lambda)},\ldots,x_{\sigma(\lambda+j-2)},x_{\lambda+j-1}),x_{\lambda+j},\ldots,x_{n}\big{)}.$
(25)

By substituting (6), (6) and (6) in the identity (6), we get that
$(\mathfrak{a},\\{\mu_{k}\\}_{k\geq 1})$ is a Leibniz∞-algebra. ∎

In the following, we introduce Rota-Baxter operators on Leibniz∞-algebras.
Note that representations of Leibniz∞-algebras are defined in [10]. The notion
of relative Rota-Baxter operators in the context of Leibniz∞-algebras can be
defined similarly.

######  6.4 Definition.

Let $(\mathfrak{g},\\{\mu_{k}\\}_{k\geq 1})$ be a Leibniz∞-algebra. A degree
$0$ linear map $T:\mathfrak{g}\rightarrow\mathfrak{g}$ is said to be a Rota-
Baxter operator if it satisfies

$\displaystyle
T(\mu_{k}(x_{1},\ldots,x_{k}))=\sum_{r=1}^{k}\mu_{k}(Tx_{1},\ldots,Tx_{r-1},x_{r},Tx_{r+1},\ldots,Tx_{k}),~{}\text{
for }k\geq 1.$

Let $(\mathfrak{g},[~{},~{}])$ be a Leibniz algebra and
$R:\mathfrak{g}\rightarrow\mathfrak{g}$ be a Rota-Baxter operator on it. In
this case, the pair $(\mathfrak{g},R)$ is often called a Rota-Baxter Leibniz
algebra. A representation of a Rota-Baxter Leibniz algebra $(\mathfrak{g},R)$
consists of a pair $(M,R_{M})$ in which $M=(M,\rho^{L},\rho^{R})$ is a
representation of the Leibniz algebra $\mathfrak{g}$ and $R_{M}:M\rightarrow
M$ is a linear map satisfying

$\displaystyle
R_{M}(\rho^{L}(x,u))=R_{M}(\rho^{L}(x,R_{M}(u))+\rho^{L}(R(x),u)),~{}~{}~{}~{}R_{M}(\rho^{R}(u,x))=R_{M}(\rho^{R}(u,R(x))+\rho^{R}(R_{M}(u),x)),$

for $x\in\mathfrak{g},u\in M.$ Let $(\mathfrak{g},R)$ be a Rota-Baxter Leibniz
algebra and $(M,R_{M})$ be a representation of it. Then it is easy to see that
$(\underbrace{M}_{-1}\oplus\underbrace{\mathfrak{g}}_{0},\\{\mu_{k}\\}_{k\geq
1})$ is a Leibniz∞-algebra, where

$\displaystyle\mu_{1}=0,~{}~{}~{}~{}\mu_{2}(x,y)=[x,y],~{}~{}~{}~{}\mu_{2}(x,u)=\rho^{L}(x,u),~{}~{}~{}~{}\mu_{2}(u,x)=\rho^{R}(u,x)~{}~{}~{}~{}\text{and
}~{}~{}\mu_{k}=0\text{ for }k\geq 3.$

Moreover, the map $T:M\oplus\mathfrak{g}\rightarrow M\oplus\mathfrak{g}$,
$T((u,x))=(R_{M}(u),R(x))$ is a Rota-Baxter operator on the Leibniz∞-algebra
$(M\oplus\mathfrak{g},\\{\mu_{k}\\}_{k\geq 1})$.

######  6.5 Theorem.

Let $T:\mathfrak{g}\rightarrow\mathfrak{g}$ be a Rota-Baxter operator on the
Leibniz∞-algebra $(\mathfrak{g},\\{\mu_{k}\\}_{k\geq 1})$. Then
$(\mathfrak{g},\\{\pi_{k}\\}_{k\geq 1})$ is a pre-Leibniz∞-algebra, where

$\displaystyle\pi_{k}([r];x_{1},\ldots,x_{k}):=\mu_{k}(Tx_{1},\ldots,Tx_{r-1},x_{r},Tx_{r+1},\ldots,Tx_{k}),\text{
for }k\geq 1\text{ and }1\leq r\leq k.$

