Homotopy Rota-Baxter operators, homotopy $\mathcal{O}$-operators and homotopy post-Lie algebras

Rota-Baxter operators, $\mathcal{O}$-operators on Lie algebras and their interconnected pre-Lie and post-Lie algebras are important algebraic structures with applications in mathematical physics. This paper introduces the notions of a homotopy Rota-Baxter operator and a homotopy $\mathcal{O}$-operator on a symmetric graded Lie algebra. Their characterization by Maurer-Cartan elements of suitable differential graded Lie algebras is provided. Through the action of a homotopy $\mathcal{O}$-operator on a symmetric graded Lie algebra, we arrive at the notion of an operator homotopy post-Lie algebra, together with its characterization in terms of Maurer-Cartan elements. A cohomology theory of post-Lie algebras is established, with an application to 2-term skeletal operator homotopy post-Lie algebras.


Introduction
This paper initiates the homotopy study of Rota-Baxter operators, O-operators and the related pre-Lie algebras and post-Lie algebras.
1.1. Background and motivation. Homotopy is a fundamental notion in topology describing continuously deforming one function to another.
The first homotopy construction in algebra is the A ∞ -algebra of Stasheff, arising from his work on homotopy characterization of connected based loop spaces [43]. Later related developments include the work of Boardman and Vogt [6] about E ∞ -spaces on the infinite loop space, the work of Schlessinger and Stasheff [40] about L ∞ -algebras on perturbations of rational homotopy types and the work of Chapoton and Livernet [10] about pre-Lie ∞ algebras, as well as homotopy Leibniz algebras [1]. See [27,30,32] for other examples.
Homotopy and operads are intimately related. In fact, operads were introduced as a tool in homotopy theory, specifically for iterated loop spaces. Vaguely speaking, the homotopy of an algebraic structure is obtained when the defining relations of the algebraic structure is relaxed to hold up to homotopy. The resulting algebraic structure is homotopy equivalent to the original algebraic structure via a Homotopy Transfer Theorem. The idea is behind much of the operadic developments. Through the work of Ginzburg-Kapranov [19], Getzler-Jones [18] and Markl [28,30], the homotopy of an operad P is in general defined to be the minimal model of P. More precisely, P ∞ is the Koszul resolution as the cobar construction ΩP i of the Koszul dual cooperad of P [27,30]. Since the latter notion makes sense only when P is quadratic, this approach does not apply to some important algebraic structures, such as the operad of Rota-Baxter algebras.
Rota-Baxter associative algebras were introduced in the probability study of G. Baxter and later found important applications in the Connes-Kreimer's algebraic approach to renormalization of quantum field theory [11], among others. A Rota-Baxter operator on a Lie algebra is naturally the operator form of a classical r-matrix [41] under certain conditions. To better understand such connection in general, the notion of an O-operator (also called a relative Rota-Baxter operator [39] or a generalized Rota-Baxter operator [46,47]) on a Lie algebra was introduced by Kupershmidt [23], which can be traced back to Bordemann [7].
Both operators have been studied extensively in recent years [20]. Their operadic theory are challenging to establish since the operads are not quadratic and have nontrivial unary operations. At the same time, they provide promising testing grounds to expand the existing operad theory.
1.2. Approach of the paper. Because of the limitation of the Koszul resolution method to study homotopy of operads, other methods to give the related resolutions have been introduced. Dotsenko and Khoroshkin [14] used shuffle operads and the Gröbner bases method to show that, for the operad ncRB of Rota-Baxter operators on associative algebras, the minimal model ncRB ∞ is a quasi-free operad. They are able to write down formulas for small arities for differentials in the free resolutions for Quillen homology computation, though "in general compact formulas are yet to be found" as noted in the paper. One can expect a similarly challenging situation for the operad of Rota-Baxter operators on Lie algebras.
This paper follows the more basic and direct approach to homotopy via differential graded Lie algebras and Maurer-Cartan elements. This is in fact the approach taken in the early developments of characterizing algebraic and homotopy algebraic structures before they are put under the more uniform and sophisticated framework of operads. These developments started with the wellknown series of work by Gerstenhaber [16,17] on deformations of associative algebras and by Nijenhuis and Richardson on Lie algebras [35] a few years later.
