Homotopy Rota-Baxter operators, homotopy O -operators and homotopy post-Lie algebras
aa r X i v : . [ m a t h . QA ] J u l HOMOTOPY ROTA-BAXTER OPERATORS, HOMOTOPY O -OPERATORS ANDHOMOTOPY POST-LIE ALGEBRAS RONG TANG, CHENGMING BAI, LI GUO, AND YUNHE SHENGA bstract . Rota-Baxter operators, O -operators on Lie algebras and their interconnected pre-Lieand 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 O -operatoron a symmetric graded Lie algebra. Their characterization by Maurer-Cartan elements of suitabledi ff erential graded Lie algebras is provided. Through the action of a homotopy O -operator on asymmetric 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 ofpost-Lie algebras is established, with an application to 2-term skeletal operator homotopy post-Liealgebras. C ontents
1. Introduction 11.1. Background and motivation 11.2. Approach of the paper 21.3. Outline of the paper 42. Homotopy O -operators of weight λ O -operators of weight λ
73. Operator homotopy post-Lie algebras 134. Classification of 2-term skeletal operator homotopy post-Lie algebras 204.1. Representations of post-Lie algebras 204.2. Cohomology groups of post-Lie algebras 224.3. Classification of 2-term skeletal operator homotopy post-Lie algebras 25References 281. I ntroduction
This paper initiates the homotopy study of Rota-Baxter operators, O -operators and the relatedpre-Lie algebras and post-Lie algebras.1.1. Background and motivation.
Homotopy is a fundamental notion in topology describingcontinuously deforming one function to another.The first homotopy construction in algebra is the A ∞ -algebra of Stashe ff , arising from his workon homotopy characterization of connected based loop spaces [43]. Later related developmentsinclude the work of Boardman and Vogt [6] about E ∞ -spaces on the infinite loop space, the workof Schlessinger and Stashe ff [40] about L ∞ -algebras on perturbations of rational homotopy typesand the work of Chapoton and Livernet [10] about pre-Lie ∞ algebras, as well as homotopy Leibnizalgebras [1]. See [27, 30, 32] for other examples. Date : August 1, 2019.2010
Mathematics Subject Classification.
Key words and phrases. homotopy Rota-Baxter operator, homotopy O -operator, homotopy post-Lie algebra,post-Lie algebra, Maurer-Cartan element, cohomology. Homotopy and operads are intimately related. In fact, operads were introduced as a tool inhomotopy theory, specifically for iterated loop spaces. Vaguely speaking, the homotopy of analgebraic structure is obtained when the defining relations of the algebraic structure is relaxedto hold up to homotopy. The resulting algebraic structure is homotopy equivalent to the originalalgebraic structure via a Homotopy Transfer Theorem. The idea is behind much of the operadicdevelopments. 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 . Moreprecisely, P ∞ is the Koszul resolution as the cobar construction Ω P i of the Koszul dual cooperadof P [27, 30]. Since the latter notion makes sense only when P is quadratic, this approach doesnot 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 andlater found important applications in the Connes-Kreimer’s algebraic approach to renormaliza-tion of quantum field theory [11], among others. A Rota-Baxter operator on a Lie algebra isnaturally the operator form of a classical r -matrix [41] under certain conditions. To better under-stand such connection in general, the notion of an O -operator (also called a relative Rota-Baxteroperator [39] or a generalized Rota-Baxter operator [46, 47]) on a Lie algebra was introduced byKupershmidt [23], which can be traced back to Bordemann [7].Both operators have been studied extensively in recent years [20]. Their operadic theory arechallenging 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 studyhomotopy of operads, other methods to give the related resolutions have been introduced.Dotsenko