###### Proof.

Since $(\mathfrak{g},\\{\mu_{k}\\}_{k\geq 1})$ is a Leibniz∞-algebra, the
identity (20) holds for $n\geq 1$ and any $x_{1},\ldots,x_{n}\in\mathfrak{g}$.
Let us replace the tuple $(x_{1},\ldots,x_{n})$ of elements in $\mathfrak{g}$
by $(Tx_{1},\ldots,x_{r},\ldots,Tx_{n})$, for some $1\leq r\leq n$. For fixed
$i,j,\lambda$ and $\sigma\in Sh(\lambda-1,j-1)$, if
$r\in\\{\sigma(1),\ldots,\sigma(\lambda-1)\\}$, then the term written in the
second line of (20) looks as

$\displaystyle\mu_{i}\big{(}T(x_{\sigma(1)}),\ldots,x_{r},\ldots,T(x_{\sigma(\lambda-1)}),\mu_{j}\big{(}T(x_{\sigma(\lambda)}),\ldots,T(x_{\sigma(\lambda+j-2)}),T(x_{\lambda+j-1})\big{)},\ldots,T(x_{n})\big{)}$
$\displaystyle=\pi_{i}\big{(}R^{\sigma}_{i;\lambda,j}[r];x_{\sigma(1)},\ldots,x_{\sigma(\lambda-1)},\pi_{j}\big{(}S^{\sigma}_{i;\lambda,j}[r];x_{\sigma(\lambda)},\ldots,x_{\sigma(\lambda+j-2)},x_{\lambda+i-1}\big{)},\ldots,x_{n}\big{)}.$

Similarly, if
$r\in\\{\sigma(\lambda),\ldots,\sigma(\lambda+j-2),\lambda+j-1\\}$ or
$r\in\\{\lambda+j,\ldots,n\\}$, then the term written in the second line of
(20) will have the same expression as above. Therefore, in any choice of $r$,
we get the identities of a pre-Leibniz∞-algebra. ∎

Skeletal and strict pre-Leibniz∞-algebras and their classifications. Here we
mainly concentrate on pre-Leibniz∞-algebras whose underlying graded vector
space is concentrated in degrees $-1$ and $0$. We call them $2$-term pre-
Leibniz∞-algebras. Two particular classes of $2$-term pre-Leibniz∞-algebras
are ‘skeletal’ and ‘strict’ algebras. Finally, we classify skeletal and strict
algebras.

Let
$(\mathfrak{a}=\oplus_{n\in\mathbb{Z}}\mathfrak{a}_{n},\\{\pi_{k}\\}_{k\geq
1})$ be a pre-Leibniz∞-algebra such that $\mathfrak{a}_{n}=0$ for $n\neq-1,0$.
Then it follows from the degree reason that $\pi_{k}=0$ for $k\geq 4$. We
write $d=\pi_{1}:\mathfrak{a}_{-1}\rightarrow\mathfrak{a}_{0}$. Then the
identities of pre-Leibniz∞-algebra can be described by the following.

######  6.6 Definition.

A $2$-term pre-Leibniz∞-algebra is a triple
$(\mathfrak{a}_{-1}\xrightarrow{d}\mathfrak{a}_{0},\pi_{2},\pi_{3})$
consisting of a $2$-term chain complex
$\mathfrak{a}_{-1}\xrightarrow{d}\mathfrak{a}_{0}$, maps
$\pi_{2}\in\mathrm{Hom}({\bf
k}[C_{2}]\otimes\mathfrak{a}_{i}\otimes\mathfrak{a}_{j},\mathfrak{a}_{i+j})$
for $-1\leq i,j,i+j\leq 0$, and a map $\pi_{3}\in\mathrm{Hom}({\bf
k}[C_{3}]\otimes\mathfrak{a}_{0}^{\otimes 3},\mathfrak{a}_{-1})$ satisfying
the following set of identities

  * (i)

$d\pi_{2}([r];x,u)=\pi_{2}([r];x,du),$
$d\pi_{2}([r];u,x)=\pi_{2}([r];du,x),$

  * (ii)

$\pi_{2}([r];du,v)=\pi_{2}([r];u,dv),$

  * (iii)

$(\pi_{2}\diamond_{1}\pi_{2}-\pi_{2}\diamond_{2}\pi_{2})([s];x,y,z)=d\pi_{3}([s];x,y,z),$

