# Compatible $L_{\infty}$-algebras

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

A compatible $L_{\infty}$-algebra is a graded vector space together with two
compatible $L_{\infty}$-algebra structures on it. Given a graded vector space,
we construct a graded Lie algebra whose Maurer-Cartan elements are precisely
compatible $L_{\infty}$-algebra structures on it. We provide examples of
compatible $L_{\infty}$-algebras arising from Nijenhuis operators, compatible
$V$-datas and compatible Courant algebroids. We define the cohomology of a
compatible $L_{\infty}$-algebra and as an application, we study formal
deformations. Next, we classify ‘strict’ and ‘skeletal’ compatible
$L_{\infty}$-algebras in terms of crossed modules and cohomology of compatible
Lie algebras. Finally, we introduce compatible Lie $2$-algebras and find their
relationship with compatible $L_{\infty}$-algebras.

2020 Mathematics Subject Classification: 17B56, 18N40, 18N25.

Keywords: Compatible Lie algebras, Compatible $L_{\infty}$-algebras,
Deformations, $V$-datas, Courant algebroids, Compatible Lie $2$-algebras.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Compatible $L_{\infty}$-algebras
  4. 4 Examples of compatible $L_{\infty}$-algebras
  5. 5 Cohomology and deformations of compatible $L_{\infty}$-algebras
  6. 6 Classification of some compatible $L_{\infty}$-algebras
  7. 7 Compatible Lie $2$-algebras

##  1 Introduction

Compatible algebraic and geometric structures appeared in various places of
mathematics and mathematical physics. Among others, compatible Lie algebras
are widely studied in the literature. A compatible Lie algebra is a triple
$(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$ consisting of a vector space
$\mathfrak{g}$ equipped with two Lie brackets $[~{},~{}]$ and
$[~{},~{}]^{\prime}$ satisfying some compatibility condition. The
compatibility is equivalent to the fact that the sum
$[~{},~{}]+[~{},~{}]^{\prime}$ is a Lie bracket on $\mathfrak{g}$. Compatible
Lie algebras are closely related to Nijenhuis deformations of Lie algebras,
classical Yang-Baxter equations, principal chiral fields, loop algebras over
Lie algebras and elliptic theta functions [8, 9, 10, 21, 22]. Similarly,
compatible associative algebras and their relations with associative Yang-
Baxter equations, quiver representations and bialgebra theory are explored in
[20, 23, 24, 32]. Recently, compatible Lie algebras and compatible associative
algebras are studied in [15, 4] from cohomological points of view. In the
geometric context, compatible Poisson structures were appeared in the
mathematical study of biHamiltonian mechanics [13, 19]. Such geometric
structures are closely related to compatible Lie algebras via dualization [3].
See [6, 29] for the operadic study of compatible structures.

The notion of $L_{\infty}$-algebras (also called strongly homotopy Lie
algebras) are the homotopy analogue of Lie algebras [17, 16]. More precisely,
an $L_{\infty}$-algebra is a pair $(L,\\{l_{k}\\}_{k\geq 1})$ consisting of a
graded vector space $L$ together with a collection of skew-symmetric
multilinear maps $l_{k}:L^{\otimes k}\rightarrow L$, for $k\geq 1$, of certain
degrees that satisfy some higher Jacobi identities. $L_{\infty}$-algebras
appeared in the study of deformation theory, multisymplectic geometry, Courant
algebroids, Gauge theory and many classical topics in mathematics and
mathematical physics [12, 25, 26, 5, 11]. In [2], the authors introduced Lie
$2$-algebras as the categorification of Lie algebras and showed that the
category of Lie $2$-algebras is equivalent to the category of
$L_{\infty}$-algebras with underlying graded vector space concentrated in two
degrees.

In this paper, we define a suitable compatibility condition between two
$L_{\infty}$-algebra structures on a graded vector space. A compatible
$L_{\infty}$-algebra is a triple $(L,\\{l_{k}\\}_{k\geq
1},\\{l_{k}^{\prime}\\}_{k\geq 1})$ in which $(L,\\{l_{k}\\}_{k\geq 1})$ and
$(L,\\{l_{k}^{\prime}\\}_{k\geq 1})$ are $L_{\infty}$-algebras satisfying the
compatibility condition. Compatible $L_{\infty}$-algebras can be seen as the
homotopy analogue of compatible (graded) Lie algebras. Given a graded vector
space $L$, we construct a graded Lie algebra whose Maurer-Cartan elements are
in one-to-one correspondence with compatible $L_{\infty}$-algebra structures
on $L$ (cf. Theorem 3.4). We construct a compatible differential graded Lie
algebra from a compatible Lie algebra via the Nijenhuis-Richardson bracket. We
provide examples of compatible $L_{\infty}$-algebras arising from compatible
$A_{\infty}$-algebras, Nijenhuis operators on $L_{\infty}$-algebras,
compatible $V$-datas and compatible Courant algebroids. An application of
compatible $V$-datas is given to relative Rota-Baxter operators on compatible
Lie algebras.

Next, we define the cohomology of a compatible $L_{\infty}$-algebra
$(L,\\{l_{k}\\}_{k\geq 1},\\{l_{k}^{\prime}\\}_{k\geq 1})$ using the
corresponding Maurer-Cartan element in the graded Lie algebra given in Theorem
3.4. As an application of cohomology, we extensively study deformations of
compatible $L_{\infty}$-algebras. We show that the vanishing of the second
cohomology group of compatible $L_{\infty}$-algebra implies the rigidity of
the structure (cf. Theorem 5.4). Moreover, given a finite order deformation,
we construct a third cohomology class, called the obstruction class, which
vanishes if and only if the given deformation extends to deformation of next
order (cf. Theorem 5.8).

$L_{\infty}$-algebras whose underlying graded vector space is concentrated in
two degrees (called $2$-term $L_{\infty}$-algebras) pay particular attention
as they are related to the categorification of Lie algebras [2]. In this
paper, we focus on $2$-term compatible $L_{\infty}$-algebras. Some particular
classes of such algebras are ‘strict algebras’ and ‘skeletal algebras’. We
introduce crossed module of compatible Lie algebras and show that there is a
one-to-one correspondence between crossed modules of compatible Lie algebras
and strict algebras (cf. Theorem 6.8). We also show that skeletal algebras can
be classified by suitable $3$-cocycles of compatible Lie algebras (cf. Theorem
6.9). Finally, we introduce compatible Lie $2$-algebras as the
categorification of compatible Lie algebras. We show that the category of
compatible Lie $2$-algebras is equivalent to the category of compatible
$L_{\infty}$-algebras whose underlying graded vector space is concentrated in
two degrees and the differentials induced by two $L_{\infty}$-algebras are the
same (cf. Theorem 7.5).

The paper is organized as follows. In Section 2, we recall some basic results
on compatible Lie algebras and $L_{\infty}$-algebras. In Section 3, we
introduce compatible $L_{\infty}$-algebras as the homotopy analogue of
compatible Lie algebras. Given a graded vector space $L$, we construct the
graded Lie algebra whose Maurer-Cartan elements correspond to compatible
$L_{\infty}$-algebra structures on $L$. Examples of compatible
$L_{\infty}$-algebras from Nijenhuis operators, compatible $V$-datas and
compatible Courant algebroids are given in Section 4. In Section 5, we
introduce the cohomology of a compatible $L_{\infty}$-algebra and study
deformations. Strict and skeletal compatible $L_{\infty}$-algebras and their
classifications are considered in Section 6. Finally, in Section 7, we
introduce compatible Lie $2$-algebras and prove Theorem 7.5.

Notations.

All (graded) vector spaces, (multi)linear maps, tensor products, wedge
products are over a field k of characteristic $0$. We denote the set of all
permutations on $k$ elements by $S_{k}$. A permutation $\sigma\in S_{k}$ is
called an $(i,k-i)$-shuffle if $\sigma(1)<\cdots<\sigma(i)$ and
$\sigma(i+1)<\cdots<\sigma(k)$. Let $Sh(i,k-i)$ denotes the set of all
$(i,k-i)$-shuffles. For any permutation $\sigma\in S_{k}$ and homogeneous
elements $v_{1},\ldots,v_{k}$ in a graded vector space $V$, the Koszul sign
$\epsilon(\sigma)=\epsilon(\sigma;v_{1},\ldots,v_{k})$ is given by
$v_{\sigma(1)}\odot\cdots\odot
v_{\sigma(k)}=\epsilon(\sigma)v_{1}\odot\cdots\odot v_{k}$, where $\odot$
denotes the product on the symmetric algebra $SV$. A graded multilinear map
$f:V^{\otimes k}\rightarrow V$ is called symmetric if
$f(v_{\sigma(1)},\ldots,v_{\sigma(k)})=\epsilon(\sigma)f(v_{1},\ldots,v_{k})$
and called skew-symmetric if
$f(v_{\sigma(1)},\ldots,v_{\sigma(k)})=(-1)^{\sigma}\epsilon(\sigma)f(v_{1},\ldots,v_{k})$,
for all $\sigma\in S_{k}$ and homogeneous elements $v_{1},\ldots,v_{k}\in V$.

##  2 Preliminaries

In this section, we recall some basic results on compatible Lie algebras and
$L_{\infty}$-algebras. Our main references are [15, 16, 17].

### Compatible Lie algebras

######  2.1 Definition.

A compatible Lie algebra is a triple
$(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$ consisting of a vector space
$\mathfrak{g}$ equipped with two Lie brackets $[~{},~{}]$ and
$[~{},~{}]^{\prime}$ on $\mathfrak{g}$ satisfying the compatibility

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

Note that the compatibility condition (1) is equivalent to say that
$[~{},~{}]+[~{},~{}]^{\prime}$ is also a Lie bracket on $\mathfrak{g}$.

######  2.2 Definition.

Let $(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$ be a compatible Lie algebra.
A representation of it consists of a triple $(V,\rho,\rho^{\prime})$ in which
$(V,\rho)$ is a representation of the Lie algebra $(\mathfrak{g},[~{},~{}])$,
and $(V,\rho^{\prime})$ is a representation of the Lie algebra
$(\mathfrak{g},[~{},~{}]^{\prime})$ satisfying additionally the following
compatibility

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

Any compatible Lie algebra $(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$ can
be regarded as a representation
$(\mathfrak{g},\rho=\mathrm{ad}_{[~{},~{}]},\rho^{\prime}=\mathrm{ad}_{[~{},~{}]^{\prime}})$
of itself. This is called the adjoint representation.

Let $(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$ be a compatible Lie algebra
and $(V,\rho,\rho^{\prime})$ be a representation of it. Consider the
Chevalley-Eilenberg cochain complex
$\\{\mathrm{Hom}(\wedge^{\bullet}\mathfrak{g},V),\partial\\}$ of the Lie
algebra $(\mathfrak{g},[~{},~{}])$ with coefficients in the representation
$(V,\rho)$, where the coboundary
$\partial:\mathrm{Hom}(\wedge^{n}\mathfrak{g},V)\rightarrow\mathrm{Hom}(\wedge^{n+1}\mathfrak{g},V)$,
for $n\geq 0$, is given by

$\displaystyle(\partial f)(x_{1},\ldots,x_{n+1})=~{}$
$\displaystyle\sum_{i=1}^{n+1}(-1)^{i+1}\rho(x_{i})f(x_{1},\ldots,\widehat{x_{i}},\ldots,x_{n+1})$
$\displaystyle+\sum_{i<j}(-1)^{i+j}f([x_{i},x_{j}],x_{1},\ldots,\widehat{x_{i}},\ldots,\widehat{x_{j}},\ldots,x_{n+1}),$

for $f\in\mathrm{Hom}(\wedge^{n}\mathfrak{g},V)$ and
$x_{1},\ldots,x_{n+1}\in\mathfrak{g}$. Similarly, we can consider the
Chevalley-Eilenberg cochain complex
$\\{\mathrm{Hom}(\wedge^{\bullet}\mathfrak{g},V),\partial^{\prime}\\}$ of the
Lie algebra $(\mathfrak{g},[~{},~{}]^{\prime})$ with coefficients in the
representation $(V,\rho^{\prime})$. It has been observed in [15] that
$\partial$ and $\partial^{\prime}$ satisfies the compatibility
$\partial\circ\partial^{\prime}+\partial^{\prime}\circ\partial=0$. Therefore,
we are now able to construct a new cochain complex
$\\{C^{\bullet}_{c}(\mathfrak{g},V),\partial_{c}\\}$, where

$\displaystyle C^{0}_{c}(\mathfrak{g},V)=\\{v\in
V|~{}\rho(x)v=\rho^{\prime}(x)v,\forall x\in\mathfrak{g}\\}~{}~{}~{}\text{ and
}~{}~{}~{}C^{n\geq
1}_{c}(\mathfrak{g},V)=\underbrace{\mathrm{Hom}(\wedge^{n}\mathfrak{g},V)\oplus\cdots\oplus\mathrm{Hom}(\wedge^{n}\mathfrak{g},V)}_{n\text{
times}}.$

The coboundary map $\partial_{c}:C^{n}_{c}(\mathfrak{g},V)\rightarrow
C^{n+1}_{c}(\mathfrak{g},V)$, for $n\geq 0$, given by

$\displaystyle\partial_{c}(f_{1},\ldots,f_{n})=(\partial
f_{1},\ldots,\underbrace{\partial f_{i}+\partial^{\prime}f_{i-1}}_{i\text{-th
place}},\ldots,\partial^{\prime}f_{n}).$

The cohomology groups of the complex
$\\{C^{\bullet}_{c}(\mathfrak{g},V),\partial_{c}\\}$ are called the cohomology
of the compatible Lie algebra $(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$
with coefficients in the representation $(V,\rho,\rho^{\prime})$.

######  2.3 Remark.

Let $n\geq 2$. Suppose $f,f^{\prime}\in\mathrm{Hom}(\wedge^{n}\mathfrak{g},V)$
are two elements that satisfies $\partial
f=\partial^{\prime}f^{\prime}=\partial f^{\prime}+\partial^{\prime}f=0$. Then
the element $(f,\underbrace{f+f^{\prime},\ldots,f+f^{\prime}}_{n-2\text{
times}},f^{\prime})\in C^{n}_{c}(\mathfrak{g},V)$ is a $n$-cocycle as

$\displaystyle\partial_{c}(f,f+f^{\prime},\ldots,f+f^{\prime},f^{\prime})=\big{(}\partial
f,\partial f+\partial
f^{\prime}+\partial^{\prime}f,\ldots,\underbrace{\partial(f+f^{\prime})+\partial^{\prime}(f+f^{\prime})}_{i\text{-th
place}},\ldots,\partial^{\prime}f^{\prime}\big{)}=0.$

Such $n$-cocycles are called induced by compatible Lie algebra cocycles. We
will use such $3$-cocycles in Section 6 to classify skeletal compatible
$L_{\infty}$-algebras.