and Khoroshkin [14] used shu ffl e operads and the Gr¨obner bases method to showthat, 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 dif-ferentials in the free resolutions for Quillen homology computation, though “in general compactformulas are yet to be found” as noted in the paper. One can expect a similarly challengingsituation for the operad of Rota-Baxter operators on Lie algebras.This paper follows the more basic and direct approach to homotopy via di ff erential graded Liealgebras and Maurer-Cartan elements. This is in fact the approach taken in the early developmentsof characterizing algebraic and homotopy algebraic structures before they are put under the moreuniform and sophisticated framework of operads. These developments started with the well-known series of work by Gerstenhaber [16, 17] on deformations of associative algebras and byNijenhuis 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 vectorspace are given by degree 1 solutions of the Maurer-Cartan equation in a suitable di ff erentialgraded Lie algebra built from the vector space. When the vector space is replaced by a gradedvector space, similar solutions give objects in the homotopy algebraic structure. To make theidea 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 = Hom( ∧ n V , V ) with thedegree 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 − ( − ( m − n − g ◦ f , with f ◦ g ∈ Hom( ∧ m + n − V , V ) being defined by(1) ( f ◦ g )( v , · · · , v m + n − ) : = X σ ∈ S ( n , m − ( − σ f ( g ( v σ (1) , · · · , v σ ( n ) ) , v σ ( n + , · · · , v σ ( m + n − ) , OMOTOPY O -OPERATORS AND HOMOTOPY POST-LIE ALGEBRAS 3 where the sum is over ( n , m − ffl es. Then (cid:0) ⊕ ∞ n = Hom( ∧ n V , V ) , [ · , · ] NR (cid:1) is a di ff erential gradedLie algebra with the trivial derivation. With this setup, a Lie algebra structure on V is precisely adegree 1 solution ω ∈ Hom( ∧ V , V ) of the Maurer-Cartan equation[ ω, ω ] NR = . In the spirit of this characterization of Lie algebra structures by solutions of the Maurer-Cartanequation, homotopy Lie algebra structures can be characterized briefly as follows, with furtherdetails provided in Section 2.1. Given a graded vector space V = ⊕ i ∈ Z V i , let S ( V ) denote thesymmetric algebra of V and let Hom n ( S ( V ) , V ) denote the space of degree n linear maps. Definethe Nijenhuis-Richardson bracket [ · , · ] NR on the graded vector space P n ∈ Z Hom n ( S ( V ) , V ) in agraded 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. Themore elementary nature of this approach makes it possible to be applied where the operadic ap-proach 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 characterizationof O -operators (of weight zero) and further to establish a deformation theory and its controllingcohomology for O -operators.To illustrate our approach in a broader context, we focus on the Rota-Baxter Lie algebra fornow 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 unaryoperation T . The homotopy of the Lie algebra ℓ ∞ = { ℓ i } ∞ i = is well-known as the L ∞ -algebra [30].Together with the natural Rota-Baxter operator action as defined in [39], we have the Rota-Baxterhomotopy Lie algebra ( ℓ ∞ , T ) or Rota-Baxter L ∞ -algebra. A Rota-Baxter Lie algebra naturallyinduces a post-Lie algebra, originated from an operadic study [48] and found applications inmathematical physics and numerical analysis [5, 34]. Likewise, a Rota-Baxter homotopy Liealgebra is expected to induce a homotopy post-Lie algebra whose construction is still not knownbeyond its conceptual definition as an operadic minimal model mentioned above, giving rise tothe commutative diagram(2) ( ℓ, T ) / / RB action (cid:15) (cid:15) ( ℓ ∞ , T ) RB action (cid:15) (cid:15) post-Lie / / homotopy post-Liewhere the horizontal arrows are taking homotopy and the vertical arrows are taking actions of theRota-Baxter