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

TL;DR

This paper extends Koszul duality theory to differential algebras with nonzero weight, constructing their minimal models and homotopy structures, and exploring the resulting deformation complexes.

Contribution

It establishes Koszul duality for weighted differential algebras, a case not previously treated due to non-quadratic operads, and develops associated homotopy and deformation theories.

Findings

Koszul duality is established for nonzero weight differential algebras.

Minimal models and homotopy cooperads are explicitly constructed.

Deformation complexes with $L_$-structure are derived.

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.

Equations

$$d(uv)=d(u)v+ud(v)+\lambda d(u)d(v),u,v\in A.$$

$$d_M(ax)$$

$$d_M(xa)$$

$$a⊢v=(a+\lambda d_A(a))x,x⊣a=x(a+\lambda d_A(a)),a\in A,x\in M.$$

$$({\mathrm{C}}_{\mathrm{Alg}}^{\bullet }(A,M):=\underset{n=0}{\overset{\infty }{⨁}}{\mathrm{C}}_{\mathrm{Alg}}^n(A,M),{\partial }_{\mathrm{Alg}}^{\bullet }),$$

$${\partial }_{\mathrm{Alg}}^n:{\mathrm{C}}_{\mathrm{Alg}}^n(A,M)⟶{\mathrm{C}}_{\mathrm{Alg}}^{n+1}(A,M),n⩾0,$$

$${\partial }_{\mathrm{Alg}}^n(f)(a_{1,n+1}):=(-1{)}^{n+1}{a}_{1}f(a_{2,n+1})+\sum _{i=1}^n(-1{)}^{n+1-i}f(a_{1,i-1}\otimes a_i{a}_{i+1}\otimes a_{i+2,n+1})+f(a_{1,n})a_{n+1}$$

$$\begin{array}{rl}{\partial }_{{\mathsf{DO}}_{\lambda }}^{n}g(a_{1,n+1}):=& (-1{)}^{n+1}{a}_1⊢g(a_{2,n+1})+\sum _{i=1}^n(-1{)}^{n+1-i}g(a_{1,i-1}\otimes a_i{a}_{i+1}\otimes a_{i+2,n+1})+g(a_{1,n})⊣a_{n+1}\\ =& (-1{)}^{n+1}(a_1+\lambda d_A(a_1))g(a_{2,n+1})+\sum _{i=1}^n(-1{)}^{n+1-i}g(a_{1,i-1}\otimes a_i{a}_{i+1}\otimes a_{i+2,n+1})\\ & +g(a_{1,n})(a_{n+1}+\lambda d_A(a_{n+1})).\end{array}$$

$$\begin{array}{rl}{\Phi }^n(f)(a_{1,n}):=& \sum _{k=1}^n{\lambda }^{k-1}\sum _{1⩽i_1<\cdots <i_k⩽n}f(a_{1,i_1-1}\otimes d_A(a_{i_1})\otimes a_{i_1+1,i_2-1}\otimes d_A(a_{i_2})\otimes \cdots \otimes d_A(a_{i_k})\otimes a_{i_k+1,n})\\ & -d_M(f(a_{1,n})),\end{array}$$

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

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

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

$${\partial }_{{\mathsf{DA}}_{\lambda }}^0(x):=({\partial }_{\mathrm{Alg}}^0(x),-{\Phi }^0(x)),x\in {\mathrm{C}}_{\mathrm{Alg}}^0(A,M)=M.$$

$$x_1\odot x_2\odot \cdots \odot x_n=ϵ(\sigma ;x_1,\dots ,x_n)x_{\sigma (1)}\odot x_{\sigma (2)}\odot \cdots \odot x_{\sigma (n)}\in S(V).$$

$$\chi (\sigma ;x_1,\dots ,x_n)=\mathrm{sgn}(\sigma )ϵ(\sigma ;x_1,\dots ,x_n),$$

$$l_n(x_{\sigma (1)}\otimes \cdots \otimes x_{\sigma (n)})=\chi (\sigma ;x_1,\dots ,x_n)l_n(x_1\otimes \cdots \otimes x_n),\sigma \in {\mathrm{S}}_n;$$

$$\sum _{i=1}^n\sum _{\sigma \in \mathrm{Sh}(i,n-i)}\chi (\sigma ;x_1,\dots ,x_n)(-1{)}^{i(n-i)}{l}_{n-i+1}(l_i(x_{\sigma (1)}\otimes \cdots \otimes x_{\sigma (i)})\otimes x_{\sigma (i+1)}\otimes \cdots \otimes x_{\sigma (n)})=0.$$