###  $L_{\infty}$-algebras

######  2.4 Definition.

An $L_{\infty}$-algebra is a pair $(L,\\{l_{k}\\}_{k\geq 1})$ consisting of a
graded vector space $L=\oplus_{i\in\mathbb{Z}}L_{i}$ together with a
collection $\\{l_{k}\\}_{k\geq 1}$ of graded skew-symmetric multilinear maps
$l_{k}:L^{\otimes k}\rightarrow L$ with $\text{deg }(l_{k})=2-k$, for $k\geq
1$, satisfying the following set of identities:

(2) $\displaystyle\sum_{i+j=n+1}\sum_{\sigma\in
Sh(i,n-i)}\chi(\sigma)(-1)^{i(j-1)}~{}l_{j}\big{(}l_{i}(x_{\sigma(1)},\ldots,x_{\sigma(i)}),x_{\sigma(i+1)},\ldots,x_{\sigma(n)}\big{)}=0,$

for $n\geq 1$ and homogeneous elements $x_{1},\ldots,x_{n}\in L.$

The identities (2) are called the higher Jacobi identities. For small values
on $n$, the meaning of the higher Jacobi identities can be found in [16].

######  2.5 Remark.

Any differential graded Lie algebra $(L,\partial,[~{},~{}])$ is an
$L_{\infty}$-algebra in which $l_{1}=\partial$, $l_{2}=[~{},~{}]$ and
$l_{k}=0$, for $k\geq 3$.

Let $(L,\\{l_{k}\\}_{k\geq 1})$ be an $L_{\infty}$-algebra. Take $V=s^{-1}L$
the desuspension of $L$, where $V_{i}=(s^{-1}L)_{i}=L_{i+1}$, for
$i\in\mathbb{Z}$. We define degree $1$ maps $\rho_{k}:V^{\otimes k}\rightarrow
V$, for $k\geq 1$, by

$\displaystyle\rho_{k}:=(-1)^{\frac{k(k-1)}{2}}~{}s^{-1}\circ l_{k}\circ
s^{\otimes k}.$

It turns out that the map $\rho_{k}$ is graded symmetric, for $k\geq 1$. Note
that the map $l_{k}$ can be reconstructed from $\rho_{k}$ by
$l_{k}=s\circ\rho_{k}\circ(s^{-1})^{\otimes k}$.

For each $k\geq 1$, the map $\rho_{k}$ uniquely lifts to a degree $1$
coderivation $\widetilde{\rho_{k}}:SV\rightarrow SV$ on the symmetric
coalgebra $SV$, given by

$\displaystyle\widetilde{\rho_{k}}(v_{1}\odot\cdots\odot
v_{n})=\sum_{\sigma\in
Sh(k,n-k)}\epsilon(\sigma)~{}\rho_{k}(v_{\sigma(1)}\odot\cdots\odot
v_{\sigma(k)})\odot v_{\sigma(k+1)}\odot\cdots\odot v_{\sigma(n)},~{}\text{
for }n\geq k$

and zero otherwise. Note that the graded space of coderivations
$\mathrm{Coder}^{\bullet}(SV)$ on the symmetric coalgebra $SV$ is a graded Lie
algebra with the commutator bracket $[~{},~{}]_{C}$.

######  2.6 Proposition.

An $L_{\infty}$-algebra structure on a graded vector space $L$ is equivalent
to a Maurer-Cartan element in the graded Lie algebra
$(\mathrm{Coder}^{\bullet}(SV),[~{},~{}]_{C})$, where $V=s^{-1}L$.

##  3 Compatible $L_{\infty}$-algebras

In this section, we introduce compatible $L_{\infty}$-algebras as the homotopy
analogue of compatible Lie algebras. Given a graded vector space $L$, we also
construct a graded Lie algebra whose Maurer-Cartan elements are given by
compatible $L_{\infty}$-algebra structures $L$.

######  3.1 Definition.

A compatible $L_{\infty}$-algebra is a triple $(L,\\{l_{k}\\}_{k\geq
1},\\{l_{k}^{\prime}\\}_{k\geq 1})$ in which $(L,\\{l_{k}\\}_{k\geq 1})$ and
$(L,\\{l_{k}^{\prime}\\}_{k\geq 1})$ are both $L_{\infty}$-algebras
additionally satisfying

(3) $\displaystyle\sum_{i+j=n+1}\sum_{\sigma\in
Sh(i,n-i)}\chi(\sigma)(-1)^{i(j-1)}~{}\bigg{\\{}$ $\displaystyle
l_{j}\big{(}l_{i}^{\prime}(x_{\sigma(1)},\ldots,x_{\sigma(i)}),x_{\sigma(i+1)},\ldots,x_{\sigma(n)}\big{)}$
$\displaystyle+l_{j}^{\prime}\big{(}l_{i}(x_{\sigma(1)},\ldots,x_{\sigma(i)}),x_{\sigma(i+1)},\ldots,x_{\sigma(n)}\big{)}\bigg{\\}}=0,$

for $n\geq 1$ and $x_{1},\ldots,x_{n}\in L.$

In this case, we often say that the $L_{\infty}$-algebras
$(L,\\{l_{k}\\}_{k\geq 1})$ and $(L,\\{l_{k}^{\prime}\\}_{k\geq 1})$ are
compatible. Note that the compatibility condition (3) is equivalent to the
fact that the pair $(L,\\{l_{k}+l_{k}^{\prime}\\}_{k\geq 1})$ is an
$L_{\infty}$-algebra.

######  3.2 Example.

(i) Any compatible Lie algebra is a compatible $L_{\infty}$-algebra whose
underlying graded vector space is concentrated in degree $0$. Any compatible
differential graded Lie algebra
$(L,\\{\partial,[~{},~{}]\\},\\{\partial^{\prime},[~{},~{}]^{\prime}\\})$ is
also a compatible $L_{\infty}$-algebra with $l_{k}=l^{\prime}_{k}=0$, for
$k\geq 3$. See also Proposition 4.1.

(ii) Let $(L,\\{l_{k}\\}_{k\geq 1})$ be an $L_{\infty}$-algebra. Then
$(L,\\{l_{k}\\}_{k\geq 1},\\{l_{k}\\}_{k\geq 1})$ is obviously a compatible
$L_{\infty}$-algebra.

(iii) Let $(L,\\{l_{k}\\}_{k\geq 1},\\{l^{\prime}_{k}\\}_{k\geq 1})$ be a
compatible $L_{\infty}$-algebra. Then $(L,\\{l^{\prime}_{k}\\}_{k\geq
1},\\{l_{k}\\}_{k\geq 1})$, $(L,\\{l_{k}\\}_{k\geq
1},\\{l_{k}+l^{\prime}_{k}\\}_{k\geq 1})$ and
$(L,\\{l_{k}+l^{\prime}_{k}\\}_{k\geq 1},\\{l^{\prime}_{k}\\}_{k\geq 1})$ are
all compatible $L_{\infty}$-algebras.

Some nontrivial examples of compatible $L_{\infty}$-algebras are provided in
Sections 4 and 6. In the next, we give some characterization results for
compatible $L_{\infty}$-algebras.

Let $(\mathfrak{g}=\oplus_{n\in\mathbb{Z}}\mathfrak{g}^{n},[~{},~{}])$ be a
graded Lie algebra and $\theta,\theta^{\prime}\in\mathfrak{g}^{1}$ be two
Maurer-Cartan elements. They are said to be compatible if they satisfies
$[\theta,\theta^{\prime}]=0$. The following result gives a characterization of
compatible $L_{\infty}$-algebra structures on a graded vector space $L$ in
terms of compatible Maurer-Cartan elements in the graded Lie algebra
$(\mathrm{Coder}^{\bullet}(SV),[~{},~{}]_{C})$, where $V=s^{-1}L$.

######  3.3 Theorem.

There is a one-to-one correspondence between compatible $L_{\infty}$-algebra
structures on a graded vector space $L$ and compatible Maurer-Cartan elements
in the graded Lie algebra $(\mathrm{Coder}^{\bullet}(SV),[~{},~{}]_{C})$,
where $V=s^{-1}L$.

###### Proof.

Let $\\{l_{k}\\}_{k\geq 1}$ and $\\{l^{\prime}_{k}\\}_{k\geq 1}$ be two
collections of multilinear maps $l_{k},l^{\prime}_{k}:L^{\otimes k}\rightarrow
L$ with $\mathrm{deg}(l_{k})=\mathrm{deg}(l^{\prime}_{k})=2-k$, for $k\geq 1$.
We consider the collections $\\{\rho_{k}\\}_{k\geq 1}$ and
$\\{\rho^{\prime}_{k}\\}_{k\geq 1}$ of degree $1$ maps
$\rho_{k},\rho^{\prime}_{k}:V^{\otimes k}\rightarrow V$ given by

$\displaystyle\rho_{k}=(-1)^{\frac{k(k-1)}{2}}~{}s^{-1}\circ l_{k}\circ
s^{\otimes k}~{}~{}~{}~{}\text{ and
}~{}~{}~{}~{}\rho^{\prime}_{k}=(-1)^{\frac{k(k-1)}{2}}~{}s^{-1}\circ
l^{\prime}_{k}\circ s^{\otimes k},~{}\text{ for }k\geq 1.$

Define two degree $1$ coderivations $D,D^{\prime}\in\mathrm{Coder}^{1}(SV)$ by
$D=\sum_{k\geq 1}\widetilde{\rho_{k}}$ and $D^{\prime}=\sum_{k\geq
1}\widetilde{\rho^{\prime}_{k}}$. Then we have

$\displaystyle[D,D]_{C}=0~{}~{}~{}\leftrightsquigarrow~{}~{}~{}(L,\\{l_{k}\\}_{k\geq
1})\text{ is an }L_{\infty}\text{-algebra},$
$\displaystyle[D^{\prime},D^{\prime}]_{C}=0~{}~{}~{}\leftrightsquigarrow~{}~{}~{}(L,\\{l^{\prime}_{k}\\}_{k\geq
1})\text{ is an }L_{\infty}\text{-algebra},$
$\displaystyle[D,D^{\prime}]_{C}=0~{}~{}~{}\leftrightsquigarrow~{}~{}~{}\text{the
compatibility condition }(\ref{hji-comp}).$

This completes the proof. ∎

Our aim now is to construct a new graded Lie algebra whose Maurer-Cartan
elements correspond to compatible $L_{\infty}$-algebra structures on $L$. We
define a graded vector space $\mathrm{Coder}^{\bullet}_{c}(SV)=\oplus_{n\geq
0}\mathrm{Coder}^{n}_{c}(SV)$, where

$\displaystyle\mathrm{Coder}^{n}_{c}(SV):=\underbrace{\mathrm{Coder}^{n}(SV)\oplus\cdots\oplus\mathrm{Coder}^{n}(SV)}_{n+1\text{
times}},\text{ for }n\geq 0.$

The graded Lie bracket $[~{},~{}]_{C}$ on $\mathrm{Coder}^{\bullet}(SV)$
induces a bracket on $\mathrm{Coder}^{\bullet}_{c}(SV)$ given by

$\displaystyle\llbracket(D_{1},\ldots,D_{m+1}),$
$\displaystyle(D^{\prime}_{1},\ldots,D^{\prime}_{n+1})\rrbracket$
$\displaystyle:=\big{(}[D_{1},D_{1}^{\prime}]_{C},\ldots,\underbrace{[D_{1},D_{i}^{\prime}]_{C}+[D_{2},D^{\prime}_{i-1}]_{C}+\cdots+[D_{i},D^{\prime}_{1}]_{C}}_{i\text{-th
place}},\ldots,[D_{m+1},D^{\prime}_{n+1}]_{C}\big{)},$

for $(D_{1},\ldots,D_{m+1})\in\mathrm{Coder}^{m}_{c}(SV)$ and
$(D^{\prime}_{1},\ldots,D^{\prime}_{n+1})\in\mathrm{Coder}^{n}_{c}(SV)$. It is
easy to verify that $\llbracket~{},~{}\rrbracket$ is a graded Lie bracket
which makes $(\mathrm{Coder}^{\bullet}_{c}(SV),\llbracket~{},~{}\rrbracket)$
into a graded Lie algebra.

######  3.4 Theorem.

There is a one-to-one correspondence between compatible $L_{\infty}$-algebra
structures on $L$ and Maurer-Cartan elements in the graded Lie algebra
$(\mathrm{Coder}^{\bullet}_{c}(SV),\llbracket~{},~{}\rrbracket)$, where
$V=s^{-1}L$.

###### Proof.

Let $D,D^{\prime}\in\mathrm{Coder}^{1}(SV)$ be two degree $1$ coderivations on
$SV$. Consider the pair $(D,D^{\prime})\in\mathrm{Coder}^{1}_{c}(SV)$. Then we
have

$\displaystyle\llbracket(D,D^{\prime}),(D,D^{\prime})\rrbracket=([D,D]_{C},~{}[D,D^{\prime}]_{C}+[D^{\prime},D]_{C},~{}[D^{\prime},D^{\prime}]_{C}).$

This shows that $D,D^{\prime}$ are two compatible Maurer-Cartan elements in
the graded Lie algebra $(\mathrm{Coder}^{\bullet}(SV),[~{},~{}]_{C})$ if and
only if the pair $(D,D^{\prime})$ is a Maurer-Cartan element in the graded Lie
algebra $(\mathrm{Coder}^{\bullet}_{c}(SV),\llbracket~{},~{}\rrbracket)$.
Hence by Theorem 3.3, we get

$\displaystyle\text{comp }L_{\infty}\text{ structure }$
$\displaystyle\leftrightsquigarrow$ comp Maurer-Cartan elements
$\displaystyle\leftrightsquigarrow$ Maurer-Cartan element
$\displaystyle(\\{l_{k}\\}_{k\geq 1},\\{l^{\prime}_{k}\\}_{k\geq 1})\text{ on
}L$ $\displaystyle
D,D^{\prime}\in(\mathrm{Coder}^{\bullet}(SV),[~{},~{}]_{C})$
$\displaystyle(D,D^{\prime})\in(\mathrm{Coder}^{\bullet}_{c}(SV),\llbracket~{},~{}\rrbracket).$

This completes the proof. ∎

##  4 Examples of compatible $L_{\infty}$-algebras

In this section, we provide some examples of compatible $L_{\infty}$-algebras
arising mainly from compatible $A_{\infty}$-algebras, Nijenhuis operators on
$L_{\infty}$-algebras, compatible $V$-datas and compatible Courant algebroids.
However, we start with an example of a compatible differential graded Lie
algebra (dgLa) from a compatible Lie algebra.

