# Koszul duality, minimal model and $L_{\infty}$-structure for differential
algebras with weight

Jun Chen, Li Guo, Kai Wang and Guodong Zhou  Jun Chen, Kai Wang and Guodong
Zhou, School of Mathematical Sciences, Shanghai Key laboratory of PMMP, East
China Normal University, Shanghai 200241, China<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Li Guo, Department of
Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, US
<EMAIL_ADDRESS>
###### Abstract.

A differential algebra with weight is an abstraction of both the derivation
(weight zero) and the forward and backward difference operators (weight $\pm
1$). In 2010 Loday established the Koszul duality for the operad of
differential algebras of weight zero. He did not treat the case of nonzero
weight, noting that new techniques are needed since the operad is no longer
quadratic. This paper continues Loday’s work and establishes the Koszul
duality in the case of nonzero weight. In the process, the minimal model and
the Koszul dual homotopy cooperad of the operad governing differential
algebras with weight are determined. As a consequence, a notion of homotopy
differential algebras with weight is obtained and the deformation complex as
well as its $L_{\infty}$-algebra structure for differential algebras with
weight are deduced.

###### Key words and phrases:

cohomology, deformation, differential algebra, homotopy cooperad, homotopy
differential algebra, Koszul dual, $L_{\infty}$-algebra, minimal model, operad

###### 2010 Mathematics Subject Classification:

18M70 12H05 18M65 12H10 16E40 16S80 18M60

###### Contents

  1. 1 Introduction
    1. 1.1 Differential algebras with weight
    2. 1.2 Cohomology and deformations of differential algebras
    3. 1.3 Our approach and layout of the paper
  2. 2 The $L_{\infty}$-algebra controlling deformations of differential algebras
    1. 2.1 Differential algebras and their cohomology
    2. 2.2  $L_{\infty}$-algebra on the deformation complex from differential algebras
    3. 2.3 Realising differential algebra structures as Maurer-Cartan elements
  3. 3 The Koszul dual homotopy cooperad
    1. 3.1 Homotopy (co)operads and $L_{\infty}$-algebras
    2. 3.2  Koszul dual homotopy cooperad of ${{}_{\lambda}\\!\mathfrak{Dif}}$
  4. 4 Operad of homotopy differential algebras with weight and minimal model
    1. 4.1 Operad of homotopy differential algebras with weight
    2. 4.2 The minimal model of the operad of differential algebras
    3. 4.3 Homotopy differential algebras with weight
  5. 5 From the minimal model to the $L_{\infty}$-algebra controlling deformations
    1. 5.1 Proof of Theorem 2.2
    2. 5.2 Another description of homotopy differential algebras

##  1\. Introduction

In this paper we develop the Koszul duality theory for differential algebras
with nonzero weight by determining the minimal model and the Koszul homotopy
dual cooperad of the corresponding operad as well as the
$L_{\infty}$-structure on the deformation complex, thereby solving a question
raised by Loday [33].

###  1.1. Differential algebras with weight

For a fixed scalar $\lambda$, a differential operator of weight $\lambda$ [20]
(called $\lambda$-derivation in [33]) is a linear operator $d=d_{\lambda}$ on
an associative algebra $A$ satisfying the operator identity

(1) $d(uv)=d(u)v+ud(v)+\lambda d(u)d(v),\quad u,v\in A.\vspace{-.1cm}$

When $\lambda=0$, this is the Leibniz rule of the derivation in analysis. When
$\lambda\neq 0$, this is the operator identity satisfied by the differential
quotient $d_{\lambda}(f)(x):=(f(x+\lambda)-f(x))/\lambda$ in the definition of
the derivation. The special cases of $\lambda=\pm 1$ gives the forward and
backward difference operators in numerical analysis and other areas. An
associative algebra with a differential operator of weight $\lambda$ is called
a differential algebra of weight $\lambda$.

Traditionally, a differential algebra is referring to a differential
commutative algebra or a field of weight zero, originated nearly a century ago
from the pioneering work of Ritt [49] in his algebraic study of differential
equations. Through the works of Kaplansky, Kolchin, Magid, Singer and many
others [24, 25, 36, 48], the subject has evolved into a well-established area
in mathematics. In theory its directions include differential Galois theory,
differential algebraic geometry and differential algebraic groups. Its
importance is also reflected by its connections with broad areas from
arithmetic geometry and logic to computer science and mathematical physics [7,
11, 41, 56].

A commutative differential algebra naturally gives rise to a Novikov algebra
by the well-known work of Gelfand and Dorfman [12]. It also gives rise to a
Novikov-Poisson algebra and a transposed Poisson algebra [3, 26, 32, 57]. More
generally, differential algebras without the commutativity condition have
attracted the interest of many recent studies. Derivations on path algebras
and universal enveloping algebras of differential Lie algebras were studied in
[21, 45, 46]. From the categorical viewpoint, differential categories were
introduced and then studies in [4, 6, 31]. In [2], the notion of a
differential algebra was generalized to non(anti)commutative superspace by
deformation in the study of instantons in string theory.

###  1.2. Cohomology and deformations of differential algebras

Beginning with the pioneering works of Gerstenhaber for associative algebras
and of Nijenhuis and Richardson for Lie algebras [13, 14, 44], deformation,
cohomology and homotopy theory have been fundamental in understanding
algebraic structures. A general philosophy, as evolved from ideas of
Gerstenhaber, Nijenhuis, Richardson, Deligne, Schlessinger, Stasheff, Goldman,
Millson etc, is that the deformation theory of any given mathematical object
can be described by a certain differential graded (dg) Lie algebra or more
generally a $L_{\infty}$-algebra associated to the mathematical object (whose
underlying complex is called the deformation complex). This philosophy has
been made into a theorem in characteristic zero by Lurie [35] and Pridham
[47], expressed in terms of infinity categories. It is an important problem to
explicitly construct the dg Lie algebra or $L_{\infty}$-algebra governing the
deformation theory of the given mathematical object. For algebraic structures
(operads) with binary operations, especially the binary quadratic operads,
there is a general theory in which the deformations and homotopy of an
algebraic structure are controlled by a dg Lie algebra naturally coming from
the operadic Koszul duality for the given algebraic structure; see [18, 19]
for the original literature and [34, 40] for general treatises.

Studies of cohomology and homotopy for algebra structures with linear
operators have become very active recently. The structures include
differential operators and Rota-Baxter operators on associative and Lie
algebras [22, 37, 52, 53]. Due to the complexity of the algebraic structures,
the $L_{\infty}$-algebras in these studies are obtained by direct
constructions without using the language of operads.

The first operadic study of algebras with linear operators is the 2010 work of
Loday [33] on differential algebras of weight zero. Since this operad is
quadratic, it can be treated applying the above-mentioned general theory of
Koszul duality [18, 19]. In particular, Loday obtained the corresponding
minimal model and cohomology theory. For further studies, see [9, 23, 30]. For
the case of nonzero weight, Loday made the following observation, where the
operad $\lambda$-AsDer is just the operad of differential algebras of weight
$\lambda$.

> If the parameter $\lambda$ is different from 0, then the operad
> $\lambda$-AsDer is not a quadratic operad since the term $d(a)d(b)$ needs
> three generating operations to be defined. So one needs new techniques to
> extend Koszul duality to this case.

The goal of this paper is to address this problem observed by Loday.

###  1.3. Our approach and layout of the paper

One of the most important consequences of the general Koszul duality theory
for a quadratic operad is that the theory directly gives the minimal model
[19, 18], that is, the cobar construction of the Koszul dual cooperad. Then
the deformation complex as well as its $L_{\infty}$-algebra structure (in fact
a dg Lie algebra in the Koszul case) could be deduced from the minimal model
[27].

With the absence of the quadratic property for differential algebras with
nonzero weight, the existing general Koszul duality theory does not apply, as
noted by Loday as above. The purpose of this work is provide such a theory for
differential algebras with nonzero weight, in hope to shed light on a general
theory beyond the quadratic operads.

To get started, we use a previous work [22], in which a cohomology theory of
differential algebras with nonzero weight was constructed from a specifically
constructed cochain complex, and was shown to control the formal deformations
as well as abelian extensions of the differential algebras. We first determine
the $L_{\infty}$-algebra structure on this deformation complex, in an ad hoc
way, such that differential algebras with weight are realized as Maurer-Cartan
elements for the $L_{\infty}$-algebra. This suggests that the
$L_{\infty}$-algebra should come from the minimal model of the operad
${{}_{\lambda}\\!\mathfrak{Dif}}$ of differential algebras with weight. Thus
instead of verifying the $L_{\infty}$ property directly, we put this
$L_{\infty}$-algebra in a suitable operadic context that is comparable to the
weight zero case obtained in Loday’s work [33]. We introduce homotopy
differential algebras with nonzero weight and show that their operad is indeed
the minimal model of the operad ${{}_{\lambda}\\!\mathfrak{Dif}}$. As a
byproduct, we obtain the Koszul dual homotopy cooperad of
${{}_{\lambda}\\!\mathfrak{Dif}}$.

The outline of the paper is as follows.

In §2, after recalling in §2.1 the needed notions and background on the
cohomology theory developed in [22], we propose an $L_{\infty}$ structure on
the deformation complex for differential algebras with nonzero weight (Theorem
2.2) in §2.2. In §2.3, we show that this $L_{\infty}$-algebra structure is the
right one that realizes differential algebras with weight as the Maurer-Cartan
elements (Proposition 2.7), while holding off the verification of the
$L_{\infty}$ property to the end of paper, until the operadic tools are
developed.

In §3, once the required basics on homotopy (co)operads are collected in §3.1,
we construct in §3.2 a homotopy cooperad
${{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}$, called the Koszul
dual homotopy cooperad of the operad ${{}_{\lambda}\\!\mathfrak{Dif}}$ of
differential algebras with weight because its cobar construction is shown to
be the minimal model of ${{}_{\lambda}\\!\mathfrak{Dif}}$ (Theorem 4.4).

In §4, we first introduce the dg operad
${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ of homotopy differential algebras
with weight in §4.1. We establish in §4.2 that the dg operad
${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ is the minimal model of
${{}_{\lambda}\\!\mathfrak{Dif}}$ (Theorem 4.4). We give the notion of
homotopy differential algebras and its explicit description (Definition 4.12
and Proposition 4.13) in §4.3.

In §5, we determine in §5.1 the the $L_{\infty}$-algebra coming from the
Koszul dual homotopy cooperad and show that it coincides with the
$L_{\infty}$-algebra of differential algebras with weight proposed in §2.2. We
also give equivalent descriptions of homotopy differential algebras
(Propositions 5.1 and 5.2) in §5.2.

Notations. Throughout this paper, let ${\bf k}$ be a field of characteristic
$0$. Unless otherwise stated, vector spaces, linear maps and tensor products
are taken over ${\bf k}$. We will use the homological grading. For two graded
spaces $V$ and $W$, let $V\otimes W$ (resp. ${\rm Hom}(V,W)$) denote the
graded tensor space (resp. the space of graded linear maps).

##  2\. The $L_{\infty}$-algebra controlling deformations of differential
algebras

In this section, we first recall the cohomology theory of differential
algebras with weight, as developed in [22] with some modification in sign
convention. We then introduce an $L_{\infty}$-algebra such that the cochain
complex of differential algebras can be realized as the shift of the
underlying complex of the $L_{\infty}$-algebra obtained by the twisting
procedure. As we will see in §5, this is exactly the $L_{\infty}$-algebra
deduced from the minimal model of the operad governing differential algebras
with weight.

###  2.1. Differential algebras and their cohomology

Let $A=(A,\mu_{A},d_{A})$ be a differential algebra of weight $\lambda$, with
a multiplication $\mu_{A}$ and a differential operator $d_{A}$ of weight
$\lambda$, as defined in Eq. (1). A bimodule $M$ over the associative algebra
$(A,\mu_{A})$ is called a differential bimodule over the differential algebra
$(A,\mu_{A},d_{A})$ if $M$ is further endowed with a linear transformation
$d_{M}:M\to M$ such that for all $a,b\in A$ and $x\in M,$

$\displaystyle d_{M}(ax)$ $\displaystyle=$ $\displaystyle
d_{A}(a)x+ad_{M}(x)+\lambda d_{A}(a)d_{M}(x),$ $\displaystyle d_{M}(xa)$
$\displaystyle=$ $\displaystyle xd_{A}(a)+d_{M}(x)a+\lambda d_{M}(x)d_{A}(a).$

The regular bimodule $A$ is obviously a differential bimodule over itself,
called the regular differential bimodule of the differential algebra $A$.

Given a differential bimodule $M=(M,d_{M})$ over the differential algebra
$A=(A,\mu_{A},d_{A})$, a new differential bimodule structure on $M$, with the
same derivation $d_{M}$, is given by

(2) $a\vdash v=(a+\lambda d_{A}(a))x,\quad x\dashv a=x(a+\lambda
d_{A}(a)),\quad a\in A,x\in M.$

For distinction, we let ${}_{\vdash}M_{\dashv}$ denote this new differential
bimodule structure over $A=(A,\mu_{A},d_{A})$; for details, see [22, Lemma
3.1].

Recall that the Hochschild cochain complex
${\mathrm{C}}^{\bullet}_{\mathrm{Alg}}(A,M)$ of an associative algebra $A$
with coefficients in a bimodule $M$ is the cochain complex

$({\mathrm{C}}^{\bullet}_{\mathrm{Alg}}(A,M):=\bigoplus_{n=0}^{\infty}{\mathrm{C}}^{n}_{\mathrm{Alg}}(A,M),\partial_{{\mathrm{Alg}}}^{\bullet}),$

where for $n\geqslant 0$, ${\mathrm{C}}^{n}_{\mathrm{Alg}}(A,M)={\rm
Hom}(A^{\otimes n},M)$ (in particular,
${\mathrm{C}}^{0}_{\mathrm{Alg}}(A,M)=M$) and the coboundary operator

$\partial_{{\mathrm{Alg}}}^{n}:{\mathrm{C}}^{n}_{\mathrm{Alg}}(A,M)\longrightarrow{\mathrm{C}}^{n+1}_{\mathrm{Alg}}(A,M),n\geqslant
0,$
$\partial_{{\mathrm{Alg}}}^{n}(f)(a_{1,n+1}):=(-1)^{n+1}a_{1}f(a_{2,n+1})+\sum_{i=1}^{n}(-1)^{n+1-i}f(a_{1,i-1}\otimes
a_{i}a_{i+1}\otimes a_{i+2,n+1})+f(a_{1,n})a_{n+1}$

for $f\in{\mathrm{C}}^{n}_{\mathrm{Alg}}(A,M),~{}a_{1},\dots,a_{n+1}\in A$,
where for $1\leqslant i\leqslant j\leqslant n+1$, we write
$a_{i,j}:=a_{i}\otimes\cdots\otimes a_{j}$ (by convention, $a_{i,j}=\emptyset$
for $i>j$). The corresponding Hochschild cohomology is denoted by
${\mathrm{H}}^{\bullet}_{\mathrm{Alg}}(A,M)$. We write
${\mathrm{H}}^{\bullet}_{\mathrm{Alg}}(A):={\mathrm{H}}^{\bullet}_{\mathrm{Alg}}(A,M)$
when $M$ is taken to be the regular bimodule $A$,

Let $M=(M,d_{M})$ be a differential bimodule over a differential algebra
$A=(A,\mu_{A},d_{A})$. The Hochschild cochain complex
$C^{\bullet}_{\mathrm{Alg}}(A,{}_{\vdash}M_{\dashv})$ of the associative
algebra $A=(A,\mu_{A})$ with coefficients in the new bimodule
${}_{\vdash}M_{\dashv}$ is called the cochain complex of the differential
operator $d_{A}$ with coefficients in the differential bimodule $M$, denoted
by
$(C^{\bullet}_{\mathsf{DO}_{\lambda}}(A,M),\partial_{{\mathsf{DO}_{\lambda}}}^{\bullet})$.
More precisely, for $g\in C^{n}_{\mathsf{DO}_{\lambda}}(A,M)\coloneqq{\rm
Hom}_{\bf k}(A^{\otimes n},M)$ and $a_{1},\dots,a_{n+1}\in A$, we have

$\begin{split}\partial_{{\mathsf{DO}_{\lambda}}}^{n}g(a_{1,n+1})\coloneqq&(-1)^{n+1}a_{1}\vdash
g(a_{2,n+1})+\sum_{i=1}^{n}(-1)^{n+1-i}g(a_{1,i-1}\otimes
a_{i}a_{i+1}a_{i+2,n+1})+g(a_{1,n})\dashv a_{n+1}\\\
=&(-1)^{n+1}(a_{1}+\lambda
d_{A}(a_{1}))g(a_{2,n+1})+\sum_{i=1}^{n}(-1)^{n+1-i}g(a_{1,i-1}\otimes
a_{i}a_{i+1}a_{i+2,n+1})\\\ &+g(a_{1,n})(a_{n+1}+\lambda
d_{A}(a_{n+1})).\end{split}$

We write
$(C^{\bullet}_{\mathsf{DO}_{\lambda}}(A),\partial_{{\mathsf{DO}_{\lambda}}}^{\bullet})\coloneqq(C^{\bullet}_{\mathsf{DO}_{\lambda}}(A,M),\partial_{{\mathsf{DO}_{\lambda}}}^{\bullet})$
by taking the differential bimodule $M$ to be the regular differential
bimodule $A$. The cohomology of the cochain complex
$(C^{\bullet}_{{\mathsf{DO}_{\lambda}}}(A,M),\partial_{{\mathsf{DO}_{\lambda}}}^{\bullet})$,
denoted by ${\mathrm{H}}^{\bullet}_{\mathsf{DO}_{\lambda}}(A,M)$, is called
the cohomology of the differential operator $d_{A}$ with coefficients in the
differential bimodule $M$. When the differential bimodule $M$ is the regular
differential bimodule $A$, write
${\mathrm{H}}^{\bullet}_{\mathsf{DO}_{\lambda}}(A):={\mathrm{H}}^{\bullet}_{\mathsf{DO}_{\lambda}}(A,M)$.