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

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

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

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

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

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

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

$$l_2(sf\otimes sg)=[sf,sg{]}_G\in {\mathfrak{C}}_{\mathrm{Alg}}(V),$$

$$l_2(sf\otimes g)=(-1{)}^{|f|+1}{s}^{-1}[sf,sg{]}_G\in {\mathfrak{C}}_{{\mathsf{DO}}_{\lambda }}(V).$$

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

$$l_{m+1}(g_1\otimes \cdots \otimes g_k\otimes sf\otimes g_{k+1}\otimes \cdots \otimes g_m)\in {\mathfrak{C}}_{{\mathsf{DO}}_{\lambda }}(V)$$

$$l_{m+1}(g_1\otimes \cdots \otimes g_k\otimes sf\otimes g_{k+1}\otimes \cdots \otimes g_m):=(-1{)}^{(|f|+1)(\sum _{j=1}^k|g_j|)+k}{l}_{n+1}(sf\otimes g_1\otimes \cdots \otimes g_m),$$

$$\sum _{n=1}^{\infty }\frac{1}{n!}(-1{)}^{\frac{n(n-1)}{2}}{l}_n({\alpha }^{\otimes n})=0$$

$$l_1(\alpha )-\frac{1}{2}{l}_2(\alpha \otimes \alpha )=0.$$

Full text

1. 1 Introduction

2. 1.1 Differential algebras with weight

3. 1.2 Cohomology and deformations of differential algebras

4. 1.3 Our approach and layout of the paper

5. 2 The $L_{\infty}$-algebra controlling deformations of differential algebras

6. 2.1 Differential algebras and their cohomology

7. 2.2 $L_{\infty}$-algebra on the deformation complex from differential algebras

8. 2.3 Realising differential algebra structures as Maurer-Cartan elements

9. 3 The Koszul dual homotopy cooperad

10. 3.1 Homotopy (co)operads and $L_{\infty}$-algebras

11. 3.2 Koszul dual homotopy cooperad of ${{}_{\lambda}\!\mathfrak{Dif}}$

12. 4 Operad of homotopy differential algebras with weight and minimal model

13. 4.1 Operad of homotopy differential algebras with weight

14. 4.2 The minimal model of the operad of differential algebras

15. 4.3 Homotopy differential algebras with weight

16. 5 From the minimal model to the $L_{\infty}$-algebra controlling deformations

17. 5.1 Proof of Theorem 2.2

18. 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 problem raised by Loday [35].

1.1. Differential algebras with weight

For a fixed scalar $\lambda$, a differential operator of weight $\lambda$ [20] (called $\lambda$-derivation in [35]) is a linear operator $d=d_{\lambda}$ on an associative algebra $A$ satisfying the operator identity

[TABLE]

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 [51] in his algebraic study of differential equations. Through the works of Kaplansky, Kolchin, Magid, Singer and many others [25, 26, 38, 50], 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, such as noncommutative Geometry, quantum Fields and motives [7], differential algebraic geometry  [11], dynamical systems  [43], and mechanical theorem-proving  [58].

A commutative differential algebra naturally gives rise to a Novikov algebra by the well-known work of Gelfand and Dorfman [12], see also [27]. Differential algebras are also closed related to newly emerged structures including Novikov-Poisson algebras [59], transposed Poisson algebras [3], anti pre-Lie (Poisson) algebras [34], and Novikov bialgebras [23].

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, 47, 48]. From the categorical viewpoint, differential categories were introduced and then studies in [4, 6, 33]. 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, 46], 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 [37] and Pridham [49], 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 [36, 42] 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. The developments began with [54] which defined the controlling cohomologie as well as the dg Lie algebra for relative Rota-Baxter operators on Lie algebras. Subsequently, the controlling cohomologie as well as the $L_{\infty}$-structure for relative Rota-Baxter Lie algebras were found in [32]; derivations (=differential operators of weight zero) on Lie algebras were considered and their cohomology theory was introduced in [55]; a cohomology theory of crossed homomorphisms (=differential operators of weight $1$) on Lie algebras was established in [39], together with a Maurer-Cartan characterisation. The reference [22] introduced a cohomology theory of differential associative algebras of arbitrary weight by an ad hoc method. The paper is a predecessor of the present one and whose results will be recalled in Subsection 2.1.