### Compatible dgLa induced from a compatible Lie algebra

We have seen in Remark 2.5 that a differential graded Lie algebra can be
considered as an $L_{\infty}$-algebra with trivial higher brackets. The
following result is a consequence of the compatibility condition (3).

######  4.1 Proposition.

Let $(L,\partial.[~{},~{}])$ and $(L,\partial^{\prime},[~{},~{}]^{\prime})$ be
two differential graded Lie algebras. Then the triple
$(L,\\{\partial,[~{},~{}]\\},\\{\partial^{\prime},[~{},~{}]^{\prime}\\})$ is a
compatible differential graded Lie algebra if and only if

  * •

$\partial\circ\partial^{\prime}+\partial^{\prime}\circ\partial=0$,

  * •

$[[x,y],z]^{\prime}+[[x,y]^{\prime},z]=(-1)^{|y||z|}[[x,z],y]^{\prime}+(-1)^{|y||z|}[[x,z]^{\prime},y]+[x,[y,z]]^{\prime}+[x,[y,z]^{\prime}],$

  * •

$\partial[x,y]^{\prime}+\partial^{\prime}[x,y]=[\partial(x),y]^{\prime}+[\partial^{\prime}(x),y]+(-1)^{|x|}[x,\partial(y)]^{\prime}+(-1)^{|x|}[x,\partial^{\prime}(y)]$,
for $x,y,z\in\mathfrak{g}$.

In the following, we construct an example of a compatible differential graded
Lie algebra from a compatible Lie algebra.

######  4.2 Example.

Let $(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$ be a compatible Lie algebra.
Consider the graded vector space $L=\oplus_{n\geq
0}\mathrm{Hom}(\wedge^{n+1}\mathfrak{g},\mathfrak{g})$ of all skew-symmetric
multilinear maps on the vector space $\mathfrak{g}$. It carries a graded Lie
bracket (Nijenhuis-Richardson bracket) given by

$\displaystyle[f,g]_{\mathsf{NR}}:=~{}$ $\displaystyle f\diamond
g-(-1)^{mn}g\diamond f,~{}\text{ where }$ $\displaystyle(f\diamond
g)(x_{1},\ldots,x_{m+n+1})=~{}$ $\displaystyle\sum_{\sigma\in
Sh(n+1,m)}(-1)^{\sigma}~{}f\big{(}g(x_{\sigma(1)},\ldots,x_{\sigma(n+1)}),x_{\sigma(n+2)},\ldots,x_{\sigma(m+n+1)}\big{)},$

for $f\in\mathrm{Hom}(\wedge^{m+1}\mathfrak{g},\mathfrak{g})$ and
$g\in\mathrm{Hom}(\wedge^{n+1}\mathfrak{g},\mathfrak{g})$. Let the compatible
Lie brackets $[~{},~{}]$ and $[~{},~{}]^{\prime}$ correspond to elements $\mu$
and $\mu^{\prime}$ in
$L_{1}=\mathrm{Hom}(\wedge^{2}\mathfrak{g},\mathfrak{g})$. Then $\mu$ and
$\mu^{\prime}$ satisfies

$\displaystyle[\mu,\mu]_{\mathsf{NR}}=0,~{}\quad~{}~{}~{}[\mu^{\prime},\mu^{\prime}]_{\mathsf{NR}}=0~{}~{}~{}~{}\text{
and }~{}~{}~{}~{}[\mu,\mu^{\prime}]_{\mathsf{NR}}=0.$

This implies that the differential graded Lie algebras
$(L,\partial_{\mu}=[\mu,-]_{\mathsf{NR}},[~{},~{}]_{\mathsf{NR}},)$ and
$(L,\partial_{\mu^{\prime}}=[\mu^{\prime},-]_{\mathsf{NR}},[~{},~{}]_{\mathsf{NR}},)$
satisfy the conditions of Proposition 4.1. In other words,
$(L,\\{\partial_{\mu},[~{},~{}]_{\mathsf{NR}}\\},\\{\partial_{\mu^{\prime}},[~{},~{}]_{\mathsf{NR}}\\})$
is a compatible differential graded Lie algebra.

### Compatible $A_{\infty}$-algebras

The notion of $A_{\infty}$-spaces and $A_{\infty}$-algebras (strongly homotopy
associative algebras) was introduced by Stasheff in the study of group-like
topological spaces [28]. They have important applications in homotopy theory
and combinatorial study of associahedra.

######  4.3 Definition.

An $A_{\infty}$-algebra is a pair $(A,\\{\mu_{k}\\}_{k\geq 1})$ consisting of
a graded vector space $A=\oplus_{i\in\mathbb{Z}}A_{i}$ together with a
collection $\\{\mu_{k}\\}_{k\geq 1}$ of graded multilinear maps
$\mu_{k}:A^{\otimes k}\rightarrow A$ with $\mathrm{deg}(\mu_{k})=2-k$, for
$k\geq 1$, satisfying for $n\geq 1$ and homogeneous elements
$a_{1},\ldots,a_{n}\in A$,

$\displaystyle\sum_{i+j=n+1}\sum_{\lambda=1}^{j}(-1)^{\lambda(i+1)+i(|a_{1}|+\cdots+|a_{i-1}|)}\mu_{j}\big{(}a_{1},\ldots,a_{i-1},\mu_{i}(a_{i},\ldots,a_{i+k-1}),\ldots,a_{n}\big{)}=0.$

Let $W=s^{-1}A$. Note that a collection $\\{\mu_{k}\\}_{k\geq 1}$ of graded
multilinear maps $\mu_{k}:A^{\otimes k}\rightarrow A$ with
$\mathrm{deg}(\mu_{k})=2-k$ induces a collection $\\{\tau_{k}:W^{\otimes
k}\rightarrow W\\}_{k\geq 1}$ of degree $1$ multilinear maps, where
$\tau_{k}=(-1)^{\frac{k(k-1)}{2}}s^{-1}\circ\mu_{k}\circ s^{\otimes k}$, for
$k\geq 1$. Hence we obtain a degree $1$ coderivation
$E=\sum_{k=1}^{\infty}\widetilde{\tau_{k}}:TW\rightarrow TW$ on the tensor
coalgebra $TW$, where $\widetilde{\tau_{k}}:TW\rightarrow TW$ is given by

$\displaystyle\widetilde{\tau_{k}}(w_{1}\otimes\cdots\otimes
w_{n})=\sum_{i=1}^{n-k+1}(-1)^{|w_{1}|+\cdots+|w_{i-1}|}~{}w_{1}\otimes\cdots\otimes\tau_{k}(w_{i}\otimes\cdots\otimes
w_{i+k-1})\otimes\cdots\otimes w_{n},$

for $n\geq k$ and zero otherwise. Consider the graded Lie algebra
$(\mathrm{Coder}^{\bullet}(TW),[~{},~{}]_{C})$ on the graded vector space
$\mathrm{Coder}^{\bullet}(TW)$ of coderivations on the tensor coalgebra $TW$.
Then $(A,\\{\mu_{k}\\}_{k\geq 1})$ is an $A_{\infty}$-algebra if and only if
$E\in\mathrm{Coder}^{1}(TW)$ satisfies $[E,E]_{C}=0$. The following result has
been proved in [16].

######  4.4 Proposition.

Let $(A,\\{\mu_{k}\\}_{k\geq 1})$ be an $A_{\infty}$-algebra. Then
$(A,\\{l_{k}\\}_{k\geq 1})$ is an $L_{\infty}$-algebra, where

$\displaystyle l_{k}(a_{1},\ldots,a_{k})=\sum_{\sigma\in
S_{k}}(-1)^{\sigma}\mu_{k}(a_{\sigma(1)},\ldots,a_{\sigma(k)}),~{}\text{ for
}k\geq 1.$

The collection $\\{l_{k}\\}_{k\geq 1}$ is called the skew-symmetrization of
$\\{\mu_{k}\\}_{k\geq 1}$.

######  4.5 Definition.

A compatible $A_{\infty}$-algebra is a triple $(A,\\{\mu_{k}\\}_{k\geq
1},\\{\mu^{\prime}_{k}\\}_{k\geq 1})$ in which $(A,\\{\mu_{k}\\}_{k\geq 1})$,
$(A,\\{\mu^{\prime}_{k}\\}_{k\geq 1})$ and
$(A,\\{\mu_{k}+\mu^{\prime}_{k}\\}_{k\geq 1})$ are all $A_{\infty}$-algebras.

It is easy to prove that (similar to Theorem 3.4) a compatible
$A_{\infty}$-algebra structure on a graded vector space $A$ is equivalent to a
Maurer-Cartan element in the graded Lie algebra
$(\mathrm{Coder}^{\bullet}_{c}(TW),\llbracket~{},~{}\rrbracket)$, where
$W=s^{-1}A$.

######  4.6 Proposition.

Let $(A,\\{\mu_{k}\\}_{k\geq 1},\\{\mu^{\prime}_{k}\\}_{k\geq 1})$ be a
compatible $A_{\infty}$-algebra. Then $(A,\\{l_{k}\\}_{k\geq
1},\\{l^{\prime}_{k}\\}_{k\geq 1})$ is a compatible $L_{\infty}$-algebra,
where $\\{l_{k}\\}_{k\geq 1}$ and $\\{l^{\prime}_{k}\\}_{k\geq 1}$ are the
skew-symmetrizations of $\\{\mu_{k}\\}_{k\geq 1}$ and
$\\{\mu^{\prime}_{k}\\}_{k\geq 1}$, respectively.

###### Proof.

Since $(A,\\{\mu_{k}\\}_{k\geq 1})$ and $(A,\\{\mu^{\prime}_{k}\\}_{k\geq 1})$
are $A_{\infty}$-algebras, it follows that $(A,\\{l_{k}\\}_{k\geq 1})$ and
$(A,\\{l^{\prime}_{k}\\}_{k\geq 1})$ are $L_{\infty}$-algebras. Further,
$(A,\\{\mu_{k}+\mu^{\prime}_{k}\\}_{k\geq 1})$ is an $A_{\infty}$-algebra
implies that $(A,\\{l_{k}+l^{\prime}_{k}\\}_{k\geq 1})$ is an
$L_{\infty}$-algebra (by Proposition 4.4). Thus, we get that
$(A,\\{l_{k}\\}_{k\geq 1},\\{l^{\prime}_{k}\\}_{k\geq 1})$ is a compatible
$L_{\infty}$-algebra. ∎

### Nijenhuis operators on $L_{\infty}$-algebras

In this subsection, we state the main result from [1] about Nijenhuis
operators on $L_{\infty}$-algebras, and observe that Nijenhuis operators
induce compatible $L_{\infty}$-algebras. Let $(L,\\{l_{k}\\}_{k\geq 1})$ be an
$L_{\infty}$-algebra. Consider the corresponding Maurer-Cartan element
$D\in\mathrm{Coder}^{1}(SV)$ in the graded Lie algebra
$(\mathrm{Coder}^{\bullet}(SV),[~{},~{}]_{C})$, where $V=s^{-1}L$.

Let $\\{N_{k}:L^{\otimes k}\rightarrow L\\}_{k\geq 1}$ be a collection of
skew-symmetric multilinear maps with $\mathrm{deg}(l_{k})=1-k$, for $k\geq 1$.
Consider the collection $\\{\mathcal{N}_{k}:V^{\otimes k}\rightarrow
V\\}_{k\geq 1}$ of degree $0$ symmetric multilinear maps on $V$ given by
$\mathcal{N}_{k}=(-1)^{\frac{k(k-1)}{2}}s^{-1}\circ N_{k}\circ s^{\otimes k}$,
for $k\geq 1$. This collection of symmetric multilinear maps lifts to a degree
$0$ coderivation $N=\sum_{k\geq
1}\widetilde{\mathcal{N}_{k}}\in\mathrm{Coder}^{0}(SV)$ on the symmetric
coalgebra $SV$.

######  4.7 Definition.

A collection $\\{N_{k}:L^{\otimes k}\rightarrow L\\}_{k\geq 1}$ of skew-
symmetric multilinear maps (with $\mathrm{deg}(N_{k})=1-k$, for $k\geq 1$) is
said to be a Nijenhuis operator on the $L_{\infty}$-algebra
$(L,\\{l_{k}\\}_{k\geq 1})$ if there exists a degree $0$ coderivation
${P}\in\mathrm{Coder}^{0}(SV)$ such that

$\displaystyle[N,[N,D]_{C}]_{C}=[P,D]_{C}~{}~{}~{}~{}\text{ and
}~{}~{}~{}~{}[N,P]_{C}=0.$

Such a $P$ is called a square of $N$. The following result has been proved in
[1, Theorem 2.4].

######  4.8 Proposition.

Let $(L,\\{l_{k}\\}_{k\geq 1})$ be an $L_{\infty}$-algebra with the
corresponding Maurer-Cartan element $D\in\mathrm{Coder}^{1}(SV)$ in the graded
Lie algebra $(\mathrm{Coder}^{\bullet}(SV),[~{},~{}]_{C})$, where $V=s^{-1}L$.
Suppose $\\{N_{k}:L^{\otimes k}\rightarrow L\\}_{k\geq 1}$ is a Nijenhuis
operator with the corresponding coderivation $N\in\mathrm{Coder}^{0}(SV)$.
Then the coderivation $D_{N}:=[N,D]_{C}\in\mathrm{Coder}^{1}(SV)$ satisfies
$[D_{N},D_{N}]_{C}=0$ and $[D,D_{N}]_{C}=0$. Hence $D_{N}$ corresponds to an
$L_{\infty}$-algebra structure $(L,\\{l_{k}^{N}\\}_{k\geq 1})$ on the same
graded vector space $L$ and $(L,\\{l_{k}\\}_{k\geq 1},\\{l_{k}^{N}\\}_{k\geq
1})$ is a compatible $L_{\infty}$-algebra.