As in [22, Proposition 3.3], define a chain map
$\Phi^{\bullet}:C^{\bullet}_{\mathrm{Alg}}(A,M)\to
C^{\bullet}_{\mathsf{DO}_{\lambda}}(A,M)$ as follows: for $f\in
C^{n}_{\mathrm{Alg}}(A,M)$ with $n\geq 1$, define

$\begin{split}\Phi^{n}(f)(a_{1,n})\coloneqq&\sum_{k=1}^{n}\lambda^{k-1}\sum_{1\leq
i_{1}<\cdots<i_{k}\leq n}f(a_{1,i_{1}-1}\otimes d_{A}(a_{i_{1}})\otimes
a_{i_{1}+1,i_{2}-1}\otimes d_{A}(a_{i_{2}})\otimes\cdots\otimes
d_{A}(a_{i_{k}})\otimes a_{i_{k}+1,n})\\\ &-d_{M}(f(a_{1,n})),\end{split}$

and

$\Phi^{0}(x)\coloneqq-d_{M}(x),\quad\ x\in C^{0}_{\mathrm{Alg}}(A,M)=M.$

Let
$(C_{{\mathsf{DA}_{\lambda}}}^{\bullet}(A,M),\partial_{{\mathsf{DA}_{\lambda}}}^{\bullet})$
be the negative shift of the mapping cone of $\Phi^{\bullet}$. More precisely,

$C_{{\mathsf{DA}_{\lambda}}}^{n}(A,M)\coloneqq\begin{cases}C^{n}_{\mathrm{Alg}}(A,M)\oplus
C^{n-1}_{\mathsf{DO}_{\lambda}}(A,M),&n\geq 1,\\\
C^{0}_{\mathrm{Alg}}(A,M)=M,&n=0,\end{cases}$

and the differentials
$\partial_{{\mathsf{DA}_{\lambda}}}^{n}:C_{{\mathsf{DA}_{\lambda}}}^{n}(A,M)\rightarrow
C_{{\mathsf{DA}_{\lambda}}}^{n+1}(A,M)$ are defined by

$\partial_{{\mathsf{DA}_{\lambda}}}^{n}(f,g):=(\partial^{n}_{{\mathrm{Alg}}}(f),-\partial_{\mathsf{DO}_{\lambda}}^{n-1}(g)-\Phi^{n}(f)),\quad
f\in C^{n}_{\mathrm{Alg}}(A,M),\,g\in
C^{n-1}_{\mathsf{DO}_{\lambda}}(A,M),n\geq 1,$
$\partial_{{\mathsf{DA}_{\lambda}}}^{0}(x):=(\partial^{0}_{{\mathrm{Alg}}}(x),-\Phi^{0}(x)),\quad
x\in C^{0}_{\mathrm{Alg}}(A,M)=M.$

The complex
$(C_{{\mathsf{DA}_{\lambda}}}^{\bullet}(A,M),\partial_{{\mathsf{DA}_{\lambda}}}^{\bullet})$
is called the cochain complex of the differential algebra $(A,d_{A})$ with
coefficients in the differential bimodule $M$. When the differential bimodule
$M$ is the regular differential bimodule $A$, we write
$(C^{\bullet}_{\mathsf{DA}_{\lambda}}(A),\partial_{{\mathsf{DA}_{\lambda}}}^{\bullet}):=(C^{\bullet}_{\mathsf{DA}_{\lambda}}(A,M),\partial_{{\mathsf{DA}_{\lambda}}}^{\bullet})$.

The cohomology of the cochain complex
$(C_{{\mathsf{DA}_{\lambda}}}^{\bullet}(A,M),\partial_{{\mathsf{DA}_{\lambda}}}^{\bullet})$,
denoted by ${\mathrm{H}}_{{\mathsf{DA}_{\lambda}}}^{\bullet}(A,M)$, is called
the cohomology of the differential algebra $A=(A,\mu_{A},d_{A})$ with
coefficients in the differential bimodule $M=(M,d_{M})$.

It is shown in [22] that this cohomology theory controls formal deformations
as well as abelian extensions of differential algebras. We will further
clarify this connection theoretically by using the $L_{\infty}$-algebra
structure on the deformation complex in §2.2; see Propositions 2.6 and 2.7.

###  2.2.  $L_{\infty}$-algebra on the deformation complex from differential
algebras

We first recall some basics on $L_{\infty}$-algebras; for more details, see
[17, 28, 29, 51].

Let $V=\oplus_{n\in\mathbb{Z}}V_{n}$ be a graded vector space. The graded
symmetric algebra $S(V)$ of $V$ is defined to be the quotient of the tensor
algebra $T(V)$ by the two-sided ideal $I$ generated by $x\otimes
y-(-1)^{|x||y|}y\otimes x$ for all homogeneous elements $x,y\in V$. For
$x_{1}\otimes\cdots\otimes x_{n}\in V^{\otimes n}\subseteq T(V)$, write
$x_{1}\odot x_{2}\odot\dots\odot x_{n}$ for its image in $S(V)$. For
homogeneous elements $x_{1},\dots,x_{n}\in V$ and $\sigma\in S_{n}$, the
Koszul sign $\epsilon(\sigma;x_{1},\dots,x_{n})$ is defined by the equality

$x_{1}\odot x_{2}\odot\dots\odot
x_{n}=\epsilon(\sigma;x_{1},\dots,x_{n})x_{\sigma(1)}\odot
x_{\sigma(2)}\odot\dots\odot x_{\sigma(n)}\in S(V).$

We also define

$\chi(\sigma;x_{1},\dots,x_{n})={\mathrm{sgn}}(\sigma)\
\epsilon(\sigma;x_{1},\dots,x_{n}),$

where ${\mathrm{sgn}}(\sigma)$ is the sign of the permutation $\sigma$.

######  Definition 2.1.

An $L_{\infty}$-algebra is a graded space
$L=\bigoplus\limits_{i\in\mathbb{Z}}L_{i}$ endowed with a family of linear
operators $l_{n}:L^{\otimes n}\rightarrow L,n\geqslant 1$ with degree
$|l_{n}|=n-2$ subject to the following conditions. For $n\geq 1$ and
homogeneous elements $x_{1},\dots,x_{n}\in L$,

  1. (i)

(generalized anti-symmetry)

$l_{n}(x_{\sigma(1)}\otimes\cdots\otimes
x_{\sigma(n)})=\chi(\sigma;x_{1},\dots,x_{n})\ l_{n}(x_{1}\otimes\cdots\otimes
x_{n}),\ \sigma\in S_{n};$

  2. (ii)

(generalized Jacobi identity)

$\sum\limits_{i=1}^{n}\sum\limits_{\sigma\in{\mathrm{Sh}}(i,n-i)}\chi(\sigma;x_{1},\dots,x_{n})(-1)^{i(n-i)}l_{n-i+1}(l_{i}(x_{\sigma(1)}\otimes\cdots\otimes
x_{\sigma(i)})\otimes x_{\sigma(i+1)}\otimes\cdots\otimes x_{\sigma(n)})=0,$

where ${\mathrm{Sh}}(i,n-i)$ is the set of $(i,n-i)$ shuffles, that is, those
$\sigma\in S_{n}$ such that

$\sigma(1)<\cdots<\sigma(i)\ \ \mathrm{and}\ \ \sigma(i+1)<\cdots<\sigma(n).$

In particular, if $l_{n}=0$ for all $n\geqslant 3$, then $(L,l_{1},l_{2})$ is
called a differential graded (=dg) Lie algebra; if $l_{n}=0$ except $n=2$,
then $(L,l_{2})$ is called a graded Lie algebra.

Let $V=\sum\limits_{i\in\mathbb{Z}}V_{i}$ be a graded space. Let $sV$ denote
the suspension of $V$ and $s^{-1}V$ the desuspension. Denote

$\mathfrak{C}_{{\mathrm{Alg}}}(V):={\rm Hom}(T(sV),sV).$

Given homogeneous elements $f\in{\rm Hom}((sV)^{\otimes m},V)$ with
$m\geqslant 1$ and $g\in{\rm Hom}((sV)^{\otimes n},V)$ with $n\geqslant 0$,
for each $1\leqslant i\leqslant m$, write

$sf\circ_{i}sg:=sf\circ({\rm Id}^{\otimes(i-1)}\otimes sg\otimes{\rm
Id}^{\otimes(m-i)}),$

and $sf\\{sg\\}:=\sum_{i=1}^{m}sf\circ_{i}sg$; when $m=0$, $sf\\{sg\\}$ is
defined to be $0$. The Gerstenhaber bracket [13] of $sf$ and $sg$ is defined
by

(3) $\displaystyle[sf,sg]_{G}:=sf\\{sg\\}-(-1)^{(|f|+1)(|g|+1)}sg\\{sf\\}.$

A celebrated result of Gerstenhaber [13] states that the graded space
$\mathfrak{C}_{{\mathrm{Alg}}}(V)$ endowed with the Gerstenhaber bracket is a
graded Lie algebra. The following facts are quoted for later applications.

  1. (i)

Let $V$ be an ungraded space considered as a graded space concentrated in
degree 0. Then there is a bijection between the set of Maurer-Cartan elements
in the graded Lie algebra $\mathfrak{C}_{{\mathrm{Alg}}}(V)$ and the set of
associative algebra structures on the space $V$, where the correspondence is
induced by

(4) $\displaystyle{\rm Hom}((sV)^{\otimes n},sV)\simeq{\rm Hom}(V^{\otimes
n},V),f\mapsto\tilde{f}:=s^{-1}\circ f\circ s^{\otimes n},\quad f\in{\rm
Hom}((sV)^{\otimes n},sV).$

  2. (ii)

Let $(A,\mu)$ be an associative algebra and $\alpha$ be the corresponding
Maurer-Cartan element in the graded Lie algebra
$\mathfrak{C}_{{\mathrm{Alg}}}(A)$. Then the underlying complex of the twisted
dg Lie algebra
$(\mathfrak{C}_{{\mathrm{Alg}}}(A),l_{1}^{\alpha},l_{2}^{\alpha})$ is exactly
$s{\mathrm{C}}^{\bullet}_{\mathrm{Alg}}(A)$, the shift of the Hochschild
cochain complex of the associative algebra $A$.

Let $V$ be a graded space. Write

$\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(V):={\rm Hom}(T(sV),V)$

and define the graded space

(5)
$\mathfrak{C}_{{\mathsf{DA}_{\lambda}}}(V):=\mathfrak{C}_{{\mathrm{Alg}}}(V)\oplus\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(V)={\rm
Hom}(T(sV),sV)\oplus{\rm Hom}(T(sV),V).$

Now we introduce an $L_{\infty}$-algebra structure on the graded space
$\mathfrak{C}_{{\mathsf{DA}_{\lambda}}}(V)$ by the following process:

  1. (i)

For $sf,sg\in\mathfrak{C}_{{\mathrm{Alg}}}(V)$, define

$l_{2}(sf\otimes sg)=[sf,sg]_{G}\in\mathfrak{C}_{{\mathrm{Alg}}}(V),$

where $[-,-]_{G}$ is the Gerstenhaber Lie bracket in Eq. (3).

  2. (ii)

Let $sf\in\mathfrak{C}_{{\mathrm{Alg}}}(V)$ and
$g\in\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(V)$. Define

$l_{2}(sf\otimes
g)=(-1)^{|f|+1}s^{-1}[sf,sg]_{G}\in\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(V).$

  3. (iii)

Let $sf\in{\rm Hom}((sV)^{\otimes
n},sV)\subseteq\mathfrak{C}_{{\mathrm{Alg}}}(V)$ and
$g_{1},\dots,g_{m}\in\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(V)$ with
$2\leqslant m\leqslant n$. Define

$l_{m+1}(sf\otimes g_{1}\otimes\cdots\otimes
g_{m})=\lambda^{m-1}\sum_{\sigma\in
S_{m}}(-1)^{\xi}s^{-1}\big{(}sf\\{sg_{\sigma(1)},\dots,sg_{\sigma(m)}\\}\big{)}\in\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(V),$

where
$(-1)^{\xi}=\chi(\sigma;g_{1},\dots,g_{m})(-1)^{m(|f|+1)+\sum\limits_{k=1}^{m-1}\sum\limits_{j=1}^{k}|g_{\sigma(j)}|}$.

  4. (iv)

Let $sf\in\mathfrak{C}_{{\mathrm{Alg}}}(V)$ and
$g_{1},\dots,g_{m}\in\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(V)$ with
$m\geqslant 1$. For $1\leqslant k\leqslant m$, define

$l_{m+1}(g_{1}\otimes\cdots\otimes g_{k}\otimes sf\otimes
g_{k+1}\otimes\cdots\otimes
g_{m})\in\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(V)$

to be

$l_{m+1}(g_{1}\otimes\cdots\otimes g_{k}\otimes sf\otimes
g_{k+1}\otimes\cdots\otimes
g_{m}):=(-1)^{(|f|+1)(\sum\limits_{j=1}^{k}|g_{j}|)+k}l_{n+1}(sf\otimes
g_{1}\otimes\cdots\otimes g_{m}),$

where the right hand side has been defined in (ii) and (iii).

  5. (v)

All other components of the operators $\\{l_{n}\\}_{n\geqslant 1}$ vanish.

Then we can state one of our main results in this paper.

######  Theorem 2.2.

Given a graded space $V$ and $\lambda\in{\bf k}$, the graded space
$\mathfrak{C}_{{\mathsf{DA}_{\lambda}}}(V)$ endowed with the operations
$\\{l_{n}\\}_{n\geqslant 1}$ defined above is an $L_{\infty}$-algebra.

The theorem can be proved by a direct but lengthy verification. Instead, we
will present an operadic proof in the spirit of [27, 54, 55] once we obtain
the minimal model. We first sketch the proof below, with details supplied
later. We then give an application in the next subsection, realizing
differential algebras with weight as the Maurer-Cartan elements in this
$L_{\infty}$-algebra.

###### Proof.

(A sketch) The minimal model of the differential algebra operad
${{}_{\lambda}\\!\mathfrak{Dif}}$ is obtained in Theorem 4.4. It is shown in
§5 that $\mathfrak{C}_{{\mathsf{DA}_{\lambda}}}(V)$ is exactly the
$L_{\infty}$-structure resulting from this minimal model. ∎

###  2.3. Realising differential algebra structures as Maurer-Cartan elements

We give an application of Theorem 2.2.

######  Definition 2.3.

An element $\alpha$ of degree $-1$ in an $L_{\infty}$-algebra
$(L,\\{l_{n}\\}_{n\geqslant 1})$ is called a Maurer-Cartan element if it
satisfies the Maurer-Cartan equation:

(6)
$\displaystyle\sum_{n=1}^{\infty}\frac{1}{n!}(-1)^{\frac{n(n-1)}{2}}l_{n}(\alpha^{\otimes
n})=0$

(in particular, this infinite sum needs to be well defined).

For a dg Lie algebra $(L,l_{1},l_{2})$, Eq. (6) reduces to

(7) $\displaystyle l_{1}(\alpha)-\frac{1}{2}l_{2}(\alpha\otimes\alpha)=0.$

######  Proposition 2.4 (twisting procedure[17]).

Let $\alpha$ be a Maurer-Cartan element in an $L_{\infty}$-algebra
$(L,\\{l_{n}\\}_{n\geqslant 1})$. A new $L_{\infty}$-structure
$\\{l_{n}^{\alpha}\\}_{n\geqslant 1}$ on the graded space $L$ is defined by

(8) $\begin{split}l_{n}^{\alpha}:\ &L^{\otimes n}\rightarrow L,\\\
l^{\alpha}_{n}(x_{1}\otimes\cdots\otimes
x_{n}):=&\sum_{i=0}^{\infty}\frac{1}{i!}(-1)^{in+\frac{i(i-1)}{2}}l_{n+i}(\alpha^{\otimes
i}\otimes x_{1}\otimes\cdots\otimes x_{n}),\ x_{1},\dots,x_{n}\in
L,\end{split}$

whenever these infinite sums exist. The new $L_{\infty}$-algebra
$(L,\\{l_{n}^{\alpha}\\}_{n\geqslant 1})$ is called the twisted
$L_{\infty}$-algebra $($by the Maurer-Cartan element $\alpha$$)$.

For a dg Lie algebra $(L,l_{1},l_{2})$ and a Maurer-Cartan element $\alpha\in
L_{-1}$, Eq. (8) reduces to

(9) $\displaystyle l_{1}^{\alpha}(x)=l_{1}(x)-l_{2}(\alpha\otimes x),\ x\in
L,\ \mathrm{and}\ l_{2}^{\alpha}=l_{2}.$

We fix the isomorphism

(10) $\displaystyle{\rm Hom}((sV)^{\otimes n},V)\simeq{\rm Hom}(V^{\otimes
n},V),g\mapsto\check{g}:=g\circ s^{\otimes n},\,g\in{\rm Hom}((sV)^{\otimes
n},V).$

The following proposition follows from general facts about
$L_{\infty}$-structures obtained from minimal models; see the discussion at
the end of §3.1.

######  Proposition 2.5.

Let $V$ be an ungraded space considered as a graded space concentrated in
degree $0$. Then a differential algebra structure of weight $\lambda$ on $V$
is equivalent to a Maurer-Cartan element in the $L_{\infty}$-algebra
$\mathfrak{C}_{{\mathsf{DA}_{\lambda}}}(V)$ in Theorem 2.2.

The following result justifies the cohomology theory introduced in [22] and
recalled in §2.1.

