# Formal deformations, cohomology theory and L∞[1]subscript𝐿delimited-[]1L_{\infty}[1]italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT [ 1 ]-structures for differential Lie algebras of arbitrary weight
**Authors**: Weiguo Lyu, Zihao Qi, Jian Yang and Guodong Zhou
> Weiguo Lyu School of Mathematical Sciences, Anhui University, Hefei 230601, China Zihao Qi School of Mathematical Sciences, Fudan University, Shanghai 200433, China Jian Yang and Guodong Zhou School of Mathematical Sciences, Key Laboratory of Mathematics and Engineering Applications (Ministry of Education), Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China
(Date: May 2, 2024)
## Abstract
Generalising a previous work of Jiang and Sheng, a cohomology theory for differential Lie algebras of arbitrary weight is introduced. The underlying $L_∞[1]$ -structure on the cochain complex is also determined via a generalised version of higher derived brackets. The equivalence between $L_∞[1]$ -structures for absolute and relative differential Lie algebras are established. Formal deformations and abelian extensions are interpreted by using lower degree cohomology groups. Also we introduce the homotopy differential Lie algebras. In a forthcoming paper, we will show that the operad of homotopy (relative) differential Lie algebras is the minimal model of the operad of (relative) differential Lie algebras.
Key words and phrases: abelian extension, cohomology, deformation, derived bracket, differential Lie algebra, $L_∞[1]$ -algebra
2010 Mathematics Subject Classification: 16E40 16S80 12H05 12H10 16W25 16S70 Contents
1. 0.1 Deformation theory and minimal models
1. 0.2 Differential Lie algebras, old and new
1. 0.3 Layout of the paper
1. 1 Differential Lie algebras of arbitrary weight and their cohomology theory
1. 1.1 Notations
1. 1.2 Differential Lie algebras with weight and their representations
1. 1.3 Cohomology of differential operators
1. 1.4 Cohomology of differential Lie algebras
1. 2 Abelian extensions of differential Lie algebras
1. 2.1 Abelian extensions
1. 2.2 Classificaiton for Abelian extensions
1. 3 Deformations of differential Lie algebras
1. 4 $L_∞[1]$ -structure for (relative and absolute) differential Lie algebras
1. 4.1 $L_∞[1]$ -algebras
1. 4.2 A generalised version of derived bracket technique
1. 4.3 $L_∞[1]$ -structure for relative differential Lie algebras with weight
1. 4.4 $L_∞[1]$ -structure for absolute differential Lie algebras
1. 4.5 Equivalences between $L_∞[1]$ -structures for absolute and relative differential Lie algebras with weight
1. 4.5.1 From relative to absolute
1. 4.5.2 From absolute to relative
1. 5 Application of $L_∞[1]$ -structure for differential Lie algebras
1. 5.1 Cohomology of differential Lie algebras from $L_∞[1]$ -structure
1. 5.2 Homotopy differential Lie algebras with weight
## Introduction
This paper studies the cohomology theory, abelian extensions, formal deformations, and $L_∞$ -structures for differential Lie algebras of arbitrary weight.
### 0.1. Deformation theory and minimal models
An important method to study a mathematical object is to investigate properties of this object under small deformations. Algebraic deformation theory originated from the deformation theory of complex structures due to Kodaira and Spencer and evolved from ideas of Grothendieck, Illusie, Gerstenhaber, Nijenhuis, Richardson, Deligne, Schlessinger, Stasheff, Goldman, Millson, etc. A fundamental idea of deformation theory is that the deformation theory of any given mathematical object can be governed by a certain differential graded Lie algebra or more generally an $L_∞$ -algebra associated to this mathematical object (whose underlying complex is called the deformation complex). This philosophy has been realised as a theorem in characteristic zero by Lurie [15] and Pridham [19]. It is an important problem to determine explicitly this differential graded Lie algebra or $L_∞$ -algebra governing deformation theory of this mathematical object.
Another important problem about algebraic structures is to study their homotopy versions, just like $A_∞$ -algebras for usual associative algebras. Expected result would be providing a minimal model of the operad governing this algebraic structure. When this operad is Koszul, the Koszul duality for operads [7, 16, 13] gives a perfect solution. More precisely, the cobar construction of the Koszul dual cooperad of the operad in question is the minimal model of this operad. However, when the operad is NOT Koszul, essential difficulties arise and few examples of minimal models have been worked out.
These two problems, say, describing controlling $L_∞$ -algebras and constructing minimal models or more generally cofibrant resolutions, are closed related. In fact, given a cofibrant resolution, in particular, a minimal model, of the operad in question, one can deduce from the cofibrant resolution the deformation complex as well as its $L_∞$ -structure as explained by Kontsevich and Soibelman [12].
However, in practice, a minimal model or a small cofibrant resolution is not known a priori. Wang and Zhou [25, 26] recently found a method to solve these two problems for a large class of non-Koszul operads. The method is in fact the original method of Gerstenhaber [4, 5]. We start by studying formal deformations of the algebraic structure in question and construct the deformation complex as well as $L_∞$ -structure on it from deformation equations. Then we consider the homotopy version which corresponds to Maurer-Cartan elements in the $L_∞$ -structure when appropriate spaces are graded. At last we show the differential graded operad governing the homotopy version is the minimal model of the operad of this algebraic structure, and we also found the Koszul dual which, in this case, is usually a homotopy cooperad.
We aim to deal with differential Lie algebras of arbitrary weight in this paper and its sequel by using the method of [25, 26] as well as derived bracket technique.
### 0.2. Differential Lie algebras, old and new
As is well known, differential operators appeared in Calculus several centuries ago. The study of differential algebras themselves, as the algebraic study of differential equations, only began in the 1930s at the hands of Ritt [20, 21]. From then on, by the work of many mathematicians in the following decades, the subject has been fully developed into a vast area in mathematics.
By definition a differential algebra is an associative commutative algebra $A$ endowed with a linear operator ${d}$ subject to the Leibniz rule:
$$
{d}(xy)={d}(x)y+x{d}(y),x,y∈ A.
$$
The discretisation of differential operators are difference operators. The action of the difference operator $Δ$ on a real function $f$ defined by:
$$
(Δ f)(x)=f(x+Δ x)-f(x),
$$
which satisfies
$$
Δ(fg)=Δ(f)g+fΔ(g)+Δ(f)Δ(g).
$$
In order to unify the study of differential operator and difference operator, Guo and Keigher introduced weighted differential operators [9].
**Definition 0.1**
*([9]) Let $λ∈{k}$ be a fixed element. A differential associative algebra of weight $λ$ is an associative algebra $A$ together with a linear operator ${d}:A→ A$ such that
$$
{d}(xy)={d}(x)y+x{d}(y)+λ{d}(x){
d}(y),x,y∈ A.
$$*
So an usual differential operator is a differential operator of weight $0 0$ and a difference operator is a differential operator of weight $1$ .
One of the motivations of Guo and his collaborators to introduce the concept of weighted differential operators is to study Rota’s program. Rota [22] proposed to classify “interesting” operators on an algebra. In order to promote this program, Guo et al. introduced the Gröbner-Shirshov basis theory for algebras endowed with linear operators [8, 1, 10, 3], and focused on two types of operator algebras: Rota-Baxter type algebras [27] and differential type algebras [10]. Weighted differential algebras fall into the second type.
Inspired by the above research, Guo and Keigher [9] introduced differential Lie algebras of arbitrary weight.
**Definition 0.2**
*([9]) Let $λ∈{k}$ be a fixed element. A differential Lie algebra of weight $λ$ is a Lie algebra $({\mathfrak{g}},[~{},~{}])$ together with a linear operator ${d}_\mathfrak{g}:{\mathfrak{g}}→{\mathfrak{g}}$ such that
$$
{d}_\mathfrak{g}([x,y])=[{d}_\mathfrak{g}(x),y]+[x,{
d}_\mathfrak{g}(y)]+λ[{d}_\mathfrak{g}(x),{\mathrm
{d}}_\mathfrak{g}(y)], ∀ x,y∈{\mathfrak{g}}.
$$*
A related notion is that of relative difference Lie algebras.
**Definition 0.3 ([2,11])**
*A LieAct triple is the triple $({\mathfrak{g}},{\mathfrak{h}},ρ)$ , where $({\mathfrak{g}},[~{},~{}]_\mathfrak{g})$ and $({\mathfrak{h}},[~{},~{}]_\mathfrak{h})$ are Lie algebras and $ρ:{\mathfrak{g}} → Der({\mathfrak{h}})$ is a homomorphism of Lie algebras, where $Der({\mathfrak{h}})$ is the space of derivations on ${\mathfrak{h}}$ . Let $({\mathfrak{g}},{\mathfrak{h}},ρ)$ be a LieAct triple. A linear map $D:{\mathfrak{g}} → {\mathfrak{h}}$ is called a relative differential operator of weight $λ$ if the following equality holds:
$$
D([x,y]_\mathfrak{g})=ρ(x)(D(y))-ρ(y)(D(x))+λ[D(x),D(y)]_
\mathfrak{h}, ∀ x,y∈{\mathfrak{g}}.
$$*
There has been some papers working on differential Lie algebras with nonzero weight before, but they mainly focus on relative differential Lie algebras of weight $1$ , also called relative difference algebras. When studying non-Abelian extensions of Lie algebras, Lue [14] introduced the concept of cross homomorphisms which are just relative difference operators of weight $1$ . Caseiro and Costa [2] studied the framework for the existence of cross homomorphisms: LieAct triple $({\mathfrak{g}},{\mathfrak{h}},ρ)$ , and gave a differential graded Lie algebra where the Maurer-Cartan elements correspond one-to-one to Lie action triples. Pei, Sheng, Tang, and Zhao [18] defined the cohomology theory of cross homomorphisms on Lie algebras, and used the derived bracket technique to find the deformation of a differential graded Lie algebra where the Maurer-Cartan elements correspond one-to-one to the cross homomorphisms. Jiang and Sheng constructed an $L_∞$ -algebra for relative difference Lie algebra using the technique of derived brackets in [11] and they also introduced cohomology theory for relative difference Lie algebras and absolute difference Lie algebras.
This paper is different from papers of Sheng and his collaborators, firstly generalising from weight $1$ to arbitrary weight. Moreover, our ultimate goal is to develop systematically the homotopy theory for both absolute and relative differential Lie algebras of arbitrary weight from an operadic viewpoint, including cohomology theory (or deformation complexes), $L_∞[1]$ -structures, homotopy versions, minimal models, and Koszul dual homotopy cooperads. This task will be completed in two papers. This paper is the elementary part of this project, which contains only cohomology theory (or deformation complexes), $L_∞[1]$ -structures and homotopy versions. In a forthcoming paper, we shall prove the operad of homotopy absolute (resp. relative) differential Lie algebras is the minimal model of the operad of absolute (resp. relative) differential Lie algebras, thus justifying the homotopy version we found is the right one.
Apart from the cohomology theory and $L_∞[1]$ -structures, in this paper we introduce a generalised version of derived bracket technique which replace Lie subalgebras by injective maps of Lie algebras. Moreover, this generalised version permits us to establish the equivalence between $L_∞[1]$ -structures for relative and absolute differential Lie algebras. This equivalence is believed by experts and firstly presented explicitly in this paper for differential Lie algebras. In the forthcoming paper, we will establish the equivalence between minimal models for relative and absolute differential Lie algebras by using a coloring and decoloring procedure for (coloured) operads.
### 0.3. Layout of the paper
The paper is organized as follows.
The first section is of preliminary nature, which contains results which are more or less known. After fixing some notations in Subsection 1.1, we recall basic notions and facts about differential Lie algebra and their representations in Subsection 1.2. In particular, a key descending property for differential representations Lemma 1.7 is presented. A differential Lie algebra is the combination of the underlying Lie algebra and the differential operator. In this light we build the cohomology theory of a differential Lie algebra by combining its components from the Lie algebra and from the differential operator. In Subsection 1.3, we establish the cohomology theory for differential operators of arbitrary weight, which is quite different from the one for the underlying algebra unless the weight is zero. In Subsection 1.4, we combine the Chevalley-Eilenberg cohomology for Lie algebras and the just established cohomology for differential operators of arbitrary weight to define the cohomology of differential Lie algebras of arbitrary weight, with the cochain maps again posing extra challenges when the weight is not zero. The proofs of this section will be given in Subsection 5.1, once the $L_∞[1]$ -structure has been established in Section 4.
As applications and further justification of our cohomology theory for differential Lie algebras, in Section 2, we apply the theory to study abelian extensions of differential Lie algebras of arbitrary weight, and show that abelian extensions are classified by the second cohomology group of the differential Lie algebras.
Further, in Section 3, we apply the above cohomology theory to study formal deformations of differential Lie algebras of arbitrary weight. In particular, we show that if the second cohomology group of a differential Lie algebra with coefficients in the regular representation is trivial, then this differential Lie algebra is rigid.
To deal with the weight case and get $L_∞[1]$ -structure in Section 4, after a reminder on $L_∞[1]$ -structures in Subsection 4.1, a generalised version of the derived bracket technique is introduced in Subsection 4.2, and the weight $λ$ is also incorporated into the statements, which permits us to deal with (absolute) differential Lie algebras of arbitrary weight as well as relative differential Lie algebras of arbitrary weight. With this generalised version at hand, we build the $L_∞[1]$ -structure of relative differential Lie algebras in Subsection 4.3, generalising the result of Jiang and Sheng [11] from weight $1$ to arbitrary weight. In Subsection 4.4, we build the $L_∞[1]$ -structure of (absolute) differential Lie algebras. The comparison with the $L_∞[1]$ -structures for relative and absolute differential Lie algebras is provided in Subsection 4.5.
At last in Section 5, based on the $L_∞[1]$ -structure of (absolute) differential Lie algebras found in Section 4, we verify that the cohomoloy theory of Section 1 is the right one in Subsection 5.1, by using a trivial extension construction in order to deal with coefficients. In Subsection 5.2, the structure of homotopy differential Lie algebra with weight is introduced as Maurer-Cartan elements in the $L_∞[1]$ -structure of (absolute) differential Lie algebras when the spaces involved are graded.
## 1. Differential Lie algebras of arbitrary weight and their cohomology theory
This section recalls some background on differential Lie algebras and their representations and introduce their cohomology theory generalsing the theory introduced by Jiang and Sheng [11] from weight $1$ to arbitrary weight.
### 1.1. Notations
Throughout this paper, let ${\bf k}$ be a field of characteristic $0 0$ . Except specially stated, vector spaces are ${\bf k}$ -vector spaces and all tensor products are taken over ${\bf k}$ . We use cohomological grading. We will employ Koszul sign rule to determine signs, that is, when exchanging the positions of two graded objects in an expression, we have to multiply the expression by a power of $-1$ whose exponent is the product of their degrees.
The suspension of a graded space $V=⊕_n∈{ℤ}V^n$ is the graded space $sV$ with $(sV)^n=V^n+1$ for any $n∈{ℤ}$ . Write $sv∈(sV)^n$ for $v∈ V^n+1$ .
Let $V=⊕_n∈ℤV^n$ be a graded vector space. Recall that 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 $x⊗ y-(-1)^|x||y|y⊗ x$ for all homogeneous elements $x,y∈ V$ . For $x_1⊗⋯⊗ x_n∈ V^⊗ n⊆ T(V)$ , write $x_1\odot x_2\odot\dots\odot x_n$ its image in $S(V)$ . For homogeneous elements $x_1,\dots,x_n∈ V$ and $σ∈ S_n$ which is the symmetric group in $n$ variables, the Koszul sign $ε(σ):=ε(σ;x_1,\dots,x_n)$ is defined by
$$
x_1\odot x_2\odot\dots\odot x_n=ε(σ)x_σ(1)\odot x_
{σ(2)}\odot\dots\odot x_σ(n)∈ S(V). \tag{1}
$$
Denote by $S^n(V)$ the image of $V^⊗ n$ in $S(V)$ .
Let $V=⊕_n∈ℤV^n$ be a graded vector space. Recall that the graded exterior algebra $\bigwedge(V)$ of $V$ is defined to be the quotient of the tensor algebra $T(V)$ by the two-sided ideal generated by $x⊗ y+(-1)^|x||y|y⊗ x$ for all homogeneous elements $x,y∈ V$ . For $x_1⊗⋯⊗ x_n∈ V^⊗ n⊆ T(V)$ , write $x_1\wedge x_2\wedge⋯\wedge x_n$ its image in $\bigwedge(V)$ . For homogeneous elements $x_1,\dots,x_n∈ V$ and $σ∈ S_n$ , the sign $χ(σ):=χ(σ;x_1,\dots,x_n)$ is defined by
$$
x_1\wedge x_2\wedge⋯\wedge x_n=χ(σ)x_σ(1)\wedge x_
σ(2)\wedge⋯\wedge x_σ(n)∈\bigwedge(V). \tag{1}
$$
Denote by $\bigwedge^n(V)$ the image of $V^⊗ n$ in $\bigwedge(V)$ . Obviously $χ(σ)=ε(σ){sgn}(σ)$ , where ${sgn}(σ)$ is the signature of $σ$ .
Recall that for a graded vector space $V$ , $S^n(sV)$ is isomorphic to $s^n\wedge^nV$ via the map
$$
sv_1\odot sv_2\odot\dots\odot sv_n↦(-1)^(n-1)|v_1|+(n-2)|v_2|
+⋯+|v_n-1|s^n(v_1\wedge v_2\wedge⋯\wedge v_n), \tag{1}
$$
for $v_1,\dots,v_n∈ V$ homogeneous elements in $V$ .
Let $n≥ 1$ . For $0≤ i_1,\dots,i_r≤ n$ with $i_1+⋯+i_r=n$ , ${Sh}(i_1,i_2,\dots,i_r)$ is the set of $(i_1,\dots,i_r)$ -shuffles, i.e., those permutation $σ∈ S_n$ such that
$$
σ(1)<σ(2)<\dots<σ(i_1), σ(i_1+1)<\dots<σ(i_1+i
_2), \dots, σ(i_1+⋯+i_r-1+1)<⋯<σ(n). \tag{1}
$$
Let $PSh(i_1,i_2,\dots,i_r)$ be the subset of ${Sh}(i_1,i_2,\dots,i_r)$ whose elements $(i_1,\dots,i_r)$ -shuffles satisfy:
$$
σ(1)<σ(i_1+1)<σ(i_1+i_2+1)<\dots<σ(i_1+·+i_r-
1+1). \tag{1}
$$
### 1.2. Differential Lie algebras with weight and their representations
In this subsection, we recall basic notions about differential Lie algebras with weight and their representations.