operators.In this paper, we will pursue the other direction, by taking homotopy of the Rota-Baxter op-erator T and obtain T ∞ : = { T i } ∞ i = , 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 men-tioned Rota-Baxter homotopy Lie algebra. The action of the operator homotopy Rota-Baxteroperator T ∞ gives rise to a variation of the homotopy post-Lie algebra which we will call the op-erator homotopy post-Lie algebra (see Remark 3.5). This gives another commutative diagramshown as the front rectangle in Eq. (3) while the diagram in Eq. (2) is embedded as the rightrectangle.Eventually, the full homotopy of the Rota-Baxter Lie algebra should come from the combinedhomotopies of both the Lie algebra structure and the Rota-Baxter operator structure, tentatively RONG TANG, CHENGMING BAI, LI GUO, AND YUNHE SHENG denoted by ( ℓ ∞ , T ∞ ) and called the full homotopy Rota-Baxter Lie algebra . A suitable actionof 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 ofthe post-Lie algebra, could be put together and form the following diagram(3) ( ℓ ∞ , T ∞ ) ( ℓ ∞ , T )( ℓ, T ∞ ) ❥❥❥❥❥❥❥ ( ℓ, T ) ❥❥❥❥❥❥❥❥ full homotopypost-Lie homotopypost-Lieoperator homotopypost-Lie ❥❥❥ post-Lie ❥❥❥❥❥❥ where going in the left and the inside directions should be taking various homotopy, and goingdownward 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 thelower half of the diagram becomes a pre-Lie algebra and the homotopy post-Lie in the back-right-lower corner is a homotopy pre-Lie algebra. We find it quite amazing that the operatorhomotopy 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-lowercorner is also the homotopy pre-Lie algebra. It would be interesting to determine this structureeven in this special case.1.3. Outline of the paper.
After some background on di ff erential 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 di ff erentialgraded 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 obtaina 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 whenthe weight of the O -operator is taken to be zero, we obtain homotopy O -operators of weightzero. Since O -operators of weight zero naturally derive pre-Lie algebras [39], it is expectedthat 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(4) O -operators homotopy / / (cid:15) (cid:15) homotopy O -operators (cid:15) (cid:15) pre-Lie homotopy / / pre-Lie ∞ In other words, the compositions of taking homotopy and taking operator action in either ordergives the pre-Lie ∞ algebras.There has been quite much interest on constructions of post-Lie algebras in the recent litera-ture [8, 9, 15, 38]. Another useful application of our general construction is the characterizationof post-Lie algebra structures on a given Lie algebra using Maurer-Cartan elements in a suitabledi ff erential 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 OMOTOPY O -OPERATORS AND HOMOTOPY POST-LIE ALGEBRAS 5 and then in [33] by resolutions of algebras from the Koszul duality theory in the framework ofoperads. An explicit cohomology theory of post-Lie algebras is not yet known. In this sectionwe 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-termskeletal operator homotopy post-Lie algebras. Notation.
We assume that all the vector spaces are over a field of characteristic zero. For ahomogeneous element x in a Z -graded vector space, we also use x in the exponent, as in ( − x , todenote its degree in order to simplify the notation.2. H omotopy O - operators of weight λ In this section, we introduce the notion of a homotopy O -operator of weight λ , where λ is aconstant. We construct a di ff erential graded Lie algebra (dgLa for short) and show that homo-topy O -operators of weight λ can be characterized as its Maurer-Cartan elements to justify ourdefinition.2.1. Maurer-Cartan elements and Nijenhuis-Richardson brackets.
We first recall some back-ground needed in later sections.