######  Proposition 2.6.

Let $(A,\mu,d)$ be a differential algebra of weight $\lambda$. Twisting the
$L_{\infty}$-algebra $\mathfrak{C}_{{\mathsf{DA}_{\lambda}}}(A)$ by the
Maurer-Cartan element corresponding to the differential algebra structure
$(A,\mu,d)$, then its underlying complex is exactly
$s{\mathrm{C}}^{\bullet}_{{\mathsf{DA}_{\lambda}}}(A)$, the suspension of the
cochain complex of the differential algebra $(A,\mu,d)$.

###### Proof.

By Proposition 2.5, the differential algebra structure $(A,\mu,d)$ is
equivalent to a Maurer-Cartan element $\alpha=(m,\tau)$ in the
$L_{\infty}$-algebra $\mathfrak{C}_{{\mathsf{DA}_{\lambda}}}(A)$ with

$m=-s\circ\mu\circ(s^{-1})^{\otimes 2}:(sV)^{\otimes 2}\rightarrow sV\
\mathrm{and}\ \tau=d\circ s^{-1}:sV\rightarrow V.$

By Proposition 2.4, the Maurer-Cartan element induces a new
$L_{\infty}$-algebra structure $\\{l_{n}^{\alpha}\\}_{n\geqslant 1}$ on the
graded space $\mathfrak{C}_{{\mathsf{DA}_{\lambda}}}(A)$.

Given $(sf,g)\in{\rm Hom}((sA)^{\otimes n},sA)\oplus{\rm Hom}((sA)^{\otimes
n-1},sA)\subset\mathfrak{C}_{{\mathrm{Alg}}}(A)\oplus\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(A)=\mathfrak{C}_{{\mathsf{DA}_{\lambda}}}(A)$,
note that by the isomorphisms (4) and (10), we get
$(\widetilde{sf},\check{g})\in{\mathrm{C}}_{{\mathsf{DA}_{\lambda}}}^{n}(A)$.
By definition, for $sf\in{\rm Hom}((sA)^{\otimes
n},sA)\subset\mathfrak{C}_{{\mathrm{Alg}}}(A)$, we have

$l_{1}^{\alpha}(sf)=\sum_{i=0}^{\infty}\frac{1}{i!}(-1)^{i+\frac{i(i-1)}{2}}l_{i+1}(\alpha^{\otimes
i}\otimes sf)=\Big{(}-l_{2}(m\otimes sf),\
\sum_{i=1}^{n}\frac{1}{i!}(-1)^{\frac{i(i+1)}{2}}l_{i+1}(\tau^{\otimes
i}\otimes sf)\Big{)}.$

The definition of $\\{l_{n}\\}_{n\geqslant 1}$ on
$\mathfrak{C}_{{\mathsf{DA}_{\lambda}}}(A)$ gives $-l_{2}(m\otimes
sf)=-[m,sf]_{G}$. On the other hand, we have

$\displaystyle\sum_{i=1}^{n}\frac{1}{i!}(-1)^{\frac{i(i+1)}{2}}l_{i+1}(\tau^{\otimes
i}\otimes sf)$ $\displaystyle=$ $\displaystyle-l_{2}(\tau\otimes
sf)+\sum_{i=2}^{n}\frac{1}{i!}(-1)^{\frac{i(i+1)}{2}}l_{i+1}(\tau^{\otimes
i}\otimes sf)$ $\displaystyle\stackrel{{\scriptstyle\rm(iv)}}{{=}}$
$\displaystyle-(-1)^{|f|}l_{2}(sf\otimes\tau)+\sum_{i=2}^{n}\frac{1}{i!}(-1)^{\frac{i(i+1)}{2}}(-1)^{i|f|}l_{i+1}(sf\otimes\tau^{\otimes
i})$ $\displaystyle\stackrel{{\scriptstyle\rm(ii)(iii)}}{{=}}$ $\displaystyle
s^{-1}[sf,s\tau]_{G}+\sum_{i=2}^{n}\frac{1}{i!}(-1)^{\frac{i(i+1)}{2}}(-1)^{i|f|}i!\lambda^{i-1}(-1)^{i(|f|+1)+\frac{i(i-1)}{2}}s^{-1}(sf\\{\underbrace{s\tau,\dots,s\tau}_{i}\\})$
$\displaystyle=$ $\displaystyle
s^{-1}[sf,s\tau]_{G}+\sum_{i=2}^{n}\lambda^{i-1}s^{-1}(sf\\{\underbrace{s\tau,\dots,s\tau}_{i}\\}).$

For $g\in{\rm
Hom}((sA)^{\otimes(n-1)},A)\subset\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(A)$,
we have

$\displaystyle l_{1}^{\alpha}(g)=$
$\displaystyle\sum_{i=0}^{\infty}\frac{1}{i!}(-1)^{\frac{i(i+1)}{2}}l_{i+1}(\alpha^{\otimes
i}\otimes g)$ $\displaystyle=$ $\displaystyle-l_{2}(m\otimes
g)-\frac{1}{2!}\Big{(}l_{3}(m\otimes\tau\otimes g)+l_{3}(\tau\otimes m\otimes
g)\Big{)}$ $\displaystyle=$ $\displaystyle s^{-1}[m,sg]+\lambda
s^{-1}m\circ(s\tau\otimes sg)+\lambda s^{-1}m\circ(sg\otimes s\tau).$

We obtain

$\begin{array}[]{rcl}l_{1}^{\alpha}(sf,g)&=&\Big{(}-[m,sf]_{G},s^{-1}[sf,s\tau]_{G}+\sum_{i=2}^{n}\lambda^{i-1}s^{-1}(sf\\{\underbrace{s\tau,\dots,s\tau}_{i}\\})\\\
&&\quad+s^{-1}[m,sg]+\lambda s^{-1}m\circ(s\tau\otimes sg)+\lambda
s^{-1}m\circ(sg\otimes s\tau)\Big{)},\end{array}$

which is easily seen to be in correspondence with
$-\partial_{{\mathsf{DA}_{\lambda}}}^{n}(\widetilde{sf},\check{g})$ via the
fixed isomorphisms (4) and (10). This completes the proof. ∎

Although $\mathfrak{C}_{{\mathsf{DA}_{\lambda}}}(A)$ is an
$L_{\infty}$-algebra, once the associative algebra structure $\mu$ over $A$ is
fixed, the graded space $\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(A)$ is indeed
a differential graded Lie algebra, as shown by the following result. Moreover,
this result justifies the cohomology theory of differential operators.

######  Proposition 2.7.

Let $(A,\mu)$ be an associative algebra. Then the graded space
$\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(A)$ can be endowed with a dg Lie
algebra structure, of which the Maurer-Cartan elements are in bijection with
the differential operators of weight $\lambda$ on $(A,\mu)$.

Given a differential operator $d$ of weight $\lambda$ on the associative
algebra $(A,\mu)$, the dg Lie algebra
$\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(A)$ twisted by the Maurer-Cartan
element corresponding to $d$ has its underlying complex being the cochain
complex ${\mathrm{C}}^{\bullet}_{\mathsf{DO}_{\lambda}}(A)$ of the
differential operator $d$.

###### Proof.

Regard $A$ as a graded space concentrated in degree 0. Define
$m=-s\circ\mu\circ(s^{-1}\otimes s^{-1}):(sA)^{\otimes 2}\rightarrow sA$. Then
it is easy to see that $\beta=(m,0)$ is naturally a Maurer-Cartan element in
the $L_{\infty}$-algebra $\mathfrak{C}_{{\mathsf{DA}_{\lambda}}}(A)$. We apply
the twisting procedure by $\beta$ on
$\mathfrak{C}_{{\mathsf{DA}_{\lambda}}}(A)$. By the construction of
$\\{l_{n}\\}_{\geqslant 1}$ on $\mathfrak{C}_{{\mathsf{DA}_{\lambda}}}(A)$,
the graded subspace $\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(A)$ is closed
under the action of the operators $\\{l_{n}^{\beta}\\}_{n\geqslant 1}$.

Since the arity of $m$ is 2, the restriction of $l_{n}^{\beta}$ on
$\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(A)$ is $0$ for $n\geqslant 3$. Thus
$(\mathfrak{C}_{{\mathsf{DO}_{\lambda}}},\\{l_{n}^{\beta}\\}_{n=1,2})$ forms a
dg Lie algebra which can be made explicit as follows. For $g\in{\rm
Hom}((sA)^{\otimes n},A)$ and $h\in{\rm Hom}((sA)^{\otimes k},A)$, we have

$\displaystyle l_{1}^{\beta}(g)=$ $\displaystyle-l_{2}(m\otimes
g)=s^{-1}[m,sg]_{G},$ $\displaystyle l_{2}^{\beta}(g\otimes h)=$
$\displaystyle l_{3}(m\otimes g\otimes h)$ $\displaystyle=$
$\displaystyle\lambda(-1)^{|g|}s^{-1}m\circ(sg\otimes
sh)+\lambda(-1)^{|g||h|+1+|h|}s^{-1}m\circ(sh\otimes sg)$ $\displaystyle=$
$\displaystyle\lambda(-1)^{n}s^{-1}m\circ(sg\otimes
sh)+\lambda(-1)^{nk+k+1}s^{-1}m\circ(sh\otimes sg).$

Since $A$ is concentrated in degree 0, we have
$\mathfrak{C}_{{\mathsf{DO}_{\lambda}}}(A)_{-1}={\rm Hom}(sA,A)$. Take an
element $\tau\in{\rm Hom}(sA,A)$. Then $\tau$ satisfies the Maurer-Cartan
equation:

$l_{1}^{\beta}(\tau)-\frac{1}{2}l_{2}^{\beta}(\tau\otimes\tau)=0$

if and only if

$s^{-1}[m,s\tau]_{G}+\lambda\ s^{-1}m\circ(s\tau\otimes s\tau)=0.$

Define $d=\tau\circ s:A\rightarrow A$. Under the isomorphisms (4) and (10),
the above equation is exactly the statement that $d$ is a differential
operator of weight $\lambda$ on the associative algebra $(A,\mu)$.

Now let $d$ be a differential operator of weight $\lambda$ on the associative
algebra $(A,\mu)$ and $\tau=d\circ s^{-1}$ be the corresponding Maurer-Cartan
element in the dg Lie algebra
$(\mathfrak{C}_{{\mathsf{DO}_{\lambda}}},\\{l_{n}^{\beta}\\}_{n=1,2})$. For
$g\in{\rm Hom}((sA)^{\otimes n},A)$, a direct computation shows that
$\partial_{\mathsf{DO}_{\lambda}}(\check{g})$ corresponds to
$(l_{1}^{\beta})^{\tau}(g)$ via the isomorphism (10). This proves the last
statement. ∎

######  Remark 2.8.

In the course of proving the above proposition, the Lie bracket
$(l_{2}^{\beta})^{\tau}$ on
${\mathrm{C}}^{\bullet}_{\mathsf{DO}_{\lambda}}(A)$ can also be given
explicitly: for $f\in{\mathrm{C}}^{n}_{\mathsf{DO}_{\lambda}}(A)$ and
$g\in{\mathrm{C}}^{k}_{\mathsf{DO}_{\lambda}}(A)$,

$[f,g](a_{1,n+k})=(-1)^{n}\lambda
f(a_{1,n})g(a_{n+1,n+k})+(-1)^{nk+k+1}\lambda g(a_{1,k})f(a_{k+1,n+k}),\quad
a_{1},\dots,a_{n+k}\in A.$

Note that when $\lambda=0$, the cochain complex
${\mathrm{C}}^{\bullet}_{\mathsf{DO}_{\lambda}}(A)$ has the trivial Lie
bracket.

##  3\. The Koszul dual homotopy cooperad

In this section, we will construct a homotopy cooperad
${{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}$, which will be called
the Koszul dual homotopy cooperad of the operad
${{}_{\lambda}\\!\mathfrak{Dif}}$ of differential algebras with weight
$\lambda$. In fact, we will show that the cobar construction of
${{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}$ is the minimal model
of ${{}_{\lambda}\\!\mathfrak{Dif}}$ (Theorem 4.4) which justifies the name.

###  3.1. Homotopy (co)operads and $L_{\infty}$-algebras

We first collect needed background on nonsymmetric homotopy (co)operads that
are scattered in several references [8, 38, 42, 43]. We also explain how to
obtain $L_{\infty}$-structures from homotopy operads and, in particular, from
convolution homotopy operads.

Recall that a graded collection
${\mathcal{O}}=\\{{\mathcal{O}}(n)\\}_{n\geqslant 1}$ is a family of graded
space indexed by positive integers, i.e. each ${\mathcal{O}}(n)$ itself being
a graded space. Elements in ${\mathcal{O}}(n)$ are said to have arity $n$. The
suspension of ${\mathcal{O}}$, denoted by $s{\mathcal{O}}$ is defined to be
the graded collection $\\{s{\mathcal{O}}(n)\\}_{n\geqslant 1}$. In the same
way, one have the desuspension $s^{-1}{\mathcal{O}}$ of the graded collection
${\mathcal{O}}$.

We need some preliminaries about trees. As we only consider planar reduced
trees (with roots at the bottom), we will simply call them trees. For a tree
$T$, denote the weight (=number of vertices) and arity (=number of leaves) of
$T$ by $\omega(T)$ and $\alpha(T)$ respectively. Write $\mathfrak{T}$ to be
the set of all trees with weight $\geqslant 1$ and, for $n\geqslant 1$, denote
by $\mathfrak{T}^{(n)}$ the set of trees of weight $n$. Since trees are
planar, each vertex in a tree has a total order on its inputs going clockwise.
By the existence of the root, there is a naturally induced total order on the
set of all vertices of a given tree $T\in\mathfrak{T}$, given by counting the
vertices starting from the root clockwise along the tree. We call this order
the planar order.

Let $T^{\prime}$ be a divisor of $T$, define $T/T^{\prime}$ to be the tree
obtained from $T$ in replacing $T^{\prime}$ by a corolla (a tree with only one
vertex) of arity $\alpha(T^{\prime})$. There is a natural permutation
$\sigma=\sigma(T,T^{\prime})\in S_{\omega(T)}$ associated to the pair
$(T,T^{\prime})$ defined as follows. Let the ordered set
$\\{v_{1}<\dots<v_{n}\\}$ be the sequence of all vertices of $T$ in the planar
order and $\omega(T^{\prime})=j$. Let $v^{\prime}$ be the vertex in
$T/T^{\prime}$ corresponding to the divisor $T^{\prime}$ in $T$ and let $i$ be
the serial number of $v^{\prime}$ in $T/T^{\prime}$ in the planar order, that
is, there are $i-1$ vertices before $T^{\prime}$. Then define
$\sigma=\sigma(T,T^{\prime})\in S_{n}$ to be the unique permutation which does
not permute the vertices $v_{1},\dots,v_{i-1}$, and such that the ordered set
$\\{v_{\sigma(i)}<\dots<v_{\sigma(i+j-1)}\\}$ is exactly the planar ordered
set of all vertices of $T^{\prime}$ and the ordered set
$\\{v_{1}<\dots<v_{i-1}<v^{\prime}<v_{\sigma(i+j)}<\dots<v_{\sigma(n)}\\}$ is
exactly the planar ordered set of all vertices in the tree $T/T^{\prime}$.

Let ${\mathcal{P}}=\\{{\mathcal{P}}(n)\\}_{n\geqslant 1}$ be a graded
collection. Let $T\in\mathfrak{T}$ and $\\{v_{1}<\dots<v_{n}\\}$ be the set of
all vertices of $T$ in the planar order. Define ${\mathcal{P}}^{\otimes T}$ to
be
${\mathcal{P}}(\alpha(v_{1}))\otimes\cdots\otimes{\mathcal{P}}(\alpha(v_{n}))$;
morally an element in ${\mathcal{P}}^{\otimes T}$ is a linear combination of
the tree $T$ with each vertex $v_{i}$ decorated by an element of
${\mathcal{P}}(\alpha(v_{i}))$.

######  Definition 3.1.

A homotopy operad structure on a graded collection
${\mathcal{P}}=\\{{\mathcal{P}}(n)\\}_{n\geqslant 1}$ consists of a family of
operations

$\\{m_{T}:{\mathcal{P}}^{\otimes
T}\rightarrow{\mathcal{P}}(\alpha(T))\\}_{T\in\mathfrak{T}}$

with $|m_{T}|=\omega(T)-2$ such that the equation

$\sum\limits_{T^{\prime}\subset{T}}(-1)^{i-1+jk}{\mathrm{sgn}}(\sigma(T,T^{\prime}))\
m_{T/T^{\prime}}\circ({\rm Id}^{\otimes{i-1}}\otimes m_{T^{\prime}}\otimes{\rm
Id}^{\otimes k})\circ r_{\sigma(T,T^{\prime})}=0$

holds for all $T\in\mathfrak{T}$, where $T^{\prime}$ runs through the set of
all subtrees of $T$, $i$ is the serial number of the vertex $v^{\prime}$ in
$T/T^{\prime}$, $j=\omega(T^{\prime})$, $k=\omega(T)-i-j$, and
$r_{\sigma(T,T^{\prime})}$ denotes the right action by
$\sigma=\sigma(T,T^{\prime})$, that is,

$r_{\sigma}(x_{1}\otimes\cdots\otimes
x_{n})=\varepsilon(\sigma;x_{1},\dots,x_{n})x_{\sigma(1)}\otimes\cdots\otimes
x_{\sigma(n)}.$

Given two homotopy operads, a strict morphism between them is a morphism of
graded collections compatible with all operations $m_{T},T\in\mathfrak{T}$.

Let ${\mathcal{I}}$ be the collection with ${\mathcal{I}}(1)=k$ and
${\mathcal{I}}(n)=0$ for $n\neq 1$. The collection ${\mathcal{I}}$ can be
endowed with a homotopy operad structure in a natural way, that is, when $T$
is the tree with two vertices and one unique leaf, then
$m_{T}:{\mathcal{I}}(1)\otimes{\mathcal{I}}(1)\to{\mathcal{I}}(1)$ is given by
the identity; for other trees $T$, $m_{T}$ vanish.