### Compatible $V$-datas

Let $L$ be a graded Lie algebra (with the bracket $[~{},~{}]$),
$\mathfrak{a}\subset L$ be an abelian Lie subalgebra, $P:L\rightarrow L$ be a
projection such that $\mathrm{im}(P)=\mathfrak{a}$ and $\mathrm{ker}(P)\subset
L$ be a Lie subalgebra. Let $\triangle\in\mathrm{ker}(P)_{1}$ be an element
that satisfies $[\triangle,\triangle]=0$. Such an element is called a ‘square-
zero odd element’. In this case, the quadruple $(L,\mathfrak{a},P,\triangle)$
is called a $V$-data [31].

Given a $V$-data $(L,\mathfrak{a},P,\triangle)$, the graded vector space
$\mathfrak{a}$ carries some degree $1$ symmetric multilinear maps given by

$\displaystyle\rho_{k}(a_{1},\ldots,a_{k}):=P[\cdots[[\triangle,a_{1}],a_{2}],\ldots,a_{k}],\text{
for }k\geq 1.$

Moreover, the pair $(s\mathfrak{a},\\{l_{k}\\}_{k\geq 1})$ is an
$L_{\infty}$-algebra [31], where $s\mathfrak{a}$ is the suspension of
$\mathfrak{a}$ and $l_{k}$’s are given by
$l_{k}=s\circ\rho_{k}\circ(s^{-1})^{\otimes k},$ for $k\geq 1$.

######  4.9 Definition.

Let $(L,\mathfrak{a},P,\triangle)$ and $(L,\mathfrak{a},P,\triangle^{\prime})$
be two $V$-datas with the same underlying structures except the square-zero
odd elements. They are said to be compatible if
$[\triangle,\triangle^{\prime}]=0$.

######  4.10 Proposition.

Let $(L,\mathfrak{a},P,\triangle)$ and $(L,\mathfrak{a},P,\triangle^{\prime})$
be two compatible $V$-datas with the corresponding $L_{\infty}$-algebras
$(s\mathfrak{a},\\{l_{k}\\}_{k\geq 1})$ and
$(s\mathfrak{a},\\{l^{\prime}_{k}\\}_{k\geq 1})$. Then
$(s\mathfrak{a},\\{l_{k}\\}_{k\geq 1},\\{l^{\prime}_{k}\\}_{k\geq 1})$ is a
compatible $L_{\infty}$-algebra.

###### Proof.

Since $(L,\mathfrak{a},P,\triangle)$ and
$(L,\mathfrak{a},P,\triangle^{\prime})$ are two compatible $V$-datas, it
follows that $(L,\mathfrak{a},P,\triangle+\triangle^{\prime})$ is a $V$-data.
If $(s\mathfrak{a},\\{l^{*}_{k}\\}_{k\geq 1})$ is the $L_{\infty}$-algebra
corresponding to the $V$-data
$(L,\mathfrak{a},P,\triangle+\triangle^{\prime})$, then it is easy to see that
$l^{*}_{k}=l_{k}+l^{\prime}_{k}$, for $k\geq 1$. This implies that
$(s\mathfrak{a},\\{l_{k}\\}_{k\geq 1},\\{l^{\prime}_{k}\\}_{k\geq 1})$ is a
compatible $L_{\infty}$-algebra. ∎

In the following, we give an example of compatible $V$-datas and the use of
the corresponding compatible $L_{\infty}$-algebra. Let $\mathfrak{g}$ and $V$
be two vector spaces. Consider the graded Lie algebra

$\displaystyle L=\big{(}\oplus_{n\geq
0}\mathrm{Hom}(\wedge^{n+1}(\mathfrak{g}\oplus V),\mathfrak{g}\oplus
V),[~{},~{}]_{\mathsf{NR}}\big{)}$

on the space of all skew-symmetric multilinear maps on $\mathfrak{g}\oplus V$
with the Nijenhuis-Richardson bracket. Then $\mathfrak{a}=\oplus_{n\geq
0}\mathrm{Hom}(\wedge^{n+1}V,\mathfrak{g})\subset L$ is an abelian Lie
subalgebra. Let $P:L\rightarrow L$ be the projection onto the subspace
$\mathfrak{a}$. Suppose there are two skew-symmetric bilinear brackets
$[~{},~{}],[~{},~{}]^{\prime}:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$
and bilinear maps $\rho,\rho^{\prime}:\mathfrak{g}\times V\rightarrow V$.
Consider the elements
$\triangle,\triangle^{\prime}\in\mathrm{Hom}(\wedge^{2}(\mathfrak{g}\oplus
V),\mathfrak{g}\oplus V)$ defined respectively by

$\displaystyle\triangle((x,u),(y,v))=([x,y],\rho(x,v)-\rho(y)u)~{}\text{ and
}~{}\triangle^{\prime}((x,u),(y,v))=([x,y]^{\prime},\rho^{\prime}(x)v-\rho^{\prime}(y)u),$

for $(x,u),(y,v)\in\mathfrak{g}\oplus V$. Then we have the following.

######  4.11 Proposition.

With the above notations, $(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$ is a
compatible Lie algebra and $(V,\rho,\rho^{\prime})$ is a representation if and
only if $[\triangle,\triangle]_{\mathsf{NR}}=0$,
$[\triangle^{\prime},\triangle^{\prime}]_{\mathsf{NR}}=0$ and
$[\triangle,\triangle^{\prime}]_{\mathsf{NR}}=0$.

###### Proof.

For any $(x,u),(y,v),(z,w)\in\mathfrak{g}\oplus V$, we have

$\displaystyle[\triangle,\triangle]_{\mathsf{NR}}((x,u),(y,v),(z,w))$
$\displaystyle=2\big{(}[[x,y],z]-[[x,z],y]+[[y,z],x],~{}\rho([x,y])w-[\rho(x),\rho(y)]w-\rho([x,z])v+[\rho(x),\rho(z)]v$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\rho([y,z])u-[\rho(y),\rho(z)]u\big{)},$
$\displaystyle[\triangle^{\prime},\triangle^{\prime}]_{\mathsf{NR}}((x,u),(y,v),(z,w))$
$\displaystyle=2\big{(}[[x,y]^{\prime},z]^{\prime}-[[x,z]^{\prime},y]^{\prime}+[[y,z]^{\prime},x]^{\prime},~{}\rho^{\prime}([x,y]^{\prime})w-[\rho^{\prime}(x),\rho^{\prime}(y)]w-\rho^{\prime}([x,z]^{\prime})v+[\rho^{\prime}(x),\rho^{\prime}(z)]v$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\rho^{\prime}([y,z]^{\prime})u-[\rho^{\prime}(y),\rho^{\prime}(z)]u\big{)}$

and

$\displaystyle[\triangle,\triangle^{\prime}]_{\mathsf{NR}}((x,u),(y,v),(z,w))$
$\displaystyle=\big{(}[[x,y]^{\prime},z]-[[x,z]^{\prime},y]+[[y,z]^{\prime},x]+[[x,y],z]^{\prime}-[[x,z],y]^{\prime}+[[y,z],x]^{\prime},$
$\displaystyle\quad\quad\rho([x,y]^{\prime})w-[\rho(x),\rho^{\prime}(y)]w-\rho([x,z]^{\prime})v+[\rho(x),\rho^{\prime}(z)]v+\rho([y,z]^{\prime})u-[\rho(y),\rho^{\prime}(z)]u$
$\displaystyle\quad\quad+\rho^{\prime}([x,y])w-[\rho^{\prime}(x),\rho(y)]w-\rho^{\prime}([x,z])v+[\rho^{\prime}(x),\rho(z)]v+\rho^{\prime}([y,z])u-[\rho^{\prime}(y),\rho(z)]u\big{)}.$

This shows that $(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$ is a compatible
Lie algebra and $(V,\rho,\rho^{\prime})$ is a representation if and only if
$[\triangle,\triangle]_{\mathsf{NR}}=0$,
$[\triangle^{\prime},\triangle^{\prime}]_{\mathsf{NR}}=0$ and
$[\triangle,\triangle^{\prime}]_{\mathsf{NR}}=0$. ∎

Let $(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$ be a compatible Lie algebra
and $(V,\rho,\rho^{\prime})$ be a representation. Then it follows from the
above Proposition that $(L,\mathfrak{a},P,\triangle)$ and
$(L,\mathfrak{a},P,\triangle^{\prime})$ are compatible $V$-datas. Therefore,
the space $s\mathfrak{a}=\oplus_{n\geq
0}\mathrm{Hom}(\wedge^{n}V,\mathfrak{g})$ carry compatible graded Lie brackets
(as higher brackets are zero for degree reason) given by

$\displaystyle\\{\\![P,Q]\\!\\}=(-1)^{m}[[\triangle,P]_{\mathsf{NR}},Q]_{\mathsf{NR}}~{}~{}~{}\text{
and
}~{}~{}~{}\\{\\![P,Q]\\!\\}^{\prime}=(-1)^{m}[[\triangle^{\prime},P]_{\mathsf{NR}},Q]_{\mathsf{NR}},$

for $P\in(s\mathfrak{a})_{m}=\mathrm{Hom}(\wedge^{m}V,\mathfrak{g})$ and
$Q\in(s\mathfrak{a})_{n}=\mathrm{Hom}(\wedge^{n}V,\mathfrak{g})$. In other
words, $(s\mathfrak{a},\\{\\![~{},~{}]\\!\\},\\{\\![~{},~{}]\\!\\}^{\prime})$
is a compatible graded Lie algebra. We will use this graded Lie algebra to
characterize relative Rota-Baxter operators on compatible Lie algebras.

######  4.12 Definition.

Let $(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$ be a compatible Lie algebra
and $(V,\rho,\rho^{\prime})$ be a representation. A linear map
$R:V\rightarrow\mathfrak{g}$ is said to be a relative Rota-Baxter operator if
it satisfies

$\displaystyle[R(v),R(w)]=R\big{(}\rho(R(v))w-\rho(R(w))v\big{)}~{}~{}\text{
and
}~{}~{}[R(v),R(w)]^{\prime}=R\big{(}\rho^{\prime}(R(v))w-\rho^{\prime}(R(w))v\big{)},~{}\text{
for }v,w\in V.$

We have the following result about the characterization of relative Rota-
Baxter operators.

######  4.13 Proposition.

A linear map $R:V\rightarrow\mathfrak{g}$ is a relative Rota-Baxter operator
if and only if $R$ is a Maurer-Cartan element in the compatible graded Lie
algebra
$(s\mathfrak{a},\\{\\![~{},~{}]\\!\\},\\{\\![~{},~{}]\\!\\}^{\prime})$, i.e.
$R$ satisfies $\\{\\![R,R]\\!\\}=\\{\\![R,R]\\!\\}^{\prime}=0$.

###### Proof.

By a straightforward calculation (see [30]), we get that

$\displaystyle\\{\\![R,R]\\!\\}(u,v)=~{}$ $\displaystyle
2\big{(}R(\rho(R(v))w)-R(\rho(R(w))v)-[R(v),R(w)]\big{)},$
$\displaystyle\\{\\![R,R]\\!\\}^{\prime}(u,v)=~{}$ $\displaystyle
2\big{(}R(\rho^{\prime}(R(v))w)-R(\rho^{\prime}(R(w))v)-[R(v),R(w)]^{\prime}\big{)},$

for $v,w\in V$. This shows that $R$ is a relative Rota-Baxter operator if and
only if $\\{\\![R,R]\\!\\}=\\{\\![R,R]\\!\\}^{\prime}=0$. ∎

### Compatible Courant algebroids

Courant algebroids were introduced in [18] as a double object of Lie
bialgebroids. A Courant algebroid over a smooth manifold $M$ consists of a
quadruple $(E,\langle~{},~{}\rangle,[~{},~{}]_{E},\rho)$ of a smooth vector
bundle $E$ over $M$, a symmetric nondegenerate $C^{\infty}(M)$-valued pairing
$\langle~{},~{}\rangle:\Gamma E\times\Gamma E\rightarrow C^{\infty}(M)$, a
bilinear bracket $[~{},~{}]_{E}:\Gamma E\times\Gamma E$ and a bundle map
$\rho:E\rightarrow TM$ satisfying the following set of identities:

  * •

$[X,[Y,Z]_{E}]_{E}=[[X,Y]_{E},Z]_{E}+[Y,[X,Z]_{E}]_{E}$ (left Leibniz
identity),

  * •

$[X,fY]_{E}=f[X,Y]_{E}+\rho(X)(f)Y,$

  * •

$\rho[X,Y]_{E}=[\rho(X),\rho(Y)],$

  * •

$[X,X]_{E}=\frac{1}{2}\mathcal{D}\langle X,X\rangle,$

  * •

$\rho(X)\langle Y,Z\rangle=\langle[X,Y]_{E},Z\rangle+\langle
Y,[X,Z]_{E}\rangle,$

for $X,Y,Z\in\Gamma E$ and $f\in C^{\infty}(M)$. Here
$\mathcal{D}:C^{\infty}(M)\rightarrow\Gamma E$ is the composition
$C^{\infty}(M)\xrightarrow{d}\Omega^{1}(M)\xrightarrow{\rho^{*}}\Gamma
E^{*}\xrightarrow{\simeq}\Gamma E$, where the last isomorphism is induced by
the pairing $\langle~{},~{}\rangle$.

In [26] Roytenberg and Weinstein showed that a Courant algebroid
$(E,\langle~{},~{}\rangle,[~{},~{}]_{E},\rho)$ associates an
$L_{\infty}$-algebra on the graded vector space
$L(E):=C^{\infty}(M)\oplus\Gamma E$ (where $C^{\infty}(M)$ is concentrated in
degree $-1$ and $\Gamma E$ is concentrated in degree $0$) whose structure maps
are given by

$\displaystyle l_{1}=\mathcal{D},$ $\displaystyle l_{2}(X,Y)=\lfloor
X,Y\rfloor_{E}:=[X,Y]_{E}-\frac{1}{2}\mathcal{D}\langle X,Y\rangle,$
$\displaystyle l_{2}(X,f)=-l_{2}(f,X)=\frac{1}{2}\langle
X,\mathcal{D}f\rangle,$ $\displaystyle l_{3}(X,Y,Z)=-\frac{1}{6}\langle\lfloor
X,Y\rfloor_{E},Z\rangle+c.p.$

(and the higher multilinear maps are vanish), for $X,Y,Z\in\Gamma E$ and $f\in
C^{\infty}(M)$. Here we followed the convension of [27]. We denote this
$L_{\infty}$-algebra by $(L(E),l_{1},l_{2},l_{3})$.