Definition 2.1. ([27]) Let ( g = ⊕ k ∈ Z g k , [ · , · ] , d ) be a dgLa. A degree 1 element θ ∈ g is called a Maurer-Cartan element of g if it satisfies the following Maurer-Cartan equation :(5) d θ +
12 [ θ, θ ] = . A permutation σ ∈ S n is called an ( i , n − i )-shu ffl e if σ (1) < · · · < σ ( i ) and σ ( i + < · · · < σ ( n ).If i = n , we assume σ = Id. The set of all ( i , n − i )-shu ffl es will be denoted by S ( i , n − i ) . Thenotion of an ( i , · · · , i k )-shu ffl e and the set S ( i , ··· , i k ) are defined analogously.Let V = ⊕ k ∈ Z V k be a Z -graded vector space. We will denote by S ( V ) the symmetric algebra of V : S ( V ) : = ⊕ ∞ i = S i ( V ) . Denote the product of homogeneous elements v , · · · , v n ∈ V in S n ( V ) by v ⊙ · · · ⊙ v n . Thedegree of v ⊙ · · · ⊙ v n is by definition the sum of the degree of v i . For a permutation σ ∈ S n and v , · · · , v n ∈ V , the Koszul sign ε ( σ ; v , · · · , v n ) ∈ {− , } is defined by v ⊙ · · · ⊙ v n = ε ( σ ; v , · · · , v n ) v σ (1) ⊙ · · · ⊙ v σ ( n ) and the antisymmetric Koszul sign χ ( σ ; v , · · · , v n ) ∈ {− , } is defined by χ ( σ ; v , · · · , v n ) : = ( − σ ε ( σ ; v , · · · , v n ) . Denote by Hom n ( S ( V ) , V ) the space of degree n linear maps from the graded vector space S ( V ) tothe 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 = P ∞ i = f i . Set C n ( V , V ) : = Hom n ( S ( V ) , V ) and C ∗ ( V , V ) : = ⊕ n ∈ Z C n ( V , V ) . Asthe 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:[ f , g ] NR : = f ◦ g − ( − mn g ◦ f , ∀ f = ∞ X i = f i ∈ C m ( V , V ) , g = ∞ X j = g j ∈ C n ( V , V ) , (6) RONG TANG, CHENGMING BAI, LI GUO, AND YUNHE SHENG where f ◦ g ∈ C m + n ( V , V ) is defined by f ◦ g = (cid:16) ∞ X i = f i (cid:17) ◦ (cid:16) ∞ X j = g j (cid:17) : = ∞ X k = (cid:16) X i + j = k + f i ◦ g j (cid:17) , (7)while f i ◦ g j ∈ Hom( S i + j − ( V ) , V ) is defined by( f i ◦ g j )( v , · · · , v i + j − ) : = X σ ∈ S ( j , i − ε ( σ ) f i ( g j ( v σ (1) , · · · , v σ ( j ) ) , v σ ( j + , · · · , v σ ( i + j − ) , with the convention that f ◦ g j : = and( f j ◦ g )( v , · · · , v j − ) : = f j ( g , v , · · · , v j − ) . The notion of a curved L ∞ -algebra was introduced in [21, 29]. See also [26] for more applica-tions. Theorem 2.2. ([2, 29])
With the above notations, ( C ∗ ( V , V ) , [ · , · ] NR ) is a graded Lie algebra( gLa for short). Its Maurer-Cartan elements P ∞ i = l i are the curved L ∞ -algebra structures onV. We denote a curved L ∞ -algebra by ( V , { l k } ∞ k = ). A curved L ∞ -algebra ( V , { l k } ∞ k = ) with l = L ∞ -algebra [24, 25, 44]. Definition 2.3. ([2]) A symmetric graded Lie algebra (sgLa) is a Z -graded vector space g equipped with a bilinear bracket [ · , · ] g : g ⊗ g → g of degree 1, satisfying(1) (graded symmetry) [ x , y ] g = ( − xy [ y , x ] g , (2) (graded Leibniz rule) [ x , [ y , z ] g ] g = ( − x + [[ x , y ] g , z ] g + ( − ( x + y + [ y , [ x , z ] g ] g .Here x , y , z are homogeneous elements in g , which also denote their degrees when in exponent.A sgLa ( g , [ · , · ] g ) is just a curved L ∞ -algebra ( g , { l k } ∞ k = ), in which l k = k ≥ k = V = ⊕ i ∈ Z V i be a gradedvector space, we define the suspension operator s : V sV by assigning V to the gradedvector space sV = ⊕ i ∈ Z ( sV ) i with ( sV ) i : = V i − . 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 − changes the grading of V according to therule ( s − V ) i : = V i + . The degree − s − : V → s − V is defined in the obvious way. Example 2.4.