A homotopy operad ${\mathcal{P}}$ is called strictly unital if there exists a
strict morphism of homotopy operads
$\eta:{\mathcal{I}}\rightarrow{\mathcal{P}}$ such that, for $n\geqslant 1$,
the compositions

${\mathcal{P}}(n)\cong{\mathcal{P}}(n)\otimes{\mathcal{I}}(1)\xrightarrow{{\rm
Id}\otimes\eta}{\mathcal{P}}(n)\otimes{\mathcal{P}}(1)\xrightarrow{m_{T_{1,i}}}{\mathcal{P}}(n)$

and

${\mathcal{P}}(n)\cong{\mathcal{I}}(1)\otimes{\mathcal{P}}(n)\xrightarrow{\eta\otimes{\rm
Id}}{\mathcal{P}}(1)\otimes{\mathcal{P}}(n)\xrightarrow{m_{T_{2}}}{\mathcal{P}}(n)$

are identity maps on ${\mathcal{P}}(n)$, where $T_{1,i}$ with $1\leqslant
i\leqslant n$ is the tree of weight $2$, arity $n$ with its second vertex
having arity $1$ and connecting to the first vertex on its $i$-th leaf, and
$T_{2}$ is the tree of weight $2$, arity $n$ whose first vertex has arity $1$.
A homotopy operad ${\mathcal{P}}$ is called augmented if there exists a strict
morphism of homotopy operads
$\varepsilon:{\mathcal{P}}\rightarrow{\mathcal{I}}$ such that
$\varepsilon\circ\eta={\rm Id}_{{\mathcal{I}}}$.

Given a dg operad ${\mathcal{P}}$, for each tree $T$, one can define the
composition $m_{\mathcal{P}}^{T}:{\mathcal{P}}^{\otimes
T}\to{\mathcal{P}}(\alpha(T))$ in ${\mathcal{P}}$ along $T$ as follows: for
$\omega(T)=1$, set $m_{\mathcal{P}}^{T}:={\rm Id}$; for $\omega(T)=2$, set
$m_{\mathcal{P}}^{T}:=m_{T}$; for $\omega(T)\geqslant 3$, write $T$ as the
grafting of a subtree $T^{\prime}$, whose vertex set is that of $T$ except the
last one, with the corolla whose unique vertex is just the last vertex of $T$
in the planar order, then set
$m_{\mathcal{P}}^{T}:=m_{T/T^{\prime}}\circ(m_{\mathcal{P}}^{T^{\prime}}\otimes{\rm
Id})$, where $m_{\mathcal{P}}^{T^{\prime}}$ is obtained by induction.

Dualizing the definition of homotopy operads, one has the notion of homotopy
cooperads.

######  Definition 3.2.

Let ${\mathcal{C}}=\\{{\mathcal{C}}(n)\\}_{n\geqslant 1}$ be a graded
collection. A homotopy cooperad structure on ${\mathcal{C}}$ consists of a
family of operations
$\\{\Delta_{T}:{\mathcal{C}}(n)\rightarrow{\mathcal{C}}^{\otimes
T}\\}_{T\in\mathfrak{T}}$ with $|\Delta_{T}|=\omega(T)-2$ such that for
$c\in{\mathcal{C}}$, $\Delta_{T}(c)=0$ for all but finitely many
$T\in{\mathfrak{T}}$, and the family of operations
$\\{\Delta_{T}\\}_{T\in{\mathfrak{T}}}$ satisfies the identity

$\sum_{T^{\prime}\subset
T}{\mathrm{sgn}}(\sigma(T,T^{\prime})^{-1})(-1)^{i-1+jk}r_{\sigma(T,T^{\prime})^{-1}}\circ({\rm
Id}^{\otimes i-1}\otimes\Delta_{T^{\prime}}\otimes{\rm Id}^{\otimes
k})\circ\Delta_{T/T^{\prime}}=0$

for all $T\in{\mathfrak{T}}$, where $T^{\prime},i,j,k$ have the same meaning
as for homotopy operads.

The graded collection ${\mathcal{I}}$ has a natural homotopy cooperad
structure. It is obtained by taking
$\Delta_{T}:{\mathcal{I}}(1)\to{\mathcal{I}}(1)\otimes{\mathcal{I}}(1)$ to be
the identity when $T$ is the tree with two vertices and one unique leaf, and
$\Delta_{T}=0$ for other $T$. Dualizing the notions of being strictly unital
and being augmented gives the nations of being strictly counital and
coaugmented. For a coaugmented homotopy cooperad ${\mathcal{C}}$, the graded
collection $\overline{{\mathcal{C}}}=\ker(\varepsilon)$ endowed with
operations $\\{\overline{\Delta}_{T}\\}_{T\in{\mathfrak{T}}}$ is naturally a
homotopy cooperad, where $\overline{\Delta}_{T}$ is the the restriction of
operation $\Delta_{T}$ on $\overline{{\mathcal{C}}}$.

A homotopy cooperad ${\mathcal{E}}=\\{{\mathcal{E}}(n)\\}_{n\geqslant 1}$ such
that $\\{\Delta_{T}\\}$ vanish for all $\omega(T)\geqslant 3$ is exactly a
noncounital dg cooperad in the sense of Markl [39].

For a (noncounital) dg cooperad ${\mathcal{E}}$, one can define the
cocomposition
$\Delta^{T}_{\mathcal{E}}:{\mathcal{E}}(\alpha(T))\to{\mathcal{E}}^{\otimes
T}$ along a tree $T$ in duality of the composition $m^{T}_{\mathcal{P}}$ along
$T$ for a dg operad ${\mathcal{P}}$.

######  Proposition-Definition 3.3.

Let ${\mathcal{C}}$ be a homotopy cooperad and ${\mathcal{E}}$ be a dg
cooperad. Then the graded collection ${\mathcal{C}}\otimes{\mathcal{E}}$ with
$({\mathcal{C}}\otimes{\mathcal{E}})(n):={\mathcal{C}}(n)\otimes{\mathcal{E}}(n),n\geqslant
1$, has a natural structure of homotopy cooperad as follows:

  1. (i)

For a tree $T\in{\mathfrak{T}}$ of weight $1$ and arity $n$, define

$\Delta_{T}^{H}(c\otimes e):=\Delta_{T}^{\mathcal{C}}(c)\otimes
e+(-1)^{|c|}c\otimes d_{{\mathcal{E}}}e$

for homogeneous elements $c\in{\mathcal{C}}(n),e\in{\mathcal{E}}(n)$;

  2. (ii)

For a tree $T$ of weight $n\geqslant 2$, define

$\Delta_{T}^{H}(c\otimes
e):=(-1)^{\sum\limits_{k=1}^{n-1}\sum_{j=k+1}^{n}|e_{k}||c_{j}|}(c_{1}\otimes
e_{1})\otimes\cdots\otimes(c_{n}\otimes
e_{n})\in({\mathcal{C}}\otimes{\mathcal{E}})^{\otimes T},$

with $c_{1}\otimes\cdots\otimes
c_{n}=\Delta_{T}^{\mathcal{C}}(c)\in{\mathcal{C}}^{\otimes T}$ and
$e_{1}\otimes\cdots\otimes
e_{n}=\Delta^{T}_{\mathcal{E}}(e)\in{\mathcal{E}}^{\otimes T}$, where
$\Delta^{T}_{\mathcal{E}}$ is the cocomposition in ${\mathcal{E}}$ along $T$.

The new homotopy cooperad is called the Hadamard product of ${\mathcal{C}}$
and ${\mathcal{E}}$, denoted by
${\mathcal{C}}\otimes_{{\mathrm{H}}}{\mathcal{E}}$.

Define ${\mathcal{S}}:=\mathrm{End}_{{\bf k}s}^{c}$ to be the graded cooperad
whose underlying graded collection is given by ${\mathcal{S}}(n)={\rm
Hom}(({\bf k}s)^{\otimes n},{\bf k}s),n\geqslant 1$. Denote $\delta_{n}$ to be
the map in ${\mathcal{S}}(n)$ sending $s^{\otimes n}$ to $s$. The cooperad
structure is given by

$\Delta_{T}(\delta_{n}):=(-1)^{(i-1)(j-1)}\delta_{n-j+1}\otimes\delta_{j}\in{\mathcal{S}}^{\otimes
T}$

for a tree $T$ of weight $2$ whose second vertex is connected with the $i$-th
leaf of its first vertex. We also define ${\mathcal{S}}^{-1}$ to be the graded
cooperad whose underlying graded collection is ${\mathcal{S}}^{-1}(n)={\rm
Hom}(({\bf k}s^{-1})^{\otimes n},s^{-1})$ for all $n\geqslant 1$ and the
cooperad structure is given by

$\Delta_{T}(\varepsilon_{n}):=(-1)^{(j-1)(n-i+1-j)}\varepsilon_{n-j+1}\otimes\varepsilon_{j}\in({\mathcal{S}}^{-1})^{\otimes
T},$

where $\varepsilon_{n}\in{\mathcal{S}}^{-1}(n)$ is the map which takes
$(s^{-1})^{\otimes n}$ to $s^{-1}$ and $T$ is the same as before. It is easy
to see that
${\mathcal{S}}\otimes_{\mathrm{H}}{\mathcal{S}}^{-1}\cong{\mathcal{S}}^{-1}\otimes_{\mathrm{H}}{\mathcal{S}}=:\mathbf{As}^{\vee}$.
Notice that for a homotopy cooperad ${\mathcal{C}}$, we have
${\mathcal{C}}\otimes_{\mathrm{H}}\mathbf{As}^{\vee}\cong{\mathcal{C}}\cong\mathbf{As}^{\vee}\otimes_{\mathrm{H}}{\mathcal{C}}.$

Let ${\mathcal{C}}$ be a homotopy cooperad. Define the operadic suspension
(resp. desuspension) of ${\mathcal{C}}$ to be the homotopy cooperad
${\mathcal{C}}\otimes_{\mathrm{H}}{\mathcal{S}}$ (resp.
${\mathcal{C}}\otimes_{\mathrm{H}}{\mathcal{S}}^{-1}$), denoted as
$\mathscr{S}{\mathcal{C}}$ (resp. $\mathscr{S}^{-1}{\mathcal{C}}$). For
cooperads, we have

######  Definition 3.4.

Let ${\mathcal{C}}=\\{{\mathcal{C}}(n)\\}_{n\geqslant 1}$ be a coaugmented
homotopy cooperad. The cobar construction of ${\mathcal{C}}$, denoted by
$\Omega{\mathcal{C}}$, is the free graded operad generated by the graded
collection $s^{-1}\overline{{\mathcal{C}}}$, endowed with the differential
$\partial$ which is lifted from

$\partial:s^{-1}\overline{{\mathcal{C}}}\to\Omega{\mathcal{C}},\
\partial(s^{-1}f):=-\sum_{T\in\mathfrak{T}}(s^{-1})^{\otimes\omega(T)}\circ\Delta_{T}(f)\
\text{for all }f\in\overline{{\mathcal{C}}}(n).$

This provides an alternative definition for homotopy operads. In fact, a
graded collection
$\overline{{\mathcal{C}}}=\\{\overline{{\mathcal{C}}}(n)\\}_{n\geqslant 1}$ is
a homotopy cooperad if and only if the free graded operad generated by
$s^{-1}\overline{{\mathcal{C}}}$ (also called the cobar construction of
${\mathcal{C}}=\overline{C}\oplus{\mathcal{I}}$) is endowed with a
differential such that it becomes a dg operad.

A natural $L_{\infty}$-algebra is associated with a homotopy operad
${\mathcal{P}}=\\{{\mathcal{P}}(n)\\}_{n\geqslant 1}$. Denote
${\mathcal{P}}^{\prod}:=\prod\limits_{n=1}^{\infty}{\mathcal{P}}(n)$. For each
$n\geqslant 1$, define operations
$m_{n}=\sum\limits_{T\in\mathfrak{T}^{(n)}}m_{T}:({\mathcal{P}}^{\prod})^{\otimes
n}\to{\mathcal{P}}^{\prod}$ and $l_{n}$ to be the anti-symmetrization of
$m_{n}$, that is,

$l_{n}(x_{1}\otimes\cdots\otimes x_{n}):=\sum\limits_{\sigma\in
S_{n}}\chi(\sigma;x_{1},\dots,x_{n})m_{n}(x_{\sigma(1)}\otimes\cdots\otimes
x_{\sigma(n)}).$

######  Proposition 3.5.

Let ${\mathcal{P}}$ be a homotopy operad. Then
$({\mathcal{P}}^{\prod},\\{l_{n}\\}_{n\geqslant 1})$ is an
$L_{\infty}$-algebra.

If a homotopy operad ${\mathcal{P}}$ satisfies $m_{T}=0$ for all
$T\in{\mathfrak{T}}$ with $\omega(T)\geqslant 2$, then ${\mathcal{P}}$ is just
a nonunital dg operad in the sense of Markl [39]. In this case,
${\mathcal{P}}^{\prod}$ is just a dg Lie algebra.

We shall need a fact about dg operads.

######  Definition 3.6.

Let ${\mathcal{P}}$ be a (nonunital) dg operad. For $f\in{\mathcal{P}}(m)$ and
$g_{1}\in{\mathcal{P}}(l_{1}),\dots,g_{n}\in{\mathcal{P}}(l_{n})$ with
$1\leqslant n\leqslant m$, define

$f\\{g_{1},\dots,g_{n}\\}:=\sum_{i_{j}\geqslant l_{j-1}+i_{j-1},n\geqslant
j\geqslant 2,i_{1}\geqslant
1}\Big{(}\big{(}(f\circ_{i_{1}}g_{1})\circ_{i_{2}}g_{2}\big{)}\dots\Big{)}\circ_{i_{n}}g_{n}.$

It is called the brace operation on ${\mathcal{P}}^{\prod}$. For
$f\in{\mathcal{P}}(m)$ and $g\in{\mathcal{P}}(m)$, define

$[f,g]_{G}:=f\\{g\\}-(-1)^{|f||g|}g\\{f\\}\in{\mathcal{P}}(m+n-1),$

called the Gerstenhaber bracket of $f$ and $g$.

The operation $l_{2}$ in the dg Lie algebra ${\mathcal{P}}^{\prod}$ is exactly
the Gerstenhaber bracket defined above.

The brace operation in a dg operad ${\mathcal{P}}$ satisfies the following
pre-Jacobi identity:

######  Proposition 3.7.

[13, 15, 16] For homogeneous elements $f,g_{1},\dots,g_{m},h_{1},\dots,h_{n}$
in ${\mathcal{P}}^{\prod}$, we have

(11)
$\displaystyle\Big{(}f\\{g_{1},\dots,g_{m}\\}\Big{)}\\{h_{1},\dots,h_{n}\\}=$
$\displaystyle\quad\sum\limits_{0\leqslant i_{1}\leqslant j_{1}\leqslant
i_{2}\leqslant j_{2}\leqslant\dots\leqslant i_{m}\leqslant j_{m}\leqslant
n}(-1)^{\sum\limits_{k=1}^{m}(|g_{k}|)(\sum\limits_{j=1}^{i_{k}}(|h_{j}|))}f\\{h_{1,i_{1}},g_{1}\\{sh_{i_{1}+1,j_{1}}\\},\dots,g_{m}\\{h_{i_{m}+1,j_{m}}\\},h_{j_{m}+1,n}\\}.$

In particular, we have

(12)
$\displaystyle(f\\{g\\})\\{h\\}=f\\{g\\{h\\}\\}+f\\{g,h\\}+(-1)^{|g||h|}f\\{h,g\\}.$

Next we introduce the notion of convolution homotopy operads.

######  Proposition-Definition 3.8.

Let ${\mathcal{C}}$ be a homotopy cooperad and ${\mathcal{P}}$ be a dg operad.
Then the graded collection $\mathbf{Hom}({\mathcal{C}},{\mathcal{P}})=\\{{\rm
Hom}({\mathcal{C}}(n),{\mathcal{P}}(n))\\}_{n\geqslant 1}$ has a natural
homotopy operad structure as follows.

  1. (i)

For $T\in\mathfrak{T}$ with $\omega(T)=1$,

$m_{T}(f):=d_{{\mathcal{P}}}\circ f-(-1)^{|f|}f\circ\Delta_{T}^{\mathcal{C}},\
f\in\mathbf{Hom}({\mathcal{C}},{\mathcal{P}})(n).$

  2. (ii)

For $T\in\mathfrak{T}$ with $\omega(T)\geqslant 2$,

$m_{T}(f_{1}\otimes\cdots\otimes
f_{r}):=(-1)^{\frac{r(r-1)}{2}+1+r\sum_{i=1}^{r}|f_{i}|}m_{{\mathcal{P}}}^{T}\circ(f_{1}\otimes\cdots\otimes
f_{r})\circ\Delta_{T}^{\mathcal{C}},$

where $m_{\mathcal{P}}^{T}$ is the composition in ${\mathcal{P}}$ along $T$.

This homotopy operad is called the convolution homotopy cooperad.

The following result explains the Maurer-Cartan elements of the
$L_{\infty}$-algebra associated to a convolution homotopy operad.

######  Proposition 3.9.

Let ${\mathcal{C}}$ be a coaugmented homotopy cooperad and ${\mathcal{P}}$ be
a unital dg operad. Then there is a natural bijection:

${\rm
Hom}_{udgOp}(\Omega{\mathcal{C}},{\mathcal{P}})\cong{\mathcal{M}}{\mathcal{C}}\Big{(}\mathbf{Hom}(\overline{{\mathcal{C}}},{\mathcal{P}})^{\prod}\Big{)},$

between the set of morphisms of unital dg operads from $\Omega C$ to
${\mathcal{P}}$ and the set of Maurer-Cartan elements in the
$L_{\infty}$-algebra
$\mathbf{Hom}(\overline{{\mathcal{C}}},{\mathcal{P}})^{\prod}$.