**Definition 1.1**
*([9]) Let $λ∈{k}$ be a fixed element. A differential Lie algebra of weight $λ$ is a Lie algebra $({\mathfrak{g}},~{}[~{},~{}])$ together with a linear operator ${d}_\mathfrak{g}:{\mathfrak{g}}→{\mathfrak{g}}$ such that
$$
{d}_\mathfrak{g}([x,y])=[{d}_\mathfrak{g}(x),y]+[x,{
d}_\mathfrak{g}(y)]+λ[{d}_\mathfrak{g}(x),{\mathrm
{d}}_\mathfrak{g}(y)], ∀ x,y∈{\mathfrak{g}}. \tag{2}
$$
Such an operator ${d}_\mathfrak{g}$ is called a differential operator of weight $λ$ or a derivation of weight $λ$ . Given two differential Lie algebras $({\mathfrak{g}},{d}_\mathfrak{g}), ({\mathfrak{h}},{d}_ \mathfrak{h})$ of the same weight $λ$ , a homomorphism of differential Lie algebras from $({\mathfrak{g}},{d}_\mathfrak{g})$ to $({\mathfrak{h}},{d}_\mathfrak{h})$ is a Lie algebra homomorphism $φ:{\mathfrak{g}} → {\mathfrak{h}}$ such that $φ∘{d}_\mathfrak{g}={d}_\mathfrak{h}∘φ$ . We denote by $DL_λ$ the category of differential Lie algebras of weight $λ$ .*
A differential operator of weight $1$ is also called a crossed homomorphism and a differential Lie algebra of weight $1$ is also called a difference Lie algebra [18, 11].
Recall that a representation of a Lie algebra ${\mathfrak{g}}$ is a pair $(V,ρ)$ , where $V$ is a vector space and $ρ:{\mathfrak{g}}→\mathfrak{gl}(V),~{}x↦(v↦ρ(x)v)$ is a homomorphism of Lie algebras for all $x,y∈{\mathfrak{g}}$ and $v∈ V$ .
**Definition 1.2**
*Let $({\mathfrak{g}},{d}_\mathfrak{g})$ be a differential Lie algebra.
- A representation over the differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ or a differential representation is a triple $(V,ρ,{d}_V)$ , where ${d}_V∈\mathfrak{gl}(V)$ , and $(V,ρ)$ is a representation over the Lie algebra ${\mathfrak{g}}$ , such that for all $x,y∈{\mathfrak{g}},v∈ V,$ the following equality holds:
| | $\displaystyle{d}_V(ρ(x)v)$ | $\displaystyle=$ | $\displaystyleρ({d}_\mathfrak{g}(x))v+ρ(x){d}_V(v)+ λρ({d}_\mathfrak{g}(x)){d}_V(v).$ | |
| --- | --- | --- | --- | --- |
- Given two representations $(U,ρ^U,{d}_U), (V,ρ^V,{d}_V)$ over $({\mathfrak{g}},{d}_\mathfrak{g})$ , a linear map $f:U → V$ is called a homomorphism of representations, if
$$
f∘ρ^U(x)=ρ^V(x)∘ f, ∀ x∈{\mathfrak{g}},
and f∘{d}_U={d}_V∘ f.
$$*
**Example 1.3**
*For a differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ , the usual adjoint representation
$$
{\rm ad}:{\mathfrak{g}}→\mathfrak{gl}({\mathfrak{g}}), x↦(y↦[x,
y])
$$
over the Lie algebra ${\mathfrak{g}}$ is also a representation of this differential Lie algebra. It is also called the adjoint representation over the differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ , denoted by ${\mathfrak{g}}_{\rm ad}$ .*
**Example 1.4**
*Let ${\mathfrak{g}}$ be a Lie algebra and $V$ be a representation of it. Then $({\mathfrak{g}},Id_\mathfrak{g})$ is a differential Lie algebra of weight $-1$ and the pair $(V,Id_V)$ gives a differential representation.*
**Example 1.5**
*Let $({\mathfrak{g}},{d}_\mathfrak{g})$ be a differential Lie algebra together with a differential representation $(V,{d}_V)$ . Then for arbitrary nonzero scalar $κ∈{k}$ , $({\mathfrak{g}},κ{d}_\mathfrak{g})$ is a differential Lie algebra of weight $\frac{λ}{κ}$ and the pair $(V,κ {d}_V)$ is a differential representation.*
It is straightforward to obtain the following result:
**Proposition 1.6**
*Let $(V,ρ,{d}_V)$ be a representation of the differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ . Then $({\mathfrak{g}}⊕ V,{d}_\mathfrak{g}+{d}_V)$ is a differential Lie algebra, where the Lie algebra structure on ${\mathfrak{g}}⊕ V$ is given by
$$
\{x+u,y+v\}:=[x,y]+ρ(x)v-ρ(y)u, ∀ x,y∈{\mathfrak{g}},~{}u,v
∈ V.
$$
This new differential Lie algebra is denoted by ${\mathfrak{g}}\ltimes V$ , called the trivial extension of ${\mathfrak{g}}$ by $V$ .*
We need the following observation which is a weighted version of [14, Proposition 3.1]:
**Lemma 1.7**
*Let $(V,ρ,{d}_V)$ be a representation of the differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ . There exists a new representation $(V,ρ_λ,{d}_V)$ over $({\mathfrak{g}},{d}_\mathfrak{g})$ , where $ρ_λ$ is given by $\colon$
$$
ρ_λ(x)v=ρ(x+λ{d}_\mathfrak{g}(x))v, ∀
x
∈{\mathfrak{g}},v∈ V.
$$
Denote the new presentation by $V_λ:=(V,ρ_λ,{d}_V)$ .*
### 1.3. Cohomology of differential operators
In this subsection, we define a cohomology theory of differential operators.
Recall that the Chevalley-Eilenberg cochain complex of a Lie algebra ${\mathfrak{g}}$ with coefficients in a representation $(V,ρ)$ is the cochain complex $({C}^*_Lie({\mathfrak{g}},V),∂_{Lie}^ *),$ where for $n≥ 0$ , ${C}^n_Lie({\mathfrak{g}},V)={\rm Hom}(\wedge^n{ \mathfrak{g}},V)$ (in particular, ${C}^0_Lie({\mathfrak{g}},V)=V$ ) and the coboundary operator
$$
∂_{Lie}^n:{C}^n_Lie({\mathfrak{g}},V
)\longrightarrow{C}^n+1_Lie({\mathfrak{g}},V),n≥ 0
$$
is given by
$$
\begin{split}∂_{Lie}^n(f)(x_1,\dots,x_n+1)&=∑_i=1
^n+1(-1)^i+nρ(x_i)f(x_1,\dots,\hat{x}_i,\dots,x_n+1)\\
&+∑_1≤ i<j≤ n+1(-1)^i+j+n+1f([x_i,x_j],x_1,\dots,\hat{x}_i
,\dots,\hat{x}_j,\dots,x_n+1),\end{split}
$$
for all $f∈{C}^n_Lie({\mathfrak{g}},V),~{}x_1,\dots,x_n+1 ∈{\mathfrak{g}}$ , where $\hat{x_i}$ means deleting this element. The corresponding Chevalley-Eilenberg cohomology is denoted by ${H}^*_Lie({\mathfrak{g}},V)$ .
When $V$ is the adjoint representation ${\mathfrak{g}}_{\rm ad}$ , we write ${H}^n_Lie({\mathfrak{g}}):={H}^n_Lie ({\mathfrak{g}},V),n≥ 0$ .
**Definition 1.8**
*Let $({\mathfrak{g}},{d}_\mathfrak{g})$ be a differential Lie algebra of weight $λ$ and $(V,{d}_V)$ be a differential representation over $({\mathfrak{g}},{d}_\mathfrak{g})$ . The cochain complex of the differential operator ${d}_\mathfrak{g}$ with coefficients in the differential representation $(V,{d}_V)$ is defined to be the Chevalley-Eilenberg cochain complex of the Lie algebra ${\mathfrak{g}}$ with coefficients in the new differential representation $V_λ$ , denoted by $({C}^*_DO_λ({\mathfrak{g}},V),∂_ DO_λ^*):=({C}^*_\rm Lie({\mathfrak{g}},V_λ), ∂_\rm Lie^*)$ . The corresponding cohomology, denoted by ${H}^*_DO_λ({\mathfrak{g}},V)$ , is called the cohomology of the differential operator ${d}_\mathfrak{g}$ with coefficients in the representation $(V_λ,{d}_V)$ .*
More precisely, for $n≥ 0$ , ${C}^n_DO_λ({\mathfrak{g}},V)={\rm Hom}(\wedge^n {\mathfrak{g}},V)$ and the coboundary operator
$∂_DO_λ^n:{C}^n_DO_λ( {\mathfrak{g}},V)→{C}^n+1_DO_λ({\mathfrak{g}},V)$ is given by
| | $\displaystyle∂_DO_λ^n(f)(x_1,\dots,x_n+1)$ | $\displaystyle=∑_i=1^n+1(-1)^i+nρ_λ(x_i)f(x_1,\dots, \hat{x}_i,\dots,x_n+1)$ | |
| --- | --- | --- | --- |
for all $f∈{C}^n_DO_λ({\mathfrak{g}},V),~{}x_1,\dots, x_n+1∈{\mathfrak{g}}$ .
### 1.4. Cohomology of differential Lie algebras
We now combine the classical Chevalley-Eilenberg cohomology of Lie algebras and the newly defined cohomology of differential operators of weight $λ$ to define the cohomology of the differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ with coefficients in the representation $(V,{d}_V)$ . In fact, we will define the cochain complex of a differential Lie algebras as, up to shift and signs, the mapping cone of a cochain map from the Chevalley-Eilenberg cochain complex of the Lie algebra to the cochain complex of the differential operators of weight $λ$ .
Notice that [11] has introduced the cohomology theory of a difference Lie algebra with coefficients in a representation. We in fact generalise their construction from weight $1$ to arbitrary weight.
Introduce the linear maps
$$
δ^n:{C}^n_Lie({\mathfrak{g}},V)→{
C}^n_DO_λ({\mathfrak{g}},V)
$$
by
| | $\displaystyleδ^nf(x_1,\dots,x_n):=∑_k=1^nλ^k-1∑_ 1≤ i_{1<⋯<i_k≤ n}f(x_1,\dots,{d}_\mathfrak{g}(x_i_ {1}),\dots,{d}_\mathfrak{g}(x_i_{k}),\dots,x_n)-{d}_ Vf(x_1,\dots,x_n),$ | |
| --- | --- | --- |
for $f∈{C}^n_Lie({\mathfrak{g}},V)$ , $n≥ 1$ and
$$
δ^0v={-}{d}_V{(v)}, ∀ v∈{C}^0_
Lie({\mathfrak{g}},V)=V.
$$
**Proposition 1.9**
*The linear map $δ$ is a cochain map from the cochain complex $({C}^*_Lie({\mathfrak{g}},V),∂_Lie^*)$ to $({C}^*_DO_λ({\mathfrak{g}},V),∂_ DO_λ^*)$ .*
Although a direct proof is possible, we shall deduce the above result from the $L_∞[1]$ -structure; see Proposition 5.2.
**Remark 1.10**
*Note that ${C}^n_DO_λ({\mathfrak{g}},V)$ equals to ${C}^n_Lie({\mathfrak{g}},V)$ as linear spaces but they are not equal as cochain complexes unless $λ=0$ . When $λ$ is not zero, a new representation structure is needed to define $∂_DO_λ^*$ which eventually leads to the rather long and technical argument in order to establish the cochain map in Proposition 1.9.*
Now we can define the cochain complex of a differential Lie algebras $({\mathfrak{g}},{d}_\mathfrak{g})$ with coefficients in the differential representation $(V,{d}_V)$ .
**Definition 1.11**
*Define the cochain complex $({C}_DL_λ^*({\mathfrak{g}},V),∂^*_ DL_λ)$ of the differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ with coefficients in the differential representation $(V,{d}_V)$ to be the negative shift of the mapping cone of the cochain map $δ^*:({C}^*_Lie({\mathfrak{g}},V),{∂}_ Lie^*)→({C}^*_DO_λ({ \mathfrak{g}},V),{∂}_DO_λ^*)$ . More precisely, the space of $n$ -cochains is given by
$$
{C}_DL_λ^n({\mathfrak{g}},V):=\begin{cases}{
C}^n_Lie({\mathfrak{g}},V)⊕{C}^n-1_
DO_λ({\mathfrak{g}},V),&n≥ 1,\\
{C}^0_Lie({\mathfrak{g}},V)=V,&n=0.\end{cases}
$$
and the differential $∂_DL_λ^n:{C}_DL_λ^n( g,V)→{C}_DL_λ^n+1(g,V)$ by
| | $\displaystyle∂_DL_λ^n(f,g)$ | $\displaystyle:=$ | $\displaystyle(∂_Lie^nf,-∂_DO_λ^ n-1g-δ^nf), ∀ f∈{C}^n_Lie({\mathfrak {g}},V), g∈{C}^n-1_DO_λ({\mathfrak{g}},V), n≥ 1,$ | |
| --- | --- | --- | --- | --- |*
**Definition 1.12**
*The cohomology of the cochain complex $({C}_DL_λ^*({\mathfrak{g}},V),∂^*_ DL_λ)$ , denoted by ${H}_DL_λ^*({\mathfrak{g}},V)$ , is called the cohomology of the differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ with coefficients in the differential representation $(V,{d}_V)$ .*
In the next two sections, we will use the following remarks.
**Remark 1.13**
*We compute 0-cocycles, 1-cocycles and 2-cocycles of the cochain complex ${C}_DL_λ^*({\mathfrak{g}},V)$ . It is obvious that for all $v∈ V$ , $∂^0_DL_λv=0$ if and only if
$$
∂^0_Liev=0, {d}_V(v)=0.
$$ For all $(f,v)∈{\rm Hom}({\mathfrak{g}},V)⊕ V$ , $∂^1_DL_λ(f,v)=0$ if and only if $∂^1_Lief=0$ and
$$
ρ_λ(x)v=f({d}_\mathfrak{g}(x))-{d}_V(f(x)),
∀ x∈{\mathfrak{g}}.
$$ For all $(f,g)∈{\rm Hom}(\wedge^2{\mathfrak{g}},V)⊕{\rm Hom}({\mathfrak{g}},V)$ , $∂^2_DL_λ(f,g)=0$ if and only if $∂^2_Lief=0,$ and
$$
\displaystyleρ_λ(x)g(y)-ρ_λ(y)g(x)-g([x,y])=-λ f({
d}_\mathfrak{g}(x),{d}_\mathfrak{g}(y))-f({d}_
\mathfrak{g}(x),y)-f(x,{d}_\mathfrak{g}(y))+{d}_V(f(x,y
)),\ \tag{3}
$$
for all $x,y∈{\mathfrak{g}}.$*
**Remark 1.14**
*we shall need a subcomplex of the cochain complex ${C}_DL_λ^*({\mathfrak{g}},V)$ . Let
$$
\tilde{{C}}_DL_λ^n({\mathfrak{g}},V):=\begin{
cases}{C}^n_Lie({\mathfrak{g}},V)⊕{C}^n-1
_DO_λ({\mathfrak{g}},V),&n≥ 2,\\
{C}^1_Lie({\mathfrak{g}},V),&n=1,\\
0,&n=0.\end{cases}
$$
Then it is obvious that $(\tilde{{C}}_DL_λ^*({\mathfrak{g}},V)=⊕_n= 0^∞\tilde{{C}}_DL_λ^n({\mathfrak{g}},V), ∂_DL_λ^*)$ is a subcomplex of the cochain complex $({C}_DL_λ^*({\mathfrak{g}},V),∂_ DL_λ^*)$ . We denote its cohomology by $\tilde{{H}}_DL_λ^*({\mathfrak{g}},V)$ . Obviously, $\tilde{{H}}_DL_λ^n({\mathfrak{g}},V)={{H }}_DL_λ^n({\mathfrak{g}},V)$ for $n>2$ .*
## 2. Abelian extensions of differential Lie algebras
In this section, we study abelian extensions of differential Lie algebras of weight $λ$ and show that they are classified by the second cohomology, as one would expect of a good cohomology theory.
### 2.1. Abelian extensions
Notice that a vector space $V$ together with a linear endomorphism ${d}_V$ can be considered as a trivial differential Lie algebra of weight $λ$ endowed with the trivial Lie bracket $[u,v]=0$ for all $u,v∈ V$ .
**Definition 2.1**
*An abelian extension of differential Lie algebras is a short exact sequence of homomorphisms of differential Lie algebras
$$
\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>{V}@>{i}>{}>\hat{{\mathfrak{g
}}}@>{p}>{}>{\mathfrak{g}}@>{}>{}>0\\
@V{{d}_V}V{}V@V{{d}_\hat{{\mathfrak{g}}}}V{}V@V{{\mathrm
{d}}_\mathfrak{g}}V{}V\\
0@>{}>{}>{V}@>{i}>{}>\hat{{\mathfrak{g}}}@>{p}>{}>{\mathfrak{g}}@>{}>{}>0\end{CD}
$$
where $(V,{d}_V)$ is a trivial differential Lie algebra. We will call $(\hat{{\mathfrak{g}}},{d}_\hat{{\mathfrak{g}}})$ an abelian extension of $({\mathfrak{g}},{d}_\mathfrak{g})$ by $(V,{d}_V)$ .*
**Definition 2.2**
*Let $(\hat{{\mathfrak{g}}}_1,{d}_\hat{{\mathfrak{g}}_1})$ and $(\hat{{\mathfrak{g}}}_2,{d}_\hat{{\mathfrak{g}}_2})$ be two abelian extensions of $({\mathfrak{g}},{d}_\mathfrak{g})$ by $(V,{d}_V)$ . They are said to be isomorphic if there exists an isomorphism of differential Lie algebras $ζ:(\hat{{\mathfrak{g}}}_1,{d}_\hat{{\mathfrak{g}}_1}) →(\hat{{\mathfrak{g}}}_2,{d}_\hat{{\mathfrak{g}}_2})$ such that the following commutative diagram holds:
$$
\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>{(V,{d}_V)}@>{i}>{
}>(\hat{{\mathfrak{g}}}_1,{d}_\hat{{\mathfrak{g}}_1})@>{p}>{}>(
{\mathfrak{g}},{d}_\mathfrak{g})@>{}>{}>0\\
\Big{\|}@V{ζ}V{}V\Big{\|}\\
0@>{}>{}>{(V,{d}_V)}@>{i}>{}>(\hat{{\mathfrak{g}}}_2,{d}
_\hat{{\mathfrak{g}}_2})@>{p}>{}>({\mathfrak{g}},{d}_\mathfrak{g
})@>{}>{}>0.\end{CD}
$$*
A section of an abelian extension $(\hat{{\mathfrak{g}}},{d}_\hat{{\mathfrak{g}}})$ of $({\mathfrak{g}},{d}_\mathfrak{g})$ by $(V,{d}_V)$ is a linear map $s:{\mathfrak{g}}→\hat{{\mathfrak{g}}}$ such that $p∘ s={\rm Id}_\mathfrak{g}$ .