Let V be a graded vector space. Then s − gl ( V ) is a sgLa where the symmetric Liebracket is given by[ s − f , s − g ] : = ( − m s − [ f , g ] , ∀ f ∈ Hom m ( V , V ) , g ∈ Hom n ( V , V ) . (8)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 φ ([ x , x ] g ) = [ φ ( x ) , φ ( x )] g ′ , ∀ x , x ∈ g . Definition 2.5.
A linear map of graded vector spaces D : g → g of degree n is called a derivationof degree n on a sgLa ( g , [ · , · ] g ) if D [ x , y ] g = ( − n [ Dx , y ] g + ( − n ( x + [ x , Dy ] g , ∀ x , y ∈ g . The linear map f is just a distinguished element Φ ∈ V . OMOTOPY O -OPERATORS AND HOMOTOPY POST-LIE ALGEBRAS 7 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. Remark 2.6.
A derivation of degree n on a sgLa ( g , [ · , · ] g ) is just a homotopy derivation { θ k } ∞ k = ofdegree n on ( g , [ · , · ] g ), in which θ k = k ≥
2. See [12, 21] for more details.By a straightforward check, we obtain
Proposition 2.7.
With the above notations, ( s − Der ( g ) , [ · , · ]) is a symmetric graded Lie subalge-bra of ( s − gl ( g ) , [ · , · ]) , where the bracket [ · , · ] is defined by (8) . Definition 2.8. An action of a sgLa ( g , [ · , · ] g ) on a sgLa ( h , [ · , · ] h ) is a homomorphism of gradedvector spaces ρ : g → Der ( h ) of degree 1 such that s − ◦ ρ : g → s − Der ( h ) is a sgLa homomor-phism.In particular, if ( h , [ · , · ] h ) is abelian, we obtain an action of a sgLa on a graded vector space. Itis 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 + ( V , V ). Moreover, there is a sgLa structure on the direct sum g ⊕ V given by[ x + v , x + v ] ρ : = [ x , x ] g + ρ ( x ) v + ( − x x ρ ( x ) v , ∀ x , x ∈ g , v , v ∈ V . 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 thispaper.
Definition 2.9.
Let ρ be an action of a sgLa ( g , [ · , · ] g ) on a sgLa ( h , [ · , · ] h ). A degree 0 element T = P + ∞ i = T i ∈ Hom( S ( h ) , g ) with T i ∈ Hom( S i ( h ) , g ) 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 ≥ v , · · · , v p ∈ V , X ≤ i < j ≤ p ( − v i ( v + ··· + v i − ) + v j ( v + ··· + v j − ) + v i v j T p − ( λ [ v i , v j ] h , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v p ) + X k + l = p + X σ ∈ S ( l , , p − l − ε ( σ ) T k − (cid:16) ρ (cid:0) T l ( v σ (1) , · · · , v σ ( l ) ) (cid:1) v σ ( l + , v σ ( l + , · · · , v σ ( p ) (cid:17) (9) = X k + l = p + X σ ∈ S ( k − , l ) ε ( σ )[ T k − ( v σ (1) , · · · , v σ ( k − ) , T l ( v σ ( k ) , · · · , v σ ( p ) )] g . Remark 2.10.
The linear map T is just a element Ω ∈ g . Below are the generalized Rota-Baxteridentities for p = , , Ω , Ω ] g = , T ( ρ ( Ω ) v ) = [ Ω , T ( v )] g , [ T ( v ) , T ( v )] g − T (cid:16) ρ ( T ( v )) v + ( − v v ρ ( T ( v )) v + λ [ v , v ] h (cid:17) = T ( ρ ( Ω ) v , v ) + ( − v v T ( ρ ( Ω ) v , v ) − [ Ω , T ( v , v )] g . RONG TANG, CHENGMING BAI, LI GUO, AND YUNHE SHENG
Remark 2.11.
If the sgLa reduces to a Lie algebra and the action reduces to an action of a Liealgebra on another Lie algebra, the above definition reduces to the definition of an O -operator ofweight λ on a Lie algebra. More precisely, the linear map T : h −→ g satisfies[ T u , T v ] = T (cid:0) ρ ( T u )( v ) − ρ ( T v )( u ) + λ [ u , v ] h (cid:1) , ∀ u , v ∈ h . Definition 2.12.