This approach is based on the principle that objects of a certain algebraic structure on a vector space are given by degree 1 solutions of the Maurer-Cartan equation in a suitable differential graded Lie algebra built from the vector space. When the vector space is replaced by a graded vector space, similar solutions give objects in the homotopy algebraic structure. To make the idea more transparent, we recall the case of Lie algebras and homotopy Lie algebras (that is, L ∞algebras). Let V be a vector space. Define the graded vector space ⊕ ∞ n=0 Hom(∧ n V, V) with the degree of elements in Hom(∧ n V, V) being n − 1. For f ∈ Hom(∧ m V, V), g ∈ Hom(∧ n V, V), define [ f, g] NR := f • g − (−1) (m−1)(n−1) g • f, with f • g ∈ Hom(∧ m+n−1 V, V) being defined by (1) ( f • g)(v 1 , · · · , v m+n−1 ) := σ∈S (n,m−1) (−1) σ f (g(v σ (1) , · · · , v σ(n) ), v σ(n+1) , · · · , v σ(m+n−1) ), where the sum is over (n, m −1)-shuffles. Then ⊕ ∞ n=0 Hom(∧ n V, V), [·, ·] NR is a differential graded Lie algebra with the trivial derivation. With this setup, a Lie algebra structure on V is precisely a degree 1 solution ω ∈ Hom(∧ 2 V, V) of the Maurer-Cartan equation [ω, ω] NR = 0.
In the spirit of this characterization of Lie algebra structures by solutions of the Maurer-Cartan equation, homotopy Lie algebra structures can be characterized briefly as follows, with further details provided in Section 2.1. Given a graded vector space V = ⊕ i∈Z V i , let S (V) denote the symmetric algebra of V and let Hom n (S (V), V) denote the space of degree n linear maps. Define the Nijenhuis-Richardson bracket [·, ·] NR on the graded vector space n∈Z Hom n (S (V), V) in a graded form of Eq. (1) (see Eq. (6)). We again have a graded Lie algebra and (curved) L ∞algebras can be characterized as degree 1 solutions of the corresponding Maurer-Cartan equation.
This approach usually agrees with the operadic approach when both approaches apply. The more elementary nature of this approach makes it possible to be applied where the operadic approach cannot be applied yet, in particular to Rota-Baxter operators and its relative generalization, the O-operators. Indeed, in [45], we took this approach to give a Maurer-Cartan characterization of O-operators (of weight zero) and further to establish a deformation theory and its controlling cohomology for O-operators.
To illustrate our approach in a broader context, we focus on the Rota-Baxter Lie algebra for now and regard its algebraic structure (operad) as a pair (ℓ, T ) consisting of a Lie bracket ℓ = [·, ·] and a Rota-Baxter operator T . Then to obtain the homotopy of the Rota-Baxter Lie algebra, one can begin with taking homotopy of the binary operation ℓ or taking homotopy of the unary operation T . The homotopy of the Lie algebra ℓ ∞ = {ℓ i } ∞ i=1 is well-known as the L ∞ -algebra [30]. Together with the natural Rota-Baxter operator action as defined in [39], we have the Rota-Baxter homotopy Lie algebra (ℓ ∞ , T ) or Rota-Baxter L ∞ -algebra. A Rota-Baxter Lie algebra naturally induces a post-Lie algebra, originated from an operadic study [48] and found applications in mathematical physics and numerical analysis [5,34]. Likewise, a Rota-Baxter homotopy Lie algebra is expected to induce a homotopy post-Lie algebra whose construction is still not known beyond its conceptual definition as an operadic minimal model mentioned above, giving rise to the commutative diagram where the horizontal arrows are taking homotopy and the vertical arrows are taking actions of the Rota-Baxter operators.
In this paper, we will pursue the other direction, by taking homotopy of the Rota-Baxter operator T and obtain T ∞ := {T i } ∞ i=0 , without taking homotopy of ℓ. We call the resulting structure (ℓ, T ∞ ) the operator homotopy Rota-Baxter Lie algebra to distinguish it from the above mentioned Rota-Baxter homotopy Lie algebra. The action of the operator homotopy Rota-Baxter operator T ∞ gives rise to a variation of the homotopy post-Lie algebra which we will call the operator homotopy post-Lie algebra (see Remark 3.5). This gives another commutative diagram shown as the front rectangle in Eq. (3) while the diagram in Eq. (2) is embedded as the right rectangle.