At last we recall the notions of minimal models and Koszul dual homotopy
cooperads of operads and explain how they are related to deformation complexes
and $L_{\infty}$-algebra structures on deformation complexes.

For a collection $M=\\{M(n)\\}_{n\geqslant 1}$ of (graded) vector spaces,
denote by $\mathcal{F}(M)$ the free graded operad generated by $M$. Recall
that a dg operad is called quasi-free if its underlying graded operad is free.

######  Definition 3.10.

[10] A minimal model for an operad $\mathcal{P}$ is a quasi-free dg operad
$(\mathcal{F}(M),d)$ together with a surjective quasi-isomorphism of operads
$(\mathcal{F}(M),\partial)\overset{\sim}{\twoheadrightarrow}\mathcal{P}$,
where the dg operad $(\mathcal{F}(M),\partial)$ satisfies the following
conditions.

  1. (i)

The differential $\partial$ is decomposable, i.e. $\partial$ takes $M$ to
$\mathcal{F}(M)^{\geqslant 2}$, the subspace of $\mathcal{F}(M)$ consisting of
elements with weight $\geqslant 2$;

  2. (ii)

The generating collection $M$ admits a decomposition
$M=\bigoplus\limits_{i\geqslant 1}M_{(i)}$ such that
$\partial(M_{(k+1)})\subset\mathcal{F}\Big{(}\bigoplus\limits_{i=1}^{k}M_{(i)}\Big{)}$
for all $k\geqslant 1$.

By [10], if an operad $\mathcal{P}$ admits a minimal model, then it is unique
up to isomorphisms.

For an operad ${\mathcal{P}}$, assume that its minimal model
${\mathcal{P}}_{\infty}$ exists. Since ${\mathcal{P}}_{\infty}$ is quasi-free,
there exists a coaugmented homotopy cooperad ${\mathcal{C}}$ such that
$\Omega{\mathcal{C}}\cong{\mathcal{P}}_{\infty}$. So ${\mathcal{C}}$ is called
the Koszul dual homotopy cooperad of ${\mathcal{P}}$, denoted by
${\mathcal{P}}^{\mathrm{\textup{!`}}}$.

Let $V$ be a complex. Denote by ${\rm End}_{V}$ the dg endomorphism operad.
Then the underlying complex of the convolution homotopy cooperad
$\mathbf{Hom}(\overline{{\mathcal{P}}^{\mathrm{\textup{!`}}}},{\rm End}_{V})$
is called the deformation complex of ${\mathcal{P}}$ on the complex $V$. An
element of ${\rm Hom}_{udgOp}({\mathcal{P}}_{\infty},{\rm End}_{V})$ is
exactly a homotopy ${\mathcal{P}}$-structure in $V$. So Proposition 3.9 gives
a bijection between the set of homotopy ${\mathcal{P}}$-structures on $V$ and
that of Maurer-Cartan elements in the $L_{\infty}$-algebra on the deformation
complex.

###  3.2.  Koszul dual homotopy cooperad of ${{}_{\lambda}\\!\mathfrak{Dif}}$

Now we introduce the operad of differential algebras with weight.

######  Definition 3.11.

The operad for differential algebras of weight $\mathbf{\lambda}$ is defined
to be the quotient of the free graded operad $\mathcal{F(M)}$ generated by a
graded collection $\mathcal{M}$ by an operadic ideal $I$, where the collection
$\mathcal{M}$ is given by $\mathcal{M}(1)={\bf k}d,\mathcal{M}(2)={\bf k}\mu$
and $\mathcal{M}(n)=0$ for $n\neq 1,2$, and where $I$ is generated by

(13) $\mu\circ_{1}\mu-\mu\circ_{2}\mu\ \ \mathrm{and}\ \
d\circ_{1}\mu-\big{(}\mu\circ_{1}d+\mu\circ_{2}d+\lambda\
(\mu\circ_{1}d)\circ_{2}d\big{)}.$

Denote this operad by ${{}_{\lambda}\\!\mathfrak{Dif}}$.

The homotopy cooperad
$\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})$ is
defined on the graded collection with arity-$n$ component

$\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})(n):={\bf
k}\widetilde{m_{n}}\oplus{\bf k}\widetilde{d_{n}}$

with $|\widetilde{m_{n}}|=0$, $|\widetilde{d_{n}}|=1$ for $n\geq 1$.

The coaugmented homotopy cooperad structure on the graded collection
$\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})$ is
defined as follows. First consider the following two types of trees:

  * (i)

where $n\geq 1,1\leq j\leq n,1\leq i\leq n-j+1$.

  * (ii)

where $2\leq p\leq n,2\leq q\leq p,1\leq k_{1}<\dots<k_{q}\leq
p,l_{1},\dots,l_{q}\geq 1,l_{1}+\cdots+l_{q}+p-q=n$.

Next, we define a family of operations
$\\{\Delta_{T}:\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})\rightarrow\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})^{\otimes
T}\\}_{T\in{\mathfrak{T}}}$ as follows:

  * (i)

For the element
$\widetilde{m_{n}}\in\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})(n)$
and a tree $T$ of type $\mathrm{(i)}$, define

  * (ii)

For the element
$\widetilde{d_{n}}\in\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})(n)$
and a tree $T$ of type $\mathrm{(i)}$, define

  * (iii)

For the element
$\widetilde{d_{n}}\in\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})(n)$
and a tree $T$ of type $\mathrm{(ii)}$, define

  * (iv)

$\Delta_{T}$ vanishes elsewhere.

Note that
$\Delta_{T}(\widetilde{m_{1}})=\widetilde{m_{1}}\otimes\widetilde{m_{1}}\in\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})^{\otimes
T}$ when $T$ is the tree of type $(i)$ with $i=j=n=1$ and
$\Delta_{T}(\widetilde{m_{1}})=0$ for every other tree $T$. So ${\bf
k}\widetilde{m_{1}}\cong{\mathcal{I}}$ as homotopy cooperads and there is a
natural projection and injection
$\varepsilon:\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})\twoheadrightarrow{\bf
k}\widetilde{m_{1}}\cong{\mathcal{I}}$ and $\eta:{\mathcal{I}}\cong{\bf
k}\widetilde{m_{1}}\hookrightarrow\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}).$

######  Proposition 3.12.

The graded collection
$\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})$ endowed
with operations $\\{\Delta_{T}\\}_{T\in{\mathfrak{T}}}$ forms a coaugmented
homotopy cooperad with strict counit
$\varepsilon:\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})\twoheadrightarrow{\bf
k}\widetilde{m_{1}}\cong{\mathcal{I}}$ and the coaugmentation
$\eta:{\mathcal{I}}\cong{\bf
k}\widetilde{m_{1}}\hookrightarrow\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}).$

###### Proof.

First of all, it is easy to check that $|\Delta_{T}|=\omega(T)-2$ for all
$T\in{\mathfrak{T}}$.

Next, we prove that $\\{\Delta_{T}\\}_{T\in{\mathfrak{T}}}$ endows
$\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})$ with a
homotopy cooperad structure. It suffices to prove that the induced derivation
$\partial$ on the cobar construction
$\Omega\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})$,
namely the free operad generated by
$s^{-1}\overline{\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})}$
is a differential: $\partial^{2}=0$. To simplify notations, we denote the
generators $s^{-1}\widetilde{m_{n}},n\geqslant 2$ and
$s^{-1}\widetilde{d_{n}},n\geqslant 1$ by $\mu_{n}$ and $\nu_{n}$ respectively
and note that $|\mu_{n}|=-1,n\geqslant 2$ and $|\nu_{n}|=0,n\geqslant 1$.

By the definition of
$\Omega\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})$
(Definition 3.4), one computes the action of $\partial$ on generators
$\mu_{n}=s^{-1}\widetilde{m_{n}}$ and $v_{n}=s^{-1}\widetilde{d_{n}}$ as

$\partial(\mu_{n})=-\sum_{j=2}^{n-1}\mu_{n-j+1}\\{\mu_{j}\\},\
\partial(\nu_{n})=\sum_{j=2}^{n}\nu_{n-j+1}\\{\mu_{j}\\}-\sum_{\begin{subarray}{c}2\leq
p\leq n,\\\ 1\leq q\leq
p\end{subarray}}\sum_{\begin{subarray}{c}l_{1}+\cdots+l_{q}+p-q=n,\\\
l_{1},\cdots,l_{q}\geq
1\end{subarray}}\lambda^{q-1}\mu_{p}\\{\nu_{l_{1}},\cdots,\nu_{l_{q}}\\}.$

As $\partial^{2}$ is also a derivation in the free operad
$\Omega\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})$,
to prove $\partial^{2}=0$ is equivalent to proving that $\partial^{2}=0$ holds
on the generators $\mu_{n},n\geq 2$ and $\nu_{n},n\geq 1$.

For $\partial^{2}(\mu_{n})$, we have

$\displaystyle\partial^{2}(\mu_{n})=$
$\displaystyle\partial(-\sum_{j=2}^{n-1}\mu_{n-j+1}\\{\mu_{j}\\})$
$\displaystyle=$
$\displaystyle-\sum_{j=2}^{n-1}\partial(\mu_{n-j+1})\\{\mu_{j}\\}+\sum_{j=2}^{n-1}\mu_{n-j+1}\\{\partial(\mu_{j})\\}$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}i+j+k-2=n,\\\ 2\leq
i,j,k\leq
n-2\end{subarray}}(\mu_{i}\\{\mu_{j}\\})\\{\mu_{k}\\}-\sum_{\begin{subarray}{c}i+j+k-2=n,\\\
2\leq i,j,k\leq n-2\end{subarray}}\mu_{i}\\{\mu_{j}\\{\mu_{k}\\}\\}$
$\displaystyle\stackrel{{\scriptstyle\eqref{Eq: pre-jacobi1}}}{{=}}$
$\displaystyle\sum_{\begin{subarray}{c}i+j+k-2=n,\\\ 2\leq i,j,k\leq
n-2\end{subarray}}\mu_{i}\\{\mu_{j}\\{\mu_{k}\\}\\}+\sum_{\begin{subarray}{c}i+j+k-2=n,\\\
2\leq i,j,k\leq
n-2\end{subarray}}\mu_{i}\\{\mu_{j},\mu_{k}\\}-\sum_{\begin{subarray}{c}i+j+k-2=n,\\\
2\leq i,j,k\leq
n-2\end{subarray}}\mu_{i}\\{\mu_{k},\mu_{j}\\}-\sum_{\begin{subarray}{c}i+j+k-2=n,\\\
2\leq i,j,k\leq n-2\end{subarray}}\mu_{i}\\{\mu_{j}\\{\mu_{k}\\}\\}$
$\displaystyle=$ $\displaystyle 0.$

The computation of $\partial^{2}(\nu_{n})$ is more involved.

$\displaystyle\partial^{2}(\nu_{n})=$
$\displaystyle\partial(\sum_{j=2}^{n}\nu_{n-j+1}\\{\mu_{j}\\}-\sum_{\begin{subarray}{c}2\leq
p\leq n,\\\ 1\leq q\leq
p\end{subarray}}\sum_{\begin{subarray}{c}l_{1}+\cdots+l_{q}+p-q=n,\\\
l_{1},\cdots,l_{q}\geq
1\end{subarray}}\lambda^{q-1}\mu_{p}\\{\nu_{l_{1}},\cdots,\nu_{l_{q}}\\})$
$\displaystyle=$
$\displaystyle\sum_{j=2}^{n}\partial(\nu_{n-j+1})\\{\mu_{j}\\}+\sum_{j=2}^{n}\nu_{n-j+1}\\{\partial(\mu_{j})\\}-\sum_{\begin{subarray}{c}2\leq
p\leq n,\\\ 1\leq q\leq
p\end{subarray}}\sum_{\begin{subarray}{c}l_{1}+\cdots+l_{q}+p-q=n,\\\
l_{1},\cdots,l_{q}\geq
1\end{subarray}}\lambda^{q-1}\partial(\mu_{p})\\{\nu_{l_{1}},\cdots,\nu_{l_{q}}\\}$
$\displaystyle+\sum_{\begin{subarray}{c}2\leq p\leq n,\\\ 1\leq q\leq
p\end{subarray}}\sum_{\begin{subarray}{c}l_{1}+\cdots+l_{q}+p-q=n,\\\
l_{1},\cdots,l_{q}\geq
1\end{subarray}}\sum_{i=1}^{q}\lambda^{q-1}\mu_{p}\\{\nu_{l_{1}},\cdots,\partial(\nu_{l_{i}}),\cdots,\nu_{l_{q}}\\}$
$\displaystyle=$
$\displaystyle\underbrace{\sum_{\begin{subarray}{c}i+j+k-2=n,\\\ 1\leq i\leq
n-2,2\leq j,k\leq
n-1\end{subarray}}(\nu_{i}\\{\mu_{j}\\})\\{\mu_{k}\\}}_{(a)}-\underbrace{\sum_{j=2}^{n}\sum_{\begin{subarray}{c}2\leq
p\leq n-j+1,\\\ 1\leq q\leq
p\end{subarray}}\sum_{\begin{subarray}{c}l_{1}+\cdots+l_{q}+p-q=n-j+1,\\\
l_{1},\cdots,l_{q}\geq
1\end{subarray}}\lambda^{q-1}(\mu_{p}\\{\nu_{l_{1}},\cdots,\nu_{l_{q}}\\})\\{\mu_{j}\\}}_{(b)}$
$\displaystyle-\underbrace{\sum_{\begin{subarray}{c}i+j+k-2=n,\\\ 1\leq i\leq
n-2,2\leq j,k\leq
n-1\end{subarray}}\nu_{i}\\{\mu_{j}\\{\mu_{k}\\}\\}}_{(c)}+\underbrace{\sum_{\begin{subarray}{c}2\leq
p\leq n,\\\ 1\leq q\leq
p\end{subarray}}\sum_{\begin{subarray}{c}l_{1}+\cdots+l_{q}+p-q=n,\\\
l_{1},\cdots,l_{q}\geq
1\end{subarray}}\sum_{j=2}^{p-1}\lambda^{q-1}(\mu_{p-j+1}\\{\mu_{j}\\})\\{\nu_{l_{1}},\cdots,\nu_{l_{q}}\\}}_{(d)}$
$\displaystyle+\underbrace{\sum_{\begin{subarray}{c}2\leq p\leq n,\\\ 1\leq
q\leq p\end{subarray}}\sum_{\begin{subarray}{c}l_{1}+\cdots+l_{q}+p-q=n,\\\
l_{1},\cdots,l_{q}\geq
1\end{subarray}}\sum_{i=1}^{q}\sum_{j=2}^{l_{i}}\lambda^{q-1}\mu_{p}\\{\nu_{l_{1}},\cdots,\nu_{l_{i}-j+1}\\{\mu_{j}\\},\cdots,\nu_{l_{q}}\\}}_{(e)}$
$\displaystyle-\underbrace{\sum_{\begin{subarray}{c}2\leq p\leq n,\\\ 1\leq
q\leq p\end{subarray}}\sum_{\begin{subarray}{c}l_{1}+\cdots+l_{q}+p-q=n,\\\
l_{1},\cdots,l_{q}\geq
1\end{subarray}}\sum_{i=1}^{q}\sum_{\begin{subarray}{c}2\leq\alpha\leq
l_{i},\\\
1\leq\beta\leq\alpha\end{subarray}}\sum_{\begin{subarray}{c}r_{1}+\cdots+r_{\beta}+\alpha-\beta=l_{i},\\\
r_{1},\cdots,r_{\beta}\geq
1\end{subarray}}\lambda^{q+\beta-2}\mu_{p}\\{\nu_{l_{1}},\cdots,\mu_{\alpha}\\{\nu_{r_{1}},\cdots,\nu_{r_{\beta}}\\},\cdots,\nu_{l_{q}}\\}}_{(f)}$

By the pre-Jacobi identity (11), we have $(a)=(c)\ \mathrm{and}\
(b)+(f)=(d)+(e).$ Thus we have $\partial^{2}(\nu_{n})=0$.

Finally, note that the maps $\varepsilon$ and $\eta$ are strict morphisms of
homotopy cooperads. This completes the proof. ∎

######  Definition 3.13.

The Hadamard product
$\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})\otimes_{\mathrm{H}}{\mathcal{S}}^{-1}$
is called the Koszul dual homotopy cooperad of
${{}_{\lambda}\\!\mathfrak{Dif}}$, and is denoted by
${{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}$.

To be precise, let $sm_{n}=\widetilde{m_{n}}\otimes\varepsilon_{n},n\geqslant
1$ and $sd_{n}=\widetilde{d_{n}}\otimes\varepsilon_{n},n\geqslant 1$. Then the
underlying graded collection of
${{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}$ is
${{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}(n)={\bf
k}sm_{n}\oplus{\bf k}sd_{n}$ with $|sm_{n}|=n-1$ and $|sd_{n}|=n$. The family
of operations $\\{\Delta_{T}\\}_{T\in{\mathfrak{T}}}$ defining its homotopy
cooperad structure is given by the following formulas.

  * (i)

For the element
$\widetilde{m_{n}}\in\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})(n)$
and $T$ of type $\mathrm{(i)}$, define

  * (ii)

For the element
$\widetilde{d_{n}}\in\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})(n)$
and $T$ of type $\mathrm{(i)}$, define

  * (iii)

For the element
$\widetilde{d_{n}}\in\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})(n)$
and tree $T$ of type $\mathrm{(ii)}$, define

where

$\alpha=\sum_{s=1}^{q}(l_{s}-1)(p-k_{s})+q(p-1)+\sum_{t=1}^{q-1}(q-t)l_{t};$

  * (iv)

All other components of $\Delta_{T},T\in{\mathfrak{T}}$ vanish.