  * (iv)

$(\pi_{2}\diamond_{1}\pi_{2}-\pi_{2}\diamond_{2}\pi_{2})([s];x,y,u)=\pi_{3}([s];x,y,du),$
$(\pi_{2}\diamond_{1}\pi_{2}-\pi_{2}\diamond_{2}\pi_{2})([s];x,u,y)=\pi_{3}([s];x,du,y),$
$(\pi_{2}\diamond_{1}\pi_{2}-\pi_{2}\diamond_{2}\pi_{2})([s];u,x,y)=\pi_{3}([s];du,x,y),$

  * (v)

$(\pi_{3}\diamond_{1}\pi_{2}-\pi_{3}\diamond_{2}\pi_{2}+\pi_{3}\diamond_{3}\pi_{2})([t];x,y,z,w)=(\pi_{2}\diamond_{1}\pi_{3}+\pi_{2}\diamond_{2}\pi_{3})([t];x,y,z,w),$

for all $x,y,z,w\in\mathfrak{a}_{0}$, $u,v\in\mathfrak{a}_{-1}$, $[r]\in
C_{2}$, $[s]\in C_{3}$ and $[t]\in C_{4}.$

######  6.7 Definition.

A $2$-term pre-Leibniz∞-algebra
$(\mathfrak{a}_{-1}\xrightarrow{d}\mathfrak{a}_{0},\pi_{2},\pi_{3})$ is called

  * •

‘skeletal’ pre-Leibniz∞-algebra if $d=0$,

  * •

‘strict’ pre-Leibniz∞-algebra if $\pi_{3}=0$.

In the following result, we classify skeletal pre-Leibniz∞-algebras in terms
of third cocycles of pre-Leibniz algebras.

######  6.8 Theorem.

There is a one-to-one correspondence between skeletal pre-Leibniz∞-algebras
and tuples $(\mathfrak{a},M,\theta)$ consisting of a pre-Leibniz algebra
$\mathfrak{a}$, a representation $M$ and a $3$-cocycle $\theta\in
Z^{3}_{\mathsf{pL}}(\mathfrak{a},M)$ of the pre-Leibniz algebra $\mathfrak{a}$
with coefficients in $M$.

###### Proof.

Let $(\mathfrak{a}_{-1}\xrightarrow{0}\mathfrak{a}_{0},\pi_{2},\pi_{3})$ be a
skeletal pre-Leibniz∞-algebra. Then it follows from Definition 6.6 (iii) that
$\mathfrak{a}_{0}$ is a pre-Leibniz algebra with structure maps

$\displaystyle x\triangleleft y=\pi_{2}([1];x,y)~{}~{}~{}~{}\text{ and
}~{}~{}~{}~{}x\triangleright y=\pi_{2}([2];x,y),~{}\text{ for
}x,y\in\mathfrak{a}_{0}.$

Moreover, the condition (iv) of Definition 6.6 implies that
$(\mathfrak{a}_{-1},\triangleleft^{L},\triangleright^{L},\triangleleft^{R},\triangleright^{R})$
is a representation of the pre-Leibniz algebra
$(\mathfrak{a}_{0},\triangleleft,\triangleright)$, where

$\displaystyle
x\triangleleft^{L}u=\pi_{2}([1];x,u),~{}~{}~{}~{}~{}~{}x\triangleright^{L}u=\pi_{2}([2];x,u),~{}~{}~{}~{}~{}~{}u\triangleleft^{R}x=\pi_{2}([1];u,x)~{}~{}~{}~{}\text{
and }~{}~{}~{}~{}u\triangleright^{R}x=\pi_{2}([2];u,x),$

for $x\in\mathfrak{a}_{0}$, $u\in\mathfrak{a}_{-1}$. Finally, the condition
(v) of Definition 6.6 is equivalent to
$(\delta_{\mathsf{pL}}(\pi_{3}))([t];x,y,z,w)=0$, where $\delta_{\mathsf{pL}}$
is the differential on the cochain complex of the pre-Leibniz algebra
$\mathfrak{a}_{0}$ with coefficients in the representation
$\mathfrak{a}_{-1}$. Hence we obtain a required tuple
$(\mathfrak{a}_{0},\mathfrak{a}_{-1},\pi_{3})$.