######  4.14 Definition.

Two Courant algebroids $(E,\langle~{},~{}\rangle,[~{},~{}]_{E},\rho)$ and
$(E,\langle~{},~{}\rangle,[~{},~{}]^{\prime}_{E},\rho^{\prime})$ on the
underlying same vector bundle $E$ and pairing $\langle~{},~{}\rangle$ are said
to be compatible if the following conditions are hold:

(4) $\displaystyle[X,[Y,Z]_{E}]^{\prime}_{E}+[X,[Y,Z]^{\prime}_{E}]_{E}=~{}$
$\displaystyle[[X,Y]_{E},Z]^{\prime}_{E}+[[X,Y]^{\prime}_{E},Z]_{E}+[Y,[X,Z]_{E}]^{\prime}_{E}+[Y,[X,Z]^{\prime}_{E}]_{E},$
(5) $\displaystyle\rho([X,Y]_{E}^{\prime})+\rho^{\prime}([X,Y]_{E})=~{}$
$\displaystyle[\rho(X),\rho^{\prime}(Y)]+[\rho^{\prime}(X),\rho(Y)],$

for $X,Y,Z\in\Gamma E$. Note that the conditions (4) and (5) is equivalent to
the fact that the new quadruple
$(E,\langle~{},~{}\rangle,[~{},~{}]_{E}+[~{},~{}]_{E}^{\prime},\rho+\rho^{\prime})$
is also a Courant algebroid.

In [14], Kosmann-Schwarzbach introduced Nijenhuis operators on a given Courant
algebroid and show that certain Nijenhuis operators induced a deformed Courant
algebroid structure. Moreover, the deformed one is compatible with the given
one. In the following, we prove that compatible Courant algebroids give rise
to compatible $L_{\infty}$-algebras.

######  4.15 Proposition.

Let $(E,\langle~{},~{}\rangle,[~{},~{}]_{E},\rho)$ and
$(E,\langle~{},~{}\rangle,[~{},~{}]^{\prime}_{E},\rho^{\prime})$ be two
compatible Courant algebroids. Consider the corresponding
$L_{\infty}$-algebras $(L(E),l_{1},l_{2},l_{3})$ and
$(L(E),l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3})$ on the graded vector
space $L(E)$. Then
$(L(E),\\{l_{1},l_{2},l_{3}\\},\\{l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3}\\})$
is a compatible $L_{\infty}$-algebra.

###### Proof.

Let $(L(E),l_{1}^{*},l_{2}^{*},l_{3}^{*})$ be the $L_{\infty}$-algebra
associated to the Courant algebroid
$(E,[~{},~{}]_{E}+[~{},~{}]_{E}^{\prime},\langle~{},~{}\rangle,\rho+\rho^{\prime})$.
It is then easy to see that

(6) $\displaystyle
l_{1}^{*}=l_{1}+l^{\prime}_{1},~{}~{}~{}~{}l_{2}^{*}=l_{2}+l^{\prime}_{2}~{}~{}\text{
and }~{}~{}l_{3}^{*}=l_{3}+l^{\prime}_{3}.$

It follows from (6) that
$(L(E),\\{l_{1},l_{2},l_{3}\\},\\{l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3}\\})$
is a compatible $L_{\infty}$-algebra. ∎

##  5 Cohomology and deformations of compatible $L_{\infty}$-algebras

In this section, we first introduce the cohomology of a compatible
$L_{\infty}$-algebra. Then we study formal and finite order deformations of a
compatible $L_{\infty}$-algebra in terms of cohomology.

### Cohomology of compatible $L_{\infty}$-algebras

Let $(L,\\{l_{k}\\}_{k\geq 1},\\{l_{k}^{\prime}\\}_{k\geq 1})$ be a compatible
$L_{\infty}$-algebra. In Theorem 3.4, we have seen that this compatible
$L_{\infty}$-algebra structure on $L$ corresponds to a Maurer-Cartan element
$(D,D^{\prime})\in\mathrm{Coder}^{1}_{c}(SV)$ in the graded Lie algebra
$(\mathrm{Coder}^{\bullet}_{c}(SV),\llbracket~{},~{}\rrbracket)$, where
$V=s^{-1}L$. We will use this Maurer-Cartan element to define the cohomology
of the compatible $L_{\infty}$-algebra $(L,\\{l_{k}\\}_{k\geq
1},\\{l_{k}^{\prime}\\}_{k\geq 1})$.

For each $n\geq 1$, we define the $n$-th cochain group $C^{n}_{c}(L,L)$ by

$\displaystyle
C^{n}_{c}(L,L):=\mathrm{Coder}^{(n-1)}_{c}(SV)=\underbrace{\mathrm{Coder}^{(n-1)}(SV)\oplus\cdots\oplus\mathrm{Coder}^{(n-1)}(SV)}_{n\text{
times}},$

and a map $\delta_{c}:C^{n}_{c}(L,L)\rightarrow C^{n+1}_{c}(L,L)$, for $n\geq
1$, defined by $\delta_{c}(-):=\llbracket(D,D^{\prime}),-\rrbracket.$ Then it
is easy to see that $(\delta_{c})^{2}=0$. In other words,
$\\{C^{\bullet}_{c}(L,L),\delta_{c}\\}$ is a cochain complex. The
corresponding cohomology groups are called the cohomology of the compatible
$L_{\infty}$-algebra $L$, and they are denoted by $H^{\bullet}_{c}(L,L).$

### Deformations of compatible $L_{\infty}$-algebras

Let $L=(L,\\{l_{k}\\}_{k\geq 1},\\{l_{k}^{\prime}\\}_{k\geq 1})$ be a
compatible $L_{\infty}$-algebra with the corresponding Maurer-Cartan element
$(D,D^{\prime})\in\mathrm{Coder}^{1}_{c}(SV)=\mathrm{Coder}^{1}(SV)\oplus\mathrm{Coder}^{1}(SV)$
in the graded Lie algebra
$(\mathrm{Coder}^{\bullet}_{c}(SV),\llbracket~{},~{}\rrbracket)$.

######  5.1 Definition.

A formal deformation of the compatible $L_{\infty}$-algebra $L$ consists of a
formal power series

(7)
$\displaystyle(D,D^{\prime})_{t}:=(D_{0},D_{0}^{\prime})+t(D_{1},D_{1}^{\prime})+t^{2}(D_{2},D_{2}^{\prime})+\cdots\text{
in }\mathrm{Coder}_{c}^{1}(SV)[[t]]$

(with $(D_{0},D_{0}^{\prime})=(D,D^{\prime})$) that satisfies
$\llbracket(D,D^{\prime})_{t},(D,D^{\prime})_{t}\rrbracket=0$.

Thus, (7) defines a formal deformation of $L$ if and only if

(8)
$\displaystyle\sum_{i+j=n}\llbracket(D_{i},D_{i}^{\prime}),(D_{j},D_{j}^{\prime})\rrbracket=0,~{}\text{
for }n\geq 0.$

This system of equations are called deformation equations. Note that (8) holds
for $n=0$ as $(D,D^{\prime})$ corresponds to the given compatible
$L_{\infty}$-algebra structure on $L$. However, for $n=1$, we get

$\displaystyle\llbracket(D,D^{\prime}),(D_{1},D_{1}^{\prime})\rrbracket=0,$

which implies that $(D_{1},D^{\prime}_{1})\in C^{2}_{c}(L,L)$ is a $2$-cocycle
in the cohomology complex of $L$. This is called the infinitesimal of the
formal deformation $(D,D^{\prime})_{t}.$ In particular, if
$(D_{1},D_{1}^{\prime})=\cdots=(D_{p-1},D_{p-1}^{\prime})=0$ and
$(D_{p},D_{p}^{\prime})$ is the first nonzero term, then
$(D_{p},D_{p}^{\prime})\in C^{2}_{c}(L,L)$ is a $2$-cocycle.

######  5.2 Definition.

Let $(D,D^{\prime})_{t}=\sum_{i=0}^{\infty}t^{i}(D_{i},D_{i}^{\prime})$ and
$\overline{(D,D^{\prime})}_{t}=\sum_{i=0}^{\infty}t^{i}(\overline{D}_{i},\overline{D}_{i}^{\prime})$
be two formal deformations of the compatible $L_{\infty}$-algebra $L$. They
are said to be equivalent if there is a formal power series

$\displaystyle\Phi_{t}:=\Phi_{0}+t\Phi_{1}+t^{2}\Phi_{2}+\cdots\text{ in
}\mathrm{Coder}^{0}_{c}(SV)[[t]]=\mathrm{Coder}^{0}(SV)[[t]]$

(with $\Phi_{0}=\mathrm{id}$) satisfying
$\overline{(D,D^{\prime})}_{t}\circ\Phi_{t}=\Phi_{t}\circ(D,D^{\prime})_{t}.$

It follows that $(D,D^{\prime})_{t}$ and $\overline{(D,D^{\prime})}_{t}$ are
equivalent if and only if

(9)
$\displaystyle\sum_{i+j=n}(\overline{D}_{i},\overline{D}_{i}^{\prime})\circ\Phi_{j}=\sum_{i+j=n}\Phi_{i}\circ(D_{j},D_{j}^{\prime}),\text{
for }n\geq 0.$

Observe that (9) holds automatically for $n=0$ as $\Phi_{0}=\mathrm{id}$. For
$n=1$, we get
$(D_{1},D_{1}^{\prime})-(D_{1},D_{1}^{\prime})=\delta_{c}(\Phi_{1})$. This
shows that the infinitesimals corresponding to equivalent deformations are
cohomologous. Hence, they gives rise to the same cohomology class in
$H^{2}_{c}(L,L)$.

Next, we consider rigidity of compatible $L_{\infty}$-algebras.

######  5.3 Definition.

(i) A deformation $(D,D^{\prime})_{t}$ of a compatible $L_{\infty}$-algebra
$L$ is said to be trivial if it is equivalent to the undeformed one
$(D,D^{\prime})$.

(ii) A compatible $L_{\infty}$-algebra $L$ is said to be rigid if any
deformation of $L$ is trivial.

The following result provides a sufficient condition for the rigidity of a
compatible $L_{\infty}$-algebra.

######  5.4 Theorem.

Let $L$ be a compatible $L_{\infty}$-algebra. If $H^{2}_{c}(L,L)=0$ then $L$
is rigid.

###### Proof.

Let $(D,D^{\prime})_{t}$ be any deformation of $L$ given by

$\displaystyle(D,D^{\prime})_{t}=(D,D^{\prime})+\sum_{i\geq
p}t^{i}(D_{i},D_{i}^{\prime}),~{}\text{ for some }p\geq 1.$

Then we know that the first nonzero term $(D_{p},D_{p}^{\prime})$ is a
$2$-cocycle. By the hypothesis, we have
$(D_{p},D_{p}^{\prime})=\delta_{c}(\Phi_{p})$, for some $\Phi_{p}\in
C^{1}_{c}(L,L)=\mathrm{Coder}^{0}(SV)$. We consider
$\Phi_{t}=\mathrm{id}+t^{p}\Phi_{p}\in\mathrm{Coder}^{0}(SV)[[t]]$, and define
$\overline{(D,D^{\prime})}_{t}=\Phi_{t}^{-1}\circ(D,D^{\prime})_{t}\circ\Phi_{t}.$
Then $\overline{(D,D^{\prime})}_{t}$ is a deformation equivalent to
$(D,D^{\prime})_{t}$, and of the form

$\displaystyle\overline{(D,D^{\prime})}_{t}=(D,D^{\prime})+\sum_{i\geq
p+1}t^{i}(\overline{D}_{i},\overline{D}_{i}^{\prime}).$

By repeating the same process, we can show that $(D,D^{\prime})_{t}$ is
equivalent to $(D,D^{\prime})$. Hence the result follows. ∎

Finite order deformations. Here we consider finite order deformations of a
compatible $L_{\infty}$-algebra and their extensions. Let
$L=(L,\\{l_{k}\\}_{k\geq 1},\\{l_{k}^{\prime}\\}_{k\geq 1})$ be a compatible
$L_{\infty}$-algebra given by a Maurer-Cartan elements
$(D,D^{\prime})\in\mathrm{Coder}^{1}_{c}(SV)$ in the graded Lie algebra
$(\mathrm{Coder}^{\bullet}_{c}(SV),\llbracket~{},~{}\rrbracket)$.

######  5.5 Definition.

Let $N\in\mathbb{N}$ be a natural number. An order $N$ deformation of $L$
consists of a finite polynomial

$\displaystyle(D,D^{\prime})_{t}=(D_{0},D_{0}^{\prime})+t(D_{1},D_{1}^{\prime})+\cdots+t^{N}(D_{N},D_{N}^{\prime})\text{
in }\text{ Coder}^{1}_{c}(SV)[[t]]/(t^{N+1})$

(with $(D_{0},D_{0}^{\prime})=(D,D^{\prime})$) that satisfies
$\llbracket(D,D^{\prime})_{t},(D,D^{\prime})_{t}\rrbracket=0$ (mod $t^{N+1}$).

Thus, in an order $N$ deformation, the equations in (8) hold for
$n=0,1,\ldots,N$. These equations can be equivalently written as

(10)
$\displaystyle\llbracket(D,D^{\prime}),(D_{n},D_{n}^{\prime})\rrbracket=-\frac{1}{2}\sum_{i+j=n;i,j\geq
1}\llbracket(D_{i},D_{i}^{\prime}),(D_{j},D_{j}^{\prime})\rrbracket,~{}\text{
for }n=0,1,\ldots,N.$

######  5.6 Definition.

An order $N$ deformation
$(D,D^{\prime})_{t}=\sum_{i=0}^{N}t^{i}(D_{i},D_{i}^{\prime})$ is said to be
extensible if there exists an element
$(D_{N+1},D_{N+1}^{\prime})\in\mathrm{Coder}^{1}_{c}(SV)$ such that
$\widetilde{(D,D^{\prime})}_{t}=(D,D^{\prime})_{t}+t^{N+1}(D_{N+1},D_{N+1}^{\prime})=\sum_{i=0}^{N+1}t^{i}(D_{i},D_{i}^{\prime})$
is an order $N+1$ deformation.