##  4\. Operad of homotopy differential algebras with weight and minimal model

Now, we are going to introduce the notion of homotopy differential algebras
and their governing dg operad and show that this dg operad is the minimal
model of the operad of differential algebras with weight.

###  4.1. Operad of homotopy differential algebras with weight

######  Definition 4.1.

The operad ${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ of homotopy differential
algebras of weight $\lambda$ is defined to be the cobar construction
$\Omega({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})$ of
${{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}$.

A direct inspection gives the following description of
${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$.

######  Proposition 4.2.

Let $\mathcal{O}=(\mathcal{O}(1),\dots,\mathcal{O}(n),\dots)$ be the graded
collection where $\mathcal{O}(1)={\bf k}d_{1}$ with $|d_{1}|=0$ and for
$n\geqslant 2$, $\mathcal{O}(n)={\bf k}d_{n}\oplus{\bf k}m_{n}$ with
$|d_{n}|=n-1,|m_{n}|=n-2$. The operad
${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ of homotopy differential algebras
of weight $\lambda$ is the differential graded operad
$(\mathcal{F(O)},\partial)$, where the underlying free graded operad is
generated by the graded collection $\mathcal{O}$ and the action of the
differential $\partial$ on generators is given by the following equations. For
each $n\geqslant 2$,

(14)
$\partial({m_{n}})=\sum_{j=2}^{n-1}\sum_{i=1}^{n-j+1}(-1)^{i+j(n-i)}m_{n-j+1}\circ_{i}m_{j}$

and for each $n\geqslant 1$,

(15) $\displaystyle\partial(d_{n})$
$\displaystyle=-\sum_{j=2}^{n}\sum_{i=1}^{n-j+1}(-1)^{i+j(n-i)}d_{n-j+1}\circ_{i}m_{j}$
$\displaystyle-\sum\limits_{\begin{subarray}{c}1\leqslant
k_{1}<\dots<k_{q}\leqslant p\\\ l_{1}+\dots+l_{q}+p-q=n\\\
l_{1},\dots,l_{q}\geqslant 1,2\leqslant p\leqslant n,1\leqslant q\leqslant
p\end{subarray}}(-1)^{\xi}\lambda^{q-1}\Big{(}\cdots\big{(}\big{(}(m_{p}\circ_{k_{1}}d_{l_{1}})\circ_{k_{2}+l_{1}-1}d_{l_{2}}\big{)}\circ_{k_{3}+l_{1}+l_{2}-2}d_{l_{3}}\big{)}\cdots\Big{)}\circ_{k_{q}+l_{1}+\dots+l_{q-1}-q+1}d_{l_{q}}$

where $\xi:=\sum_{s=1}^{q}(l_{s}-1)(p-k_{s}).$

######  Remark 4.3.

  1. (i)

When $\lambda=0$, the operad ${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ is
precisely the operad $\mathrm{AsDer}_{\infty}$ constructed by Loday in [33] by
using Koszul duality for operads.

  2. (ii)

It is easy to see that $\partial(m_{2})=0=\partial(d_{1})$. So the degree zero
part of ${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ is just the free non-graded
operad generated by $m_{2}$ and $d_{1}$. This fact will be used in Lemma 4.5.

As free operads are constructed from planar rooted trees, we use planar rooted
trees to display the dg operad ${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ for
later applications. First, we use the following two kinds of corollas to
represent generators of ${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$:

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ m_{n}(n\geqslant 2)$1in

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ d_{n}(n\geqslant 1)$1in

For example, the element
$(((m_{3}\circ_{1}d_{4})\circ_{3}m_{4})\circ_{9}m_{2})\circ_{10}d_{2}$ can be
represented by the following tree

$\small m_{3}$$d_{4}$$m_{4}$$m_{2}$$d_{2}$

Eqs. (14) and (15) can be represented by

###  4.2. The minimal model of the operad of differential algebras

The following result is the main result of this paper.

######  Theorem 4.4.

The dg operad ${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ is the minimal model
of the operad ${{}_{\lambda}\\!\mathfrak{Dif}}$.

The proof of Theorem 4.4 will be carried out in the rest of this subsection.

######  Lemma 4.5.

The natural surjection

(16)
$p:{{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}\rightarrow{{}_{\lambda}\\!\mathfrak{Dif}}$

induces an isomorphism
${\mathrm{H}}_{0}({{}_{\lambda}\\!\mathfrak{Dif}_{\infty}})\cong{{}_{\lambda}\\!\mathfrak{Dif}}$.

###### Proof.

By Remark 4.3 (ii), the degree zero part of
${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ is the free (non-graded) operad
generated by $m_{2}$ and $d_{1}$. By definition, one has

$\partial(m_{3})=-\mu\circ_{1}\mu+\mu\circ_{2}\mu,$
$\partial(d_{2})=d_{1}\circ_{1}m_{2}-(m_{2}\circ_{1}d_{1}+m_{2}\circ_{2}d_{1}+\lambda(m_{2}\circ_{1}d_{1})\circ_{2}d_{1}).$

So the surjection
$p:{{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}\rightarrow{{}_{\lambda}\\!\mathfrak{Dif}}$
sending $m_{2}$ to $\mu$ and $d_{1}$ to $d$ induces
${\mathrm{H}}_{0}({{}_{\lambda}\\!\mathfrak{Dif}_{\infty}})\cong{{}_{\lambda}\\!\mathfrak{Dif}}$.
∎

Obviously, the differential $\partial$ of
${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ satisfies Definition 3.10 (i) and
(ii), so in order to complete the proof of Theorem 4.4, we only need to show
that
$p:{{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}\rightarrow{{}_{\lambda}\\!\mathfrak{Dif}}$
is a quasi-isomorphism. For this purpose, we are going to build a homotopy
map, i.e., a degree $1$ map
$H:{{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}\rightarrow{{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$
such that in each positive degree

$\partial\circ H+H\circ\partial={\rm Id}.$

We need the following notion of graded path-lexicographic ordering on
${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$.

Each tree monomial gives rise to a path sequence [5, Chapter 3]. More
precisely, to a tree monomial $T$ with $n$ leaves (written as
$\mbox{arity}(T)=n$), we can associate a sequence $(x_{1},\dots,x_{n})$ where
$x_{i}$ is the word formed by generators of
${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ corresponding to the vertices along
the unique path from the root of $T$ to its $i$-th leaf. We define an order on
the graded tree monomials as follows. For graded tree monomials $T$ and
$T^{\prime}$, define $T>T^{\prime}$ if

  1. (i)

either $\mbox{arity}(T)>\mbox{arity}(T^{\prime})$;

  2. (ii)

or $\mbox{arity}(T)=\mbox{arity}(T^{\prime})$, and $\deg(T)>\deg(T^{\prime})$,
where $\deg(T)$ is the sum of the degrees of all generators of
${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ appearing in the tree monomial $T$;

  3. (iii)

or $\mbox{arity}(T)=\mbox{airty}(T^{\prime})(=n),\deg(T)=\deg(T^{\prime})$,
and the path sequences $(x_{1},\dots,x_{n})$ and
$(x^{\prime}_{1},\dots,x_{n}^{\prime})$ associated to $T$ and $T^{\prime}$
satisfy $(x_{1},\dots,x_{n})>(x_{1}^{\prime},\dots,x_{n}^{\prime})$ with
respect to the length-lexicographic order of words induced by

$m_{2}<d_{1}<m_{3}<d_{2}<\dots<m_{n}<d_{n-1}<m_{n+1}<d_{n}<\dots.$

It is readily seen that $>$ is a well order.

Given a linear combination of tree monomials, its leading monomial is the
largest tree monomial appearing with nonzero coefficient in this linear
combination, and the coefficient of the leading monomial is called the leading
coefficient.

Let $S$ be a generator of degree $\geqslant 1$ in
${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$. Denote the leading monomial of
$\partial S$ by $\widehat{S}$ and the coefficient of $\widehat{S}$ in
$\partial S$ by $c_{S}$. It is easily seen that the coefficient $c_{S}$ is
always $\pm 1$. A tree monomial of the form $\widehat{S}$ with the degree of
$S$ being positive is called typical. More precisely, typical tree monomials
are of the following forms:

$m_{n}\quad(n\geqslant 2)$$m_{2}$ $d_{n}\quad(n\geqslant 1)$$m_{2}$

where the first one is the leading monomial of $\partial m_{n+1},n\geqslant
2$, the second being that of $\partial d_{n+1},n\geqslant 1$.

######  Definition 4.6.

We call a tree monomial  effective if it satisfies the the following
conditions:

  1. (i)

There exists a typical divisor $\widehat{S}$ in $T$;

  2. (ii)

On the path from the root $v$ of $\widehat{S}$ to the leftmost leaf $l$ of $T$
above $v$, there are no other typical divisors and there are no vertices of
positive degree except possibly $v$ itself;

  3. (iii)

For a leaf $l^{\prime}$ of $T$ which is located on the left of $l$, there are
no vertices of positive degree and no typical divisors on the path from the
root of $T$ to $l^{\prime}$.

The divisor $\widehat{S}$ is unique if exists, and is called the effective
divisor of $T$ and $l$ is called the effective leaf of $T$.

Morally, the effective divisor of a tree monomial $T$ is the left-upper-most
typical divisor of $T$. It follows by the definition that for an effective
divisor $T^{\prime}$ in $T$ with the effective leaf $l$, no vertex in
$T^{\prime}$ belongs to the path from the root of $T$ to any leaf $l^{\prime}$
located on the left of $l$.

######  Example 4.7.

Consider the following three tree monomials with the same underlying tree:

  1. (i)

The tree monomial $T_{1}$ is effective and the divisor in the red/dashed
circle is its effective divisor. Note that there is a vertex of positive
degree on the path from the root of the tree to the root of the effective
divisor.

  2. (ii)

The tree monomial $T_{2}$ is not effective, because there is a vertex of
degree $1$ (the root itself) on the path from the root of the entire tree to
the first leaf.

  3. (iii)

For the tree monomial $T_{3}$, although there is a typical divisor
$m_{3}\circ_{1}m_{2}$ on the path from the root to the second leaf, there is a
vertex $d_{2}$ of positive degree on the path from the root of this divisor to
the second leaf of $T_{3}$. Thus $T_{3}$ is not effective.

######  Definition 4.8.

Let $T$ be an effective tree monomial in
${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ and $T^{\prime}$ be its effective
divisor. Assume that $T^{\prime}=\widehat{S}$, where $S$ is a generator of
positive degree. Then define

$\overline{H}(T):=(-1)^{\omega}\frac{1}{c_{S}}m_{T^{\prime},S}(T),$

where $m_{T^{\prime},S}(T)$ is the tree monomial obtained from $T$ by
replacing the effective divisor $T^{\prime}$ by $S$, $\omega$ is the sum of
degrees of all the vertices on the path from the root of $T^{\prime}$ to the
root of $T$ (except the root vertex of $T^{\prime}$) and on the left of this
path.

Now, we construct the homotopy map $H$ via the following inductive procedure:

  1. (i)

For a non-effective tree monomial $T$, define $H(T)=0$;

  2. (ii)

For an effective tree monomial $T$, define
$H(T)=\overline{H}(T)+H(\overline{T})$, where $\overline{T}$ is obtained from
$T$ by replacing $\widehat{S}$ in $T$ by $\widehat{S}-\frac{1}{c_{S}}\partial
S$. Then each tree monomials appearing in $\overline{T}$ is strictly smaller
than $T$, and we define $H(\overline{T})$ by taking induction on the leading
terms.

We explain more on the definition of $H$. Denote $T$ also by $T_{1}$ and take
$I_{1}:=\\{1\\}$ for convenience. Then by the definition above,
$H(T)=\overline{H}(T_{1})+H(\overline{T_{1}})$. Since $H$ vanishes on non-
effective tree monomials, we have $H(\overline{T}_{1})=H(\sum_{i_{2}\in
I_{2}}T_{i_{2}})$ where $\\{T_{i_{2}}\\}_{i_{2}\in I_{2}}$ is the set of
effective tree monomials together with their nonzero coefficients appearing in
the expansion of $\overline{T_{1}}$. Then by the definition of $H$, we have
$H\Big{(}\sum_{i_{2}\in
I_{2}}T_{i_{2}}\Big{)}=\overline{H}\Big{(}\sum_{i_{2}\in
I_{2}}T_{i_{2}}\Big{)}+H\Big{(}\sum_{i_{2}\in
I_{2}}\overline{T_{i_{2}}}\Big{)},$ giving

$H(T)=\overline{H}\Big{(}\sum_{i_{1}\in
I_{1}}T_{i_{1}}\Big{)}+\overline{H}\Big{(}\sum_{i_{2}\in
I_{2}}T_{i_{2}}\Big{)}+H\Big{(}\sum_{i_{2}\in
I_{2}}\overline{T_{i_{2}}}\Big{)}.$

An induction on the leading terms shows that $H(T)$ is the following series:

(17) $\displaystyle H(T)=\sum_{k=1}^{\infty}\overline{H}\Big{(}\sum_{i_{k}\in
I_{k}}T_{i_{k}}\Big{)},$

where $\\{T_{i_{k}}\\}_{i_{k}\in I_{k}},k\geq 1,$ is the set of effective tree
monomials appearing in the expansion of $\sum_{i_{k-1}\in
I_{k-1}}\overline{T_{i_{k-1}}}$ with nonzero coefficients.

######  Lemma 4.9.

For an effective tree monomial $T$, the expansion of $H(T)$ in Eq. (17) is
always a finite sum, that is, there exists an integer $n$ such that none of
the tree monomials in $\sum_{i_{n}\in I_{n}}\overline{T_{i_{n}}}$ is
effective.

###### Proof.

It is easy to see that $\max\\{T_{i_{k}}|i_{k}\in
I_{k}\\}>\max\\{T_{i_{k+1}}|i_{k+1}\in I_{k+1}\\}$ for all $k\geq 1$ (by
convention, $i_{1}\in I_{1}=\\{1\\}$). So Eq. (17) is a finite sum, as $>$ is
a well order. ∎

######  Lemma 4.10.

Let $T$ be an effective tree monomial in
${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$. Then

$\partial\overline{H}(T)+H\partial(T-\overline{T})=T-\overline{T}.$

###### Proof.

We can write $T$ via partial compositions as follows:

$(\cdots(((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\widehat{S})\circ_{j_{1}}Y_{1})\circ_{j_{2}}Y_{2})\circ_{j_{3}}\cdots)\circ_{j_{q}}Y_{q},$

where $\widehat{S}$ is the effective divisor of $T$ and $X_{1},\cdots,X_{p}$
are generators of ${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ corresponding to
the vertices that live on the path from the root of $T$ to the root of
$\widehat{S}$ (except the root of $\widehat{S}$) and on the left of this path.

By definition,

$\displaystyle\partial\overline{H}(T)=$
$\displaystyle\frac{1}{c_{S}}(-1)^{\sum\limits_{j=1}^{p}|X_{j}|}\partial\Big{(}(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{q}}Y_{q}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{c_{S}}\sum\limits_{k=1}^{p}(-1)^{\sum\limits_{j=1}^{p}|X_{j}|+\sum\limits_{j=1}^{k-1}|X_{j}|}$
$\displaystyle(\cdots((((\cdots((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{k-1}}\partial
X_{k})\circ_{i_{k}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{q}}Y_{q}$
$\displaystyle+\frac{1}{c_{S}}(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\partial
S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{q}}Y_{q}$
$\displaystyle+\frac{1}{c_{S}}\sum\limits_{k=1}^{q}(-1)^{|S|+\sum\limits_{j=1}^{k-1}|Y_{j}|}$
$\displaystyle(\cdots((\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots\cdots)\circ_{j_{k}}\partial
Y_{k})\circ_{j_{k+1}}\cdots)\circ_{j_{q}}Y_{q}$

and

$\displaystyle H\partial(T-\overline{T})$ $\displaystyle=$
$\displaystyle\frac{1}{c_{S}}H\partial\Big{(}(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\partial
S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{q}}Y_{q}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{c_{S}}\sum\limits_{k=1}^{p}(-1)^{\sum\limits_{j=1}^{p}|X_{j}|+\sum\limits_{j=1}^{k-1}|X_{j}|}$
$\displaystyle
H\big{(}(\cdots((((\cdots((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{k-1}}\partial
X_{k})\circ_{i_{k}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\partial
S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{q}}Y_{q}\big{)}$
$\displaystyle+\frac{1}{c_{S}}\sum\limits_{k=1}^{q}(-1)^{\sum\limits_{j=1}^{p}|X_{j}|+|S|-1+\sum\limits_{j=1}^{k-1}|Y_{j}|}$
$\displaystyle
H\Big{(}(\cdots((\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\partial
S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{k}}\partial
Y_{k})\circ_{j_{k+1}}\cdots)\circ_{j_{q}}Y_{q}\Big{)}$ $\displaystyle=$
$\displaystyle\sum\limits_{k=1}^{p}(-1)^{\sum\limits_{j=1}^{p}|X_{j}|+\sum\limits_{j=1}^{k-1}|X_{j}|}$
$\displaystyle
H\big{(}(\cdots((((\cdots((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{k-1}}\partial
X_{k})\circ_{i_{k}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\widehat{S})\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{q}}Y_{q}\big{)}$
$\displaystyle+\sum\limits_{k=1}^{p}(-1)^{\sum\limits_{j=1}^{p}|X_{j}|+\sum\limits_{j=1}^{k-1}|X_{j}|}$
$\displaystyle
H\big{(}(\cdots((((\cdots((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{k-1}}\partial
X_{k})\circ_{i_{k}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}(\frac{1}{c_{S}}\partial
S-\widehat{S}))\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{q}}Y_{q}\big{)}$
$\displaystyle+\sum\limits_{k=1}^{q}(-1)^{\sum\limits_{j=1}^{p}|X_{j}|+|S|-1+\sum\limits_{j=1}^{k-1}|Y_{j}|}$
$\displaystyle
H\Big{(}(\cdots((\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\widehat{S})\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{k}}\partial
Y_{k})\circ_{j_{k+1}}\cdots)\circ_{j_{q}}Y_{q}\Big{)}$
$\displaystyle+\sum\limits_{k=1}^{q}(-1)^{\sum\limits_{j=1}^{p}|X_{j}|+|S|-1+\sum\limits_{j=1}^{k-1}|Y_{j}|}$
$\displaystyle
H\Big{(}(\cdots((\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}(\frac{1}{c_{S}}\partial
S-\widehat{S}))\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{k}}\partial
Y_{k})\circ_{j_{k+1}}\cdots)\circ_{j_{q}}Y_{q}\Big{)}.$