Eventually, the full homotopy of the Rota-Baxter Lie algebra should come from the combined homotopies of both the Lie algebra structure and the Rota-Baxter operator structure, tentatively denoted by (ℓ ∞ , T ∞ ) and called the full homotopy Rota-Baxter Lie algebra. A suitable action of T ∞ on ℓ ∞ should give the full homotopy post-Lie algebra whose structure is still mysterious. These various homotopies of the Rota-Baxter Lie algebra, as well as their derived homotopies of the post-Lie algebra, could be put together and form the following diagram where going in the left and the inside directions should be taking various homotopy, and going downward should be taking the actions of (homotopy) Rota-Baxter operators. We note that when the weight of the Rota-Baxter operator is zero, the post-Lie algebra in the lower half of the diagram becomes a pre-Lie algebra and the homotopy post-Lie in the backright-lower corner is a homotopy pre-Lie algebra. We find it quite amazing that the operator homotopy post-Lie algebra in the front-left-lower corner is also the homotopy pre-Lie algebra (Corollary 3.12). This of course does not mean that the algebraic structure in the back-left-lower corner is also the homotopy pre-Lie algebra. It would be interesting to determine this structure even in this special case.
1.3. Outline of the paper. After some background on differential graded Lie algebras and Maurer-Cartan equations summarized in Section 2.1, we introduce in Section 2.2 the notion of a homotopy O-operator with any weight (Definition 2.9). Using derived bracket, we construct a differential graded Lie algebra which characterizes homotopy O-operators of any weight as their Maurer-Cartan elements (Theorem 2.16).
In Section 3, by applying a homotopy O-operator to a symmetric graded Lie algebra we obtain a variation of the homotopy post-Lie algebra, called the operator homotopy post-Lie algebra. From here we can specialize in several directions and obtain interesting applications. First when the weight of the O-operator is taken to be zero, we obtain homotopy O-operators of weight zero. Since O-operators of weight zero naturally derive pre-Lie algebras [39], it is expected that homotopy O-operators of weight zero derive homotopy pre-Lie algebras, that is pre-Lie ∞algebras. We confirm this in Corollary 3.12, yielding the commutative diagram In other words, the compositions of taking homotopy and taking operator action in either order gives the pre-Lie ∞ algebras. There has been quite much interest on constructions of post-Lie algebras in the recent literature [8,9,15,38]. Another useful application of our general construction is the characterization of post-Lie algebra structures on a given Lie algebra using Maurer-Cartan elements in a suitable differential graded Lie algebra (Corollary 3.7).
In Section 4 we first consider the cohomology theory of post-Lie algebras. In the "abelian" case of pre-Lie algebras, the cohomology groups were first defined in [13] by derived functors and then in [33] by resolutions of algebras from the Koszul duality theory in the framework of operads. An explicit cohomology theory of post-Lie algebras is not yet known. In this section we establish such a theory which reduces to the existing cohomology theory of pre-Lie algebras. The third cohomology group of a post-Lie algebra are applied in Section 4.3 to classify 2-term skeletal operator homotopy post-Lie algebras. Notation. We assume that all the vector spaces are over a field of characteristic zero. For a homogeneous element x in a Z-graded vector space, we also use x in the exponent, as in (−1) x , to denote its degree in order to simplify the notation.

Homotopy O-operators of weight λ
In this section, we introduce the notion of a homotopy O-operator of weight λ, where λ is a constant. We construct a differential graded Lie algebra (dgLa for short) and show that homotopy O-operators of weight λ can be characterized as its Maurer-Cartan elements to justify our definition.
2.1. Maurer-Cartan elements and Nijenhuis-Richardson brackets. We first recall some background needed in later sections.
Let V = ⊕ k∈Z V k be a Z-graded vector space. We will denote by S (V) the symmetric algebra of V: . Denote the product of homogeneous elements v 1 , · · · , v n ∈ V in S n (V) by v 1 ⊙ · · · ⊙ v n . The degree of v 1 ⊙ · · · ⊙ v n is by definition the sum of the degree of v i . For a permutation σ ∈ S n and v 1 , · · · , v n ∈ V, the Koszul sign ε(σ; v 1 , · · · , v n ) ∈ {−1, 1} is defined by and the antisymmetric Koszul sign χ(σ; v 1 , · · · , v n ) ∈ {−1, 1} is defined by Denote by Hom n (S (V), V) the space of degree n linear maps from the graded vector space S (V) to the graded vector space V. Obviously, an element f ∈ Hom n (S (V), V) is the sum of f i : S i (V) → V. We will write f = ∞ i=0 f i . Set C n (V, V) := Hom n (S (V), V) and C * (V, V) := ⊕ n∈Z C n (V, V). As the graded version of the classical Nijenhuis-Richardson bracket given in [35,36], the Nijenhuis-Richardson bracket [·, ·] NR on the graded vector space C * (V, V) is given by: with the convention that f 0 • g j := 0 1 and The notion of a curved L ∞ -algebra was introduced in [21,29]. See also [26] for more applications. We denote a curved L ∞ -algebra by ( [24,25,44].