A degree 0 element R = P + ∞ i = R i ∈ Hom( S ( g ) , g ) with R i ∈ Hom( S i ( g ) , g ) iscalled a homotopy Rota-Baxter operator of weight λ on a sgLa ( g , [ · , · ] g ) if the following equal-ities hold for all p ≥ x , · · · , x p ∈ g , X ≤ i < j ≤ p ( − x i ( x + ··· + x i − ) + x j ( x + ··· + x j − ) + x i x j R p − ( λ [ x i , x j ] g , x , · · · , ˆ x i , · · · , ˆ x j , · · · , x p ) + X k + l = p + X σ ∈ S ( l , , p − l − ε ( σ ) R k − (cid:0) [ R l ( x σ (1) , · · · , x σ ( l ) ) , x σ ( l + ] g , x σ ( l + , · · · , x σ ( p ) (cid:1) = X k + l = p + X σ ∈ S ( k − , l ) ε ( σ )[ R k − ( x σ (1) , · · · , x σ ( k − ) , R l ( x σ ( k ) , · · · , x σ ( p ) )] g . Remark 2.13.
A homotopy Rota-Baxter operator R = P + ∞ i = R i ∈ Hom( S ( g ) , g ) of weight λ ona sgLa ( g , [ · , · ] g ) is a homotopy O -operator of weight λ with respect to the adjoint action ad . Ifmoreover 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[ R ( x ) , R ( y )] g = R (cid:0) [ R ( x ) , y ] g + [ x , R ( y )] g + λ [ x , y ] g (cid:1) , ∀ x , y ∈ g . In the sequel, we construct a dgLa and show that homotopy O -operators of weight λ can becharacterized as its Maurer-Cartan elements to justify our definition of homotopy O -operators ofweight 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 s g by[ sx , sy ] d : = ( − x s [ dx , y ] g , ∀ x , y ∈ g . (10)The new bracket is called the derived bracket [22, 49]. It is well known that the derived bracketis a graded Leibniz bracket on the shifted graded space s g . Note that the derived bracket is notgraded 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 isabelian, i.e. [ h , h ] g = . If the derived bracket is closed on s h , then ( s h , [ · , · ] d ) is a gLa . Let ρ be an action of a sgLa ( g , [ · , · ] g ) on a sgLa ( h , [ · , · ] h ). Consider the graded vector space C ∗ ( h , g ) : = ⊕ n ∈ Z Hom n ( S ( h ) , g ). Define a linear map d : Hom n ( S ( h ) , g ) −→ Hom n + ( S ( h ) , g ) by(d g ) p ( v , · · · , v p ) = X ≤ i < j ≤ p ( − n + + v i ( v + ··· + v i − ) + v j ( v + ··· + v j − ) + v i v j g p − ( λ [ v i , v j ] h , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v p ) . (11)Also define a graded bracket operation ~ · , · (cid:127) : Hom m ( S ( h ) , g ) × Hom n ( S ( h ) , g ) −→ Hom m + n + ( S ( h ) , g )by (cid:18) f , g (cid:19) p ( v , · · · , v p ) = − X k + l = p + X σ ∈ S ( l , , p − l − ε ( σ ) f k − (cid:16) ρ (cid:0) g l ( v σ (1) , · · · , v σ ( l ) ) (cid:1) v σ ( l + , v σ ( l + , · · · , v σ ( p ) (cid:17) (12) OMOTOPY O -OPERATORS AND HOMOTOPY POST-LIE ALGEBRAS 9 + ( − ( m + n + X k + l = s + X σ ∈ S ( k − , , p − k ) ε ( σ ) g l (cid:16) ρ (cid:0) f k − ( v σ (1) , · · · , v σ ( k − ) (cid:1) v σ ( k ) , v σ ( k + , · · · , v σ ( p ) (cid:17) − X k + l = p + X σ ∈ S ( k − , l ) ( − n ( v σ (1) + ··· + v σ ( k − ) + m + ε ( σ )[ f k − ( v σ (1) , · · · , v σ ( k − ) , g l ( v σ ( k ) , · · · , v σ ( p ) )] g for all f = P i f i ∈ Hom m ( S ( h ) , g ), g = P g i ∈ Hom n ( S ( h ) , g ) with f i , g i ∈ Hom( S i ( h ) , g ) and v , · · · , v p ∈ h . Here we write d g = P i (d g ) i with (d g ) i ∈ Hom( S i ( h ) , g ), and (cid:18) f , g (cid:19) = P i (cid:18) f , g (cid:19) i with (cid:18) f , g (cid:19) i ∈ Hom( S i ( h ) , g ). Theorem 2.15.