By the definition of effective divisors in an effective tree monomial, it can
be easily seen that each tree monomial in the expansion of

$T_{k}^{\prime}:=(\cdots((((\cdots((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{k-1}}\partial
X_{k})\circ_{i_{k}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\widehat{S})\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{q}}Y_{q}$

and

$T_{k}^{\prime\prime}:=(\cdots((\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\widehat{S})\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{k}}\partial
Y_{k})\circ_{j_{k+1}}\cdots)\circ_{j_{q}}Y_{q}$

is still an effective tree monomial with $\widehat{S}$ as the effective
divisor. We thus have

$\displaystyle H\partial(T-\overline{T})$ $\displaystyle=$
$\displaystyle\frac{1}{c_{S}}\sum\limits_{k=1}^{p}(-1)^{\sum\limits_{j=1}^{p}|X_{j}|+\sum\limits_{j=1}^{k-1}|X_{j}|}(H(T_{k}^{\prime})-H(\overline{T_{k}^{\prime}}))+\sum\limits_{k=1}^{q}(-1)^{\sum\limits_{j=1}^{p}|X_{j}|+|S|-1+\sum\limits_{j=1}^{k-1}|Y_{j}|}(H(T_{k}^{\prime\prime})-H(\overline{T_{k}^{\prime\prime}}))$
$\displaystyle=$
$\displaystyle\frac{1}{c_{S}}\sum\limits_{k=1}^{p}(-1)^{\sum\limits_{j=1}^{p}|X_{j}|+\sum\limits_{j=1}^{k-1}|X_{j}|}\overline{H}(T_{k}^{\prime})+\sum\limits_{k=1}^{q}(-1)^{\sum\limits_{j=1}^{p}|X_{j}|+|S|-1+\sum\limits_{j=1}^{k-1}|Y_{j}|}\overline{H}(T_{k}^{\prime\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{c_{S}}\sum\limits_{k=1}^{p}(-1)^{\sum\limits_{j=1}^{p}|X_{j}|+\sum\limits_{j=1}^{k-1}|X_{j}|}$
$\displaystyle(\cdots((((\cdots((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{k-1}}\partial
X_{k})\circ_{i_{k}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{q}}Y_{q}$
$\displaystyle+\frac{1}{c_{S}}\sum\limits_{k=1}^{q}(-1)^{|S|-1+\sum\limits_{j=1}^{k-1}|Y_{j}|}$
$\displaystyle(\cdots((\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots\cdots)\circ_{j_{k}}\partial
Y_{k})\circ_{j_{k+1}}\cdots)\circ_{j_{q}}Y_{q}.$

Therefore, adding the expansions of $\partial\overline{H}(T)$ and
$H\partial(T-\overline{T})$, we obtain

$\hskip
128.0374pt\partial\overline{H}(T)+H\partial(T-\overline{T})=T-\overline{T}.\hskip
128.0374pt\qed$

######  Proposition 4.11.

The degree $1$ map $H$ defined in Eq. (17) satisfies $\partial
H+H\partial=\mathrm{id}$ on ${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ in each
positive degree.

###### Proof.

We first prove that for a non-effective tree monomial $T$, the equation
$\partial H(T)+H\partial(T)=T$ holds. By the definition of $H$, since $T$ is
not effective, $H(T)=0$. Thus we just need to check $H\partial(T)=T$. Since
$T$ has positive degree, there exists at least one vertex of positive degree.
We pick such a vertex $S$ that satisfies the following additional conditions:

  1. (i)

On the path from $S$ to the leftmost leaf $l$ of $T$ above $S$, there are no
other vertices of positive degree;

  2. (ii)

For a leaf $l^{\prime}$ of $T$ located on the left of $l$, the vertices on the
path from the root of $T$ to $l^{\prime}$ are all of degree 0.

Such a vertex always exists. Then the element in
${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$ corresponding to $T$ can be written
as

$(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}Y_{2}\cdots)\circ_{j_{q}}Y_{q},$

where $X_{1},\cdots,X_{p}$ correspond to the vertices along the path from the
root of $T$ to $S$ and the vertices on the left of this path. Thus, all
$X_{i},i=1,\dots,p$ are of degree zero.

Then by definition,

$\displaystyle H\partial T=$ $\displaystyle H\
\Big{(}(-1)^{|X_{1}|+\cdots+|X_{p}|}(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\partial
S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{q}}Y_{q}\Big{)}$
$\displaystyle+\sum\limits_{k=1}^{q}(-1)^{\sum\limits_{t=1}^{p}|X_{t}|+|S|+\sum\limits_{t=1}^{k-1}|Y_{t}|}$
$\displaystyle\quad H\
\Big{(}(\cdots((\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{k}}\partial
Y_{k})\circ_{j_{k+1}}\cdots\circ_{j_{q}}Y_{q}\Big{)}.$

By the assumption, the divisor consisting of the path from $S$ to $l$ must be
of the following forms

By the assumption that $T$ is not effective and the additional properties of
the position of $S$ stated above, one can see that the effective tree
monomials in $\partial T$ will only appear in the expansion of

$(-1)^{|X_{1}|+\cdots+|X_{p}|}(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\partial
S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}Y_{2}\cdots)\circ_{j_{q}}Y_{q}.$

Consider the tree monomial
$(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\widehat{S})\circ_{j_{1}}Y_{1})\circ_{j_{2}}Y_{2}\cdots)\circ_{j_{q}}Y_{q}$
in $\partial T$. The path connecting the root of $\widehat{S}$ and $l$ must be
one of the following forms:

By the assumption that $T$ is not effective and the choice of $S$, there
exists no effective divisor on the left of the path from the root of $T$ to
$l$. So the tree monomial

$(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\widehat{S})\circ_{j_{1}}Y_{1})\circ_{j_{2}}Y_{2}\cdots)\circ_{j_{q}}Y_{q}$

is effective and its effective divisor is simply $\widehat{S}$ itself. Then we
have

$\displaystyle H\partial T$ $\displaystyle=$ $\displaystyle
H\Big{(}(-1)^{|X_{1}|+\cdots+|X_{p}|}(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\partial
S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{q}}Y_{q}\Big{)}$
$\displaystyle=$ $\displaystyle
c_{S}H\Big{(}(-1)^{|X_{1}|+\cdots+|X_{p}|}(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\widehat{S})\circ_{j_{1}}Y_{1})\circ_{j_{2}}Y_{2}\cdots)\circ_{j_{q}}Y_{q}\Big{)}$
$\displaystyle+H\Big{(}(-1)^{|X_{1}|+\cdots+|X_{p}|}(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}(\partial
S-c_{S}\widehat{S}))\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{q}}Y_{q}\Big{)}$
$\displaystyle=$ $\displaystyle
c_{S}\overline{H}\Big{(}(-1)^{|X_{1}|+\cdots+|X_{p}|}(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\widehat{S})\circ_{j_{1}}Y_{1})\circ_{j_{2}}\cdots)\circ_{j_{q}}Y_{q}\Big{)}$
$\displaystyle+c_{S}H\Big{(}(-1)^{|X_{1}|+\cdots+|X_{p}|}(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}(\widehat{S}-\frac{1}{c_{S}}\partial
S))\circ_{j_{1}}Y_{1})\circ_{j_{2}}Y_{2}\cdots)\circ_{j_{q}}Y_{q}\Big{)}$
$\displaystyle+H\Big{(}(-1)^{|X_{1}|+\cdots+|X_{p}|}(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}(\partial
S-c_{S}\widehat{S}))\circ_{j_{1}}Y_{1})\circ_{j_{2}}Y_{2}\cdots)\circ_{j_{q}}Y_{q}\Big{)}$
$\displaystyle=$ $\displaystyle
c_{S}\overline{H}\Big{(}(-1)^{|X_{1}|+\cdots+|X_{p}|}(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}\widehat{S})\circ_{j_{1}}Y_{1})\circ_{j_{2}}Y_{2}\cdots)\circ_{j_{q}}Y_{q}\Big{)}$
$\displaystyle=$
$\displaystyle(\cdots((((\cdots(X_{1}\circ_{i_{1}}X_{2})\circ_{i_{2}}\cdots)\circ_{i_{p-1}}X_{p})\circ_{i_{p}}S)\circ_{j_{1}}Y_{1})\circ_{j_{2}}Y_{2}\cdots)\circ_{j_{q}}Y_{q}$
$\displaystyle=$ $\displaystyle T.$

Let $T$ be an effective tree monomial. By Lemma 4.10, we have
$\partial\overline{H}(T)+H\partial(T-\overline{T})=T-\overline{T}$. Moreover,
since each tree monomial of $\overline{T}$ is strictly smaller than $T$, by
induction, we obtain $H\partial(\overline{T})+\partial
H(\overline{T})=\overline{T}$. By $H(T)=\overline{H}(T)+H(\overline{T})$, we
get

$\hskip 28.45274pt\partial H(T)+H\partial(T)=\partial\overline{H}(T)+\partial
H(\overline{T})+H\partial(T-\overline{T})+H\partial(\overline{T})=T-\overline{T}+\overline{T}=T.\hskip
28.45274pt\qed$

Combining Lemma 4.5 and Proposition 4.11, we obtain a surjective quasi-
isomorphism
$p:{{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}\to{{}_{\lambda}\\!\mathfrak{Dif}}$.
Notice that conditions (i) and (ii) of Definition 3.10 hold by the
construction of the dg operad ${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}$. Thus
the proof of Theorem 4.4 is completed.

###  4.3. Homotopy differential algebras with weight

######  Definition 4.12.

Let $V=(V,m_{1})$ be a complex. A homotopy differential algebra structure of
weight $\lambda$ on $V$ is a homomorphism
${{}_{\lambda}\\!\mathfrak{Dif}_{\infty}}\to\mathrm{End}_{V}$ of unital dg
operads.

Definition 4.12 translates to the following explicit characterization of
homotopy differential algebras.

######  Proposition 4.13.

A homotopy differential algebra structure of weight $\lambda$ on a graded
space $V$ is equivalently given by two family of operators
$\\{m_{n}:V^{\otimes n}\to V\\}_{n\geqslant 1}$ and $\\{d_{n}:V^{\otimes n}\to
V\\}_{n\geqslant 1}$ with $|m_{n}|=n-2$ and $|d_{n}|=n-1$, fulfilling the
following identities: for each $n\geqslant 1$,

(18) $\displaystyle\sum_{i+j+k=n\atop i,k\geqslant 0,j\geqslant
1}(-1)^{i+jk}m_{i+1+k}\circ({\rm Id}^{\otimes i}\otimes m_{j}\otimes{\rm
Id}^{\otimes k})=0,$ (19) $\displaystyle\sum_{i+j+k=n\atop i,k\geqslant
0,j\geqslant 1}(-1)^{i+jk}d_{i+1+k}\circ({\rm Id}^{\otimes i}\otimes
m_{j}\otimes{\rm Id}^{\otimes k})$
$\displaystyle\quad\quad\quad\quad\quad=\sum_{\begin{subarray}{c}l_{1}+\dots+l_{q}+j_{1}+\dots+j_{q+1}=n\\\
j_{1}+\dots+j_{q+1}+q=p\\\ j_{1},\dots,j_{q+1}\geqslant
0,l_{1},\dots,l_{q}\geqslant 1\\\ n\geqslant p\geqslant q\geqslant
1\end{subarray}}(-1)^{\eta}\lambda^{q-1}m_{p}\circ({\rm Id}^{\otimes
j_{1}}\otimes d_{l_{1}}\otimes{\rm Id}^{\otimes j_{2}}\otimes\cdots\otimes
d_{l_{q}}\otimes{\rm Id}^{\otimes j_{q+1}}),$

where

$\eta:=\frac{n(n-1)}{2}+\frac{p(p-1)}{2}+\sum\limits_{k=1}^{q}\frac{l_{k}(l_{k}-1)}{2}+\sum\limits_{k=1}^{q}\big{(}l_{k}-1\big{)}\big{(}\sum\limits_{r=1}^{k}j_{r}+\sum\limits_{r=1}^{k-1}l_{r}\big{)}=\sum_{k=1}^{q}(l_{k}-1)(q-k+\sum_{r=k+1}^{q+1}j_{r}).$

Notice that Eq. (18) is exactly the Stasheff identity defining
$A_{\infty}$-algebras [50]. As is well known, the homology
${\mathrm{H}}_{*}(V,m_{1})$ endowed with the associative product induced by
$m_{2}$ is a graded algebra.

Expanding Eq. (19) for small $n$’s, one obtains

  1. (i)

when $n=1$, $|d_{1}|=0$ and $d_{1}\circ m_{1}=m_{1}\circ d_{1},$ that is,
$d_{1}:(V,m_{1})\to(V,m_{1})$ is a chain map;

  2. (ii)

when $n=2$, $|d_{2}|=1$ and

$\displaystyle d_{1}\circ m_{2}-\Big{(}m_{2}\circ(d_{1}\otimes{\rm
Id})+m_{2}\circ({\rm Id}\otimes d_{1})+\lambda m_{2}\circ(d_{1}\otimes
d_{1})\Big{)}$ $\displaystyle=d_{2}\circ({\rm Id}\otimes
m_{1}+m_{1}\otimes{\rm Id})+m_{1}\circ d_{2},$

which shows that $d_{1}$ is, up to a homotopy given by $d_{2}$, a differential
operator of weight $\lambda$ with respect to the “multiplication” $m_{2}$.

As a consequence, the homology ${\mathrm{H}}_{*}(V,m_{1})$, endowed with the
associative product induced by $m_{2}$ and the differential operator induced
by $d_{1}$, is a differential algebra of weight $\lambda$. This indicates that
the notion of homotopy differential algebras is a “higher homotopy” version of
that of differential algebras with weight.

##  5\. From the minimal model to the $L_{\infty}$-algebra controlling
deformations

To finish the paper, we will use the homotopy cooperad
${{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}$ to deduce the
$L_{\infty}$-structure controlling the deformations of differential algebras
with weight, as promised in Theorem 2.2. As a consequence, homotopy
differential algebras can be described as the Maurer-Catan elements in (a
reduced version) of this $L_{\infty}$-algebra.

###  5.1. Proof of Theorem 2.2

By Proposition 3.5 and Proposition-Definition 3.8, for a graded space $V$, the
$L_{\infty}$-algebra from the deformation complex of differential algebras is

$\mathfrak{C}_{{{}_{\lambda}\\!\mathfrak{Dif}}}(V):=\mathbf{Hom}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}},{\rm
End}_{V})^{\prod}.$

Now, we determine the $L_{\infty}$-algebra
${\mathfrak{C}}_{{{}_{\lambda}\\!\mathfrak{Dif}}}(V)$ explicitly and thus give
the promised operadic proof of Theorem 2.2.

The sign rules in ${{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}$ are
complicated, so we use some transformations. Notice that there is a natural
isomorphism of operads: $\mathbf{Hom}({\mathcal{S}},{\rm End}_{sV})\cong{\rm
End}_{V}$. Then we have the following isomorphisms of homotopy operads:

$\begin{array}[]{rcl}\mathbf{Hom}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}},{\rm
End}_{V})&\cong&\mathbf{Hom}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}},\mathbf{Hom}({\mathcal{S}},{\rm
End}_{sV}))\cong\mathbf{Hom}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}\otimes_{\mathrm{H}}{\mathcal{S}},{\rm
End}_{sV})\\\
&=&\mathbf{Hom}(\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}),{\rm
End}_{sV}).\end{array}$

Recall that for $n\geq 1$,
$\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})(n)={\bf
k}\widetilde{m_{n}}\oplus{\bf k}\widetilde{d_{n}}$ with
$|\widetilde{m_{n}}|=0$ and $|\widetilde{d_{n}}|=1$. By definition, we have

$\begin{array}[]{rcl}\mathbf{Hom}(\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}),{\rm
End}_{sV})(n)&=&{\rm Hom}({\bf k}\widetilde{m_{n}}\oplus{\bf
k}\widetilde{d_{n}},{\rm Hom}((sV)^{\otimes n},sV))\\\ &\cong&{\rm Hom}({\bf
k}\widetilde{m_{n}},{\rm Hom}((sV)^{\otimes n},sV))\oplus{\rm Hom}({\bf
k}\widetilde{d_{n}},{\rm Hom}((sV)^{\otimes n},sV)).\end{array}$

For a homogeneous element $f\in{\rm Hom}((sV)^{\otimes n},sV)$, define

$\widehat{f}:{\bf k}\widetilde{m_{n}}\to{\rm Hom}((sV)^{\otimes
n},sV)),\quad\widetilde{m_{n}}\mapsto f,$

and for a homogeneous element $g\in{\rm Hom}((sV)^{\otimes n},V)$, define

$\overline{g}:{\bf k}\widetilde{d_{n}}\to{\rm Hom}((sV)^{\otimes
n},sV)),\quad\widetilde{d_{n}}\mapsto(-1)^{|g|}sg.$

The resulting bijections, sending $f$ and $g$ to $\widehat{f}$ and
$\overline{g}$ respectively, allow us to identify
${\mathfrak{C}}_{{\mathsf{DA}_{\lambda}}}(V)$ with
$\mathbf{Hom}(\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}),{\rm
End}_{sV})$.