Now for an abelian extension $(\hat{{\mathfrak{g}}},{d}_\hat{{\mathfrak{g}}})$ of $({\mathfrak{g}},{d}_\mathfrak{g})$ by $(V,{d}_V)$ with a section $s:{\mathfrak{g}}→\hat{{\mathfrak{g}}}$ , then there exist a unique linear map $t:\hat{{\mathfrak{g}}}→ V$ such that:
$$
t∘ i=Id_V,t∘ s=0,i∘ t+s∘ p=Id_\hat{{
\mathfrak{g}}},
$$
then we can define
$$
ρ(x)v:=t[s(x),i(v)]_\hat{{\mathfrak{g}}}, ∀ x∈{\mathfrak{g}},
v∈ V.
$$
For convenience, we denote it by $ρ(s(x))v,$ and observe that $[s(x),i(v)]_\hat{{\mathfrak{g}}}∈ i(V)$ , and we identify $V$ with $i(V).$
Now we get a linear map $ρ:{\mathfrak{g}}→\mathfrak{gl}(V),~{}x↦(v↦ρ(x)v).$
**Proposition 2.3**
*With the above notations, $(V,ρ,{d}_V)$ is a representation over the differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ .*
* Proof*
For arbitrary $x,y∈{\mathfrak{g}},v∈ V$ , since $s([x,y]_{\mathfrak{g}})-[s(x),s(y)]_\hat{{\mathfrak{g}}}∈ V$ implies $s([x,y]_{\mathfrak{g}})v=[s(x),s(y)]_\hat{{\mathfrak{g}}}v$ , we have
$$
ρ([x,y]_{\mathfrak{g}})(v)=ρ(s([x,y]_{\mathfrak{g}}))v=ρ([s(x),s(
y)]_\hat{{\mathfrak{g}}})v\xlongequal{\mbox{Jacobi identity}}ρ(x)ρ(y)(
v)-ρ(y)ρ(x)(v).
$$
Hence, $ρ$ is a Lie algebra homomorphism. Moreover, ${d}_\hat{{\mathfrak{g}}}(s(x))-s({d}_\mathfrak{g}(x))∈ V$ means that
$$
ρ({d}_\hat{{\mathfrak{g}}}(s(x)))v=ρ(s({d}_
\mathfrak{g}(x)))v.
$$
Thus we have
| | $\displaystyle{d}_V(ρ(x)v)$ | $\displaystyle={d}_V(ρ(s(x))v)={d}_\hat{{\mathfrak{g}} }(ρ(s(x))v)$ | |
| --- | --- | --- | --- |
Hence, $(V,ρ,{d}_V)$ is a representation over $({\mathfrak{g}},{d}_\mathfrak{g})$ . ∎
We further define linear maps $ψ:{\mathfrak{g}}⊗{\mathfrak{g}}→ V$ and $χ:{\mathfrak{g}}→ V$ respectively by
| | $\displaystyleψ(x,y)$ | $\displaystyle=[s(x),s(y)]_\hat{{\mathfrak{g}}}-s([x,y]_{\mathfrak{g}}), ∀ x,y∈{\mathfrak{g}},$ | |
| --- | --- | --- | --- |
We transfer the differential Lie algebra structure on $\hat{{\mathfrak{g}}}$ to ${\mathfrak{g}}⊕ V$ by endowing ${\mathfrak{g}}⊕ V$ with a multiplication $[·,·]_ψ$ and a differential operator ${d}_χ$ defined by
$$
\displaystyle[x+u,y+v]_ψ \displaystyle=[x,y]+ρ(x)v-ρ(y)u+ψ(x,y), ∀ x,y∈{\mathfrak{g}}
, u,v∈ V, \displaystyle{d}_χ(x+v) \displaystyle={d}_\mathfrak{g}(x)+χ(x)+{d}_V(v),
∀ x∈{\mathfrak{g}}, v∈ V. \tag{4}
$$
**Proposition 2.4**
*The triple $({\mathfrak{g}}⊕ V,[·,·]_ψ,{d}_χ)$ is a differential Lie algebra if and only if $(ψ,χ)$ is a 2-cocycle of the differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ with the coefficient in $(V,{d}_V)$ .*
* Proof*
If $({\mathfrak{g}}⊕ V,[·,·]_ψ,{d}_χ)$ is a differential Lie algebra, then $[·,·]_ψ$ implies
$$
ψ(x,[y,z])+ρ(x)ψ(y,z)+ψ(y,[z,x])\\
+ρ(y)ψ(z,x)+ψ(z,[x,y])+ρ(z)ψ(x,y)=0, \tag{6}
$$
for arbitrary $x,y,z∈{\mathfrak{g}}$ . Since ${d}_χ$ satisfies (2), we deduce that
$$
χ([x,y])-ρ_λ(x)χ(y)+ρ_λ(y)χ(x)+{d}_V
(ψ(x,y))-ψ({d}_\mathfrak{g}(x),y)-ψ(x,{d}_
\mathfrak{g}(y))-λψ({d}_\mathfrak{g}(x),{d}_
\mathfrak{g}(y))=0. \tag{7}
$$
Hence, $(ψ,χ)$ is a 2-cocycle, see Eq. (3). Conversely, if $(ψ,χ)$ satisfies equalities (6) and (7), one can easily check that $({\mathfrak{g}}⊕ V,[·,·]_ψ,{d}_χ)$ is a differential Lie algebra. ∎
### 2.2. Classificaiton for Abelian extensions
Now we are ready to classify abelian extensions of a differential Lie algebra.
**Theorem 2.5**
*Let $V$ be a vector space and ${d}_V∈{\rm End}_k(V)$ . Then abelian extensions of a differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ by $(V,{d}_V)$ are classified by the second cohomology group ${\rm\tilde{H}}_DL_λ^2({\mathfrak{g}},V)$ of $({\mathfrak{g}},{d}_\mathfrak{g})$ with coefficients in the representation $(V,{d}_V)$ .*
* Proof*
Let $(\hat{{\mathfrak{g}}},{d}_\hat{{\mathfrak{g}}})$ be an abelian extension of $({\mathfrak{g}},{d}_\mathfrak{g})$ by $(V,{d}_V)$ . We choose a section $s:{\mathfrak{g}}→\hat{{\mathfrak{g}}}$ to obtain a 2-cocycle $(ψ,χ)$ by Proposition 2.4. We first show that the cohomological class of $(ψ,χ)$ does not depend on the choice of sections. Indeed, let $s_1$ and $s_2$ be two distinct sections providing 2-cocycles $(ψ_1,χ_1)$ and $(ψ_2,χ_2)$ respectively. We define $φ:{\mathfrak{g}}→ V$ by $φ(x)=s_1(x)-s_2(x)$ . Then
| | $\displaystyleψ_1(x,y)$ | $\displaystyle=[s_1(x),s_1(y)]-s_1([x,y])$ | |
| --- | --- | --- | --- |
and
| | $\displaystyleχ_1(x)$ | $\displaystyle={d}_\hat{{\mathfrak{g}}}(s_1(x))-s_1({d} _\mathfrak{g}(x))$ | |
| --- | --- | --- | --- |
That is, $(ψ_1,χ_1)=(ψ_2,χ_2)+∂^1_DL_λ(φ)$ . Thus $(ψ_1,χ_1)$ and $(ψ_2,χ_2)$ are in the same cohomological class in $\tilde{\rm{H}}_DL_λ^2({\mathfrak{g}},V)$ . Next we prove that isomorphic abelian extensions give rise to the same element in $\tilde{\rm{H}}_DL_λ^2({\mathfrak{g}},V)$ . Assume that $(\hat{{\mathfrak{g}}}_1,{d}_\hat{{\mathfrak{g}}_1})$ and $(\hat{{\mathfrak{g}}}_2,{d}_\hat{{\mathfrak{g}}_2})$ are two isomorphic abelian extensions of $({\mathfrak{g}},{d}_\mathfrak{g})$ by $(V,{d}_V)$ with the associated homomorphism $ζ:(\hat{{\mathfrak{g}}}_1,{d}_\hat{{\mathfrak{g}}_1}) →(\hat{{\mathfrak{g}}}_2,{d}_\hat{{\mathfrak{g}}_2})$ . Let $s_1$ be a section of $(\hat{{\mathfrak{g}}}_1,{d}_\hat{{\mathfrak{g}}_1})$ . As $p_2∘ζ=p_1$ , we have
$$
p_2∘(ζ∘ s_1)=p_1∘ s_1={\rm Id}_{\mathfrak{g}}.
$$
Therefore, $ζ∘ s_1$ is a section of $(\hat{{\mathfrak{g}}}_2,{d}_\hat{{\mathfrak{g}}_2})$ . Denote $s_2:=ζ∘ s_1$ . Since $ζ$ is a homomorphism of differential Lie algebras such that $ζ|_V={\rm Id}_V$ , we have
| | $\displaystyleψ_2(x,y)$ | $\displaystyle=[s_2(x),s_2(y)]-s_2([x,y])=[ζ(s_1(x)),ζ(s_1(y) )]-ζ(s_1([x,y]))$ | |
| --- | --- | --- | --- |
and
| | $\displaystyleχ_2(x)$ | $\displaystyle={d}_\hat{{\mathfrak{g}}_2}(s_2(x))-s_2({\mathrm {d}}_\mathfrak{g}(x))={d}_\hat{{\mathfrak{g}}_2}(ζ(s_1(x) ))-ζ(s_1({d}_\mathfrak{g}(x)))$ | |
| --- | --- | --- | --- |
Consequently, all isomorphic abelian extensions give rise to the same element in $\tilde{\rm{H}}_DL_λ^2({\mathfrak{g}},V)$ . Conversely, given two 2-cocycles $(ψ_1,χ_1)$ and $(ψ_2,χ_2)$ , we can construct two abelian extensions $({\mathfrak{g}}⊕ V,[·,·]_ψ_1,{d}_χ_1)$ and $({\mathfrak{g}}⊕ V,[·,·]_ψ_2,{d}_χ_2)$ via equalities (4) and (5). If they represent the same cohomological class in $\tilde{\rm{H}}_DL_λ^2({\mathfrak{g}},V)$ , then there exists a linear map $φ:{\mathfrak{g}}→ V$ such that
$$
(ψ_1,χ_1)=(ψ_2,χ_2)+∂^1_DL_λ(
φ).
$$
Define $ζ:{\mathfrak{g}}⊕ V→{\mathfrak{g}}⊕ V$ by
$$
ζ(x,v):=x+φ(x)+v.
$$
Then $ζ$ is an isomorphism of these two abelian extensions. ∎
**Remark 2.6**
*In particular, any vector space $V$ with linear endomorphism ${d}_V$ can serve as a trivial representation of $({\mathfrak{g}},{d}_\mathfrak{g})$ . In this situation, central extensions of $({\mathfrak{g}},{d}_\mathfrak{g})$ by $(V,{d}_V)$ are classified by the second cohomology group ${\rm{H}}_DL_λ^2({\mathfrak{g}},V)$ of $({\mathfrak{g}},{d}_\mathfrak{g})$ with the coefficient in the trivial representation $(V,{d}_V)$ . Note that for a trivial representation $(V,{d}_V)$ , since $∂^1_DO_λv=0$ for all $v∈ V$ , we have
$$
{\rm{H}}_DL_λ^2({\mathfrak{g}},V)=\tilde{\rm{H}}_\mathsf
{DL_λ}^2({\mathfrak{g}},V).
$$*
## 3. Deformations of differential Lie algebras
In this section, we study formal deformations of a differential Lie algebra of weight $λ$ . In particular, we show that if the second cohomology group $\tilde{\rm{H}}^2_DL_λ({\mathfrak{g}},{\mathfrak{g}})=0$ , then the differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ is rigid.
Let $({\mathfrak{g}},{d}_\mathfrak{g})$ be a differential Lie algebra. Denote by $μ_\mathfrak{g}$ the multiplication of ${\mathfrak{g}}$ . Consider the 1-parameterized family
$$
μ_t=∑_i=0^∞μ_it^i, μ_i∈{C}^2_{
Lie}({\mathfrak{g}},{\mathfrak{g}}), d_t=∑_i=0^∞d
_it^i, d_i∈{C}^1_DO_λ({\mathfrak{g}},
{\mathfrak{g}}).
$$
**Definition 3.1**
*A 1-parameter formal deformation of a differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ is a pair $(μ_t,d_t)$ , which endows the ${k}[[t]]$ -module $({\mathfrak{g}}[[t]],μ_t,d_t)$ with a differential Lie algebra structure such that $(μ_0,d_0)=(μ_\mathfrak{g},{d}_\mathfrak{g})$ .*
The pair $(μ_t,d_t)$ generates a 1-parameter formal deformation of the differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ if and only if for all $x,y,z∈{\mathfrak{g}}$ , the following equalities hold:
$$
\displaystyle 0 \displaystyle= \displaystyleμ_t(x,μ_t(y,z))+μ_t(y,μ_t(z,x))+μ_t(z,μ_t
(x,y)), \displaystyle d_t(μ_t(x,y)) \displaystyle= \displaystyleμ_t(d_t(x),y)+μ_t(x,d_t(y))+λμ_t(d_t(x),d
_t(y)). \tag{8}
$$
Expanding these equations and collecting coefficients of $t^n$ , we see that Eqs. (8) and (9) are equivalent to the systems of equations: for each $n≥ 0$ ,
$$
\displaystyle 0 \displaystyle= \displaystyle∑_i=0^nμ_i(x,μ_n-i(y,z))+μ_i(y,μ_n-i(z,x))
+μ_i(z,μ_n-i(x,y)), \displaystyle∑_k,l≥ 0\atop k+l=nd_lμ_k(x,y) \displaystyle= \displaystyle∑_k,l≥ 0\atop k+l=n≤ft(μ_k(d_l(x),y)+μ_k(x,d_
{l}(y))\right)+λ∑_k,l,m≥ 0\atop k+l+m=nμ_k(d_l(x),d_m(y)). \tag{10}
$$
**Remark 3.2**
*For $n=0$ , Eq. (10) is equivalent to the Jabobi identity of $μ_\mathfrak{g}$ , and Eq. (11) is equivalent to the fact that ${d}_\mathfrak{g}$ is a $λ$ -derivation. For $n=1$ , Eq. (10) has the form:
$$
\displaystyle 0 \displaystyle=μ_\mathfrak{g}(x,μ_1(y,z))+μ_\mathfrak{g}(y,μ_1(
z,x))+μ_\mathfrak{g}(z,μ_1(x,y)) \displaystyle +μ_1(x,μ_\mathfrak{g}(y,z))+μ_1(y,μ_\mathfrak
{g}(z,x))+μ_1(z,μ_\mathfrak{g}(x,y)). \tag{12}
$$
And for $n=1$ , Eq. (11) has the form:
$$
\displaystyle d_1μ_\mathfrak{g}(x,y)+{d}_\mathfrak{g}μ_1(
x,y) \displaystyle=μ_1(d_\mathfrak{g}(x),y)+μ_1(x,{d}_\mathfrak
{g}(y))+μ_\mathfrak{g}(d_1(x),y)+μ_\mathfrak{g}(x,d_1(y)) \displaystyle +λμ_1({d}_\mathfrak{g}(x),{d}_
{\mathfrak{g}}(y))+λμ_\mathfrak{g}(d_1(x),{d}_\mathfrak{
g}(y))+λμ_\mathfrak{g}({d}_\mathfrak{g}(x),d_1(y)). \tag{13}
$$*
**Proposition 3.3**
*Let $({\mathfrak{g}}[[t]],μ_t,d_t)$ be a $1$ -parameter formal deformation of a differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ . Then $(μ_1,d_1)$ is a 2-cocycle of the differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ with the coefficient in the adjoint representation ${\mathfrak{g}}_\rm ad$ .*
* Proof*
For $n=1$ , Eq. (12) is equivalent to $∂_Lie^2μ_1=0$ , and Eq. (13) is equivalent to Eq. (3), that is
$$
∂^1_DO_λd_1+δ^2μ_1=0.