Here x, y, z are homogeneous elements in g, which also denote their degrees when in exponent.
We recall the notion of the suspension and desuspension operators. Let V = ⊕ i∈Z V i be a graded vector space, we define the suspension operator s : V → sV by assigning V to the graded vector space sV = ⊕ i∈Z (sV) i with (sV) i := V i−1 . There is a natural degree 1 map s : V → sV that is the identity map of the underlying vector space, sending v ∈ V to its suspended copy sv ∈ sV. Likewise, the desuspension operator s −1 changes the grading of V according to the rule (s −1 V) i := V i+1 . The degree −1 map s −1 : V → s −1 V is defined in the obvious way.
Example 2.4. Let V be a graded vector space. Then s −1 gl(V) is a sgLa where the symmetric Lie bracket is given by Let (g, [·, ·] g ) and (g ′ , [·, ·] g ′ ) be sgLa's. A homomorphism from g to g ′ is a linear map φ : g → g ′ of degree 0 such that Definition 2.5. A linear map of graded vector spaces D : g → g of degree n is called a derivation of degree n on a sgLa (g, [·, ·] g ) if D[x, y] g = (−1) n [Dx, y] g + (−1) n(x+1) [x, Dy] g , ∀x, y ∈ g. 1 The linear map f 0 is just a distinguished element Φ ∈ V 0 . We denote the vector space of derivations of degree n by Der n (g). Denote by Der(g) = ⊕ n∈Z Der n (g), which is a graded vector space.
In particular, if (h, [·, ·] h ) is abelian, we obtain an action of a sgLa on a graded vector space. It is obvious that ad : g → Der(g) is an action of the sgLa (g, [·, ·] g ) on itself, which is called the adjoint action.
Let ρ be an action of a sgLa (g, [·, ·] g ) on a graded vector space V. For x ∈ g i , we have ρ(x) ∈ Hom i+1 (V, V). Moreover, there is a sgLa structure on the direct sum g ⊕ V given by This sgLa is called the semidirect product of the sgLa (g, [·, ·] g ) and (V; ρ), and denoted by g⋉ ρ V.

2.2.
Homotopy O-operators of weight λ. Now we are ready to give the main notion of this paper. Definition 2.9. Let ρ be an action of a sgLa (g, [·, ·] g ) on a sgLa (h, [·, ·] h ). A degree 0 element is called a homotopy O-operator of weight λ on a sgLa (g, [·, ·] g ) with respect to the action ρ if the following equalities hold for all p ≥ 0 and all homogeneous elements v 1 , · · · , v p ∈ V, Remark 2.10. The linear map T 0 is just a element Ω ∈ g 0 . Below are the generalized Rota-Baxter identities for p = 0, 1, 2 : [Ω, Ω] g = 0, Remark 2.11. If the sgLa reduces to a Lie algebra and the action reduces to an action of a Lie algebra on another Lie algebra, the above definition reduces to the definition of an O-operator of weight λ on a Lie algebra. More precisely, the linear map T : h −→ g satisfies is called a homotopy Rota-Baxter operator of weight λ on a sgLa (g, [·, ·] g ) if the following equalities hold for all p ≥ 0 and all homogeneous elements Remark 2.13. A homotopy Rota-Baxter operator R = +∞ i=0 R i ∈ Hom(S (g), g) of weight λ on a sgLa (g, [·, ·] g ) is a homotopy O-operator of weight λ with respect to the adjoint action ad. If moreover the sgLa reduces to a Lie algebra, then the resulting linear operator R : g −→ g is a Rota-Baxter operator of weight λ in the sense that In the sequel, we construct a dgLa and show that homotopy O-operators of weight λ can be characterized as its Maurer-Cartan elements to justify our definition of homotopy O-operators of weight λ. For this purpose, we recall the derived bracket construction of graded Lie algebras. Let (g, [·, ·] g , d) be a dgLa. We define a new bracket on sg by The new bracket is called the derived bracket [22,49]. It is well known that the derived bracket is a graded Leibniz bracket on the shifted graded space sg. Note that the derived bracket is not graded skew-symmetric in general. We recall a basic result. Proposition 2.14. ( [22]) Let (g, [·, ·] g , d) be a dgLa, and let h ⊂ g be a subalgebra which is abelian, i.e. [h, h] g = 0. If the derived bracket is closed on sh, then (sh, [·, ·] d ) is a gLa.