Thus, together with equations in (10), one more equation need to be satisfied,
namely,

(11)
$\displaystyle\llbracket(D,D^{\prime}),(D_{N+1},D_{N+1}^{\prime})\rrbracket=-\frac{1}{2}\sum_{i+j=N+1;i,j\geq
1}\llbracket(D_{i},D_{i}^{\prime}),(D_{j},D_{j}^{\prime})\rrbracket.$

The right hand side of the above equation does not contain the term
$(D_{N+1},D_{N+1}^{\prime})$. Therefore, it depends only on the given orden
$N$ deformation $(D,D^{\prime})_{t}$. It is called the obstruction to extend
the deformation $(D,D^{\prime})_{t}$.

######  5.7 Proposition.

The obstruction is a $3$-cocycle in the cohomology complex of $L$.

###### Proof.

We have

$\displaystyle\delta_{c}\big{(}-\frac{1}{2}\sum_{i+j=N+1;i,j\geq
1}\llbracket(D_{i},D_{i}^{\prime}),(D_{j},D_{j}^{\prime})\rrbracket\big{)}$
$\displaystyle=-\frac{1}{2}\sum_{i+j=N+1;i,j\geq
1}\llbracket(D,D^{\prime}),\llbracket(D_{i},D^{\prime}_{i}),(D_{j},D^{\prime}_{j})\rrbracket\rrbracket$
$\displaystyle=-\frac{1}{2}\sum_{i+j=N+1;i,j\geq
1}\big{(}\llbracket\llbracket(D,D^{\prime}),(D_{i},D^{\prime}_{i})\rrbracket,(D_{j},D^{\prime}_{j})\rrbracket~{}-~{}\llbracket(D_{i},D^{\prime}_{i}),\llbracket(D,D^{\prime}),(D_{j},D^{\prime}_{j})\rrbracket\rrbracket\big{)}$
$\displaystyle=\frac{1}{4}\sum_{i_{1}+i_{2}+j=N+1;i_{1},i_{2},j\geq
1}\llbracket\llbracket(D_{i_{1}},D^{\prime}_{i_{1}}),(D_{i_{2}},D^{\prime}_{i_{2}})\rrbracket,(D_{j},D^{\prime}_{j})\rrbracket$
$\displaystyle\qquad-\frac{1}{4}\sum_{i+j_{1}+j_{2}=N+1;i,j_{1},j_{2}\geq
1}\llbracket(D_{i},D^{\prime}_{i}),\llbracket(D_{j_{1}},D^{\prime}_{j_{1}}),(D_{j_{2}},D^{\prime}_{j_{2}})\rrbracket\rrbracket\qquad(\text{form
}(\ref{fin-def-eq}))$ $\displaystyle=\frac{1}{2}\sum_{i+j+k=N+1;i,j,k\geq
1}\llbracket\llbracket(D_{i},D^{\prime}_{i}),(D_{j},D^{\prime}_{j})\rrbracket,(D_{k},D^{\prime}_{k})\rrbracket=0.$

Hence the proof. ∎

Therefore, it follows that the obstruction induces a cohomology class in
$H^{3}_{c}(L,L)$. This is called the obstruction class. As a consequence of
(11), we get the following.

######  5.8 Theorem.

A finite order deformation of a compatible $L_{\infty}$-algebra $L$ is
extensible if and only if the corresponding class in $H^{3}_{c}(L,L)$ is
trivial.

######  5.9 Corollary.

Let $L$ be a compatible $L_{\infty}$-algebra. If $H^{3}_{c}(L,L)=0$ then any
finite order deformation of $L$ is extensible.

##  6 Classification of some compatible $L_{\infty}$-algebras

In this section, we will consider compatible $L_{\infty}$-algebras whose
underlying graded vector space is concentrated in degrees $-1$ and $0$. We
call them $2$-term compatible $L_{\infty}$-algebras. We first recall $2$-term
$L_{\infty}$-algebras [2].

######  6.1 Definition.

(i) A $2$-term $L_{\infty}$-algebra is a tuple $(L_{-1}\oplus
L_{0},l_{1},l_{2},l_{3})$ consisting of a graded vector space $L_{-1}\oplus
L_{0}$ (concentrated in degrees $-1$ and $0$) together with a linear map
$l_{1}:L_{-1}\rightarrow L_{0}$, skew-symmetric bilinear maps
$l_{2}:L_{i}\times L_{j}\rightarrow L_{i+j}$ ($-1\leq i,j,i+j\leq 0$) and a
skew-symmetric trilinear map $l_{3}:L_{0}\times L_{0}\times L_{0}\rightarrow
L_{-1}$ satisfying for $w,x,y,z\in L_{0}$ and $h,k\in L_{-1},$

(12) $\displaystyle l_{1}(l_{2}(x,h))=l_{2}(x,l_{1}(h)),$ (13) $\displaystyle
l_{2}(l_{1}(h),k)=l_{2}(h,l_{1}(k)),$ (14) $\displaystyle
l_{1}(l_{3}(x,y,z))=-l_{2}(l_{2}(x,y),z)+l_{2}(l_{2}(x,z),y)+l_{2}(x,l_{2}(y,z)),$
(15) $\displaystyle
l_{3}(l_{1}(h),x,y)=-l_{2}(l_{2}(x,y),h)+l_{2}(l_{2}(x,h),y)+l_{2}(x,l_{2}(y,h)),$
(16) $\displaystyle
l_{2}(l_{3}(w,x,y),z)+l_{2}(l_{3}(w,y,z),x)-l_{2}(l_{3}(w,x,z),y)-l_{2}(l_{3}(x,y,z),w)=l_{3}(l_{2}(w,x),y,z)$
$\displaystyle-
l_{3}(l_{2}(w,y),x,z)+l_{3}(l_{2}(w,z),x,y)+l_{3}(l_{2}(x,y),w,z)-l_{3}(l_{2}(x,z),w,y)+l_{3}(l_{2}(y,z),w,x).$

(ii) Let $(L_{-1}\oplus L_{0},l_{1},l_{2},l_{3})$ and $(M_{-1}\oplus
M_{0},m_{1},m_{2},m_{3})$ be $2$-term $L_{\infty}$-algebras. A morphism
between them is a triple $(f_{-1},f_{0},f)$ consisting of linear maps
$f_{-1}:L_{-1}\rightarrow M_{-1}$ and $f_{0}:L_{0}\rightarrow M_{0}$, and a
skew-symmetric bilinear map $f:L_{0}\times L_{0}\rightarrow M_{-1}$ satisfying
for $x,y,z\in L_{0}$ and $h\in L_{-1}$,

(17) $\displaystyle f_{0}\circ l_{1}=m_{1}\circ f_{-1},$ (18) $\displaystyle
m_{1}(f(x,y))=f_{0}(l_{2}(x,y))-m_{2}(f_{0}(x),f_{0}(y)),$ (19) $\displaystyle
f(x,l_{1}(h))=f_{-1}(l_{2}(x,h))-m_{2}(f_{0}(x),f_{-1}(h)),$ (20)
$\displaystyle
m_{2}(f(x,y),f_{0}(z))+f(l_{2}(x,y),z)+f_{-1}(l_{3}(x,y,z))=m_{3}(f_{0}(x),f_{0}(y),f_{0}(z))$
$\displaystyle+m_{2}(f_{0}(x),f(y,z))+m_{2}(f(x,z),f_{0}(y))+f(x,l_{2}(y,z))+f(l_{2}(x,z),y)).$

######  6.2 Definition.

A $2$-term compatible $L_{\infty}$-algebra is a tuple $(L_{-1}\oplus
L_{0},\\{l_{1},l_{2},l_{3}\\},\\{l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3}\\})$
in which $(L_{-1}\oplus L_{0},l_{1},l_{2},l_{3})$ and $(L_{-1}\oplus
L_{0},l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3})$ are $2$-term
$L_{\infty}$-algebras satisfying for $w,x,y,z\in L_{0}$ and $h,k\in L_{-1},$

(21) $\displaystyle
l_{1}(l^{\prime}_{2}(x,h))+l^{\prime}_{1}(l_{2}(x,h))=l_{2}(x,l^{\prime}_{1}(h))+l^{\prime}_{2}(x,l_{1}(h)),$
(22) $\displaystyle
l_{2}(l^{\prime}_{1}(h),k)+l^{\prime}_{2}(l_{1}(h),k)=l_{2}(h,l^{\prime}_{1}(k))+l^{\prime}_{2}(h,l_{1}(k)),$
(23) $\displaystyle
l_{1}(l^{\prime}_{3}(x,y,z))+l^{\prime}_{1}(l_{3}(x,y,z))=-l_{2}(l^{\prime}_{2}(x,y),z)-l^{\prime}_{2}(l_{2}(x,y),z)+l_{2}(l^{\prime}_{2}(x,z),y)+l^{\prime}_{2}(l_{2}(x,z),y)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+l_{2}(x,l^{\prime}_{2}(y,z))+l^{\prime}_{2}(x,l_{2}(y,z)),$
(24) $\displaystyle
l_{3}(l^{\prime}_{1}(h),x,y)+l^{\prime}_{3}(l_{1}(h),x,y)=-l_{2}(l^{\prime}_{2}(x,y),h)-l^{\prime}_{2}(l_{2}(x,y),h)+l_{2}(l^{\prime}_{2}(x,h),y)+l^{\prime}_{2}(l_{2}(x,h),y)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+l_{2}(x,l^{\prime}_{2}(y,h))+l^{\prime}_{2}(x,l_{2}(y,h)),$
(25) $\displaystyle
l_{2}(l^{\prime}_{3}(w,x,y),z)+l^{\prime}_{2}(l_{3}(w,x,y),z)+l_{2}(l^{\prime}_{3}(w,y,z),x)+l^{\prime}_{2}(l_{3}(w,y,z),x)-l_{2}(l^{\prime}_{3}(w,x,z),y)$
$\displaystyle-l^{\prime}_{2}(l_{3}(w,x,z),y)-l_{2}(l^{\prime}_{3}(x,y,z),w)-l^{\prime}_{2}(l_{3}(x,y,z),w)=l_{3}(l^{\prime}_{2}(w,x),y,z)+l^{\prime}_{3}(l_{2}(w,x),y,z)$
$\displaystyle-
l_{3}(l^{\prime}_{2}(w,y),x,z)-l^{\prime}_{3}(l_{2}(w,y),x,z)+l_{3}(l^{\prime}_{2}(w,z),x,y)+l^{\prime}_{3}(l_{2}(w,z),x,y)+l_{3}(l^{\prime}_{2}(x,y),w,z)$
$\displaystyle+l^{\prime}_{3}(l_{2}(x,y),w,z)-l_{3}(l^{\prime}_{2}(x,z),w,y)-l^{\prime}_{3}(l_{2}(x,z),w,y)+l_{3}(l^{\prime}_{2}(y,z),w,x)+l^{\prime}_{3}(l_{2}(y,z),w,x).$

######  6.3 Remark.

Let $(L_{-1}\oplus
L_{0},\\{l_{1},l_{2},l_{3}\\},\\{l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3}\\})$
be a $2$-term compatible $L_{\infty}$-algebra with $l_{1}=l^{\prime}_{1}=l.$
Then the compatibility conditions (21) - (25) is equivalent to the fact that
$(L_{-1}\oplus L_{0},l,l_{2}+l^{\prime}_{2},2(l_{3}+l^{\prime}_{3}))$ is a
$2$-term $L_{\infty}$-algebra.

######  6.4 Definition.

Let $(L_{-1}\oplus
L_{0},\\{l_{1},l_{2},l_{3}\\},\\{l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3}\\})$
and $(M_{-1}\oplus
M_{0},\\{m_{1},m_{2},m_{3}\\},\\{m^{\prime}_{1},m^{\prime}_{2},m^{\prime}_{3}\\})$
be $2$-term compatible $L_{\infty}$-algebras. A morphism between them is a
triple $(f_{-1},f_{0},f)$ which is an $L_{\infty}$-algebra morphism from
$(L_{-1}\oplus L_{0},l_{1},l_{2},l_{3})$ to $(M_{-1}\oplus
M_{0},m_{1},m_{2},m_{3})$ and an $L_{\infty}$-algebra morphism from
$(L_{-1}\oplus L_{0},l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3})$ to
$(M_{-1}\oplus M_{0},m^{\prime}_{1},m^{\prime}_{2},m^{\prime}_{3})$.

Let $(f_{-1},f_{0},f):(L_{-1}\oplus
L_{0},\\{l_{1},l_{2},l_{3}\\},\\{l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3}\\})\rightsquigarrow(M_{-1}\oplus
M_{0},\\{m_{1},m_{2},m_{3}\\},\\{m^{\prime}_{1},m^{\prime}_{2},m^{\prime}_{3}\\})$
and $(g_{-1},g_{0},g):(M_{-1}\oplus
M_{0},\\{m_{1},m_{2},m_{3}\\},\\{m^{\prime}_{1},m^{\prime}_{2},m^{\prime}_{3}\\})\rightsquigarrow(N_{-1}\oplus
N_{0},\\{n_{1},n_{2},n_{3}\\},\\{n^{\prime}_{1},n^{\prime}_{2},n^{\prime}_{3}\\})$
be two morphisms of compatible $L_{\infty}$-algebras. Their composition is
defined by the triple $(g_{-1}\circ f_{-1},~{}g_{0}\circ
f_{0},~{}g\circ(f_{0}\times f_{0})+g_{-1}\circ f)$.

######  6.5 Notation.

We denote by ${\bf 2CompL_{\infty}}$ the category of $2$-term compatible
$L_{\infty}$-algebras and morphisms between them. Let ${\bf
2CompL^{1}_{\infty}}$ be the subcategory of ${\bf 2CompL_{\infty}}$ whose
objects are $2$-term compatible $L_{\infty}$-algebras of the form
$(L_{-1}\oplus
L_{0},\\{l_{1},l_{2},l_{3}\\},\\{l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3}\\})$
with $l_{1}=l^{\prime}_{1}.$

The following definitions are motivated by the standard $L_{\infty}$ case [2].

######  6.6 Definition.

A $2$-term compatible $L_{\infty}$-algebra $(L_{-1}\oplus
L_{0},\\{l_{1},l_{2},l_{3}\\},\\{l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3}\\})$
is said to be

  * (i)

‘strict’ if $l_{3}=l^{\prime}_{3}=0$,

  * (ii)

‘skeletal’ if $l_{1}=l^{\prime}_{1}=0$.