Conversely, let $(\mathfrak{a},M,\theta)$ be a triple consisting of a pre-
Leibniz algebra $\mathfrak{a}=(\mathfrak{a},\triangleleft,\triangleright)$, a
representation
$M=(M,\triangleleft^{L},\triangleright^{L},\triangleleft^{R},\triangleright^{R})$
and a $3$-cocycle $\theta\in Z^{3}_{\mathsf{pL}}(\mathfrak{a},M)$. Then it can
be easily checked that $(M\xrightarrow{0}\mathfrak{a},\pi_{2},\pi_{3})$ is a
skeletal pre-Leibniz∞-algebra, where

$\displaystyle\pi_{2}([1];x,y)=$ $\displaystyle x\triangleleft
y,\quad\pi_{2}([2];x,y)=x\triangleright
y,\quad\pi_{2}([1];x,u)=x\triangleleft^{L}u,\quad\pi_{2}([2];x,u)=x\triangleright^{L}u,$
$\displaystyle\pi_{2}([1];u,x)=u\triangleleft^{R}x,\quad\pi_{2}([2];u,x)=u\triangleright^{R}x~{}~{}~{}~{}~{}\text{
and }~{}~{}~{}~{}\pi_{3}=\theta.$

Finally, the above two correspondences are inverses to each other. This
completes the proof. ∎

In the following, we introduce crossed module of pre-Leibniz algebras and
using them, we classify strict pre-Leibniz∞-algebras.

######  6.9 Definition.

A crossed module of pre-Leibniz algebras is a tuple
$(\mathfrak{a},\mathfrak{b},d,\pi^{L},\pi^{R})$ consisting of two pre-Leibniz
algebras $\mathfrak{a}$ and $\mathfrak{b}$, a pre-Leibniz algebra morphism
$d:\mathfrak{a}\rightarrow\mathfrak{b}$ and maps

$\displaystyle\pi^{L}\in\mathrm{Hom}(\mathbf{k}[C_{2}]\otimes\mathfrak{b}\otimes\mathfrak{a},\mathfrak{a}),\qquad\quad\pi^{R}\in\mathrm{Hom}(\mathbf{k}[C_{2}]\otimes\mathfrak{a}\otimes\mathfrak{b},\mathfrak{a})$

which makes $\mathfrak{a}$ into a representation of the pre-Leibniz algebra
$\mathfrak{b}$, satisfying additionally for $x,y\in\mathfrak{a}$,
$b\in\mathfrak{b}$, $[r]\in C_{2}$ and $[s]\in C_{3}$,

  * (i)

$d(\pi^{L}([r];b,x))=\pi_{\mathfrak{b}}([r];b,d(x)),$
$d(\pi^{R}([r];x,b))=\pi_{\mathfrak{b}}([r];d(x),b),$

  * (ii)

$\pi^{L}([r];d(x),y)=\pi_{\mathfrak{a}}([r];x,y),$
$\pi^{R}([r];x,d(y))=\pi_{\mathfrak{a}}([r];x,y),$

  * (iii)

$(\pi^{R}\diamond_{1}\pi_{\mathfrak{a}})([s];x,y,b)=(\pi_{\mathfrak{a}}\diamond_{2}\pi^{R})([s];x,y,b),$
$(\pi_{\mathfrak{a}}\diamond_{1}\pi^{R})([s];x,b,y)=(\pi_{\mathfrak{a}}\diamond_{2}\pi^{L})([s];x,b,y),$
$(\pi_{\mathfrak{a}}\diamond_{1}\pi^{L})([s];b,x,y)=(\pi^{L}\diamond_{2}\pi_{\mathfrak{a}})([s];b,x,y).$

######  6.10 Theorem.

There is a one-to-one correspondence between strict pre-Leibniz∞-algebras and
crossed module of pre-Leibniz algebras.