Due to the complexity of the algebraic structures, the $L_{\infty}$-algebras in these studies so far have been obtained by direct constructions or by using the derived bracket technique without using the language of operads.

The first operadic study of algebras with linear operators is the 2010 work of Loday [35] 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 works [9, 24, 31]. For the case of nonzero weight, Loday made the following observation in [35], 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 [28].

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 [35]. 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 ${{}_{\lambda}\!\mathfrak{Dif}}$.

Her is an outline of the paper.

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 [math]. 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 $x\in M,$

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

[TABLE]

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 ${\mathrm{C}}^{\bullet}_{\mathrm{Alg}}(A,{}_{\vdash}M_{\dashv})$ 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 $({\mathrm{C}}^{\bullet}_{\mathsf{DO}_{\lambda}}(A,M),\partial_{{\mathsf{DO}_{\lambda}}}^{\bullet})$. More precisely, for $g\in{\mathrm{C}}^{n}_{\mathsf{DO}_{\lambda}}(A,M)\coloneqq{\rm Hom}(A^{\otimes n},M)$ and $a_{1},\dots,a_{n+1}\in A$, we have

[TABLE]

We write $({\mathrm{C}}^{\bullet}_{\mathsf{DO}_{\lambda}}(A),\partial_{{\mathsf{DO}_{\lambda}}}^{\bullet})\coloneqq({\mathrm{C}}^{\bullet}_{\mathsf{DO}_{\lambda}}(A,M),\partial_{{\mathsf{DO}_{\lambda}}}^{\bullet})$ by taking the differential bimodule $M$ to be the regular differential bimodule $A$. The cohomology of the cochain complex $({\mathrm{C}}^{\bullet}_{{\mathsf{DO}_{\lambda}}}(A,M),\partial_{{\mathsf{DO}_{\lambda}}}^{\bullet})$, 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 $\Phi^{\bullet}:{\mathrm{C}}^{\bullet}_{\mathrm{Alg}}(A,M)\to{\mathrm{C}}^{\bullet}_{\mathsf{DO}_{\lambda}}(A,M)$ as follows: for $f\in{\mathrm{C}}^{n}_{\mathrm{Alg}}(A,M)$ with $n\geqslant 1$, define

[TABLE]

and

[TABLE]

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,

[TABLE]

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

[TABLE]

[TABLE]

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

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, 29, 30, 53].

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 $\sigma\in{\rm S}_{n}$, the Koszul sign $\epsilon(\sigma;x_{1},\dots,x_{n})$ is defined by the equality

[TABLE]

We also define

[TABLE]

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 $n\geqslant 1$ and homogeneous elements $x_{1},\dots,x_{n}\in L$,

1. (i)

(generalized anti-symmetry)

[TABLE] 2. (ii)

(generalized Jacobi identity)

[TABLE]

Here ${\mathrm{Sh}}(i,n-i)$ is the set of $(i,n-i)$ shuffles, consisting of $\sigma\in{\rm S}_{n}$ such that

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE] 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

[TABLE]

and define the graded space

[TABLE]

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

[TABLE]

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

[TABLE] 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

[TABLE]

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

[TABLE]

to be

[TABLE]

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 [28, 56, 57] 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:

[TABLE]

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

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

[TABLE]

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 $\{l_{n}^{\alpha}:L^{\otimes n}\rightarrow L\}_{n\geqslant 1}$ on the graded space $L$ is defined by

[TABLE]

provided the infinite sums exist. The resulting $L_{\infty}$-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

[TABLE]

We fix the isomorphism

[TABLE]

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 [math]. 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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

We obtain

[TABLE]

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 [math] 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

[TABLE]

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:

[TABLE]

if and only if

[TABLE]

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 $g\in{\mathrm{C}}^{k}_{\mathsf{DO}_{\lambda}}(A)$,

[TABLE]

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, 40, 44, 45]. We also explain how $L_{\infty}$-structures can be obtained from homotopy operads, in particular from convolution homotopy operads.

A graded collection is a family ${\mathcal{O}}=\{{\mathcal{O}}(n)\}_{n\geqslant 1}$ of graded space indexed by positive integers. So each ${\mathcal{O}}(n)$ itself is 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 $\{s{\mathcal{O}}(n)\}_{n\geqslant 1}$. The desuspension $s^{-1}{\mathcal{O}}$ of the graded collection ${\mathcal{O}}$ is defined in the same way.