$$
Thus $(μ_1,d_1)$ is a 2-cocycle by Remark 1.13. ∎
If $μ_t=μ_\mathfrak{g}$ in the above $1$ -parameter formal deformation of the differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ , we obtain a $1$ -parameter formal deformation of the differential operator ${d}_\mathfrak{g}$ . Consequently, we have
**Corollary 3.4**
*Let $d_t$ be a $1$ -parameter formal deformation of the differential operator ${d}_\mathfrak{g}$ . Then $d_1$ is a 1-cocycle of the differential operator ${d}_\mathfrak{g}$ with coefficients in the adjoint representation ${\mathfrak{g}}_\rm ad$ .*
* Proof*
In the special case when $n=1$ , Eq. (11) is equivalent to $∂^1_DO_λd_1=0$ , which implies that $d_1$ is a 1-cocycle of the differential operator ${d}_\mathfrak{g}$ with coefficients in the adjoint representation ${\mathfrak{g}}_\rm ad$ . ∎
**Definition 3.5**
*The $2$ -cocycle $(μ_1,d_1)$ is called the infinitesimal of the $1$ -parameter formal deformation $({\mathfrak{g}}[[t]],μ_t,d_t)$ of $({\mathfrak{g}},{d}_\mathfrak{g})$ .*
**Definition 3.6**
*Let $({\mathfrak{g}}[[t]],μ_t,d_t)$ and $({\mathfrak{g}}[[t]],\bar{μ}_t,\bar{d}_t)$ be $1$ -parameter formal deformations of $({\mathfrak{g}},{d}_\mathfrak{g})$ . A formal isomorphism from $({\mathfrak{g}}[[t]],\bar{μ}_t,\bar{d}_t)$ to $({\mathfrak{g}}[[t]],μ_t,d_t)$ is a power series $Φ_t=∑_i≥ 0φ_it^i:{\mathfrak{g}}[[t]] → { \mathfrak{g}}[[t]]$ , where $φ_i:{\mathfrak{g}}→{\mathfrak{g}}$ are linear maps with $φ_0={\rm Id}_\mathfrak{g}$ , such that
| | $\displaystyleΦ_t∘\bar{μ}_t$ | $\displaystyle=$ | $\displaystyleμ_t∘(Φ_t×Φ_t),$ | |
| --- | --- | --- | --- | --- | Two $1$ -parameter formal deformations $({\mathfrak{g}}[[t]],μ_t,d_t)$ and $({\mathfrak{g}}[[t]],\bar{μ}_t,\bar{d}_t)$ are said to be equivalent if there exists a formal isomorphism $Φ_t:({\mathfrak{g}}[[t]],\bar{μ}_t,\bar{d}_t) → ({ \mathfrak{g}}[[t]],μ_t,d_t)$ .*
**Theorem 3.7**
*The infinitesimals of two equivalent $1$ -parameter formal deformations of $({\mathfrak{g}},{d}_\mathfrak{g})$ are in the same cohomology class in $\tilde{\rm{H}}_DL_λ^2({\mathfrak{g}},{\mathfrak{g}})$ .*
* Proof*
Let $Φ_t:({\mathfrak{g}}[[t]],\bar{μ}_t,\bar{d}_t) → ({ \mathfrak{g}}[[t]],μ_t,d_t)$ be a formal isomorphism. For all $x,y∈{\mathfrak{g}}$ , we have
| | $\displaystyleΦ_t∘\bar{μ}_t(x,y)$ | $\displaystyle=$ | $\displaystyleμ_t∘(Φ_t×Φ_t)(x,y),$ | |
| --- | --- | --- | --- | --- |
Expanding the above identities and comparing the coefficients of $t$ , we obtain
| | $\displaystyle\bar{μ}_1(x,y)$ | $\displaystyle=$ | $\displaystyleμ_1(x,y)+[φ_1(x),y]+[x,φ_1(y)]-φ_1([x,y]),$ | |
| --- | --- | --- | --- | --- |
Thus, we have
$$
(\bar{μ}_1,\bar{d}_1)=(μ_1,d_1)+∂_DL_λ^
1(φ_1),
$$
which implies that $[(\bar{μ}_1,\bar{d}_1)]=[(μ_1,d_1)]$ in $\tilde{\rm{H}}^2_DL_λ({\mathfrak{g}},{\mathfrak{g}})$ . ∎
Given any differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ , interpret $μ_\mathfrak{g}$ and ${d}_\mathfrak{g}$ as the formal power series $μ_t$ and $d_t$ with $μ_i=δ_i,0μ_\mathfrak{g}$ and $d_i=δ_i,0{d}_\mathfrak{g}$ respectively for all $i≥ 0$ , where $δ_i,0$ is the Kronecker sign. Then $({\mathfrak{g}}[[t]],μ_\mathfrak{g},{d}_\mathfrak{g})$ is a 1-parameter formal deformation of $({\mathfrak{g}},{d}_\mathfrak{g})$ .
**Definition 3.8**
*A $1$ -parameter formal deformation $({\mathfrak{g}}[[t]],μ_t,d_t)$ of $({\mathfrak{g}},{d}_\mathfrak{g})$ is said to be trivial if it is equivalent to the deformation $({\mathfrak{g}}[[t]],μ_\mathfrak{g},{d}_\mathfrak{g})$ , that is, there exists $Φ_t=∑_i≥ 0φ_it^i:{\mathfrak{g}}[[t]] → { \mathfrak{g}}[[t]]$ , where $φ_i:{\mathfrak{g}}→{\mathfrak{g}}$ are linear maps with $φ_0={\rm Id}_\mathfrak{g}$ , such that
| | $\displaystyleΦ_t∘μ_t$ | $\displaystyle=$ | $\displaystyleμ_\mathfrak{g}∘(Φ_t×Φ_t),$ | |
| --- | --- | --- | --- | --- |*
**Definition 3.9**
*A differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ is said to be rigid if every $1$ -parameter formal deformation is trivial.*
**Theorem 3.10**
*If $\tilde{\rm{H}}^2_DL_λ({\mathfrak{g}},{\mathfrak{g}}_\rm ad )=0$ , then the differential Lie algebra $({\mathfrak{g}},{d}_\mathfrak{g})$ is rigid.*
* Proof*
Let $({\mathfrak{g}}[[t]],μ_t,d_t)$ be a $1$ -parameter formal deformation of $({\mathfrak{g}},{d}_\mathfrak{g})$ . By Proposition 3.3, $(μ_1,d_1)$ is a 2-cocycle. By $\tilde{\rm{H}}^2_DL_λ({\mathfrak{g}},{\mathfrak{g}}_\rm ad )=0$ , there exists a 1-cochain $φ_1∈{C}^1_Lie({\mathfrak{g}},{\mathfrak{g}})$ such that
$$
\displaystyle(μ_1,d_1)=-∂_DL_λ^1(φ_1). \tag{14}
$$
Then setting $Φ_t={\rm Id}_\mathfrak{g}+φ_1t$ , we have a 1-parameter formal deformation $({\mathfrak{g}}[[t]],\bar{μ}_t,\bar{d}_t)$ , where
| | $\displaystyle\bar{μ}_t(x,y)$ | $\displaystyle=$ | $\displaystyle\big{(}Φ_t^-1∘μ_t∘(Φ_t×Φ_t)\big {)}(x,y),$ | |
| --- | --- | --- | --- | --- |
Thus, $({\mathfrak{g}}[[t]],\bar{μ}_t,\bar{d}_t)$ is equivalent to $({\mathfrak{g}}[[t]],μ_t,d_t)$ . Moreover, we have
| | $\displaystyle\bar{μ}_t(x,y)$ | $\displaystyle=$ | $\displaystyle({\rm Id}_\mathfrak{g}-φ_1t+φ_1^2t^2+⋯+(-1)^ {i}φ_1^it^i+⋯)(μ_t(x+φ_1(x)t,y+φ_1(y)t)),$ | |
| --- | --- | --- | --- | --- |
Therefore,
| | $\displaystyle\bar{μ}_t(x,y)$ | $\displaystyle=$ | $\displaystyle[x,y]+(μ_1(x,y)+[x,φ_1(y)]+[φ_1(x),y]-φ_1([x,y ]))t+\bar{μ}_2(x,y)t^2+⋯,$ | |
| --- | --- | --- | --- | --- |
By Eq. (14), we have
| | $\displaystyle\bar{μ}_t(x,y)$ | $\displaystyle=$ | $\displaystyle[x,y]+\bar{μ}_2(x,y)t^2+⋯,$ | |
| --- | --- | --- | --- | --- |
Then by repeating the argument, we can show that $({\mathfrak{g}}[[t]],μ_t,d_t)$ is equivalent to $({\mathfrak{g}}[[t]],μ_\mathfrak{g},{d}_\mathfrak{g})$ . Thus, $({\mathfrak{g}},{d}_\mathfrak{g})$ is rigid. ∎
## 4. $L_∞[1]$ -structure for (relative and absolute) differential Lie algebras
In this section, we study the $L_∞[1]$ -structure for differential Lie algebras of weight $λ$ . In order to deal with absolute differential Lie algebras, we introduce a generalised version of derived bracket technique; moreover, to consider the weight case, we incorporate the weight $λ$ into the statements. These are the main differences of this paper with [18, 11], since the latter papers consider difference operators, say, relative differential operators of weight $1$ .
### 4.1. $L_∞[1]$ -algebras
In this subsection, we recall some preliminaries on $L_∞[1]$ -algebras.
**Definition 4.1**
*An $L_∞[1]$ -algebra is a graded vector space ${\mathfrak{L}}=\bigoplus\limits_i∈ℕ{\mathfrak{L}}^i$ endowed with a family of graded linear maps $l_n:{\mathfrak{L}}^⊗ n→{\mathfrak{L}},n≥ 1$ of degree $1$ satisfying the following equations: for arbitrary $n≥ 1$ , $σ∈ S_n$ and $x_1,\dots,x_n∈{\mathfrak{L}}$ ,
1. (graded symmetry)
$$
l_n(x_σ(1),\dots,x_σ(n))=ε(σ)l_n(x_1,\dots,
x_n), \tag{1}
$$
1. (generalised Jacobi identity)
$$
∑_i=1^n∑_σ∈{Sh(i,n-i)}ε(σ)l_n-i+1
(l_i(x_σ(1),\dots,x_σ(i)),x_σ(i+1),\dots,x_σ(n))
=0. \tag{1}
$$*
**Remark 4.2**
*Let us consider the generalised Jacobi identity for $n≤ 3$ with the assumption of generalised symmetry. 1. For $n=1$ , then $l_1∘ l_1=0$ , that is, $l_1$ is a differential.
1. For $n=2$ , then $l_1∘ l_2+l_2∘(l_1⊗{\rm Id}+{\rm Id}⊗ l_1)=0$ , that is , $l_1$ is a derivation with respect to $l_2$ .
1. For $n=3$ and arbitrary homogeneous elements $x_1,x_2,x_3∈ L$ , we have
$$
\begin{array}[]{ll}&l_2(l_2(x_1⊗ x_2)⊗ x_3)+(-1)^|x_1
|(|x_2|+|x_3|)l_2(l_2(x_2⊗ x_3)⊗ x_1)+(-1)^|x_3|
(|x_1|+|x_2|)l_2(l_2(x_3⊗ x_1)⊗ x_2)\\
=&-\Big{(}l_1(l_3(x_1⊗ x_2⊗ x_3))+l_3(l_1(x_1)
⊗ x_2⊗ x_3)+(-1)^|x_1|l_3(x_1⊗ l_1(x_2)
⊗ x_3)+\\
&(-1)^|x_1|+|x_2|l_3(x_1⊗ x_2⊗ l_1(x_3))\Big{)},
\end{array}
$$
that is, $l_2$ satisfies the Jacobi identity up to homotopy.*
**Definition 4.3**
*A Maurer-Cartan element of an $L_∞[1]$ -algebra $({\mathfrak{L}},\{l_n\}_n≥ 1)$ is an element $α∈{\mathfrak{L}}^0$ satisfying the Maurer-Cartan equation:
| | $\displaystyle∑_n=1^∞\frac{1}{n!}l_n(α^⊗ n)=0,$ | |
| --- | --- | --- |
whenever this infinite sum exists. Denote $MC({\mathfrak{L}}):=\{\mbox{Maurer-Cartan elements of}~{}{\mathfrak{ L}}\}$ .*
**Proposition 4.4**
*[6, Twisting procedure] Let $α$ be a Maurer-Cartan element of $L_∞[1]$ -algebra ${\mathfrak{L}}$ , The twisted $L_∞[1]$ -algebra is given by $l_n^α:{\mathfrak{L}}^⊗ n→{\mathfrak{L}}$ which is defined as follows $\colon$
| | $\displaystyle l^α_n(x_1⊗⋯⊗ x_n)=∑_i=0^ ∞\frac{1}{i!}l_n+i(α^⊗ i⊗ x_1⊗⋯⊗ x _n), ∀ x_1,\dots,x_n∈{\mathfrak{L}},$ | |
| --- | --- | --- |
whenever these infinite sums exist.*
The following simple observation with immediate proof will be very useful while considering differential operators of a given weight $λ$ .
**Proposition 4.5**
*Let ${\mathfrak{L}}=({\mathfrak{L}},\{l_n\}_n≥ 1)$ be an $L_∞[1]$ -algebra and $λ∈{\bf k}$ . Consider $l_n^\prime:{\mathfrak{L}}^⊗ n→{\mathfrak{L}},n≥ 1$ given by
| | $\displaystyle l^\prime_n(x_1⊗⋯⊗ x_n)=λ^n-1l_ n(x_1⊗⋯⊗ x_n), ∀ x_1,\dots,x_n∈{\mathfrak{L }},$ | |
| --- | --- | --- |
then $({\mathfrak{L}},\{l_n^\prime\}_n≥ 1)$ is also an $L_∞[1]$ -algebra. If $l_1=0$ , then imposing $l_1^\prime=0$ and $l_n^\prime=λ^n-2l_n$ for $n≥ 2$ also gives a new $L_∞[1]$ -structure on ${\mathfrak{L}}$ .*
### 4.2. A generalised version of derived bracket technique
Now we introduce a generalised version of the derived bracket technique invented by Voronov [23, 24]. Since we consider differential Lie algebras with weight $λ∈{\bf k}$ , we insert $λ$ in the statements whenever needed by Proposition 4.5, thus modify corresponding results in [2, 11].