Now we compute the $L_{\infty}$-structure $\\{\ell_{n}\\}_{n\geq 1}$ of
${\mathfrak{C}}_{{{}_{\lambda}\\!\mathfrak{Dif}}}(V)$. Recall that for
homogeneous elements
$x_{i}\in{\mathfrak{C}}_{{{}_{\lambda}\\!\mathfrak{Dif}}}(V),1\leq i\leq n$,

$\ell_{n}(x_{1},\cdots,x_{n})=\sum_{\sigma\in
S_{n}}\chi(\sigma;x_{1},\dots,x_{n})m_{n}(x_{\sigma(1)}\otimes\cdots\otimes
x_{\sigma(n)})$

with

$m_{n}(x_{1}\otimes\cdots\otimes
x_{n})=\sum_{T\in{\mathfrak{T}},w(T)=n}m_{T}(x_{1}\otimes\cdots\otimes
x_{n}),$

where $\\{m_{T}\\}_{T\in{\mathfrak{T}}}$ is given by the homotopy operad
structure of
$\mathbf{Hom}(\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}),{\rm
End}_{sV})$. One computes the maps $m_{n}$ as follows.

  * (i)

For homogeneous elements $f,g\in{\rm End}_{sV}$,
$m_{2}(\widehat{f}\otimes\widehat{g})=\widehat{f\\{g\\}};$

  * (ii)

For homogeneous elements $f,g\in{\rm End}_{sV}$,
$m_{2}(\overline{f}\otimes\widehat{g})=(-1)^{|\widehat{g}|}\overline{f\\{g\\}};$

  * (iii)

For homogeneous elements $f_{0},f_{1},\dots,f_{n}\in{\rm End}_{sV},n\geq 1$,

$m_{n+1}(\widehat{f_{0}}\otimes\overline{f_{1}}\otimes\cdots\otimes\overline{f_{n}})=(-1)^{(n+1)|\widehat{f_{0}}|+\sum_{k=1}^{n}(n-k)|\overline{f_{k}}|}\lambda^{n-1}\overline{f_{0}\\{f_{1},\dots,f_{n}\\}};$

  * (iv)

All other components of operators $\\{m_{n}\\}_{n\geq 1}$ vanish.

Furthermore, the maps $\\{\ell_{n}\\}_{n\geqslant 1}$ are given as follows.

  * (i)

For homogeneous elements $f,g\in{\rm End}_{sV}$,

$\begin{array}[]{rcl}\ell_{2}(\widehat{f}\otimes\widehat{g})&=&m_{2}(\widehat{f}\otimes\widehat{g})-(-1)^{|\widehat{f}||\widehat{g}|}m_{2}(\widehat{g}\otimes\widehat{f})\\\
&=&\widehat{f\\{g\\}}-(-1)^{|f||g|}\widehat{g\\{f\\}}\\\
&=&\widehat{[f,g]_{G}};\end{array}$

  * (ii)

For homogeneous elements $f,g\in{\rm End}_{sV}$,

$\begin{array}[]{rcl}\ell_{2}(\widehat{f}\otimes\overline{g})&=&m_{2}(\widehat{f}\otimes\overline{g})-(-1)^{|\widehat{f}||\overline{g}|}m_{2}(\overline{g}\otimes\widehat{f})\\\
&=&\overline{f\\{g\\}}-(-1)^{|f||g|}\overline{g\\{f\\}}\\\
&=&\overline{[f,g]_{G}};\end{array}$

  * (iii)

For homogeneous elements $f_{0},f_{1},\dots,f_{n}\in{\rm End}_{sV}$ with
$n\geq 1$,

$\displaystyle\ell_{n+1}(\widehat{f_{0}}\otimes\overline{f_{1}}\otimes\cdots\otimes\overline{f_{n}})=$
$\displaystyle\sum_{\sigma\in
S_{n}}\chi(\sigma;\overline{f_{1}},\dots,\overline{f_{n}})m_{n}(\widehat{f_{0}}\otimes\overline{f_{\sigma(1)}}\otimes\cdots\otimes\overline{f_{\sigma(n)}})$
$\displaystyle=$ $\displaystyle\sum_{\sigma\in
S_{n}}(-1)^{(n+1)|\widehat{f_{0}}|+\sum_{k=1}^{n-1}(n-k)|\overline{f_{\sigma(k)}}|}\lambda^{n-1}\overline{f_{0}\\{f_{\sigma(1)},\dots,f_{\sigma(n)}\\}};$

  * (iv)

For homogeneous elements $f_{0},f_{1},\dots,f_{n}\in{\rm End}_{sV}$ with
$n\geq 1$ and $1\leq i\leq n$,

$\ell_{n+1}(\overline{f_{1}}\otimes\cdots\otimes\overline{f_{i}}\otimes\widehat{f_{0}}\otimes\overline{f_{i+1}}\otimes\cdots\otimes\overline{f_{n}})=(-1)^{i+|\widehat{f_{0}}|(\sum_{k=1}^{i}|\overline{f_{k}}|)}\ell_{n+1}(\widehat{f_{0}}\otimes\overline{f_{1}}\otimes\cdots\otimes\overline{f_{n}});$

  * (v)

All other components of the operators $\\{\ell_{n}\\}_{n\geq 1}$ vanish.

Via the bijections $f\mapsto\widehat{f}$ and $g\mapsto\overline{g}$, it is
readily verified that the above $L_{\infty}$-structure on
$\mathbf{Hom}(\mathscr{S}({{}_{\lambda}\\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}}),{\rm
End}_{sV})$ is exactly the one on
${\mathfrak{C}}_{{\mathsf{DA}_{\lambda}}}(V)$ defined in Eq (5), thereby
proving Theorem 2.2.

###  5.2. Another description of homotopy differential algebras

We end the paper with a characterization of the key notion of homotopy
differential algebras and give a generalization of a work of Kajura and
Stasheff [23].

By the last paragraph of §3.1, there exists another definition of homotopy
differential algebras in terms of the Maurer-Cartan elements in the
$L_{\infty}$-algebra on the deformation complex. Let us make it precise.

Let $V$ be a graded space. Denote

$\overline{T}(V)\coloneqq\oplus_{n\geq 1}V^{\otimes n},\
\overline{{\mathfrak{C}}_{\mathrm{Alg}}}(V):={\rm Hom}(\overline{T}(sV),sV),\
\overline{{\mathfrak{C}}_{\mathsf{DO}_{\lambda}}}(V):={\rm
Hom}(\overline{T}(sV),V).$

We consider the subspace

$\overline{{{\mathfrak{C}}}_{{\mathsf{DA}_{\lambda}}}}(V):=\overline{{\mathfrak{C}}_{\mathrm{Alg}}}(V)\oplus\overline{{\mathfrak{C}}_{\mathsf{DO}_{\lambda}}}(V)$

of ${\mathfrak{C}}_{\mathsf{DA}_{\lambda}}(V)$ in Eq. (5). It is easy to see
that $\overline{{{\mathfrak{C}}}_{{\mathsf{DA}_{\lambda}}}}(V)$ is an
$L_{\infty}$-subalgebra of ${\mathfrak{C}}_{{\mathsf{DA}_{\lambda}}}(V)$.

By the remark at the end of §3.1, Definitions 4.12 can be rephrased as
follows.

######  Proposition 5.1.

Let $V$ be a graded space. A homotopy differential algebra structure of weight
$\lambda$ on $V$ is defined to be a Maurer-Cartan element in the
$L_{\infty}$-algebra
$\overline{{{\mathfrak{C}}}_{{\mathsf{DA}_{\lambda}}}}(V)$.

Notice that if we take the whole space
${\mathfrak{C}}_{{\mathsf{DA}_{\lambda}}}(V)$ instead of the reduced part
$\overline{{\mathfrak{C}}_{\mathrm{Alg}}}(V)$, we get the curved homotopy
differential algebra structure.

Solving the Maurer-Cartan equation in the $L_{\infty}$-algebra
$\overline{{{\mathfrak{C}}}_{{\mathsf{DA}_{\lambda}}}}(V)$, Proposition 5.1
can be explicated as follows.

######  Proposition 5.2.

A homotopy differential algebra structure of weight $\lambda$ on a graded
space $V$ is equivalent to two families of linear maps

$b_{n}:(sV)^{\otimes n}\rightarrow sV,\quad R_{n}:(sV)^{\otimes n}\rightarrow
V,\quad n\geq 1,$

both of degree $-1$ and subject to the following equations:

$\sum_{i+j-1=n\atop i,j\geqslant 1}b_{i}\\{b_{j}\\}=0\ \mathrm{and}\
\sum_{u+j-1=n\atop u,j\geqslant
1}sR_{u}\\{b_{j}\\}=\sum_{l_{1}+\dots+l_{q}+p-q=n\atop{l_{1},\dots,l_{q}\geqslant
1\atop 1\leqslant q\leqslant p\leqslant
n}}\lambda^{q-1}b_{p}\\{sR_{l_{1}},\dots,sR_{l_{q}}\\}.$

######  Remark 5.3.

  1. (i)

The equivalence between Definition 4.13 and Proposition 5.2 is given by

$m_{n}:=s^{-1}\circ b_{n}\circ s^{\otimes n}:V^{\otimes n}\rightarrow V,\quad
d_{n}:=R_{n}\circ s^{\otimes n}:V^{\otimes n}\rightarrow V,\ n\geqslant 1.$

  2. (ii)

Thanks to Proposition 5.2, Definition 4.13 generalizes the notion of homotopy
derivations on $A_{\infty}$-algebras introduced by Kajura and Stasheff [23]
from zero weight to nonzero weight.

Acknowledgments.  This work is supported by NSFC (12071137, 11971460) and by
STCSM (22DZ2229014).

Declaration of interests.  The authors have no conflicts of interest to
disclose.

Data availability.  No new data were created or analyzed in this study.

## References

  * [1]
  * [2] T. Araki, T. Takashima, S. Watamura, Instantons in non(anti)commutative gauge theory via deformed ADHM construction, in: Noncommutative Geometry and Physics 2005, pp. 253-269, World Scientific, 2007.
  * [3] C. Bai, R. Bai, L. Guo, Y. Wu, Transposed Poisson algebras, Novikov-Poisson algebras and $3$-Lie algebras, arXiv:2005.01110.
  * [4] R. F. Blute, J. R. B. Cockett, J.-S. P. Lemay, R. A. G. Seely, Differential categories revisited. Appl. Categ. Structures 28 (2020) 171-235.
  * [5] M. R. Bremner, V. Dotsenko, Algebraic Operads. An Algorithmic Companion, CRC Press, 2016.
  * [6] J. R. B. Cockett, J.-S. Lemay, Integral categories and calculus categories, Math. Structures Comput. Sci. 29 (2019) 243-308.
  * [7] A. Connes, M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives, Amer. Math. Soc. 2019
  * [8] V. Dotsenko, N. Poncin, A tale of three homotopies, Appl. Categ. Structures 24 (2016) 845-873.
  * [9] M. Doubek, T. Lada, Homotopy derivations. J. Homotopy Relat. Struct. 11 (2016) 599-630.
  * [10] G. Drummond-Cole, B. Vallette, The minimal model for the Batalin-Vilkovisky operad, Selecta Math. 19 (2013) 1-47.
  * [11] J. Freitag, W. Li, T. Scanlon, Differential Chow varieties exist, J. Lond. Math. Soc. 95 (2017) 128-156.
  * [12] I. M. Gelfand, I. Ya. Dorfman, Hamiltonian operators and algebraic structures related to them, Funct. Anal. Appl. 13 (1979) 248-262.
  * [13] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math. 78 (1963) 267-288.
  * [14] M. Gerstenhaber, On the deformation of rings and algebras, Ann. Math. 79 (1964) 59-103.
  * [15] M. Gerstenhaber, A. Voronov, Homotopy G-algebras and moduli space operad, Int. Math. Res. Not. (1995) 141-153.
  * [16] E. Getzler, Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology, Israel Math. Conf. Proc. 7 (1993) 65-78.
  * [17] E. Getzler, Lie theory for nilpotent $L_{\infty}$-algebras, Ann. Math. 170 (2009) 271-301.
  * [18] E. Getzler, D. S. J. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, arXiv:9403055.
  * [19] V. Ginzburg, M. Kapranov, Koszul duality for operads. Duke Math. J. 76 (1994) 203-272.
  * [20] L. Guo, W. Keigher, On differential Rota-Baxter algebras. J. Pure Appl. Algebra 212 (2008) 522-540.
  * [21] L. Guo, F. Li, Structure of Hochschild cohomology of path algebras and differential formulation of Euler’s polyhedron formula, Asian J. Math. 18 (2014) 545-572.
  * [22] L. Guo, Y. Li, Y. Sheng, G. Zhou, Cohomologies, extensions and deformations of differential algebras with arbitrary weight, Theory Appl. Categ. 8 (2022) 1409-1433.
  * [23] H. Kajiura, J. Stasheff, Homotopy algebras inspired by classical open-closed string field theory. Comm. Math. Phys. 263 (2006) 553-581.
  * [24] I. Kaplansky, An Introduction to Differential Algebra, Hermann, 1976.
  * [25] E. R. Kolchin, Differential Algebras and Algebraic Groups, Academic Press, New York, 1973.
  * [26] P. S. Kolesnikov, B. Sartayev, A. Orazgaliev, Gelfand-Dorfman algebras, derived identities, and the Manin product of operads, J. Algebra 539 (2019) 260-284.
  * [27] M. Kontsevich, Y. Soibelman, Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato 1999, Vol. I (Dijon), Math. Phys. Stud. 21 (2000) 255-307.
  * [28] T. Lada, J. Stasheff, Introduction to sh Lie algebras for physicists, Internat. J. Theoret. Phys. 3(2) (1993) 1087-1103.
  * [29] T. Lada, M. Markl, Strongly homotopy Lie algebras, Comm. Algebra 23 (1995) 2147-2161.
  * [30] T. Lada, M. Tolley, Derivations of homotopy algebras, Arch. Math. (Brno) 49 (2013) 309-315.
  * [31] J. S. Pacaud Lemay, Differential algebras in codifferential categories, J. Pure Appl. Algebra 223 (2019) 4191-4225.
  * [32] G. Liu, C. Bai, Anti-pre-Lie algebras, Novikov algebras and commutative $2$-cocycles on Lie algebras, J. Algebra 609 (2022) 337-379.
  * [33] J.-L. Loday, On the operad of associative algebras with derivation, Georgian Math. J. 17 (2010) 347-372.
  * [34] J.-L. Loday, B. Vallette, Algebraic Operads, Springer, 2012.
  * [35] J. Lurie, DAG X: Formal moduli problems, https://www.math.ias.edu/ lurie/.
  * [36] A. R. Magid, Lectures on Differential Galois Theory, Amer. Math. Soc. 1994.
  * [37] Y. Pei, Y. Sheng, R. Tang, K. Zhao, Actions of monoidal categories and representations of Cartan type Lie algebras, J. Inst. Math. Jussieu (2022), https://doi.org/10.1017/S147474802200007X.
  * [38] M. Markl, Cotangent cohomology of a category and deformations, J. Pure Appl. Algebra 113 (1996) 195-218.
  * [39] M. Markl, Operads and PROPs, Handbook of algebra. 5, 87-140, Elsevier/North-Holland, 2008.
  * [40] M. Markl, S. Shnider, J. D. Stasheff, Operads in Algebra, Topology and Physics, Amer Math Soc. 2002.
  * [41] A. Medvedev, T. Scanlon, Invariant varieties for polynomial dynamical systems, Ann. Math. 179 (2014) 81-177.
  * [42] S. Merkulov, B. Vallette, Deformation theory of representations of prop(erad)s. I. J. Reine Angew. Math. 634 (2009) 51-106.
  * [43] S. Merkulov, B. Vallette, Deformation theory of representations of prop(erad)s. II. J. Reine Angew. Math. 636 (2009) 123-174.
  * [44] A. Nijenhuis, R. Richardson, Cohomology and deformations in graded Lie algebras, Bull. Amer. Math. Soc. 72 (1966) 1-29.
  * [45] L. Poinsot, Differential (monoid) algebra and more, Lecture Notes in Comput. Sci. 8372, Springer, (2014) 164-189.
  * [46] L. Poinsot, Differential (Lie) algebras from a functorial point of view, Adv. Appl. Math. 72 (2016) 38-76.
  * [47] J. P. Pridham, Unifying derived deformation theories, Adv. Math. 224 (2010) 772-826.
  * [48] M. Van der Put, M. Singer, Galois Theory of Linear Differential Equations, Springer, 2003.
  * [49] J. F. Ritt, Differential equations from the algebraic standpoint, Amer. Math. Soc., 1934.
  * [50] J. Stasheff, Homotopy associativity of H-spaces I, Trans. Amer. Math. Soc. 108 (1963) 275-292.
  * [51] J. Stasheff, Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras, Lecture Notes in Mathematics 1510, Springer (1992) 120-137.
  * [52] R. Tang, C. Bai, L. Guo, Y. Sheng, Deformations and their controlling cohomologies of $O$-operators, Comm. Math. Phys. 368 (2019) 665-700.
  * [53] R. Tang, Y. Frégier, Y. Sheng, Cohomologies of a Lie algebra with a derivation and applications, J. Algebra 534 (2019) 65-99.
  * [54] P. Van der Laan, Operads up to homotopy and deformations of operad maps, arXiv 0208041.
  * [55] P. Van der Laan, Coloured Koszul duality and strongly homotopy operads, arXiv 0312147.
  * [56] W.-T. Wu, A constructive theory of differential algebraic geometry based on works of J. F. Ritt with particular applications to mechanical theorem-proving of differential geometries, Lecture Notes in Math. 1255, Springer-Verlag, (1987) 173-189.
  * [57] X. Xu, Novikov-Poisson algebras, J. Algebra 190 (1997) 253-279.