Let ρ be an action of a sgLa (g, [·, ·] g ) on a sgLa (h, [·, ·] h ). Consider the graded vector space Also define a graded bracket operation Proof. By Theorem 2.2, the graded Nijenhuis-Richardson bracket [·, ·] NR associated to the direct sum vector space g ⊕ h gives rise to a gLa ( is an abelian subalgebra. We denote the symmetric graded Lie brackets [·, ·] g and [·, ·] h by µ g and µ h respectively. Since ρ is an action of the sgLa (g, [·, ·] g ), µ g + ρ is a semidirect product sgLa structure on g ⊕ h. By Theorem 2.2, we deduce that µ g + ρ and λµ h are Maurer-Cartan elements of the gLa (C By (7), for all k ≥ 2, x 1 , · · · , x k ∈ g and v 1 · · · , v k ∈ h, we have By straightforward computations, we have ). For any σ ∈ S (l,p−l) , we define τ = τ σ ∈ S (s−l,l) by ) . In fact, the elements of S (l,p−l) are in bijection with the elements of S (p−l,l) . Moreover, by k + l = p + 1, we have For any σ ∈ S (l,p−l) and l + 1 ≤ j ≤ p, we define τ = τ σ, j ∈ S (l,1,p−l−1) by Therefore, we obtain On the other hand, By (13), we obtain that the derived bracket ·, · is closed on sC * (h, g), and given by (12). Therefore, (sC * (h, g), ·, · ) is a gLa.
Homotopy O-operators of weight λ can be characterized as Maurer-Cartan elements of the above dgLa. Note that an element T = +∞ i=0 T i ∈ Hom(S (h), g) is of degree 0 if and only if the corresponding element T ∈ sHom(S (h), g) is of degree 1.
If the Lie algebra h is abelian in the above corollary, we recover the gLa that controls the deformations of O-operators of weight 0 given in [45,Proposition 2.3].

Operator homotopy post-Lie algebras
In this section, we first recall the notion of a post-Lie algebra, and then give the definition of an operator homotopy post-Lie algebra as a variation of a homotopy post-Lie algebra. We construct a dgLa and show that operator homotopy post-Lie algebras can be characterized as its Maurer-Cartan elements to justify the notion. We also show that operator homotopy post-Lie algebras naturally arise from homotopy O-operators of weight 1.
Define L ⊲ : g → gl(g) by L ⊲ (x)(y) = x ⊲ y. Then by (14), L ⊲ is a linear map from g to Der(g). In the sequel, we will say that L ⊲ is a post-Lie algebra structure on the Lie algebra (g, [·, ·] g ).
Remark 3.2. Let (g, [·, ·] g , ⊲) be a post-Lie algebra. If the Lie bracket [·, ·] g = 0, then (g, ⊲) becomes a pre-Lie algebra. Thus, a post-Lie algebra can be viewed as a nonabelian version of a pre-Lie algebra. See [8,9] for the classifications of post-Lie algebras on certain Lie algebras, and [34] for applications of post-Lie algebras in numerical integration.
The following well-known result is a special case of splitting of operads [4,39,48].
is a Lie bracket.
Remark 3.5. The explicit definition of a homotopy post-Lie algebra is still not known though its operad should be the minimal model of the post-Lie operad by general construction as noted in the introduction [27]. Since a Rota-Baxter Lie algebra of weight 1 gives a post-Lie algebra, we expect that a Rota-Baxter homotopy Lie algebra of weight 1 (a homotopy Lie algebra with a Rota-Baxter operator of weight 1) induces a homotopy post-Lie algebra. Based on preliminary computations in this regards, there should be more terms in the right hand side of (19) in the definition of a homotopy post-Lie algebra, suggesting that the homotopy post-Lie algebra is different from the operator homotopy post-Lie algebra just defined. So we have chosen a different name for distinction. • Define a graded linear map ∂ : • Define a graded bracket operation Proof. Setting λ = 1, g = s −1 Der(h) and ρ = s in Theorem 2.15, we find that (sC * (h, h), [·, ·] c , ∂) is a dgLa.
When the sgLa h is abelian, we characterize pre-Lie ∞ -algebras as Maurer-Cartan elements, which was originally given in [10]. .