**Definition 4.6**
*A generalised V-data is a septuple $({\mathfrak{L}},{\mathfrak{M}},ι_\mathfrak{M},{\mathfrak{A}},ι_ \mathfrak{A},P,Δ)$ , where
- $({\mathfrak{L}},[-,-])$ is a graded Lie algebra,
- $({\mathfrak{M}},[-,-]_\mathfrak{M})$ is a graded Lie algebra together with an injective linear map $ι_\mathfrak{M}:{\mathfrak{M}}→{\mathfrak{L}}$ which is a homomorphism of graded Lie algebras,
- ${\mathfrak{A}}$ is an abelian graded Lie algebra endowed with an injective linear map $ι_\mathfrak{A}:{\mathfrak{A}}→{\mathfrak{L}}$ which is a homomorphism of graded Lie algebras,
- $P:{\mathfrak{L}} → {\mathfrak{A}}$ is a linear map such that $P∘ι_\mathfrak{A}={\rm Id}_\mathfrak{A}$ and ${\rm Ker}(P)$ is a graded Lie subalgebra of ${\mathfrak{L}}$ ,
- $Δ∈Ker(P)^1$ satisfying $[Δ,Δ]=0$ and $[Δ,ι_\mathfrak{M}({\mathfrak{M}})]⊂ι_\mathfrak{M}({ \mathfrak{M}})$ .*
**Theorem 4.7**
*Let $({\mathfrak{L}},{\mathfrak{M}},ι_\mathfrak{M},{\mathfrak{A}},ι_ \mathfrak{A},P,Δ)$ be a generalised V-datum and $λ∈{\bf k}$ . Then the graded vector space $s{\mathfrak{L}}⊕{\mathfrak{A}}$ has an $L_∞[1]$ -algebra structure which is defined as follows $\colon$
$$
\begin{array}[]{rcl}l_1(sf)&=&(-s[Δ,f],P(f)),\\
l_1(ξ)&=&P[Δ,ι_{\mathfrak{A}}(ξ)],\\
l_2(sf,sg)&=&(-1)^|f|λ s[f,g],\\
l_i(sf,ξ_1,⋯,ξ_i-1)&=&λ^i-1P[⋯[f,ι_{\mathfrak
{A}}(ξ_1)],⋯,ι_{\mathfrak{A}}(ξ_i-1)],i≥ 2,\\
l_i(ξ_1,⋯,ξ_i)&=&λ^i-1P[⋯[Δ,ι_{\mathfrak
{A}}(ξ_1)],⋯,ι_{\mathfrak{A}}(ξ_i)],i≥ 2,\end{array}
$$
for homogeneous elements $f,g∈{\mathfrak{L}}$ , $ξ,ξ_1,⋯,ξ_i∈{\mathfrak{A}}$ and all other components of $\{l_i\}_i=1^+∞$ vanish. Similarly, the graded vector space $s{\mathfrak{M}}⊕{\mathfrak{A}}$ has also an $L_∞[1]$ -algebra structure which is defined as follows $\colon$
$$
\begin{array}[]{rcl}l_1(sf)&=&(-sι^-1_\mathfrak{M}[Δ,ι_
\mathfrak{M}(f)],Pι_\mathfrak{M}(f)),\\
l_1(ξ)&=&P[Δ,ι_{\mathfrak{A}}(ξ)],\\
l_2(sf,sg)&=&(-1)^|f|λ s[f,g]_\mathfrak{M},\\
l_i(sf,ξ_1,⋯,ξ_i-1)&=&λ^i-1P[⋯[ι_\mathfrak{M
}(f),ι_{\mathfrak{A}}(ξ_1)],⋯,ι_{\mathfrak{A}}(ξ_i-1)
],i≥ 2,\\
l_i(ξ_1,⋯,ξ_i)&=&λ^i-1P[⋯[Δ,ι_{\mathfrak
{A}}(ξ_1)],⋯,ι_{\mathfrak{A}}(ξ_i)],i≥ 2,\end{array}
$$
for homogeneous elements $f,g∈{\mathfrak{M}}$ , $ξ,ξ_1,⋯,ξ_i∈{\mathfrak{A}}$ and all other components of $\{l_i\}_i=1^+∞$ vanish. Moreover, there exists an injective $L_∞[1]$ -algebra homomorphism $ι:s\mathfrak{M}⊕{\mathfrak{A}} → s{\mathfrak{L}}⊕{ \mathfrak{A}}$ induced by $ι_\mathfrak{M}$ .*
* Proof*
Obviously, the quadruple $({\mathfrak{L}},\mathfrak{ι_A(A)},ι_{\mathfrak{A}}P,Δ)$ is a V-data in the sense of Voronov [23], by [23, Section 3], there is an $L_∞[1]$ -algebra $\{l_i\}_i=1^+∞$ on $s{\mathfrak{L}}⊕\mathfrak{ι_A{(A)}}$ , where for homogeneous elements $f,g∈{\mathfrak{L}}$ , $ξ,ξ_1,⋯,ξ_i∈{\mathfrak{A}}$ ,
$$
\begin{array}[]{rcl}l_1(sf)&=&(-s[Δ,f],ι_{\mathfrak{A}}P(f)),\\
l_1(ι_{\mathfrak{A}}(ξ))&=&ι_{\mathfrak{A}}P([Δ,ι_{
\mathfrak{A}}(ξ)]),\\
l_2(sf,sg)&=&(-1)^|f|s[f,g],\\
l_i(sf,ι_{\mathfrak{A}}(ξ_1),⋯,ι_{\mathfrak{A}}(ξ_i-
1))&=&ι_{\mathfrak{A}}P[⋯[f,ι_{\mathfrak{A}}(ξ_1)],
⋯,ι_{\mathfrak{A}}(ξ_i-1)],i≥ 2,\\
l_i(ι_{\mathfrak{A}}(ξ_1),⋯,ι_{\mathfrak{A}}(ξ_i))&
=&ι_{\mathfrak{A}}P[⋯[Δ,ι_{\mathfrak{A}}(ξ_1)],
⋯,ι_{\mathfrak{A}}(ξ_i)],i≥ 2,\end{array}
$$
and all other components of $\{l_i\}_i=1^+∞$ vanish. A similar $L_∞[1]$ -algebra structure $\{l_i\}_i=1^+∞$ on $sι_\mathfrak{M}({\mathfrak{M}})⊕\mathfrak{ι_A{(A)}}$ exists, as well as an inclusion $sι_\mathfrak{M}({\mathfrak{M}})⊕\mathfrak{ι_A{(A)}} \hookrightarrow s{\mathfrak{L}}⊕\mathfrak{ι_A{(A)}}$ of $L_∞[1]$ -algebras. By Proposition 4.5, we could insert $λ$ into the construction of the higher Lie brackets. The $L_∞[1]$ -algebra structure on $s{\mathfrak{L}}⊕\mathfrak{ι_A{(A)}}$ are given by: for homogeneous elements $f,g∈{\mathfrak{L}}$ , $ξ,ξ_1,⋯,$ $ξ_i∈{\mathfrak{A}}$ ,
$$
\begin{array}[]{rcl}l_1(sf)&=&(-s[Δ,f],ι_{\mathfrak{A}}P(f)),\\
l_1(ι_{\mathfrak{A}}(ξ))&=&ι_{\mathfrak{A}}P([Δ,ι_{
\mathfrak{A}}(ξ)]),\\
l_2(sf,sg)&=&(-1)^|f|λ s[f,g],\\
l_i(sf,ι_{\mathfrak{A}}(ξ_1),⋯,ι_{\mathfrak{A}}(ξ_i-
1))&=&λ^i-1ι_{\mathfrak{A}}P[⋯[f,ι_{\mathfrak{A}}(
ξ_1)],⋯,ι_{\mathfrak{A}}(ξ_i-1)],i≥ 2,\\
l_i(ι_{\mathfrak{A}}(ξ_1),⋯,ι_{\mathfrak{A}}(ξ_i))&
=&λ^i-1ι_{\mathfrak{A}}P[⋯[Δ,ι_{\mathfrak{A}}(
ξ_1)],⋯,ι_{\mathfrak{A}}(ξ_i)],i≥ 2,\end{array}
$$
and all other components of higher Lie brackets vanish. The $L_∞[1]$ -algebra structure on $sι_\mathfrak{M}({\mathfrak{M}})⊕\mathfrak{ι_A{(A)}}$ are given by: for homogeneous elements $f,g∈{\mathfrak{M}}$ , $ξ,ξ_1,⋯,ξ_i∈{\mathfrak{A}}$ ,
$$
\begin{array}[]{rcl}l_1(sι_\mathfrak{M}(f))&=&(-sι_\mathfrak{M}^
{-1}([Δ,ι_\mathfrak{M}(f)]),ι_{\mathfrak{A}}Pι_\mathfrak
{M}(f)),\\
l_1(ι_{\mathfrak{A}}(ξ))&=&ι_{\mathfrak{A}}P([Δ,ι_{
\mathfrak{A}}(ξ)]),\\
l_2(sι_\mathfrak{M}(f),sι_\mathfrak{M}(g))&=&(-1)^|f|λ s
ι_\mathfrak{M}([f,g]_\mathfrak{M}),\\
l_i(sι_\mathfrak{M}(f),ι_{\mathfrak{A}}(ξ_1),⋯,ι_{
\mathfrak{A}}(ξ_i-1))&=&λ^i-1ι_{\mathfrak{A}}P[⋯[ι
_\mathfrak{M}(f),ι_{\mathfrak{A}}(ξ_1)],⋯,ι_{\mathfrak{A
}}(ξ_i-1)],i≥ 2,\\
l_i(ι_{\mathfrak{A}}(ξ_1),⋯,ι_{\mathfrak{A}}(ξ_i))&
=&λ^i-1ι_{\mathfrak{A}}P[⋯[Δ,ι_{\mathfrak{A}}(
ξ_1)],⋯,ι_{\mathfrak{A}}(ξ_i)],i≥ 2,\end{array}
$$
and all other components of higher Lie brackets vanish. There exists also an inclusion $sι_\mathfrak{M}({\mathfrak{M}})⊕\mathfrak{ι_A{(A)}} \hookrightarrow s{\mathfrak{L}}⊕\mathfrak{ι_A{(A)}}$ of $L_∞[1]$ -algebras. Then the theorem holds by the following commutative diagram
$$
\textstyle{s{\mathfrak{L}}⊕{\mathfrak{A}}\ignorespaces\ignorespaces
\ignorespaces\ignorespaces} \scriptstyle{≅} \scriptstyle{≤ft(\begin{matrix}\rm{Id}&0\\
0&ι_\mathfrak{A}\end{matrix}\right)} \textstyle{s{\mathfrak{L}}⊕ι_\mathfrak{A}{({\mathfrak{A}})}} \textstyle{s{\mathfrak{M}}⊕{\mathfrak{A}}\ignorespaces\ignorespaces
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} \scriptstyle{ι} \scriptstyle{≅} \scriptstyle{≤ft(\begin{matrix}ι_\mathfrak{M}&0\\
0&ι_\mathfrak{A}\end{matrix}\right)} \textstyle{sι_\mathfrak{M}{({\mathfrak{M}})}⊕ι_\mathfrak{A}{(
{\mathfrak{A}})}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces} \scriptstyle{\rm{inc}}
$$ ∎
**Definition 4.8**
*Let $({\mathfrak{L}},{\mathfrak{M}},ι_\mathfrak{M},{\mathfrak{A}},ι_ \mathfrak{A},P,Δ)$ and $(\mathfrak{L^\prime},\mathfrak{M^\prime},ι_{\mathfrak{M}^\prime}, \mathfrak{A^\prime},ι_{\mathfrak{A}^\prime},P^\prime,Δ^ \prime)$ be two generalised V-data. A morphism between them is a triple $f=(f_\mathfrak{L},f_\mathfrak{M},f_\mathfrak{A})$ , where $f_\mathfrak{L}:{\mathfrak{L}}→\mathfrak{L^\prime}$ , $f_\mathfrak{M}:{\mathfrak{M}}→\mathfrak{M^\prime}$ and $f_\mathfrak{A}:{\mathfrak{A}}→\mathfrak{A^\prime}$ are three homomorphisms of graded Lie algebras such that $f∘ι_\mathfrak{M}=ι_{\mathfrak{M}^\prime}∘ f_\mathfrak{M}$ , $f∘ι_\mathfrak{A}=ι_{\mathfrak{A}^\prime}∘ f_\mathfrak{A}$ , $f_\mathfrak{A}∘ P=P^\prime∘ f_\mathfrak{L}$ and $f_\mathfrak{L}(Δ)=Δ^\prime$ .*
The following result is obvious.
**Proposition 4.9**
*Given a morphism of generalised V-data
$$
f:({\mathfrak{L}},{\mathfrak{M}},ι_\mathfrak{M},{\mathfrak{A}},ι_
\mathfrak{A},P,Δ)→(\mathfrak{L^\prime},\mathfrak{M^\prime},ι_
{\mathfrak{M}^\prime},\mathfrak{A^\prime},ι_{\mathfrak{A}^\prime}
,P^\prime,Δ^\prime),
$$
there exists an $L_∞[1]$ -algebra homomorphism
$$
\tilde{f}:s{\mathfrak{M}}⊕{\mathfrak{A}}→ s\mathfrak{M^\prime}⊕
\mathfrak{A^\prime}
$$
induced by $f$ .*
Obviously we have the modified version of Theorem 4.7 as well as a characterisation of their Maurer-Cartan elements.
**Proposition 4.10**
*Let $({\mathfrak{L}},{\mathfrak{M}},ι_\mathfrak{M},{\mathfrak{A}},ι_ \mathfrak{A},P,Δ)$ be a generalised V-datum and $λ∈{\bf k}$ . If $P∘ι_\mathfrak{M}=0$ and $Δ=0$ , then there exists an $L_∞[1]$ -algebra structure on $s{\mathfrak{M}}⊕{{\mathfrak{A}}}$ , where the higher Lie bracket $\{l_i\}_i=1^+∞)$ are given by
$$
\begin{array}[]{rcl}l_2(sf,sg)&=&(-1)^|f|s[f,g]_\mathfrak{M},\\
l_i(sf,ξ_1,⋯,ξ_i-1)&=&λ^i-2P[⋯[ι_\mathfrak{M
}(f),ι_{{\mathfrak{A}}}(ξ_1)]_{\mathfrak{L}},⋯,ι_{{
\mathfrak{A}}}(ξ_i-1)]_{\mathfrak{L}},i≥ 2,\\
\end{array}
$$
for homogeneous elements $f,g∈{\mathfrak{M}}$ , $ξ_1,⋯,ξ_i∈{{\mathfrak{A}}}$ and all other components vanish. Let $f∈{\mathfrak{M}}^1,ξ∈{\mathfrak{A}}^0$ . The pair $(sf,ξ)$ is a Maurer-Cartan element in the $L_∞[1]$ -algebra $s{\mathfrak{M}}⊕{{\mathfrak{A}}}$ if and only if
$$
f∈MC({\mathfrak{M}}) and∑_n=2^∞\frac{1}{(n-1
)!}λ^n-2P[⋯[ι_\mathfrak{M}(f),\underbrace{ι_{{
\mathfrak{A}}}(ξ)]_{\mathfrak{L}},⋯,ι_{{\mathfrak{A}}}(ξ)}_
(n-1) times]_{\mathfrak{L}}=0.
$$*
* Proof*
The first statement follows from Proposition 4.5. For the second statement, $(sf,ξ)$ is a Maurer-Cartan element in the $L_∞[1]$ -algebra $s{\mathfrak{M}}⊕{{\mathfrak{A}}}$ if and only if
| | $\displaystyle 0$ | $\displaystyle=∑_n=1^∞\frac{1}{n!}l_n((sf,ξ)^⊗ n)$ | |
| --- | --- | --- | --- |
if and only if
$$
[f,f]_{\mathfrak{M}}=0 and ∑_n=2^∞\frac{1}{(n-1)!}
λ^n-2P[⋯[ι_\mathfrak{M}(f),\underbrace{ι_{{\mathfrak{A
}}}(ξ)]_{\mathfrak{L}},⋯,ι_{{\mathfrak{A}}}(ξ)}_(n-1)
times]_{\mathfrak{L}}=0.
$$
∎
### 4.3. $L_∞[1]$ -structure for relative differential Lie algebras with weight
In this subsection, by using the generalised version of derived bracket technique, we found an $L_∞[1]$ -algebra whose Maurer-Cartan elements are in bijection with the set of structures of relative differential Lie algebras of weight $λ$ , thus generalising the result of Jiang and Sheng [11] from weight $1$ case to arbitrary weight cases.
We recall the classical Nijenhuis-Richardson brackets and basic facts about LieAct triples; for details, see [2, 11].
Given a vector space $V$ , its exterior algebra is $\bigwedge(V):=\bigoplus_k=0^∞\wedge^kV$ and the reduced version is $\bar{\bigwedge}(V):=\bigoplus_k=1^∞\wedge^kV$ . We consider the graded vector space $Hom(\bar{\bigwedge}(V),V)$ , so for $f∈Hom(\wedge^n+1V,V)$ , its degree is $n$ . The graded space $Hom(\bar{\bigwedge}(V),V)$ is a graded Lie algebra [17] under the Nijenhuis-Richardson bracket $[~{},~{}]_{\rm NR}$ which is defined as follows: for arbitrary $f∈Hom(\wedge^p+1V,V)$ and $g∈Hom(\wedge^q+1V,V)$ ,
$$
[f,g]_{\rm NR}:=f\bar{∘}g-(-1)^pqg\bar{∘}f,
$$
where $f\bar{∘}g∈Hom(\wedge^p+q+1V,V)$ is given by
$$
\displaystyle f\bar{∘}g(x_1,⋯,x_p+q+1) \displaystyle=∑_σ∈{Sh(q+1,p)}{sgn}(σ)f(g(x_
{σ(1)},⋯,x_σ(q+1)),x_σ(q+2),⋯,x_σ(p+q+1)). \tag{1}
$$
**Definition 4.11**
*A LieAct triple is a triple $({\mathfrak{g}},{\mathfrak{h}},ρ)$ , where $({\mathfrak{g}},[~{},~{}]_\mathfrak{g})$ and $({\mathfrak{h}},[~{},~{}]_\mathfrak{h})$ are Lie algebras and $ρ:{\mathfrak{g}} → Der({\mathfrak{h}})$ is a homomorphism of Lie algebras.*
**Remark 4.12**
*Given a LieAct triple $({\mathfrak{g}},{\mathfrak{h}},ρ)$ , the Lie algebra homomorphism $ρ:{\mathfrak{g}} → Der({\mathfrak{h}})$ means that ${\mathfrak{h}}$ is a Lie ${\mathfrak{g}}$ -module and there exists an action ${\mathfrak{g}}⊗{\mathfrak{h}} → {\mathfrak{h}}$ given by $x· u:=ρ(x)(u)$ , subject to the Leibniz rule
$$
x·[u,v]_\mathfrak{h}=[x· u,v]_\mathfrak{h}+[u,x· v]_\mathfrak
{h}, ∀ x∈{\mathfrak{g}}~{}\mbox{and}~{}u,v∈{\mathfrak{h}}.
$$*
Given two vector spaces $V$ and $W$ , by the isomorphism
$$
\begin{array}[]{rcl}φ:\wedge^n(V⊕ W)&\longrightarrow&\bigoplus_k
+l=n,k,l≥ 0\wedge^kV⊗\wedge^lW\\
(v_1+w_1)\wedge⋯\wedge(v_n+w_n)&↦&∑\limits_σ∈{
Sh(k,n-k)}{sgn}(σ)v_σ(1)\wedge⋯\wedge v_
σ(k)⊗ w_σ(k+1)\wedge⋯\wedge w_σ(n),\end{array} \tag{1}
$$
we have an isomorphism
$$
Hom(\wedge^n(V⊕ W),V⊕ W)≅\big{(}⊕_k+l=n\mathrm
{Hom}(\wedge^kV⊗\wedge^lW,V)⊕(⊕_k+l=nHom(\wedge
^kV⊗\wedge^lW,W))\big{)}.
$$
With this isomorphism in mind, we have the following result.
**Proposition 4.13**
*([2]) Let ${\mathfrak{g}}$ and ${\mathfrak{h}}$ be two vector spaces. Let
$$
{\mathfrak{L}}^\prime=Hom(\bar{\wedge}({\mathfrak{g}}⊕{
\mathfrak{h}}),{\mathfrak{g}}⊕{\mathfrak{h}})
$$
and denote
$$
\mathfrak{M}^\prime:=Hom(\bar{\wedge}{\mathfrak{g}},{\mathfrak{g}})
⊕Hom(\bar{\wedge}{\mathfrak{g}}⊗\bar{\wedge}{\mathfrak{h}}
,{\mathfrak{h}})⊕Hom(\bar{\wedge}{\mathfrak{h}},{\mathfrak{h}}).
$$
Let $ι_{\mathfrak{M}^\prime}:{{\mathfrak{M}}^\prime}→{{\mathfrak{L}}^ \prime}$ be the natural injection. Then the Nijenhuis-Richardson bracket on $({{\mathfrak{L}}^\prime},[-,-]_{\rm NR})$ induces a Lie bracket on $\mathfrak{M}$ such that $ι_{\mathfrak{M}^\prime}:{{\mathfrak{M}}^\prime}→{{\mathfrak{L}}^ \prime}$ is an injective homomorphism of graded Lie algebras. Moreover, Maurer-Cartan elements of the graded Lie algebra ${\mathfrak{M}}^\prime$ are in bijection with the set of LieAct triple structures on $({\mathfrak{g}},{\mathfrak{h}})$ .*
**Definition 4.14 ([2,11])**
*Let $({\mathfrak{g}},{\mathfrak{h}},ρ)$ be a LieAct triple. A linear map $D:{\mathfrak{g}} → {\mathfrak{h}}$ is called a relative differential operator of weight $λ$ if the following equality holds:
$$
D([x,y]_\mathfrak{g})=ρ(x)(D(y))-ρ(y)(D(x))+λ[D(x),D(y)]_
\mathfrak{h}, ∀ x,y∈{\mathfrak{g}}.
$$
The quadruple $({\mathfrak{g}},{\mathfrak{h}},ρ,D)$ is called a relative differential Lie algebra of weight $λ$ .*
Now we exhibit an $L_∞[1]$ -algebra whose Maurer-Cartan elements are in bijection with the set of structures of relative differential Lie algebras of weight $λ$ structures on $({\mathfrak{g}},{\mathfrak{h}})$ , thus generalising the result of Jiang and Sheng [11] from weight $1$ to arbitrary weight.
Let ${\mathfrak{g}}$ and ${\mathfrak{h}}$ be two vector spaces. As above, let
$$
\begin{array}[]{rcl}{{\mathfrak{L}}^\prime}&=&Hom(\bar{\wedge}({
\mathfrak{g}}⊕{\mathfrak{h}}),{\mathfrak{g}}⊕{\mathfrak{h}}),\\
{{\mathfrak{M}}^\prime}&=&Hom(\bar{\wedge}{\mathfrak{g}},{\mathfrak
{g}})⊕Hom(\bar{\wedge}{\mathfrak{g}}⊗\bar{\wedge}{
\mathfrak{h}},{\mathfrak{h}})⊕Hom(\bar{\wedge}{\mathfrak{h}},{
\mathfrak{h}}),\end{array}
$$
and let
$$
{{\mathfrak{A}}^\prime}=Hom(\bar{\wedge}{\mathfrak{g}},{\mathfrak{h
}}).
$$
Denote $ι_{\mathfrak{A}^\prime}:{{\mathfrak{A}}^\prime}→{{\mathfrak{L}}^ \prime}$ to be the natural injection and $P^\prime:{{\mathfrak{L}}^\prime} → {{\mathfrak{A}}^\prime}$ to be the natural surjection. Obviously with the induced Lie bracket, ${{\mathfrak{A}}^\prime}$ is an abelian graded Lie algebra, $ι_{\mathfrak{A}^\prime}:{{\mathfrak{A}}^\prime}→{{\mathfrak{L}}^ \prime}$ is also a homomorphism of graded Lie algebras, ${\rm Ker}(P^\prime)$ is a graded Lie subalgebra of ${{\mathfrak{L}}^\prime}$ , $P∘ι_{\mathfrak{A}^\prime}={\rm Id}_{\mathfrak{A}^\prime}$ . Now let $Δ^\prime=0$ . Then the following result is easy by direct inspection and by Proposition 4.10.