In the rest of this section, we will be interested in strict and skeletal
compatible $L_{\infty}$-algebras and their classifications.

### Crossed modules of compatible Lie algebras and strict compatible
$L_{\infty}$-algebras

In this subsection, we introduce crossed module of compatible Lie algebras and
show that they are closely related to strict compatible $L_{\infty}$-algebras.

######  6.7 Definition.

A crossed module of compatible Lie algebras is tuple
$(\mathfrak{g},\mathfrak{h},t,t^{\prime},\alpha,\alpha^{\prime})$ consisting
of compatible Lie algebras
$\mathfrak{g}=(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$ and
$\mathfrak{h}=(\mathfrak{h},[~{},~{}]_{\mathfrak{h}},[~{},~{}]^{\prime}_{\mathfrak{h}})$,
homomorphisms
$t:(\mathfrak{g},[~{},~{}])\rightarrow(\mathfrak{h},[~{},~{}]_{\mathfrak{h}})$
and
$t^{\prime}:(\mathfrak{g},[~{},~{}]^{\prime})\rightarrow(\mathfrak{h},[~{},~{}]^{\prime}_{\mathfrak{h}})$
of Lie algebras, and bilinear maps
$\alpha,\alpha^{\prime}:\mathfrak{h}\times\mathfrak{g}\rightarrow\mathfrak{g}$
satisfying for $m,n\in\mathfrak{g}$ and $x,y\in\mathfrak{h}$,

  * (i)

$t(\alpha(x,m))=[x,t(m)]_{\mathfrak{h}}$,
$t^{\prime}(\alpha^{\prime}(x,m))=[x,t^{\prime}(m)]^{\prime}_{\mathfrak{h}}$,
$t(\alpha^{\prime}(x,m))+t^{\prime}(\alpha(x,m))=[x,t^{\prime}(m)]_{\mathfrak{h}}+[x,t(m)]^{\prime}_{\mathfrak{h}},$

  * (ii)

$\alpha(t(m),n)=[m,n]$,
$\alpha^{\prime}(t^{\prime}(m),n)=[m,n]^{\prime}$,
$\alpha(t^{\prime}(m),n)+\alpha^{\prime}(t(m),n)=\alpha(m,t^{\prime}(n))+\alpha^{\prime}(m,t(n))$,

  * (iii)

$\alpha(x,[m,n])=[\alpha(x,m),n]+[m,\alpha(x,n)],$
$\alpha^{\prime}(x,[m,n]^{\prime})=[\alpha^{\prime}(x,m),n]^{\prime}+[m,\alpha^{\prime}(x,n)]^{\prime},$
$\alpha(x,[m,n]^{\prime})+\alpha^{\prime}(x,[m,n])=[\alpha(x,m),n]^{\prime}+[\alpha^{\prime}(x,m),n]+[m,\alpha(x,n)]^{\prime}+[m,\alpha^{\prime}(x,n)],$

  * (iv)

$\alpha([x,y],m)=\alpha(x,\alpha(y,m))-\alpha(y,\alpha(x,m)),$
$\alpha^{\prime}([x,y]^{\prime},m)=\alpha^{\prime}(x,\alpha^{\prime}(y,m))-\alpha^{\prime}(y,\alpha^{\prime}(x,m)),$
$\alpha([x,y]^{\prime},m)+\alpha^{\prime}([x,y],m)=\alpha(x,\alpha^{\prime}(y,m))+\alpha^{\prime}(x,\alpha(y,m))-\alpha(y,\alpha^{\prime}(x,m))-\alpha^{\prime}(y,\alpha(x,m))$.

######  6.8 Theorem.

There is a one-to-one correspondence between strict compatible
$L_{\infty}$-algebras and crossed module of compatible Lie algebras.

###### Proof.

Let $(L_{-1}\oplus
L_{0},\\{l_{1},l_{2},0\\},\\{l^{\prime}_{1},l^{\prime}_{2},0\\})$ be a strict
compatible $L_{\infty}$-algebra. Since $(L_{-1}\oplus L_{0},l_{1},l_{2},0)$ is
a strict $L_{\infty}$-algebra [2], it follows that $\mathfrak{g}=L_{-1}$ and
$\mathfrak{h}=L_{0}$ both carries Lie brackets given by

$\displaystyle[h,k]:=l_{2}(l_{1}(h),k)=l_{2}(h,l_{1}(k)),~{}\text{ for }h,k\in
L_{-1}~{}~{}~{}\text{ and }~{}~{}~{}[x,y]_{\mathfrak{h}}:=l_{2}(x,y),~{}\text{
for }x,y\in L_{0}.$

Similarly, $(L_{-1}\oplus L_{0},l^{\prime}_{1},l^{\prime}_{2},0)$ is a strict
$L_{\infty}$-algebra implies that $\mathfrak{g}=L_{-1}$ and
$\mathfrak{h}=L_{0}$ equipped with another Lie brackets

$\displaystyle[h,k]^{\prime}:=l_{2}^{\prime}(l^{\prime}_{1}(h),k)=l^{\prime}_{2}(h,l^{\prime}_{1}(k)),\text{
for }h,k\in L_{-1}~{}~{}~{}\text{ and
}~{}~{}~{}[x,y]^{\prime}_{\mathfrak{h}}:=l^{\prime}_{2}(x,y),~{}\text{ for
}x,y\in L_{0}.$

The compatibility conditions (23) and (24) implies that
$(\mathfrak{g}=L_{-1},[~{},~{}],[~{},~{}]^{\prime})$ and
$(\mathfrak{h}=L_{0},[~{},~{}]_{\mathfrak{h}},[~{},~{}]^{\prime}_{\mathfrak{h}})$
are compatible Lie algebras. Finally, we define maps

$\displaystyle
t=l_{1}:\mathfrak{g}\rightarrow\mathfrak{h},~{}~{}~{}~{}t^{\prime}=l^{\prime}_{1}:\mathfrak{g}\rightarrow\mathfrak{h},~{}~{}~{}~{}\alpha=l_{2}:\mathfrak{h}\times\mathfrak{g}\rightarrow\mathfrak{g}~{}~{}~{}\text{
and
}~{}~{}~{}\alpha^{\prime}=l^{\prime}_{2}:\mathfrak{h}\times\mathfrak{g}\rightarrow\mathfrak{g}.$

Then it is easy to verify that
$(\mathfrak{g},\mathfrak{h},t,t^{\prime},\alpha,\alpha^{\prime})$ is a crossed
module of compatible Lie algebras.

Conversely, let
$(\mathfrak{g},\mathfrak{h},t,t^{\prime},\alpha,\alpha^{\prime})$ be a crossed
module of compatible Lie algebras. Then it is easy to verify that
$(\underbrace{\mathfrak{g}}_{-1}\oplus\underbrace{\mathfrak{h}}_{0},\\{l_{1},l_{2},0\\},\\{l^{\prime}_{1},l^{\prime}_{2},0\\})$
is a strict compatible $L_{\infty}$-algebra, where

$\displaystyle
l_{1}=t,~{}~{}~{}~{}l_{2}(x,y)=[x,y]_{\mathfrak{h}},~{}~{}~{}~{}l_{2}(x,h)=-l_{2}(h,x):=\alpha(x,h),~{}\text{
for }x,y\in\mathfrak{h},~{}h\in\mathfrak{g},$ $\displaystyle
l^{\prime}_{1}=t^{\prime},~{}~{}~{}~{}l^{\prime}_{2}(x,y)=[x,y]^{\prime}_{\mathfrak{h}},~{}~{}~{}~{}l^{\prime}_{2}(x,h)=-l^{\prime}_{2}(h,x):=\alpha^{\prime}(x,h),~{}\text{
for }x,y\in\mathfrak{h},~{}h\in\mathfrak{g},$

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

### Skeletal compatible $L_{\infty}$-algebras

######  6.9 Theorem.

There is a one-to-one correspondence between skeletal compatible
$L_{\infty}$-algebras and tuples
$(\mathfrak{g},V,(\theta,\theta+\theta^{\prime},\theta^{\prime}))$, where
$\mathfrak{g}=(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$ is a compatible Lie
algebra, $V=(V,\rho,\rho^{\prime})$ is a representation and
$(\theta,\theta+\theta^{\prime},\theta^{\prime})\in C^{3}_{c}(\mathfrak{g},V)$
is a $3$-cocycle induced by compatible Lie algebra cocycles.

###### Proof.

Let $(L_{-1}\oplus
L_{0},\\{0,l_{2},l_{3}\\},\\{0,l^{\prime}_{2},l^{\prime}_{3}\\})$ be a
skeletal compatible $L_{\infty}$-algebra. Since $(L_{-1}\oplus
L_{0},0,l_{2},l_{3})$ is a skeletal $L_{\infty}$-algebra [2], it follows from
(14) that $\mathfrak{g}=L_{0}$ is a Lie algebra with the bracket
$[x,y]:=l_{2}(x,y)$, for $x,y\in\mathfrak{g}=L_{0}$. Moreover, if follows from
(15) that $V=L_{-1}$ is a representation of the Lie algebra
$(\mathfrak{g},[~{},~{}])$ with the action map
$\rho:\mathfrak{g}\rightarrow\mathrm{End}(V)$, $\rho(x)(h):=l_{2}(x,h)$, for
$x\in\mathfrak{g}$ and $h\in V$. Finally, the condition (16) says that the
skew-symmetric trilinear map $l_{3}:L_{0}\times L_{0}\times L_{0}\rightarrow
L_{-1}$ (i.e. $l_{3}\in\mathrm{Hom}(\wedge^{3}\mathfrak{g},V)$) is a
$3$-cocycle in the Chevalley-Eilenberg cochain complex of the Lie algebra
$(\mathfrak{g},[~{},~{}])$ with coefficients in the representation $(V,\rho)$,
i.e. $\delta(l_{3})=0$. Similarly, $(L_{-1}\oplus
L_{0},0,l^{\prime}_{2},l^{\prime}_{3})$ is a skeletal $L_{\infty}$-algebra
implies that $(\mathfrak{g}=L_{0},[~{},~{}]^{\prime})$ is a Lie algebra,
$(V=L_{-1},\rho^{\prime})$ is a representation, where

$\displaystyle[x,y]^{\prime}:=l^{\prime}_{2}(x,y)~{}~{}~{}~{}\text{ and
}~{}~{}~{}~{}\rho^{\prime}(x)(h)=l^{\prime}_{2}(x,h),~{}\text{ for
}x,y\in\mathfrak{g}=L_{0}\text{ and }h\in V=L_{-1}.$

And the skew-symmetric trilinear map $l^{\prime}_{3}:L_{0}\times L_{0}\times
L_{0}\rightarrow L_{-1}$ (i.e.
$l_{3}\in\mathrm{Hom}(\wedge^{3}\mathfrak{g},V)$) is a $3$-cocycle in the
Chevalley-Eilenberg complex of the Lie algebra
$(\mathfrak{g},[~{},~{}]^{\prime})$ with coefficients in the representation
$(V,\rho^{\prime})$, i.e. $\delta^{\prime}(l^{\prime}_{3})=0$.

Finally, the condition (23) implies that
$(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$ is a compatible Lie algebra, the
condition (24) implies that $(V,\rho,\rho^{\prime})$ is a representation of
the compatible Lie algebra. Lastly, the condition (25) is equivalent to
$\delta(l^{\prime}_{3})+\delta^{\prime}(l_{3})=0$. Therefore, by Remark 2.3,
we have $\delta_{c}(l_{3},l_{3}+l^{\prime}_{3},l^{\prime}_{3})=0$. As a
summary, we get a required triple
$(\mathfrak{g},V,(l_{3},l_{3}+l^{\prime}_{3}+l^{\prime}_{3}))$.

Conversely, let
$(\mathfrak{g},V,(\theta,\theta+\theta^{\prime},\theta^{\prime}))$ be a triple
consisting of a compatible Lie algebra
$\mathfrak{g}=(\mathfrak{g},[~{},~{}],[~{},~{}]^{\prime})$, a representation
$V=(V,\rho,\rho^{\prime})$ and a $3$-cocycle
$(\theta,\theta+\theta^{\prime},\theta^{\prime})\in C^{3}_{c}(\mathfrak{g},V)$
induced by compatible Lie algebra cocycles. Then it is easy to verify that
$(\underbrace{V}_{-1}\oplus\underbrace{\mathfrak{g}}_{0},\\{0,l_{2},l_{3}\\},\\{0,l^{\prime}_{2},l^{\prime}_{3}\\})$
is a skeletal compatible $L_{\infty}$-algebra, where

$\displaystyle
l_{2}(x,y):=[x,y],~{}~{}~{}~{}l_{2}(x,h)=-l_{2}(h,x)=\rho(x)(h),~{}~{}~{}~{}l_{3}=\theta,$
$\displaystyle
l^{\prime}_{2}(x,y):=[x,y]^{\prime},~{}~{}~{}~{}l^{\prime}_{2}(x,h)=-l^{\prime}_{2}(h,x)=\rho^{\prime}(x)(h),~{}~{}~{}~{}l^{\prime}_{3}=\theta^{\prime},$

for $x,y\in\mathfrak{g}$ and $h\in V$. The above two correspondences are
inverses to each other. Hence the proof. ∎

##  7 Compatible Lie $2$-algebras

In this section, we introduce compatible Lie $2$-algebras as the
categorification of compatible Lie algebras. We show that the category of
compatible Lie $2$-algebras is equivalent to category ${\bf
2CompL^{1}_{\infty}}$ introduced in Section 6 (Notation 6.5).

Like a (compatible) Lie algebra has an underlying vector space, a (compatible)
Lie $2$-algebra has an underlying $2$-vector space. A $2$-vector space is
basically a category internal to the category Vect of vector spaces. More
precisely, a $2$-vector space $\mathcal{C}$ is a category with a vector space
of objects $\mathcal{C}_{0}$ and a vector space of morphisms $\mathcal{C}_{1}$
such that the source and target maps
$s,t:\mathcal{C}_{1}\rightarrow\mathcal{C}_{0}$, the identity assigning map
$i:\mathcal{C}_{0}\rightarrow\mathcal{C}_{1}$, and the composition map
$\circ:C_{1}\times_{\mathcal{C}_{0}}\mathcal{C}_{1}$ are all linear maps.
Given $2$-vector spaces $\mathcal{C}$ and $\mathcal{D}$, a morphism
$F:\mathcal{C}\rightarrow\mathcal{D}$ is just a functor in the category Vect.