We also recall preliminaries about trees. As we only consider planar reduced rooted trees, we will simply call them trees. For a tree $T\in\mathfrak{T}$, denote the weight (=number of vertices) and arity (=number of leaves) of $T$ by $\omega(T)$ and $\alpha(T)$ respectively. Let $\mathfrak{T}$ be the set of all trees with weight $\geqslant 1$ and, for $n\geqslant 1$, let $\mathfrak{T}^{(n)}$ be the set of trees of weight $n$. Since the trees are planar, the inputs of each vertex in a tree have total order by going clockwise. Furthermore by the existence of the root, there is a naturally induced total order on the set of all vertices of the tree $T$, 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 $T$. Let $T/T^{\prime}$ denote 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 $\sigma=\sigma(T,T^{\prime})\in{\rm S}_{\omega(T)}$ associated to the pair $(T,T^{\prime})$ defined as follows. Let $\{v_{1}<\dots<v_{n}\}$ be the sequence of all vertices of $T$ listed in the planar order and let $\omega(T^{\prime})=j$. Let $v^{\prime}$ be the vertex in $T/T^{\prime}$ corresponding to the divisor $T^{\prime}$ of $T$ 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 $\sigma=\sigma(T,T^{\prime})\in{\rm S}_{n}$ to be the unique permutation with the properties that it does not permute the vertices $v_{1},\dots,v_{i-1}$, 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 let $\{v_{1}<\dots<v_{n}\}$ be the sequence of all vertices of $T$ in the planar order. Let ${\mathcal{P}}^{\otimes T}$ denote ${\mathcal{P}}(\alpha(v_{1}))\otimes\cdots\otimes{\mathcal{P}}(\alpha(v_{n}))$. Its element can be pictured as 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 ${\mathcal{P}}=\{{\mathcal{P}}(n)\}_{n\geqslant 1}$ is a family of operations

[TABLE]

with $|m_{T}|=\omega(T)-2$, such that, for all $T\in\mathfrak{T}$, the equation

[TABLE]

holds. Here $T^{\prime}$ runs through subtrees of $T$, $i$ is the serial number of the vertex $v^{\prime}$ in $T/T^{\prime}$ as above, $j=\omega(T^{\prime})$, $k=\omega(T)-i-j$, and $r_{\sigma}$ denotes the right action by $\sigma=\sigma(T,T^{\prime})$:

[TABLE]

A strict morphism between two homotopy operads is a morphism of graded collections that is 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 naturally endowed with a homotopy operad structure: when $T$ is the tree with two vertices and a 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

[TABLE]

and

[TABLE]

are identity maps on ${\mathcal{P}}(n)$, where for $1\leqslant i\leqslant n$, $T_{1,i}$ 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 $1$. Moreover, for a tree $T$ of weight at least three, $m_{T}({\rm Id}^{\otimes i-1}\otimes\eta\otimes{\rm Id}^{\otimes\omega(T)-i})=0$ for each $1\leqslant i\leqslant\omega(T)$.

Also a homotopy operad ${\mathcal{P}}$ is called augmented if there is 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 $\{\Delta_{T}:{\mathcal{C}}(\alpha(T))\rightarrow{\mathcal{C}}^{\otimes T}\}_{T\in\mathfrak{T}}$ 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

[TABLE]

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 notions 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 [41].

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

This provides an alternative definition for homotopy cooperads. 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 define $l_{n}$ to be the anti-symmetrization of $m_{n}$, that is,

[TABLE]

The we have the following conclusion.

Proposition 3.5**.**

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

For a homotopy operad ${\mathcal{P}}$ satisfying $m_{T}=0$ for all $T\in{\mathfrak{T}}$ with $\omega(T)\geqslant 3$, the operad ${\mathcal{P}}$ reduces to the nonunital dg operad in the sense of Markl [41] and ${\mathcal{P}}^{\prod}$ is just a dg Lie algebra. The operation $l_{2}$ in the dg Lie algebra ${\mathcal{P}}^{\prod}$ recovers the Gerstenhaber bracket defined in (3).

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

[TABLE]

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

[TABLE]

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

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

[TABLE]

In particular, the following equation holds.

[TABLE]

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 $\omega(T)=1$,

[TABLE] 2. (ii)

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

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)

[TABLE]

where $n\geqslant 1,1\leqslant j\leqslant n,1\leqslant i\leqslant n-j+1$.