**Proposition 4.15 (Compare with[11, Proposition 3.7])**
*The data $({{\mathfrak{L}}^\prime},{{\mathfrak{M}}^\prime},ι_{\mathfrak{M}^ \prime},{{\mathfrak{A}}^\prime},ι_{\mathfrak{A}^\prime},P^\prime, Δ^\prime=0)$ introduced above is a generalised V-datum, and an $L_∞[1]$ -algebra structure on $s{{\mathfrak{M}}^\prime}⊕{{\mathfrak{A}}^\prime}$ is given by
$$
\begin{array}[]{rcl}l_2(sf,sg)&=&(-1)^|f|s[f,g]_{\rm NR},\\
l_i(sf,ξ_1,⋯,ξ_i-1)&=&λ^i-2P[⋯[f,ξ_1]_{\rm NR
},⋯,ξ_i-1]_{\rm NR},i≥ 2,\end{array}
$$
for homogeneous elements $f,g∈{{\mathfrak{M}}^\prime}$ , $ξ_1,⋯,ξ_i-1∈{{\mathfrak{A}}^\prime}$ and the other components vanish.*
**Theorem 4.16 (Compare with[11, Theorem 3.8])**
*Let ${\mathfrak{g}},{\mathfrak{h}}$ be two vector spaces. Let $π∈Hom(\wedge^2{\mathfrak{g}},{\mathfrak{g}})$ , $ρ∈Hom({\mathfrak{g}}⊗{\mathfrak{h}},{\mathfrak{h}})$ , $μ∈Hom(\wedge^2{\mathfrak{h}},{\mathfrak{h}})$ and $D∈Hom({\mathfrak{g}},{\mathfrak{h}})$ , then $(s(π+ρ+μ),D)∈MC(s{\mathfrak{M}}^\prime⊕{\mathfrak{A} }^\prime)$ if and only if $({\mathfrak{g}},{\mathfrak{h}},ρ)$ is a LieAct triple and $D$ is a relative differential operator of weight $λ∈{\bf k}$ .*
* Proof*
Let $χ:=π+ρ+μ$ . By Proposition 4.10, $(sχ,D)∈MC(s\mathfrak{M^\prime}⊕\mathfrak{A^\prime})$ if and only if $χ∈MC({\mathfrak{M}}^\prime)$ (that is, $({\mathfrak{g}},{\mathfrak{h}},ρ)$ is a LieAct triple by Proposition 4.13) and $∑\limits_k=2^∞\frac{λ^k-2}{(k-1)!}P[\dots,[χ, \underbrace{D]_{\rm NR},\dots,D}_(k-1) times]_{\rm NR}=0$ . Since $[χ,D]_{\rm NR}∈Hom(\wedge^2{\mathfrak{g}},{\mathfrak{h}}) ⊕Hom({\mathfrak{g}}⊗{\mathfrak{h}},{\mathfrak{h}})$ ,
$$
[[χ,D]_{\rm NR},D]_{\rm NR}∈Hom(\wedge^2{\mathfrak{g}},{
\mathfrak{h}}) and [[[χ,D]_{\rm NR},D]_{\rm NR},D]_{\rm NR
}=0,
$$
then we obtain
| | $\displaystyle 0=$ | $\displaystyle∑_k=2^∞\frac{λ^k-2}{(k-1)!}P[\dots,[χ, \underbrace{D]_{\rm NR},\dots,D}_(k-1) times]_{\rm NR}$ | |
| --- | --- | --- | --- |
For arbitrary $(x,y)∈\wedge^2{\mathfrak{g}}$ , we have
| | $\displaystyle 0$ | $\displaystyle=P[χ,D]_{\rm NR}(x,y)+\frac{λ}{2}P[[χ,D]_{\rm NR} ,D]_{\rm NR}(x,y)$ | |
| --- | --- | --- | --- |
Hence $D$ is a relative differential operator of weight $λ$ . ∎
### 4.4. $L_∞[1]$ -structure for absolute differential Lie algebras
Let $\mathfrak{g}$ be a vector space. We consider the graded Lie algebra
$$
{\mathfrak{L}}:=Hom(\bar{\wedge}({\mathfrak{g}}⊕\mathfrak{g}),{
\mathfrak{g}}⊕\mathfrak{g})
$$
endowed with the Nijenhuis-Richardson bracket $[~{},~{}]_{\rm NR}$ . For convenience of presentation, we shall write ${\mathfrak{g}}^\prime$ for the second ${\mathfrak{g}}$ in ${\mathfrak{g}}⊕{\mathfrak{g}}$ , that is,
$$
{\mathfrak{L}}:=Hom(\bar{\wedge}({\mathfrak{g}}⊕\mathfrak{g}^
\prime),{\mathfrak{g}}⊕\mathfrak{g}^\prime).
$$
Let
$$
{\mathfrak{M}}:=Hom(\bar{\wedge}{\mathfrak{g}},{\mathfrak{g}})
and {\mathfrak{A}}:={\rm Hom}(\bar{\wedge}{\mathfrak{g}},{\mathfrak{
g}}).
$$
Endow ${\mathfrak{M}}$ with the Nijenhuis-Richardson bracket and ${\mathfrak{A}}$ with the trivial bracket. Consider two linear maps $ι_\mathfrak{M}:{\mathfrak{M}}→{\mathfrak{L}}$ and $ι_\mathfrak{A}:{\mathfrak{A}}→{\mathfrak{L}}$ defined as follows: For given $f:\wedge^n+1{\mathfrak{g}} → {\mathfrak{g}}∈{\mathfrak{M}}$ , $ι_\mathfrak{M}(f)=∑\limits_i=0^n+1f_i$ where $f_i:\wedge^n+1-i{\mathfrak{g}}⊗\wedge^i{\mathfrak{g}}^\prime → {\mathfrak{g}}^\prime$ , $0≤ i≤ n+1$ is given by
$$
f_i(x_1\wedge\dots\wedge x_n+1-i⊗ y_n+2-i\wedge\dots\wedge y_n+
1):=f(x_1\wedge\dots\wedge x_n+1-i\wedge y_n+2-i\wedge\dots\wedge y_n+
1),
$$
for $x_1\wedge\dots\wedge x_n+1-i⊗ y_n+2-i\wedge\dots\wedge y_n+1∈ \wedge^n+1-i{\mathfrak{g}}⊗\wedge^i{\mathfrak{g}}^\prime$ ; $ι_\mathfrak{A}$ identifies ${\mathfrak{A}}={\rm Hom}(\bar{\wedge}{\mathfrak{g}},{\mathfrak{g}})$ with the subspace ${\rm Hom}(\bar{\wedge}{\mathfrak{g}},{\mathfrak{g}}^\prime)$ of ${\mathfrak{L}}$ . Let $P:{\mathfrak{L}}→{\mathfrak{A}}$ be the natural projection identifying the subspace ${\rm Hom}(\bar{\wedge}{\mathfrak{g}},{\mathfrak{g}}^\prime)$ with ${\mathfrak{A}}={\rm Hom}(\bar{\wedge}{\mathfrak{g}},{\mathfrak{g}})$ .
**Proposition 4.17**
*Let ${\mathfrak{g}}$ be a vector space and $λ∈{\bf k}$ . The datum $({\mathfrak{L}},{\mathfrak{M}},ι_\mathfrak{M},{\mathfrak{A}},ι_ \mathfrak{A},P,Δ=0)$ defined above is a generalised V-datum.*
* Proof*
Obviously, $ι_\mathfrak{M}:{\mathfrak{M}}→{\mathfrak{L}}$ and $ι_\mathfrak{A}:{\mathfrak{A}}→{\mathfrak{L}}$ are injective. The only unclear statement is that $ι_\mathfrak{M}:{\mathfrak{M}}→{\mathfrak{L}}$ is a homomorphism of graded Lie algebras. In fact, for arbitrary $f:\wedge^n+1{\mathfrak{g}} → {\mathfrak{g}}$ and $g:\wedge^m+1{\mathfrak{g}} → {\mathfrak{g}}$ in ${\mathfrak{M}}$ , we have
| | | $\displaystyle[ι(f),ι(g)]_{\rm NR}$ | |
| --- | --- | --- | --- |
Thus $ι_\mathfrak{M}$ is a graded Lie algebra homomorphism. ∎
**Lemma 4.18**
*Keep the above notations, for arbitrary $f=∑\limits_i=0^n+1f_i∈{\rm Hom}(\wedge^n+1{\mathfrak{g}},{ \mathfrak{g}})$ and $1≤ r≤ n+1$ , take $ξ_i∈{\rm Hom}(\wedge^m_i+1{\mathfrak{g}},{\mathfrak{g}}^\prime)$ , $1≤ i≤ r$ . We have
| | $\displaystyle(⋯((f\bar{∘}ξ_1)\bar{∘}ξ_2)⋯)\bar{∘ }ξ_r=∑\limits_τ∈{Sh(m_r+1,\dots,m_1+1,n+1-r)}(-1)^ ∑\limits_j=1^{r(m_1+⋯+m_j-1)m_j}f_r\big{(}ξ_r⊗ \dots⊗ξ_1⊗{\rm Id}^⊗ n+1-r\big{)}τ^-1,$ | |
| --- | --- | --- |
where, denote $t=n+∑\limits_j=1^rm_j+1$ , $τ^-1$ acts on $\wedge^tV$ via $τ^-1(x_1,\dots,x_t):={sgn}(τ)(x_τ(1),\dots,x_τ(t ))$ .*
* Proof*
In fact, by definition of $\bar{∘}$ , we have
$$
\displaystyle \big{(}(⋯((f\bar{∘}ξ_1)\bar{∘}ξ_2)
⋯)\bar{∘}ξ_r\big{)}(x_1,\dots,x_t) \displaystyle= \displaystyle∑_σ_{1∈{Sh}(m_r+1,t-1-m_r)}{sgn
}(σ_1)\big{(}(⋯(f_r\bar{∘}ξ_1)⋯)\bar{∘}ξ_r-1
\big{)}\Big{(}ξ_r(x_σ_{1(1)},\dots,x_σ_{1(m_r+1)}),x_
σ_{1(m_r+2)},\dots,x_σ_{1(t)}\Big{)} \displaystyle= \displaystyle∑_\begin{subarray{c}σ_1∈{Sh}(m_r+1,t-1-m
_r)\\
σ_2∈{Sh}(m_r-1+1,t-1-m_r-m_r-1)\end{subarray}}{\mathrm
{sgn}}(σ_2){sgn}(σ_1) \displaystyle\big{(}(⋯(f_r\bar{∘}ξ_1)⋯)\bar{∘}ξ_r-2
\big{)}\Big{(}ξ_r-1(x_σ_{1(m_r+σ_2(1))},\dots,x_σ_{
1(m_r+σ_2(m_r-1+1))}),ξ_r(x_σ_{1(1)},\dots,x_σ_{1
(m_r+1)}), \displaystyle x_σ_{1(m_r+σ_2(m_r-1+3))},\dots,x_σ_{1(
m_r+σ_2(t-m_r))}\Big{)} \displaystyle= \displaystyle⋯⋯⋯ \displaystyle= \displaystyle∑_\begin{subarray{c}σ_1∈{Sh}(m_r+1,t-1-m
_r)\\
σ_2∈{Sh}(m_r-1+1,t-1-m_r-m_r-1)\\
\dots\\
σ_r∈{Sh}(m_1+1,n)\end{subarray}}{sgn}(σ_r)
⋯{sgn}(σ_2){sgn}(σ_1) \displaystyle f_r\Big{(}ξ_1(x_σ_{1(m_r+σ_2(m_r-1+\dots
+σ_r-1(m_2+σ_r(1))⋯))},\dots,x_σ_{1(m_r+σ_
2(m_r-1+\dots+σ_r-1(m_2+σ_r(m_1+1))⋯))}),\dots, \displaystyleξ_r-1(x_σ_{1(m_r+σ_2(1))},\dots,x_σ_{1
(m_r+σ_2(m_r-1+1))}),ξ_r(x_σ_{1(1)},\dots,x_σ_{1(
m_r+1)}), \displaystyle x_σ_{1(m_r+σ_2(m_r-1+\dots+σ_r-1(m_2+
σ_r(m_1+r+1))⋯))},\dots,x_σ_{1(m_r+σ_2(m_r-1+
\dots+σ_r-1(m_2+σ_r(m_1+n+1))⋯))}\Big{)} \displaystyle= \displaystyle∑_\begin{subarray{c}σ_1∈{Sh}(m_r+1,t-1-m
_r)\\
σ_2∈{Sh}(m_r-1+1,t-2-m_r-m_r-1)\\
\dots\\
σ_r∈{Sh}(m_1+1,n)\end{subarray}}(-1)^\frac{r(r-1){2}}{
sgn}(σ_1){sgn}(σ_2)⋯{sgn}(σ
_r) \displaystyle f_r\Big{(}ξ_r(x_σ_{1(1)},\dots,x_σ_{1(m_r+
1)}),ξ_r-1(x_σ_{1(m_r+σ_2(1))},\dots,x_σ_{1(m_r+
σ_2(m_r-1+1))}),\dots, \displaystyleξ_1(x_σ_{1(m_r+σ_2(m_r-1+\dots+σ_r-1
(m_2+σ_r(1))⋯))},\dots,x_σ_{1(m_r+σ_2(m_r-1+
\dots+σ_r-1(m_2+σ_r(m_1+1))⋯))}), \displaystyle x_σ_{1(m_r+σ_2(m_r-1+\dots+σ_r-1(m_2+
σ_r(m_1+r+1))⋯))},\dots,x_σ_{1(m_r+σ_2(m_r-1+
\dots+σ_r-1(m_2+σ_r(m_1+n+1))⋯))}\Big{)} \displaystyle= \displaystyle∑_\begin{subarray{c}σ_1∈{Sh}(m_r+1,t-1-m
_r)\\
σ_2∈{Sh}(m_r-1+1,t-2-m_r-m_r-1)\\
\dots\\
σ_r∈{Sh}(m_1+1,n)\end{subarray}}(-1)^\frac{r(r-1){2}+
κ}{sgn}(σ_1){sgn}(σ_2)⋯{sgn
}(σ_r)f_r\big{(}ξ_r⊗\dots⊗ξ_1⊗{\rm Id}^
⊗ n+1-r\big{)} \displaystyle\Big{(}x_σ_{1(1)},\dots,x_σ_{1(m_r+1)}|,x_σ
_{1(m_r+σ_2(1))},\dots,x_σ_{1(m_r+σ_2(m_r-1+1))}|,\dots, \displaystyle|x_σ_{1(m_r+σ_2(m_r-1+\dots+σ_r-1(m_2+
σ_r(1))⋯))},\dots,x_σ_{1(m_r+σ_2(m_r-1+\dots+
σ_r-1(m_2+σ_r(m_1+1))⋯))}|, \displaystyle|x_σ_{1(m_r+σ_2(m_r-1+\dots+σ_r-1(m_2+
σ_r(m_1+r+1))⋯))},\dots,x_σ_{1(m_r+σ_2(m_r-1+
\dots+σ_r-1(m_2+σ_r(m_1+n+1))⋯))}\Big{)} \displaystyle(\mbox{where $κ$ is given by Koszul sign}) \displaystyle= \displaystyle∑\limits_τ∈{Sh(m_r+1,\dots,m_1+1,n+1-r)}(-
1)^∑\limits_j=1^{r(m_1+⋯+m_j-1)m_j}f_r\big{(}ξ_r
⊗\dots⊗ξ_1⊗{\rm Id}^⊗ n+1-r\big{)}τ^-1(x_1
,\dots,x_t). \tag{1}
$$
∎
By Proposition 4.10, we have:
**Proposition 4.19**
*Keep the above notations, there exists an $L_∞[1]$ -algebra structure on $s\mathfrak{M}⊕{\mathfrak{A}}$ , where $l_i$ are given by
$$
l_2(sf,sg)=(-1)^|f|s[f,g]_{\rm NR}, l_2(sf,ξ)=[f,ξ]_{\rm NR},
$$
and for $3≤ i≤ n+2$ ,
$$
l_i(sf,ξ_1,⋯,ξ_i-1)=∑\limits_τ∈{Sh(m_i-1+1
,\dots,m_1+1,n+2-i)}(-1)^∑\limits_j=1^{i-1(m_1+⋯+m_j-1)m_j
}λ^i-2f\big{(}ξ_i-1⊗\dots⊗ξ_1⊗{\rm Id}^
⊗ n+2-i\big{)}τ^-1,
$$
for homogeneous elements $f∈{\rm Hom}(\wedge^n+1{\mathfrak{g}},{\mathfrak{g}})⊆{\mathfrak{M}}$ , $g∈{\rm Hom}(\wedge^m+1{\mathfrak{g}},{\mathfrak{g}})⊆{\mathfrak{M}}$ , $ξ∈{\rm Hom}(\wedge^m+1{\mathfrak{g}},{\mathfrak{g}})⊆{\mathfrak{ A}}$ , and $ξ_j∈{\rm Hom}(\wedge^m_j+1{\mathfrak{g}},{\mathfrak{g}})⊆{ \mathfrak{A}}$ , $1≤ j≤ i-1$ , and all others components vanish.*
* Proof*
The assertion follows from Proposition 4.17, Proposition 4.10, and Lemma 4.18. Indeed, by Propositions 4.17 and 4.10,
$$
\begin{array}[]{rcl}l_2(sf,sg)&=&(-1)^|f|s[f,g]_{\rm NR},\\
l_i(sf,ξ_1,⋯,ξ_i-1)&=&λ^i-2P[⋯[ι_\mathfrak{M
}(f),ι_\mathfrak{A}(ξ_1)]_{\rm NR},⋯,ι_\mathfrak{A}{(
ξ_i-1)}]_{\rm NR},i≥ 2.\end{array}
$$
On one hand, for $i=2$ , we have
| | $\displaystyle l_2(sf,ξ)$ | $\displaystyle=P[ι_\mathfrak{M}(f),ι_\mathfrak{A}(ξ)]_{\rm NR}$ | |
| --- | --- | --- | --- |
On the other hand, for $i≥ 3$ , we have
| | | $\displaystyle P[⋯[ι_\mathfrak{M}(f),ι_\mathfrak{A}(ξ_ 1)]_{\rm NR},\dots,ι_\mathfrak{A}(ξ_i-1)]_{\rm NR}$ | |
| --- | --- | --- | --- |
where the last equality follows from Lemma 4.18. ∎
**Example 4.20**
*Consider a special case where $n=1,i=3,t=2,m_1=m_2=0$ . For homogeneous elements $π∈{\rm Hom}(\wedge^2{\mathfrak{g}},{\mathfrak{g}})$ , $ξ_1=ξ_2=D∈{\rm Hom}({\mathfrak{g}},{\mathfrak{g}})$ , we have
| | $\displaystyle l_3(sπ,D,D)(x,y)$ | $\displaystyle=∑_τ∈{Sh(1,1)}λπ(D⊗ D\big{)}~{} τ^-1(x,y)$ | |
| --- | --- | --- | --- |*
**Theorem 4.21**
*Let ${\mathfrak{g}}$ be a vector space. Let $π∈Hom(\wedge^2{\mathfrak{g}},{\mathfrak{g}})$ , $D∈Hom({\mathfrak{g}},\mathfrak{g})$ , then $(sπ,D)∈MC(s\mathfrak{M}⊕{\mathfrak{A}})$ if and only if $({\mathfrak{g}},π,D)$ is a differential Lie algebra of weight $λ∈{\bf k}$ .*
* Proof*
By Proposition 4.10, $(sπ,D)∈MC(s\mathfrak{M}⊕{\mathfrak{A}})$ if and only if $[π,π]_{\rm NR}=0$ (that is, $π$ is a Lie bracket) and
| | $\displaystyle 0$ | $\displaystyle=∑_k=2^∞\frac{1}{(k-1)!}l_k(sπ,\underbrace{D, \dots,D}_(k-1) times)$ | |
| --- | --- | --- | --- |
For arbitrary $(x,y)∈\wedge^2{\mathfrak{g}}$ , we have
| | $\displaystyle 0$ | $\displaystyle=\big{(}π\bar{∘}D-D\bar{∘}π+\frac{λ}{2}l_3(s π,D,D)\big{)}(x,y)$ | |
| --- | --- | --- | --- |
Hence $D$ is a differential operator of weight $λ$ . ∎
### 4.5. Equivalences between $L_∞[1]$ -structures for absolute and relative differential Lie algebras with weight
#### 4.5.1. From relative to absolute
Let $({\mathfrak{g}},μ,{d})$ be a differential Lie algebra of weight $λ$ .