######  7.1 Definition.

[2] A Lie $2$-algebra is a $2$-vector space $\mathcal{C}$ together with a
skew-symmetric bilinear functor
$[~{},~{}]:\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$ and a skew-
symmetric trilinear natural isomorphism (called the Jacobiator)

$\displaystyle J_{x,y,z}:[[x,y],z]\rightarrow[[x,z],y]+[x,[y,z]],\text{ for
}x,y,z\in\mathcal{C}_{0}$

which makes the following diagram commutative: for
$w,x,y,z\in\mathcal{C}_{0}$,

A Lie $2$-algebra as above may be denoted by the triple
$(\mathcal{C},[~{},~{}],J)$.

######  7.2 Definition.

Let $(\mathcal{C},[~{},~{}],J)$ and $(\mathcal{D},\lfloor~{},~{}\rfloor,K)$ be
Lie $2$-algebras. A morphism between them consists of a pair $(F,G)$ of a
linear functor $F:\mathcal{C}\rightarrow\mathcal{D}$ on the underlying
$2$-vector spaces and a skew-symmetric bilinear natural transformation
$G_{x,y}:\lfloor F(x),F(y)\rfloor\rightarrow F[x,y]$, for
$x,y\in\mathcal{C}_{0}$, satisfying

In the following, we introduce compatible Lie $2$-algebras and morphism
between them.

######  7.3 Definition.

A compatible Lie $2$-algebra is a tuple
$(\mathcal{C},[~{},~{}],J,[~{},~{}]^{\prime},J^{\prime},J^{c})$ in which
$(\mathcal{C},[~{},~{}],J)$ and $(\mathcal{C},[~{},~{}]^{\prime},J^{\prime})$
are Lie $2$-algebras and $J^{c}$ is a trilinear skew-symmetric natural
isomorphism

$\displaystyle
J^{c}_{x,y,z}:[[x,y],z]^{\prime}+[[x,y]^{\prime},z]\rightarrow[[x,z],y]^{\prime}+[[x,z]^{\prime},y]+[x,[y,z]]^{\prime}+[x,[y,z]^{\prime}]$

which makes the triple
$(\mathcal{C},[~{},~{}]+[~{},~{}]^{\prime},J+J^{\prime}+J^{c})$ into a Lie
$2$-algebra.

######  7.4 Definition.

Let $(\mathcal{C},[~{},~{}],J,[~{},~{}]^{\prime},J^{\prime},J^{c})$ and
$(\mathcal{D},\lfloor~{},~{}\rfloor,K,\lfloor~{},~{}\rfloor^{\prime},K^{\prime},K^{c})$
be two compatible Lie $2$-algebras. A morphism between them is a triple
$(F,G,G^{\prime})$ consisting of a linear funtor
$F:\mathcal{C}\rightarrow\mathcal{D}$ on the underlying $2$-vector spaces, and
two skew-symmetric bilinear natural transformations

$\displaystyle G_{x,y}:\lfloor F(x),F(y)\rfloor\rightarrow
F[x,y]~{}~{}~{}~{}\text{ and }~{}~{}~{}G^{\prime}_{x,y}:\lfloor
F(x),F(y)\rfloor^{\prime}\rightarrow F[x,y]^{\prime}$

such that $(F,G)$ is a morphism of Lie $2$-algebras from
$(\mathcal{C},[~{},~{}],J)$ to $(\mathcal{D},\lfloor~{},~{}\rfloor,K)$, and
$(F,G^{\prime})$ is a morphism of Lie $2$-algebras from
$(\mathcal{C},[~{},~{}]^{\prime},J^{\prime})$ to
$(\mathcal{D},\lfloor~{},~{}\rfloor^{\prime},K^{\prime}).$

Compatible Lie $2$-algebras and morphisms between them forms a category. We
denote this category by CompLie2.

######  7.5 Theorem.

The categories 2CompL${}^{1}_{\infty}$ and CompLie2 are equivalent.

###### Proof.

Let $L=(L_{-1}\oplus
L_{0},\\{l,l_{2},l_{3}\\},\\{l,l^{\prime}_{2},l^{\prime}_{3}\\})$ be an
element in 2CompL${}^{1}_{\infty}$. We define a $2$-vector space $\mathcal{C}$
whose set of objects $\mathcal{C}_{0}=L_{0}$ and the set of morphisms
$\mathcal{C}_{1}=L_{-1}\oplus L_{0}$, and structure maps

$\displaystyle s(h,x)=x,~{}~{}~{}~{}t(h,x)=l(h)+x~{}~{}\text{ and
}~{}~{}i(x)=(0,x).$

Define a skew-symmetric bilinear functor
$[~{},~{}]:\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$ and a skew-
symmetric trilinear natural isomorphism $J$ by

$\displaystyle[(h,x),(k,y)]:=(l_{2}(h,y)+l_{2}(x,k)+l_{2}(dh,k),~{}l_{2}(x,y))~{}~{}\text{
and }~{}~{}J_{x,y,z}:=\big{(}l_{3}(x,y,z),~{}l_{2}(l_{2}(x,y),z)\big{)}.$

Since $(L_{-1}\oplus L_{0},l,l_{2},l_{3})$ is a $2$-term $L_{\infty}$-algebra,
it follows from [2] that $(\mathcal{C},[~{},~{}],J)$ is a Lie $2$-algebra.
Similarly, $(L_{-1}\oplus L_{0},l,l^{\prime}_{2},l^{\prime}_{3})$ is a
$2$-term $L_{\infty}$-algebra implies that
$(\mathcal{C},[~{},~{}]^{\prime},J^{\prime})$ is a Lie $2$-algebra, where

$\displaystyle[(h,x),(k,y)]^{\prime}:=(l^{\prime}_{2}(h,y)+l^{\prime}_{2}(x,k)+l^{\prime}_{2}(dh,k),~{}l^{\prime}_{2}(x,y))~{}~{}\text{
and
}~{}~{}J^{\prime}_{x,y,z}:=\big{(}l^{\prime}_{3}(x,y,z),~{}l^{\prime}_{2}(l^{\prime}_{2}(x,y),z)\big{)}.$

Finally, we define another skew-symmetric trilinear natural isomorphism
$J^{c}$ by

$\displaystyle
J^{c}_{x,y,z}:=(l_{3}(x,y,z)+l^{\prime}_{3}(x,y,z),~{}l_{2}(l^{\prime}_{2}(x,y),z)+l^{\prime}_{2}(l_{2}(x,y),z)).$

By various conditions (21)-(25) of the compatibility of the
$L_{\infty}$-algebras $(L_{-1}\oplus L_{0},l,l_{2},l_{3})$ and $(L_{-1}\oplus
L_{0},l,l^{\prime}_{2},l^{\prime}_{3})$, we deduce that
$(\mathcal{C},[~{},~{}],J,[~{},~{}]^{\prime},J^{\prime},J^{c})$ is a
compatible Lie $2$-algebra. We denote this compatible Lie $2$-algebra by
$\Phi(L).$

Next, let $(f_{-1},f_{0},f):\underbrace{(L_{-1}\oplus
L_{0},\\{l,l_{2},l_{3}\\},\\{l,l^{\prime}_{2},l^{\prime}_{3}\\})}_{L}\rightsquigarrow\underbrace{(M_{-1}\oplus
M_{0},\\{m,m_{2},m_{3}\\},\\{m,m^{\prime}_{2},m^{\prime}_{3}\\})}_{M}$ be a
morphism of $2$-term compatible $L_{\infty}$-algebras. We define a morphism
$(F,G,G^{\prime}):\Phi(L)\rightsquigarrow\Phi(M)$ of compatible Lie
$2$-algebras by

$\displaystyle
F((h,x))=(f_{-1}(h),f_{0}(x)),~{}~{}G_{x,y}=(f(x,y),~{}l_{2}(f_{0}(x),f_{0}(y)))~{}~{}\text{
and }~{}~{}G^{\prime}_{x,y}=(f(x,y),~{}l^{\prime}_{2}(f_{0}(x),f_{0}(y))).$

One can also show that such construction preserves identity morphisms and the
composition of morphisms. Thus, we obtain a functor $\Phi:{\bf
2CompL^{1}_{\infty}}\rightarrow{\bf CompLie2}$.

Conversely, let
$\mathcal{C}=(\mathcal{C},[~{},~{}],J,[~{},~{}]^{\prime},J^{\prime},J^{c})$ be
a compatible Lie $2$-algebra. We define two vector spaces $L_{-1},L_{0}$ and a
linear map $l:L_{-1}\rightarrow L_{0}$ by

$\displaystyle
L_{-1}=\mathrm{ker}(s),~{}~{}~{}L_{0}=\mathcal{C}_{0}~{}~{}~{}\text{ and
}~{}~{}~{}l=t|_{\mathrm{ker}(s)}.$

Define a skew-symmetric bilinear map $l_{2}:L_{i}\times L_{j}\rightarrow
L_{i+j}$ $(-1\leq i,j,i+j\leq 0)$ and a skew-symmetric trilinear map
$l_{3}:L_{0}\times L_{0}\times L_{0}\rightarrow L_{-1}$ by

$\displaystyle
l_{2}(x,y)=[x,y],~{}~{}~{}l_{2}(x,h)=-l_{2}(h,x)=[i(x),h]~{}~{}\text{ and
}~{}~{}l_{3}(x,y,z)=\mathrm{pr}_{1}(J_{x,y,z}).$

Since $(\mathcal{C},[~{},~{}],J)$ is a Lie $2$-algebra, it follows from [2]
that $(L_{-1}\oplus L_{0},l,l_{2},l_{3})$ is a $2$-term $L_{\infty}$-algebra.
Similarly, $(\mathcal{C},[~{},~{}]^{\prime},J^{\prime})$ is a Lie $2$-algebra
implies that $(L_{-1}\oplus L_{0},l,l^{\prime}_{2},l^{\prime}_{3})$ is a
$2$-term $L_{\infty}$-algebra, where

$\displaystyle
l^{\prime}_{2}(x,y)=[x,y]^{\prime},~{}~{}~{}l^{\prime}_{2}(x,h)=-l^{\prime}_{2}(h,x)=[i(x),h]^{\prime}~{}~{}\text{
and }~{}~{}l^{\prime}_{3}(x,y,z)=\mathrm{pr}_{1}(J^{\prime}_{x,y,z}).$

Finally, the compatibility of Lie $2$-algebras $(\mathcal{C},[~{},~{}],J)$ and
$(\mathcal{C},[~{},~{}]^{\prime},J^{\prime})$ implies that $(L_{-1}\oplus
L_{0},\\{l,l_{2},l_{3}\\},\\{l,l^{\prime}_{2},l^{\prime}_{3}\\})$ is a
$2$-term compatible $L_{\infty}$-algebra. We denote this algebra by
$\Psi(\mathcal{C})$.

Next, let $(F,G,G^{\prime})$ be a morphism from
$\underbrace{(\mathcal{C},[~{},~{}],J,[~{},~{}]^{\prime},J^{\prime},J^{c})}_{\mathcal{C}}$
to
$\underbrace{(\mathcal{D},\lfloor~{},~{}\rfloor,K,\lfloor~{},~{}\rfloor^{\prime},K^{\prime},K^{c})}_{\mathcal{D}}$.
Then it follows from [2] that
$(f_{-1},f_{0},f):\Psi(\mathcal{C})\rightsquigarrow\Psi(\mathcal{D})$ is a
morphism of compatible $L_{\infty}$-algebras, where

$\displaystyle
f_{-1}:=F|_{\mathrm{ker}(s)},~{}~{}~{}f_{0}:=F|_{\mathcal{C}_{0}}~{}~{}\text{
and }~{}~{}f(x,y):=G(x,y)-i(s(G(x,y)))=G^{\prime}(x,y)-i(s(G^{\prime}(x,y))).$

This construction preserves the identity morphisms and the composition of
morphisms. Hence we get a functor $\Psi:{\bf CompLie2}\rightarrow{\bf
2CompL^{1}_{\infty}}.$

We are now left to show that there are natural isomorphisms
$\alpha:\Psi\circ\Phi\Rightarrow 1_{{\bf 2CompL^{1}_{\infty}}}$ and
$\beta:\Phi\circ\Psi\Rightarrow 1_{\bf CompLie2}$. Let $L=(L_{-1}\oplus
L_{0},\\{l,l_{2},l_{3}\\},\\{l,l^{\prime}_{2},l^{\prime}_{3}\\})$ be a
compatible $L_{\infty}$-algebra in ${\bf 2CompL^{1}_{\infty}}$. By applying
$\Psi\circ\Phi$, we get that $(\Psi\circ\Phi)(L)$ is same as $L$. Hence
$\alpha$ is the identity transformation.

On the other hand, let
$\mathcal{C}=(\mathcal{C},[~{},~{}],J,[~{},~{}]^{\prime},J^{\prime},J^{c})$ be
a compatible Lie $2$-algebra. If we apply $\Psi$, we get the $2$-term
compatible $L_{\infty}$-algebra

$\displaystyle\Psi(\mathcal{C})=(\mathrm{ker}(s)\oplus\mathcal{C}_{0},\\{l=t|_{\mathrm{ker}(s)},l_{2},l_{3}\\},\\{l=t|_{\mathrm{ker}(s)},l^{\prime}_{2},l^{\prime}_{3}\\}).$

By applying $\Phi$ to it, we get the compatible Lie $2$-algebra
$\dot{\mathcal{C}}=(\dot{\mathcal{C}},\dot{[~{},~{}]},\dot{J},\dot{[~{},~{}]^{\prime}},\dot{J^{\prime}},\dot{J^{c}})$,
where the space of objects and the space of morphisms are respectively given
by $\dot{\mathcal{C}}_{0}=\mathcal{C}_{0}$ and
$\dot{\mathcal{C}}_{1}=\mathrm{ker}(s)\oplus\mathcal{C}_{0}$. Define
$\beta_{\mathcal{C}}:\dot{\mathcal{C}}\rightarrow\mathcal{C}$ by
$\beta_{\mathcal{C}}(h,x)=h+i(x)$. Then it is easy to verify that
$\beta_{\mathcal{C}}$ is an isomorphism of compatible Lie $2$-algebras. This
is infact a natural isomorphism. This completes the proof. ∎

Acknowledgements. Some parts of the work was carried out when the author was a
postdoctoral fellow at Indian Institute of Technology (IIT) Kanpur. The author
thanks the Institute for financial support.

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