###### Proof.

Let $(\mathfrak{a}_{-1}\xrightarrow{d}\mathfrak{a}_{0},\pi_{2},\pi_{3}=0)$ be
a strict pre-Leibniz∞-algebra. Define an element

$\displaystyle\pi_{\mathfrak{a}_{-1}}\in\mathrm{Hom}({\bf
k}[C_{2}]\otimes(\mathfrak{a}_{-1})^{\otimes 2},\mathfrak{a}_{-1})~{}~{}\text{
by
}~{}~{}\pi_{\mathfrak{a}_{-1}}([r];u,v):=\pi_{2}([r];du,v)=\pi_{2}([r];u,dv),$

for $[r]\in C_{2}$ and $u,v\in\mathfrak{a}_{-1}$. Then
$\pi_{\mathfrak{a}_{-1}}$ defines a pre-Leibniz algebra structure on
$\mathfrak{a}_{-1}$. We also define an element

$\displaystyle\pi_{\mathfrak{a}_{0}}\in\mathrm{Hom}({\bf
k}[C_{2}]\otimes(\mathfrak{a}_{0})^{\otimes 2},\mathfrak{a}_{0})~{}~{}\text{
by }~{}~{}\pi_{\mathfrak{a}_{0}}([r];x,y):=\pi_{2}([r];x,y),\text{ for }[r]\in
C_{2}\text{ and }x,y\in\mathfrak{a}_{0}.$

It follows from Definition 6.6 (iii) that $\pi_{\mathfrak{a}_{0}}$ defines a
pre-Leibniz algebra structure on $\mathfrak{a}_{0}$. Moreover, the map
$d:\mathfrak{a}_{-1}\rightarrow\mathfrak{a}_{0}$ is a pre-Leibniz algebra
morphism as

$\displaystyle
d\big{(}\pi_{\mathfrak{a}_{-1}}([r];u,v)\big{)}=d(\pi_{2}([r];du,v))=\pi_{2}([r];du,dv)=\pi_{\mathfrak{a}_{0}}([r];du,dv).$

Finally, we define maps $\pi^{L}\in\mathrm{Hom}({\bf
k}[C_{2}]\otimes\mathfrak{a}_{0}\otimes\mathfrak{a}_{-1},\mathfrak{a}_{-1})$
and $\pi^{R}\in\mathrm{Hom}({\bf
k}[C_{2}]\otimes\mathfrak{a}_{-1}\otimes\mathfrak{a}_{0},\mathfrak{a}_{-1})$
by

$\displaystyle\pi^{L}([r];x,u)=\pi_{2}([r];x,u)~{}~{}\text{ and
}~{}~{}\pi^{R}([r];u,x)=\pi_{2}([r];u,x),~{}\text{ for
}x\in\mathfrak{a}_{0},u\in\mathfrak{a}_{-1}.$

It is easy to see that
$(\mathfrak{a}_{-1},\mathfrak{a}_{0},d,\pi^{L},\pi^{R})$ is a crossed module
of pre-Leibniz algebras.

Conversely, let $(\mathfrak{a},\mathfrak{b},d,\pi^{L},\pi^{R})$ be a crossed
module of pre-Leibniz algebras. Then it is straightforward to verify that
$(\mathfrak{a}\xrightarrow{d}\mathfrak{b},\pi_{2},\pi_{3}=0)$ is a strict pre-
Leibniz∞-algebra, where

$\displaystyle\pi_{2}([r];b,b^{\prime})=\pi_{\mathfrak{b}}([r];b,b^{\prime}),~{}~{}~{}~{}\pi_{2}([r];b,x)=\pi^{L}([r];b,x)~{}~{}\text{
and }~{}~{}\pi_{2}([r];x,b)=\pi^{R}([r];x,b),$

for $b,b^{\prime}\in\mathfrak{b}$ and $x\in\mathfrak{a}.$ This completes the
proof. ∎

######  6.11 Example.

Let $\mathfrak{a}$ be a pre-Leibniz algebra, and let $\pi\in\mathrm{Hom}({\bf
k}[C_{2}]\otimes\mathfrak{a}^{\otimes 2},\mathfrak{a})$ be the corresponding
Maurer-Cartan element. Then
$(\mathfrak{a}\xrightarrow{\mathrm{id}}\mathfrak{a},\pi_{2}=\pi,\pi_{3}=0)$ is
a strict pre-Leibniz∞-algebra. The corresponding crossed module of pre-Leibniz
algebras is given by $(\mathfrak{a},\mathfrak{a},\mathrm{id},\pi,\pi).$

Acknowledgements. The author would like to thank Indian Institute of
Technology (IIT) Kharagpur for providing the beautiful academic environment
where the research has been carried out.