Let ${\mathfrak{h}}={\mathfrak{g}}$ , $ρ:{\mathfrak{g}}→Der({\mathfrak{h}})$ be the adjoint representation. Consider ${d}$ as a map ${d}:{\mathfrak{g}}→{\mathfrak{h}}$ . Then $({\mathfrak{g}},{\mathfrak{h}},ρ,{d})$ is a relative differential Lie algebra of weight $λ$ . As seen in Subsection 4.3, let
$$
\begin{array}[]{rcl}{{\mathfrak{L}}^\prime}&=&Hom(\bar{\wedge}({
\mathfrak{g}}⊕{\mathfrak{h}}),{\mathfrak{g}}⊕{\mathfrak{h}}),\\
{{\mathfrak{M}}^\prime}&=&Hom(\bar{\wedge}{\mathfrak{g}},{\mathfrak
{g}})⊕Hom(\bar{\wedge}{\mathfrak{g}}⊗\bar{\wedge}{
\mathfrak{h}},{\mathfrak{h}})⊕Hom(\bar{\wedge}{\mathfrak{h}},{
\mathfrak{h}}),\\
{{\mathfrak{A}}^\prime}&=&Hom(\bar{\wedge}{\mathfrak{g}},{\mathfrak
{h}}).\end{array}
$$
Denote $ι_{\mathfrak{A}^\prime}:{{\mathfrak{A}}^\prime}→{{\mathfrak{L}}^ \prime}$ and $ι_{\mathfrak{M}^\prime}:{{\mathfrak{M}}^\prime}→{{\mathfrak{L}}^ \prime}$ to be the natural injection and $P:{{\mathfrak{L}}^\prime} → {{\mathfrak{A}}^\prime}$ to be the natural surjection. Then $({{\mathfrak{L}}^\prime},{{\mathfrak{M}}^\prime},ι_{\mathfrak{M}^ \prime},{{\mathfrak{A}}^\prime},ι_{\mathfrak{A}^\prime},P,Δ^ \prime=0)$ is a generalised V-datum.
Recall that in Subsection 4.4, let ${\mathfrak{g}}^\prime={\mathfrak{g}}$ and let
$$
\begin{array}[]{rcl}{\mathfrak{L}}&=&Hom(\bar{\wedge}({\mathfrak{g}}
⊕\mathfrak{g}^\prime),{\mathfrak{g}}⊕\mathfrak{g}^\prime),\\
{\mathfrak{M}}&=&Hom(\bar{\wedge}{\mathfrak{g}},{\mathfrak{g}}),\\
{\mathfrak{A}}&=&{\rm Hom}(\bar{\wedge}{\mathfrak{g}},{\mathfrak{g}}).\end{array}
$$
Let two linear maps $ι_\mathfrak{M}:{\mathfrak{M}}→{\mathfrak{L}}$ and $ι_\mathfrak{A}:{\mathfrak{A}}→{\mathfrak{L}}$ be defined as follows: For given $f:\wedge^n+1{\mathfrak{g}} → {\mathfrak{g}}∈{\mathfrak{M}}$ , $ι_\mathfrak{M}(f)=∑\limits_i=0^n+1f_i$ where $f_i:\wedge^n+1-i{\mathfrak{g}}⊗\wedge^i{\mathfrak{g}}^\prime → {\mathfrak{g}}^\prime$ , $0≤ i≤ n+1$ is given by
$$
f_i(x_1\wedge\dots\wedge x_n+1-i⊗ y_n+2-i\wedge\dots\wedge y_n+
1):=f(x_1\wedge\dots\wedge x_n+1-i\wedge y_n+2-i\wedge\dots\wedge y_n+
1),
$$
for $x_1\wedge\dots\wedge x_n+1-i⊗ y_n+2-i\wedge\dots\wedge y_n+1∈ \wedge^n+1-i{\mathfrak{g}}⊗\wedge^i{\mathfrak{g}}^\prime$ ; $ι_\mathfrak{A}$ identifies ${\mathfrak{A}}={\rm Hom}(\bar{\wedge}{\mathfrak{g}},{\mathfrak{g}})$ with the subspace ${\rm Hom}(\bar{\wedge}{\mathfrak{g}},{\mathfrak{g}}^\prime)$ of ${\mathfrak{L}}$ . Let $P:{\mathfrak{L}}→{\mathfrak{A}}$ be the natural projection identifying the subspace ${\rm Hom}(\bar{\wedge}{\mathfrak{g}},{\mathfrak{g}}^\prime)$ with ${\mathfrak{A}}={\rm Hom}(\bar{\wedge}{\mathfrak{g}},{\mathfrak{g}})$ . By Proposition 4.17, $({\mathfrak{L}},{\mathfrak{M}},ι_\mathfrak{M},{\mathfrak{A}},ι_ \mathfrak{A},P,Δ=0)$ is a generalised V-datum.
By identifying ${\mathfrak{g}}^\prime$ with ${\mathfrak{h}}$ , let $f_\mathfrak{L}:{\mathfrak{L}}→{\mathfrak{L}}^\prime$ and $f_\mathfrak{A}:{\mathfrak{A}}→\mathfrak{A^\prime}$ be the identity maps, let $f_\mathfrak{M}:{\mathfrak{M}}→\mathfrak{M^\prime}$ be exactly defined as $ι_\mathfrak{M}$ .
The following result is clear.
**Proposition 4.22**
*The triple $f=(f_\mathfrak{L},f_\mathfrak{M},f_\mathfrak{A})$ is a morphism of generalised V-data from $({\mathfrak{L}},{\mathfrak{M}},ι_\mathfrak{M},{\mathfrak{A}},$ $ι_\mathfrak{A},P,Δ=0)$ to $({{\mathfrak{L}}^\prime},{{\mathfrak{M}}^\prime},ι_{\mathfrak{M}^ \prime},{{\mathfrak{A}}^\prime},ι_{\mathfrak{A}^\prime},P,Δ^ \prime=0)$ . It induces an injective homomorphism of $L_∞[1]$ -algebras from $s\mathfrak{M}⊕{\mathfrak{A}}$ introduced in Proposition 4.19 to $s\mathfrak{M}^\prime⊕{\mathfrak{A}}^\prime$ introduced in Proposition 4.15.*
The above result means that one can deduce the $L_∞[1]$ -structure of absolute differential Lie algebras from that of relative differential Lie algebras.
#### 4.5.2. From absolute to relative
Let $({\mathfrak{g}},{\mathfrak{h}},ρ)$ be a LieAct triple and $D:{\mathfrak{g}} → {\mathfrak{h}}$ be a linear map.
By [18], there exists a Lie algebra ${\mathfrak{g}}\ltimes_ρ{\mathfrak{h}}$ , where the Lie bracket is given by
$$
[x+u,y+v]_\ltimes=[x,y]_\mathfrak{g}+ρ(x)v-ρ(y)u+λ[u,v]_
\mathfrak{h},∀ x,y∈{\mathfrak{g}},u,v∈{\mathfrak{h}}.
$$
The map $D$ extends to $\tilde{D}:{\mathfrak{g}}\ltimes_ρ{\mathfrak{h}}→{\mathfrak{g}}\ltimes_ ρ{\mathfrak{h}}$ by $\tilde{D}(x+u)=D(x)-u,∀ x∈{\mathfrak{g}},u∈{\mathfrak{h}}$ . Furthermore, we have the following observation which seems to be new.
**Proposition 4.23**
*$\tilde{D}$ is a differential operator of weight $λ$ if and only if $D$ is a relative differential operator of weight $λ$ .*
* Proof*
According to the definition of $\tilde{D}$ , we have
$$
\displaystyle\tilde{D}([x+u,y+v]_\ltimes) \displaystyle= \displaystyle\tilde{D}([x,y]_\mathfrak{g}+ρ(x)v-ρ(y)u+λ[u,v]_
\mathfrak{h}) \displaystyle= \displaystyle D([x,y]_\mathfrak{g})-ρ(x)v+ρ(y)u-λ[u,v]_
\mathfrak{h}. \tag{15}
$$
On the other hand,
$$
\displaystyle[x+u,\tilde{D}(y+v)]_\ltimes+[\tilde{D}(x+u),y+v]_\ltimes+
λ[\tilde{D}(x+u),\tilde{D}(y+v)]_\ltimes \displaystyle= \displaystyle[x+u,D(y)-v]_\ltimes+[D(x)-u,y+v]_\ltimes+λ[D(x)-u,D(y)
-v]_\mathfrak{h} \displaystyle= \displaystyleρ(x)(D(y)-v)+λ[u,D(y)-v]_\mathfrak{h}-ρ(y)(D(x)-u)-
λ[v,D(x)-u]_\mathfrak{h} \displaystyle+λ[D(x)-u,D(y)-v]_\mathfrak{h} \displaystyle= \displaystyleρ(x)(D(y))-ρ(x)(v)+λ[u,D(y)]_\mathfrak{h}-λ[u
,v]_\mathfrak{h}-ρ(y)(D(x))+ρ(y)(u)+λ[D(x),v]_\mathfrak{h} \displaystyle-λ[u,v]_\mathfrak{h}+λ[D(x),D(y)]_\mathfrak{h}-
λ[D(x),v]_\mathfrak{h}-λ[u,D(y)]_\mathfrak{h}+λ[u,v]_
\mathfrak{h}. \tag{16}
$$
Compare the Eq. (15) with Eq. (16), it is easy to see $\tilde{D}$ is a differential operator of weight $λ$ if and only if $D$ is a relative differential operator of weight $λ$ . ∎
Now let $({\mathfrak{g}},{\mathfrak{h}},ρ,D)$ be a relative differential Lie algebra of weight $λ$ , then $({\mathfrak{g}}\ltimes_ρ{\mathfrak{h}},[~{},~{}]_\ltimes,\tilde{D})$ is a differential Lie algebra of weight $λ$ by Proposition 4.23. As seen in Subsection 4.4, let $({\mathfrak{g}}\ltimes_ρ{\mathfrak{h}})^\prime={\mathfrak{g}}\ltimes_ ρ{\mathfrak{h}}$ and
$$
\begin{array}[]{rcl}{{\mathfrak{L}}}&=&Hom(\bar{\wedge}({\mathfrak{g}
}\ltimes_ρ{\mathfrak{h}}⊕({\mathfrak{g}}\ltimes_ρ{\mathfrak{h}}
)^\prime),{\mathfrak{g}}\ltimes_ρ{\mathfrak{h}}⊕({\mathfrak{g}}
\ltimes_ρ{\mathfrak{h}})^\prime),\\
{{\mathfrak{M}}}&=&Hom(\bar{\wedge}({\mathfrak{g}}\ltimes_ρ{
\mathfrak{h}}),{\mathfrak{g}}\ltimes_ρ{\mathfrak{h}}),\\
{{\mathfrak{A}}}&=&Hom(\bar{\wedge}({\mathfrak{g}}\ltimes_ρ{
\mathfrak{h}}),{\mathfrak{g}}\ltimes_ρ{\mathfrak{h}}).\end{array}
$$
The two linear maps $ι_\mathfrak{M}:{\mathfrak{M}}→{\mathfrak{L}}$ and $ι_\mathfrak{A}:{\mathfrak{A}}→{\mathfrak{L}}$ defined as that in Subsection 4.4. Then $({{\mathfrak{L}}},{{\mathfrak{M}}},ι_\mathfrak{M},{{\mathfrak{A}}},ι _\mathfrak{A},P,Δ=0)$ is a generalised V-datum.
Recall in Subsection 4.3, let
$$
\begin{array}[]{rcl}{{\mathfrak{L}}^\prime}&=&Hom(\bar{\wedge}({
\mathfrak{g}}⊕{\mathfrak{h}}),{\mathfrak{g}}⊕{\mathfrak{h}}),\\
{{\mathfrak{M}}^\prime}&=&Hom(\bar{\wedge}{\mathfrak{g}},{\mathfrak
{g}})⊕Hom(\bar{\wedge}{\mathfrak{g}}⊗\bar{\wedge}{
\mathfrak{h}},{\mathfrak{h}})⊕Hom(\bar{\wedge}{\mathfrak{h}},{
\mathfrak{h}}),\\
{{\mathfrak{A}}^\prime}&=&Hom(\bar{\wedge}{\mathfrak{g}},{\mathfrak
{h}}).\end{array}
$$
Denote $ι_{\mathfrak{A}^\prime}:{{\mathfrak{A}}^\prime}→{{\mathfrak{L}}^ \prime}$ to be the natural injection and $P:{{\mathfrak{L}}^\prime} → {{\mathfrak{A}}^\prime}$ to be the natural surjection. Then $({{\mathfrak{L}}^\prime},{{\mathfrak{M}}^\prime},ι_{\mathfrak{M}^ \prime},{{\mathfrak{A}}^\prime},ι_{\mathfrak{A}^\prime},P,Δ^ \prime=0)$ is a generalised V-datum.
Let $f_{\mathfrak{L}^\prime}:{\mathfrak{L}}^\prime→{\mathfrak{L}}$ , $f_{\mathfrak{M}^\prime}:{\mathfrak{M}}^\prime→\mathfrak{M}$ and $f_{\mathfrak{A}^\prime}:{\mathfrak{A}}^\prime→\mathfrak{A}$ be natural injection maps.
The following result is clear.
**Proposition 4.24**
*The triple $f=(f_{\mathfrak{L}^\prime},f_{\mathfrak{M}^\prime},f_{\mathfrak{A}^ \prime})$ is a morphism of generalised V-data from $({{\mathfrak{L}}^\prime},{{\mathfrak{M}}^\prime},ι_{\mathfrak{M}^ \prime},$ ${{\mathfrak{A}}^\prime},ι_{\mathfrak{A}^\prime},P,Δ^\prime=0)$ to $({\mathfrak{L}},{\mathfrak{M}},ι_\mathfrak{M},{\mathfrak{A}},$ $ι_\mathfrak{A},P,Δ=0)$ . It induces an injective homomorphism of $L_∞[1]$ -algebras from $s\mathfrak{M}^\prime⊕{\mathfrak{A}}^\prime$ introduced in Proposition 4.15 to $s\mathfrak{M}⊕{\mathfrak{A}}$ introduced in Proposition 4.19.*
The above result means that one can deduce the $L_∞[1]$ -structure of relative differential Lie algebras from that of absolute differential Lie algebras.
## 5. Application of $L_∞[1]$ -structure for differential Lie algebras
In this section, we will derive from the $L_∞[1]$ -structure the cohomology theory of differential Lie algebras of arbitrary weight and the notion of homotopy differential Lie algebras of arbitrary weight.
### 5.1. Cohomology of differential Lie algebras from $L_∞[1]$ -structure
Let ${\mathfrak{g}}$ be a vector space. Let
$$
{\mathfrak{M}}:=Hom(\bar{\wedge}{\mathfrak{g}},{\mathfrak{g}})
and {\mathfrak{A}}:={\rm Hom}(\bar{\wedge}{\mathfrak{g}},{\mathfrak{
g}}).
$$
Recall that we have constructed an $L_∞[1]$ -structure on $s{\mathfrak{M}}⊕{\mathfrak{A}}$ in Proposition 4.19.
Let $({\mathfrak{g}},μ,{d})$ be a differential Lie algebra. By Theorem 4.21, $(sμ,{d})$ is a Maurer-Cartan element in the $L_∞[1]$ -algebra $s{\mathfrak{M}}⊕{\mathfrak{A}}$ . By Proposition 4.4, twisting $s{\mathfrak{M}}⊕{\mathfrak{A}}$ by $(sμ,{d})$ gives a new $L_∞[1]$ -algebra, whose new differential is denoted by $l_1^(sμ,{d)}$ .