• (ii)

[TABLE]

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

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

[TABLE]

• (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

[TABLE]

• (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

[TABLE]

• (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 $\mathscr{S}({{}_{\lambda}\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})$ (Definition 3.11 $\mathrm{(i)-(iv)}$ ) and $\Omega\mathscr{S}({{}_{\lambda}\!\mathfrak{Dif}}^{\mathrm{\textup{!`}}})$ (Definition 3.4), the action of $\partial$ on generators $\mu_{n}=s^{-1}\widetilde{m_{n}}$ and $v_{n}=s^{-1}\widetilde{d_{n}}$ can be writted in terms of trees as:

[TABLE]

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 $\mu_{n},n\geqslant 2$ and $\nu_{n},n\geqslant 1$.

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

[TABLE]

where we use the pre-Jacobi identity (12) implicitly in the second-to-last equality.

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

[TABLE]

Here, the pre-Jacobi identity (11) is implicitly used in the last equation above. Notice that all trees appear twice with opposite signs. 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

[TABLE]

• (ii)

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

[TABLE]

• (iii)

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

[TABLE]

where

[TABLE]

• (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 $n\geqslant 2$,

[TABLE]

and for each $n\geqslant 1$,

[TABLE]

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 [35] 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 ${{}_{\lambda}\!\mathfrak{Dif}_{\infty}}$:

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

Eqs. (16) and (17) can be represented by

[TABLE]

[TABLE]

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

[TABLE]

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

[TABLE]

[TABLE]

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

[TABLE]

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

[TABLE]

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:

[TABLE]

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 $T$.

Intuitively, 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:

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

where $\{T_{i_{k}}\}_{i_{k}\in I_{k}},k\geqslant 1,$ 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. (19) 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 easily seen that $\max\{T_{i_{k}}|i_{k}\in I_{k}\}>\max\{T_{i_{k+1}}|i_{k+1}\in I_{k+1}\}$ for all $k\geqslant 1$, where by convention, $i_{1}\in I_{1}=\{1\}$. Then Eq. (19) is a finite sum since $>$ is a well order. ∎

Lemma 4.10**.**

For an effective tree monomial $T\in{{}_{\lambda}\!\mathfrak{Dif}_{\infty}}$, we have

[TABLE]

Proof.

Express $T$ via partial compositions as follows:

[TABLE]

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,

[TABLE]

and

[TABLE]

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

[TABLE]

and

[TABLE]

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

[TABLE]

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

[TABLE]

Proposition 4.11**.**

The degree $1$ map $H$ defined in Eq. (19) 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

[TABLE]

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,

[TABLE]

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

[TABLE]

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

[TABLE]

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:

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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 $n\geqslant 1$,

[TABLE]

[TABLE]

where

[TABLE]

Notice that Eq. (20) is exactly the Stasheff identity defining $A_{\infty}$-algebras [52]. 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. (21) 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

[TABLE]

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

[TABLE]

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:

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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 $\{\ell_{n}\}_{n\geqslant 1}$ of ${\mathfrak{C}}_{{{}_{\lambda}\!\mathfrak{Dif}}}(V)$. Recall that for homogeneous elements $x_{i}\in{\mathfrak{C}}_{{{}_{\lambda}\!\mathfrak{Dif}}}(V),1\leqslant i\leqslant n$,

[TABLE]

with

[TABLE]

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 $f_{0},f_{1},\dots,f_{n}\in{\rm End}_{sV},n\geqslant 1$,

[TABLE]

• (iv)

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

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

• (i)

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

[TABLE]

• (ii)

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

[TABLE]

• (iii)

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

[TABLE]

• (iv)

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

[TABLE]

• (v)

All other components of the operators $\{\ell_{n}\}_{n\geqslant 1}$ 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 [24].

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

[TABLE]

We consider the subspace

[TABLE]

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

[TABLE]

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

[TABLE]

Remark 5.3**.**

1. (i)

The equivalence between Definition 4.13 and Proposition 5.2 is given by

[TABLE] 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 [24] from zero weight to nonzero weight.

**Acknowledgments. ** This work was supported by the National Natural Science Foundation of China (No. 12071137, 11971460), by Key Laboratory of MEA (Ministry of Education), by Shanghai Key Laboratory of PMMP (No. 22DZ2229014), and by Fundamental Research Funds for the Central Universities.

The authors would like to express our sincere gratitude to the referee for helpful suggestions which improve much the presentation of the paper.

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

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