## References

  * [1] M. Ammar and N. Poncin, Coalgebraic approach to the Loday Infinity Category, Stem differential for $2n$-ary graded and homotopy algebras, Ann. Inst. Fourier, Grenoble 60 (1) (2010) 355-387.
  * [2] C. Bai, Left-symmetric algebras from linear functions, J. Algebra 281 (2004) 651-665.
  * [3] C. Bai, Left-symmetric bialgebras and an analogue of the classical Yang-Baxter equation, Comm. Contemp. Math. 10 (2008) 221-260.
  * [4] D. Balavoine, Deformation of algebras over a quadratic operad, Operads: Proceedings of Renaissance Conferences (Hartford, CT/ Luminy, 1995), 207-234, Contemp. Math., 202, Amer. Math. Soc., Providence, RI, 1997.
  * [5] J. C. Baez and A. S. Crans, Higher-dimensional algebra VI: Lie $2$-algebras, Theor. Appl. Cat. 12 (2004) 492-538.
  * [6] A. Bloh, A generalization of the concept of a Lie algebra, Dokl. Akad. Nauk SSSR 165 (3) (1965) 471-473.
  * [7] A. Bloh, Cartan-Eilenberg homology theory for a generalized class of Lie algebras, Dokl. Akad. Nauk SSSR 175 (8) (1967) 824-826.
  * [8] D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Centr. Eur. J. Math. 4 (2006) 323-357.
  * [9] F. Chapoton and M. Livernet, Pre-Lie algebras and the rooted trees operad, International Mathematics Research Notices, Volume 2001, Issue 8 (2001) 395-408.
  * [10] Z. Chen, M. Stiénon and P. Xu, From Atiyah Classes to Homotopy Leibniz algebras, Commun. Math. Phys. 341 (2016) 309-349.
  * [11] A. Diatta and A. Medina, Classical Yang-Baxter equation and left-symmetric affine geometry on Lie groups, Manuscripta Math. 114 (2004) 477-486.
  * [12] V. Dotsenko, S. Shadrin and B. Vallette, Pre-Lie deformation theory, Mosc. Math. J. 16 (2016) no. 3, 505-543.
  * [13] A. Dzhumadil’daev, Cohomologies and deformations of right-symmetric algebras, J. Math. Sci. vol. 93, no. 6 (1999) 836-876.
  * [14] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963) 267-288.
  * [15] M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964) 59-103.
  * [16] G. Hochschild, On the cohomology of associative algebras, Ann. of Math. (2) 46 (1945) 58-67.
  * [17] D. Khudaverdyan, N. Poncin and J. Qiu, On the infinity category of homotopy Leibniz algebras, Theory Appl. Categ. 29 (2014) 332-370.
  * [18] B. A. Kupershmidt, What a classical r-matrix really is, J. Nonlinear Math. Phys. 6 (1999) no. 4, 448-488.
  * [19] J.-L. Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math. (2) 39 (1993) no. 3-4, 269-293.
  * [20] J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296 (1993) no. 1, 139-158.
  * [21] A. Nijenhuis and R. Richardson, Deformation of Lie algebra structures, J. Math. Mech. 17 (1967) 89-105.
  * [22] A. Rezaei-Aghdam, L. Sedghi-Ghadim and G. Haghighatdoost, Leibniz bialgebras, classical Yang-Baxter equations and dynamical systems, Adv. Appl. Clifford Algebras (2021) 31:77.
  * [23] D. Segal, Free left-symmetrical algebras and an analogue of the Poincare-Birkhoff-Witt theorem, J. Algebra 164 (1994) 750-772.
  * [24] Y. Sheng and R. Tang, Leibniz bialgebras, relative Rota-Baxter operators and the classical Leibniz Yang-Baxter equation, J. Noncommutative Geom. to appear.
  * [25] R. Tang, Y. Sheng and Y. Zhou, Deformations of relative Rota-Baxter operators on Leibniz algebras, Int. J. Geom. Methods Mod. Phys. Vol. 17, No. 12, 2050174 (2020).
  * [26] K. Uchino, Derived bracket and sh Leibniz algebras, J. Pure Appl. Algebra 215 (2011) 1102-1111.