Consider the cochain complex ${C}_DL_λ^*({\mathfrak{g}},{\mathfrak{g}}_\rm ad)$ of the differential Lie algebra $({\mathfrak{g}},μ,{d})$ with coefficients in the adjoint representation ${\mathfrak{g}}_\rm ad$ . Note that for each $n≥ 1$ ,
$$
{C}_DL_λ^n({\mathfrak{g}},{\mathfrak{g}}_\rm ad
)={C}^n_Lie({\mathfrak{g}},{\mathfrak{g}}_\rm ad)
⊕{C}^n-1_DO_λ({\mathfrak{g}},{\mathfrak{g}}
_\rm ad)=Hom(\wedge^n{\mathfrak{g}},{\mathfrak{g}})⊕
Hom(\wedge^n-1{\mathfrak{g}},{\mathfrak{g}})=(s{\mathfrak{M}}⊕{
\mathfrak{A}})^n-2
$$
is exactly the degree $n-2$ part of $s{\mathfrak{M}}⊕{\mathfrak{A}}$ .
**Proposition 5.1**
*The underlying complex of the twisted $L_∞[1]$ -algebra $s{\mathfrak{M}}⊕{\mathfrak{A}}$ is exactly the double shift of the cochain complex ${C}_DL_λ^*({\mathfrak{g}},{\mathfrak{g}}_\rm ad)$ , up to signs.*
* Proof*
It suffices to make explicit the differential $l_1^(sμ,{d)}$ . For $n≥ 1$ , $f∈Hom(\wedge^n{\mathfrak{g}},{\mathfrak{g}}),g∈Hom( \wedge^n-1{\mathfrak{g}},{\mathfrak{g}})$ ,
| | $\displaystyle l_1^(sμ,{d)}(sf,g)$ | $\displaystyle=$ | $\displaystyle∑_k=0^∞\frac{1}{k!}l_k+1(\underbrace{(sμ,{\mathrm {d}}),⋯,(sμ,{d})}_k times,(sf,g))$ | |
| --- | --- | --- | --- | --- |
| $\displaystyle=$ | $\displaystyle l_2((sμ,{d}),(sf,g))+∑_k=2^∞\frac{1}{k! }l_k+1(\underbrace{(sμ,{d}),⋯,(sμ,{d})}_k times,(sf,g))$ | | | |
| $\displaystyle=$ | $\displaystyle\big{(}l_2(sμ,sf),l_2(sμ,g)+l_3(sμ,{d},g)+l_ {2}(sf,{d})+∑_k=2^n\frac{1}{k!}l_k+1(sf,\underbrace{{\mathrm {d}},⋯,{d}}_k times)\big{)}.$ | | | | It is easy to see that $l_2(sμ,sf)=-s[μ,f]_{\rm NR}$ is exactly $-s∂_\rm Lie^n(f)$ . Let us compute $l_2(sμ,g)+l_3(sμ,{d},g)+l_2(sf,{d})+∑\limits_k =2^n\frac{1}{k!}l_k+1(sf,\underbrace{{d},⋯,{d}}_k times)$ . For $x_1,\dots,x_n∈{\mathfrak{g}}$ , we have
| | | | $\displaystyle(l_2(sμ,g)+l_3(sμ,{d},g))(x_1,\dots,x_n)$ | |
| --- | --- | --- | --- | --- |
| $\displaystyle=$ | $\displaystyle[μ,g]_\rm NR(x_1,\dots,x_n)+λ∑\limits_τ∈{ Sh(1,n-1)}μ\big{(}g⊗{d}\big{)}τ^-1(x_1,\dots ,x_n)$ | | | |
| $\displaystyle=$ | $\displaystyle∑_i=1^n(-1)^i+n-1ρ(x_i)g(x_1,\dots,\hat{x}_i, \dots,x_n)$ | | | |
| $\displaystyle+∑_1≤ i<j≤ n^n-1(-1)^i+j+ng([x_i,x_j],x_1, \dots,\hat{x}_i,\dots,\hat{x}_j,\dots,x_n)$ | | | | |
| $\displaystyle+λ∑_i=1^n(-1)^i+n-1ρ({d}(x_i))g(x_1 ,\dots,\hat{x}_i,\dots,x_n)$ | | | | |
| $\displaystyle=$ | $\displaystyle∂_DO_λ^n-1(g)(x_1,\dots,x_n).$ | | | |
On the other hand,
| | | | $\displaystyle l_2(sf,{d})+∑\limits_k=2^n\frac{1}{k!}l_k+1( sf,\underbrace{{d},⋯,{d}}_k times)(x_1, \dots,x_n)$ | |
| --- | --- | --- | --- | --- |
| $\displaystyle=$ | $\displaystyle\big{(}-{d}∘ f+∑_k=1^n\frac{1}{k!}λ^k- 1f\bar{∘}\{\underbrace{{d},⋯,{d}}_k times\}\big{)}(x_1,\dots,x_n)$ | | | |
| $\displaystyle=$ | $\displaystyle-{d}(f(x_1,⋯,x_n))+∑_k=1^n\frac{1}{k!} ∑_τ∈{Sh(\underbrace{1,⋯,1}_k times,n-k)} λ^k-1f(\underbrace{{d}⊗\dots⊗{d}}_k times⊗{\rm Id}^⊗ n-k\big{)}(x_1,\dots,x_n)$ | | | |
| $\displaystyle=$ | $\displaystyle∑_σ∈{Sh(k,n-k)}{sgn}(σ)f_k(x _1,⋯,\hat{x_i_1},⋯,\hat{x_i_{k}},⋯,x_n,{d}(x _i_1),⋯,{d}(x_i_{k}))-{d}(f(x_1,⋯,x_n))$ | | | |
| $\displaystyle=$ | $\displaystyleδ^n(f).$ | | | | That is, we have
$$
l_1^(sμ,{d)}(sf,g)=\big{(}-s∂_\rm Lie^n(f),∂_
{DO_λ}^n-1(g)+δ^n(f)\big{)}=-∂^n_
DL_λ(f,g).
$$
∎
Let $(V,{d}_V)$ be a representation of the differential Lie algebra $(\mathfrak{{\mathfrak{g}}},{d}_{\mathfrak{g}})$ with weight $λ$ . Now we justify the cochain complex $({C}_DL_λ^*({\mathfrak{g}},V),∂_ DL_λ^*)$ of the differential Lie algebra $(\mathfrak{{\mathfrak{g}}},{d}_{\mathfrak{g}})$ with coefficients in the representation $(V,{d}_V)$ introduced in Definition 1.11 and Proposition 1.9.
To this end, we consider the trivial extension ${\mathfrak{g}}\ltimes V$ of the differential Lie algebra $({\mathfrak{g}},{d}_{\mathfrak{g}})$ by the representation $(V,{d}_V)$ in Proposition 1.6, and we get the complex $({C}^*_DL_λ({\mathfrak{g}}\ltimes V,({\mathfrak{ g}}\ltimes V)_\rm ad),∂_DL_λ^*)$ , by Proposition 5.1. The following result can be proved by direct inspection.
**Proposition 5.2**
*The cochain complex $({C}_DL_λ^*({\mathfrak{g}},V),∂_ DL_λ^*)$ of the differential Lie algebra $(\mathfrak{{\mathfrak{g}}},{d}_{\mathfrak{g}})$ with coefficients in the representation $(V,{d}_V)$ introduced in Definition 1.11 is a subcomplex of $({C}^*_DL_λ({\mathfrak{g}}\ltimes V,({\mathfrak{ g}}\ltimes V)_\rm ad),∂_DL_λ^*)$ .*
* Proof*
For vector spaces $U$ and $W$ , we have an isomorphism
$$
\wedge^p(U⊕ W)≃\wedge^pU⊕(⊕_i=1^p-1\wedge^iU
⊗\wedge^p-iW)⊕\wedge^pW.
$$
Hence, there exists a natural injection from
$$
{C}_DL_λ^n({\mathfrak{g}},V)=Hom(\wedge
^n{\mathfrak{g}},V)⊕Hom(\wedge^n-1{\mathfrak{g}},V)
$$
to
$$
{C}^n_DL_λ({\mathfrak{g}}\ltimes V,({\mathfrak{g
}}\ltimes V)_\rm ad)=Hom(\wedge^n({\mathfrak{g}}⊕ V),{
\mathfrak{g}}⊕ V)⊕Hom(\wedge^n-1({\mathfrak{g}}⊕ V),
{\mathfrak{g}}⊕ V).
$$
Now it suffices to show that this map commutes with differentials. ∎
### 5.2. Homotopy differential Lie algebras with weight
In this section, we need to use graded Nijenhuis-Richardson brackets.
Let ${\mathfrak{g}}$ be a graded vector space and denote ${\mathfrak{l}}=s{\mathfrak{g}}$ . Consider the graded vector space ${\mathfrak{C}}_Lie({\mathfrak{g}},{\mathfrak{g}}):=Hom(S({ \mathfrak{l}}),{\mathfrak{l}})$ .
Recall that by Eq. (1), $S^n(s{\mathfrak{g}})$ is isomorphic to $s^n\wedge^n{\mathfrak{g}}$ . When ${\mathfrak{g}}={\mathfrak{g}}^0=V$ is ungraded, then ${\rm Hom}(S^n(sV),sV)$ , which has degree $n-1$ , is isomorphic to ${\rm Hom}(\wedge^nV,V)$ as vector spaces. This justifies why we have imposed degree $n-1$ on the latter in Subsection 4.3.
The graded Nijenhuis-Richardson bracket $[-,-]_{\rm NR}$ on the graded vector space ${\mathfrak{C}}_Lie({\mathfrak{g}},{\mathfrak{g}})$ can be defined as follows: for $f∈Hom(S^n({\mathfrak{l}}),{\mathfrak{l}})$ of degree $p$ and $g∈Hom(S^m({\mathfrak{l}}),{\mathfrak{l}})$ of degree $q$ ,
$$
[f,g]_{\rm NR}:=f\bar{∘}g-(-1)^pqg\bar{∘}f,
$$
where $f\bar{∘}g∈Hom(S^m+n-1({\mathfrak{l}}),{\mathfrak{l}})$ of degree $p+q$ is defined by
$$
\displaystyle f\bar{∘}g(v_1,\dots,v_m+n-1) \displaystyle=∑_σ∈{Sh(m,n-1)}ε(σ)f(g(v_
σ(1),\dots,v_σ(m)),v_σ(m+1),\dots,v_σ(m+n-1)), \tag{1}
$$
for $v_1,\dots,v_m+n-1∈{\mathfrak{l}}$ . It is well known that $({\mathfrak{C}}_Lie({\mathfrak{g}},{\mathfrak{g}}),[~{},~{}]_\rm NR)$ is a graded Lie algebra whose Maurer-Cartan elements correspond bijectively to $L_∞[1]$ -algebra structures on ${\mathfrak{l}}$ .
We consider the graded Lie algebra
$$
{\mathfrak{L}}:={\mathfrak{C}}_Lie({\mathfrak{g}}⊕\mathfrak{g}
,{\mathfrak{g}}⊕\mathfrak{g})
$$
endowed with the graded Nijenhuis-Richardson bracket $[~{},~{}]_{\rm NR}$ .
Let
$$
{\mathfrak{M}}:=Hom(\bar{S}({\mathfrak{l}}),{\mathfrak{l}})
and {\mathfrak{A}}:={\rm Hom}(\bar{S}({\mathfrak{l}}),{\mathfrak{l}}).
$$
Endow ${\mathfrak{M}}$ with the graded Nijenhuis-Richardson bracket and ${\mathfrak{A}}$ with the trivial bracket. With the same analysis in Subsection 4.4, we have:
**Proposition 5.3**
*Let ${\mathfrak{g}}$ be a graded vector space. Then we get a generalised V-datum $({\mathfrak{L}},{\mathfrak{M}},ι_\mathfrak{M},{\mathfrak{A}}$ , $ι_\mathfrak{A},P,Δ=0)$ and there is an $L_∞$ [1]-algebra $s{\mathfrak{M}}⊕{\mathfrak{A}}$ given below:
$$
l_2(sf,sg)=(-1)^|f|s[f,g]_{\rm NR}, l_2(sf,ξ)=[f,ξ]_{\rm NR},
$$
and for $3≤ i≤ n+2$ ,
$$
l_i(sf,ξ_1,⋯,ξ_i-1)=∑\limits_τ∈{Sh(m_i-1+1
,\dots,m_1+1,n+2-i)}(-1)^∑\limits_j=1^{i-1(|ξ_1|+⋯+|ξ_j-1
|)|ξ_j|}λ^i-2f\big{(}ξ_i-1⊗\dots⊗ξ_1⊗{
\rm Id}^⊗ n+2-i\big{)}τ^-1,
$$
for homogeneous elements $f∈{\rm Hom}(S^n+1({\mathfrak{l}}),{\mathfrak{l}})⊆{\mathfrak{M}}$ , $g∈{\rm Hom}(S^m+1({\mathfrak{l}}),{\mathfrak{l}})⊆{\mathfrak{M}}$ , $ξ∈{\rm Hom}(S^m+1({\mathfrak{l}}),{\mathfrak{l}})⊆{\mathfrak{A}}$ , and $ξ_j∈{\rm Hom}(S^m_j+1({\mathfrak{l}}),{\mathfrak{l}})⊆{ \mathfrak{A}}$ , $1≤ j≤ i-1$ , and all others components vanish.*
**Definition 5.4**
*Let ${\mathfrak{g}}$ be a graded vector space. A structure of homotopy differential Lie algebra with weight $λ$ on ${\mathfrak{g}}$ is defined to be a Maurer-Cartan element of the $L_∞$ [1]-algebra $s{\mathfrak{M}}⊕{\mathfrak{A}}$ introduced in Proposition 5.3.*
**Theorem 5.5**
*Let ${\mathfrak{g}}$ be a graded vector space and denote ${\mathfrak{l}}=s{\mathfrak{g}}$ . A homotopy differential Lie algebra with weight $λ∈{\bf k}$ on ${\mathfrak{g}}$ is equivalent to the pair $(μ=\{μ_i\}_i≥ 1,D=\{D_i\}_i≥ 1)$ , where for each $i≥ 1$ , $μ_i:S^i({\mathfrak{l}})→{\mathfrak{l}}$ is a degree $1$ map such that $({\mathfrak{g}},μ)$ is an $L_∞[1]$ -algebra and for each $i≥ 1$ , $D_i:S^i({\mathfrak{l}})→{\mathfrak{l}}$ is of degree $0 0$ which form a homotopy differential operator of weight $λ$ . More precisely, for $n≥ 1$ and $x_1,\dots,x_n∈{\mathfrak{l}}$ , we have
$$
∑_i=1^n∑_σ∈{Sh(i,n-i)}ε(σ)μ_n-i+
1(μ_i(x_σ(1),⋯,x_σ(i)),x_σ(i+1),⋯,x_
σ(n))=0, \tag{17}
$$
and
$$
\displaystyle∑_p≥ 2∑_t=p-1^n∑_m_{1+⋯+m_p-1=t}∑_
\begin{subarray{c}σ∈Sh(m_p-1,⋯,m_1,n-t)\\
σ|_t∈PSh(m_p-1,⋯,m_1)\end{subarray}}ε(
σ)λ^p-2μ_n-t+p-1(D_m_{p-1}(x_σ(1),⋯,x_σ(m
_{p-1)}),⋯, \displaystyle ⋯,D_m_1(x_σ(m_{2+⋯+m_p-1+1)},⋯,
x_σ(t)),x_σ(t+1)⋯,x_σ(n)) \displaystyle-∑_j=1^n∑_σ∈{Sh(j,n-j)}ε(
σ)D_n-j+1(μ_j(x_σ(1),⋯,x_σ(j)),x_σ(j+1),
⋯,x_σ(n))=0. \tag{18}
$$*
* Proof*
Let $μ=∑\limits_i=1^∞μ_i:\bar{S}({\mathfrak{l}}) → { \mathfrak{l}}$ , with $μ_i:S^i({\mathfrak{l}}) → {\mathfrak{l}}$ and $D=∑\limits_i=1^∞D_i:\bar{S}({\mathfrak{l}}) → { \mathfrak{l}}$ , with $D_i:S^i({\mathfrak{l}}) → {\mathfrak{l}}$ . Then by Proposition 4.10, $(sμ,D)∈MC(s\mathfrak{M}⊕{\mathfrak{A}})$ is equivalent to
$$
[μ,μ]_{\rm NR}=0 and ∑\limits_p=2^∞\frac{1}{(p-1)
!}l_p(sμ,\underbrace{D,\dots,D}_(p-1) times)=0.
$$
It is well known that, see for instance, [2, Theorem 4.2], that $[μ,μ]_{\rm NR}=0$ if and only if Eq. (17) holds if and only if ${\mathfrak{g}}$ is an $L_∞[1]$ -algebra. On the other hand, we have
| | $\displaystyle 0=$ | $\displaystyle∑_p=2^∞\frac{1}{(p-1)!}l_p(sμ,\underbrace{D,\dots ,D}_(p-1) times)$ | |
| --- | --- | --- | --- |
| | $\displaystyle=$ | $\displaystyle∑_m=1^∞∑_j=1^∞[μ_m,D_j]_\rm NR$ | |
| --- | --- | --- | --- |
which is equivalent to Eq. (18). ∎
**Example 5.6**
*Since Eq. (17) is the well known $L_∞[1]$ -structure, we only need to focus on homotopy differential operators. When $n=1$ , Eq. (18) gives
| | $\displaystyleμ_1D_1=D_1μ_1.$ | |
| --- | --- | --- |
which means that $D_1$ is a cochain map. When $n=2$ , Eq. (18) gives
| | $\displaystyle D_1(μ_2(x,y))-μ_2(D_1(x),y)-μ_2(x,D_1(y))- λμ_2(D_1(x),D_1(y))$ | |
| --- | --- | --- |
which means that $D_1$ is a differential operator (with respect to the multiplication $μ_2$ ), but only up to the homotopy given by $D_2$ .*
Acknowledgments. The first author was supported by National Natural Science Foundation of China (No. 11871071). The fourth author was supported by the National Natural Science Foundation of China (No. 12071137), by Key Laboratory of Ministry of Education, by Shanghai Key Laboratory of PMMP (No. 22DZ2229014), and by Fundamental Research Funds for the Central Universities.
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