Moreover, L = ∞ i=0 L i ∈ Hom 1 (S (h), gl(h)) defines a pre-Lie ∞ algebra structure by on the graded vector space V if and only if L = ∞ i=0 L i is a Maurer-Cartan element of the gLa (C * (V, V), [·, ·] c ).
In the above corollary, if the graded vector space V reduces to a usual vector space, we characterize pre-Lie algebra structures as Maurer-Cartan elements. See [10,37,50] for more details.
It is known that post-Lie algebras naturally arise from O-operators of weight 1 as follows.
In the sequel, we generalize the above relation to homotopy O-operators of weight 1 and operator homotopy post-Lie algebras.
Define a graded linear map Ψ : C * (h, g) →C * (h, h) of degree 0 by In the following, we set λ = 1 in Theorem 2.15 for notational simplicity. Proof. For all f = i f i ∈ Hom m (S (h), g), g = i g i ∈ Hom n (S (h), g) with f i , g i ∈ Hom(S i (h), g), (1) , · · · , v σ(p−l) ) , ρ g l (v σ(p−l+1) , · · · , v σ(p) ) ] Moreover, for all g ∈ Hom n (S (h), g), we have Thus, we deduce that Ψ is a homomorphism. Now we are ready to show that the homotopy O-operators of weight 1 induce operator homotopy post-Lie algebras.
It is straightforward to obtain the following result.
Similarly, the above result also holds for homotopy O-operators of weight 0.

Classification of 2-term skeletal operator homotopy post-Lie algebras
In general, it is expected that the 2-term homotopy of an algebra structure is equivalent to the categorification of this algebraic structure, and the 2-term homotopy algebras are quasiisomorphic to the 2-term skeletal homotopy algebras, which are classified by the third cohomological group. Baez and Crans [3] accomplished these for Lie algebras. In this spirit, we classify 2-term skeletal operator homotopy post-Lie algebras by the third cohomology group of a post-Lie algebra. For this purpose, we first define representations of post-Lie algebras and then develop the corresponding cohomology theory.

4.1.
Representations of post-Lie algebras. Here we introduce the notion of a representation of a post-Lie algebra (g, [·, ·] g , ⊲) on a vector space V. We show that there is naturally an induced representation of the subadjacent Lie algebra g C on Der(g, V). This fact plays a crucial role in our study of cohomology groups of post-Lie algebras in the next subsection.
To prove that the operator δ is indeed a coboundary operator, i.e. δ • δ = 0, we need some preparations.
Also f ∈ C 2 Der (g, V) is closed if and only if Φ( f ) ∈ Hom(g, Der(g, V)) and There is a close relationship between the cohomology groups of post-Lie algebras and those of the corresponding sub-adjacent Lie algebras.
In the above theorem, if [·, ·] g and ρ are zero, then the post-Lie algebra is a pre-Lie algebra and we recover the result of [13] as follows.
Corollary 4.11. Let (V; µ, ν) be a representation of a pre-Lie algebra (g, ⊲). Then the cohomology group H n (g, V) of the pre-Lie algebra (g, ⊲) and the cohomology group H n−1 (g C , Hom(g, V)) of the subadjacent Lie algebra g C are isomorphic for all n ≥ 1.

4.3.
Classification of 2-term skeletal operator homotopy post-Lie algebras. In this subsection, we first give an equivalent definition of an operator homotopy post-Lie algebra and then classify 2-term skeletal operator homotopy post-Lie algebras using the third cohomology group given in Section 4.2 For all i ≥ 1, let Θ i : ∧ i−1 g ⊗ g → g be a graded linear map of degree 2 − i. Define D(Θ i ) : ⊙ i−1 s −1 g ⊗ s −1 g → s −1 g by which is a graded linear map of degree 1. This is illustrated by the following commutative diagram: − −−−− → s −1 g Using this process, we can give an equivalent definition of an operator homotopy post-Lie algebra as follows.
Now we will show that the corresponding cohomology space H * (g) of the complex (g, Θ 1 ) enjoys a graded post-Lie algebra structure and this justifies our definition of an "operator homotopy post-Lie algebra". Proof. For any homogeneous element x ∈ ker(Θ 1 ), we denote by x ∈ H * (g) its cohomological class. First we define a graded bracket operation [·, ·] on the graded vector space H * (g) by [x,ȳ] := [x, y] g , ∀x,ȳ ∈ H * (g).
By truncation, we obtain the definition of a 2-term operator homotopy post-Lie algebra.