Let ρ be an action of a sgLa ( g , [ · , · ] g ) on a sgLa ( h , [ · , · ] h ) . Then ( sC ∗ ( h , g ) , ~ · , · (cid:127) , d) is a dgLa .Proof. By Theorem 2.2, the graded Nijenhuis-Richardson bracket [ · , · ] NR associated to the directsum vector space g ⊕ h gives rise to a gLa ( C ∗ ( g ⊕ h , g ⊕ h ) , [ · , · ] NR ). Obviously C ∗ ( h , g ) = M n ∈ Z Hom n ( S ( h ) , g )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 productsgLa structure on g ⊕ h . By Theorem 2.2, we deduce that µ g + ρ and λµ h are Maurer-Cartanelements of the gLa ( C ∗ ( g ⊕ h , g ⊕ h ) , [ · , · ] NR ). Define a di ff erential d µ g + ρ on ( C ∗ ( g ⊕ h , g ⊕ h ) , [ · , · ] NR )via d µ g + ρ : = [ µ g + ρ, · ] NR . Further, we define the derived bracket on the graded vector space ⊕ n ∈ Z Hom n ( S ( h ) , g ) by (cid:18) f , g (cid:19) : = ( − m [ d µ g + ρ f , g ] NR = ( − m [[ µ g + ρ, f ] NR , g ] NR , (13)for all f = P f i ∈ Hom m ( S ( h ) , g ) , g = P i g i ∈ Hom n ( S ( h ) , g ) . Write[ µ g + ρ, f ] NR = ∞ X i = [ µ g + ρ, f ] iNR with [ µ g + ρ, f ] iNR ∈ Hom( S i ( h ) , g ) . By (7), for all k ≥ x , · · · , x k ∈ g and v · · · , v k ∈ h , we have[ µ g + ρ, f ] kNR (cid:16) ( x , v ) , · · · , ( x k , v k ) (cid:17) = (cid:16) ( µ g + ρ ) ◦ f k − − ( − m f k − ◦ ( µ g + ρ ) (cid:17)(cid:16) ( x , v ) , · · · , ( x k , v k ) (cid:17) = k X i = ( − α ( µ g + ρ ) (cid:16) f k − (cid:0) ( x , v ) , · · · , [ ( x i , v i ) , · · · , ( x k , v k ) (cid:1) , ( x i , v i ) (cid:17) − ( − m X ≤ i < j ≤ k ( − β f k − (cid:16) ( µ g + ρ ) (cid:0) ( x i , v i ) , ( x j , v j ) (cid:1) , ( x , v ) , · · · , [ ( x i , v i ) , · · · , [ ( x j , v j ) , · · · , ( x k , v k ) (cid:17) = k X i = ( − α ( µ g + ρ ) (cid:16)(cid:0) f k − ( v , · · · , ˆ v i , · · · , v k ) , (cid:1) , ( x i , v i ) (cid:17) − ( − m X ≤ i < j ≤ k ( − β f k − (cid:16)(cid:0) [ x i , x j ] g , ρ ( x i ) v j + ( − v i v j ρ ( x j ) v i (cid:1) , ( x , v ) , · · · , [ ( x i , v i ) , · · · , [ ( x j , v j ) , · · · , ( x k , v k ) (cid:17) = k X i = ( − α (cid:0) [ f k − ( v , · · · , ˆ v i , · · · , v k ) , x i ] g , ρ ( f k − ( v , · · · , ˆ v i , · · · , v k )) v i (cid:1) − ( − m X ≤ i < j ≤ k ( − β (cid:0) f k − ( ρ ( x i ) v j + ( − v i v j ρ ( x j ) v i , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v k ) , (cid:1) , here α = v i ( v i + + · · · + v k ) and β = v i ( v + · · · + v i − ) + v j ( v + · · · + v j − ) + v i v j . On the other hand,we have [ µ g + ρ, f ] NR = µ g + ρ, f ] NR ( x , v ) = (cid:0) [ f , x ] g , ρ ( f ) v (cid:1) .Moreover, we obtain[[ µ g + ρ, f ] NR , g ] pNR (cid:16) ( x , v ) , · · · , ( x p , v p ) (cid:17) = (cid:16) X k + l = p + [ µ g + ρ, f ] kNR ◦ g l − ( − ( m + n X k + l = p + g l ◦ [ µ g + ρ, f ] kNR (cid:17)(cid:16) ( x , v ) , · · · , ( x p , v p ) (cid:17) . By straightforward computations, we have([ µ g + ρ, f ] kNR ◦ g l ) (cid:16) ( x , v ) , · · · , ( x p , v p ) (cid:17) = X σ ∈ S ( l , p − l ) ε ( σ )[ µ g + ρ, f ] kNR (cid:16) g l (cid:0) ( x σ (1) , v σ (1) ) , · · · , ( x σ ( l ) , v σ ( l ) ) (cid:1) , ( x σ ( l + , v σ ( l + ) , · · · , ( x σ ( p ) , v σ ( p ) ) (cid:17) = X σ ∈ S ( l , p − l ) ε ( σ )[ µ g + ρ, f ] kNR (cid:16)(cid:0) g l ( v σ (1) , · · · , v σ ( l ) ) , (cid:1) , ( x σ ( l + , v σ ( l + ) , · · · , ( x σ ( p ) , v σ ( p ) ) (cid:17) = X σ ∈ S ( l , p − l ) ε ( σ )( − ¯ α (cid:0) [ f k − ( v σ ( l + , · · · , v σ ( p ) ) , g l ( v σ (1) , · · · , v σ ( l ) )] g , (cid:1) − ( − m X σ ∈ S ( l , p − l ) ε ( σ ) p X j = l + ( − ¯ β (cid:16) f k − (cid:0) ρ ( g l ( v σ (1) , · · · , v σ ( l ) )) v σ ( j ) , v σ ( l + , · · · , ˆ v σ ( j ) , · · · , v σ ( p ) (cid:1) , (cid:17) , where ¯ α = ( v σ (1) + · · · + v σ ( l ) + n )( v σ ( l + + · · · + v σ ( p ) ) and ¯ β = v σ ( j ) ( v σ ( l + + · · · + v σ ( j − ). For any σ ∈ S ( l , p − l ) , we define τ = τ σ ∈ S ( s − l , l ) by τ ( i ) = ( σ ( i + l ) , ≤ i ≤ p − l ; σ ( i − p + l ) , p − l + ≤ i ≤ s . Thus ε ( τ ; v , · · · , v p ) = ε ( σ ; v , · · · , v p )( − ( v σ (1) + ··· + v σ ( l ) )( v σ ( l + + ··· + v σ ( p ) ) . 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 X σ ∈ S ( l , p − l ) ε ( σ )( − ¯ α (cid:0) [ f k − ( v σ ( l + , · · · , v σ ( p ) ) , g l ( v σ (1) , · · · , v σ ( l ) )] g , (cid:1) = X τ ∈ S ( k − , l ) ε ( τ )( − n ( v τ (1) + ··· + v τ ( k − ) (cid:0) [ f k − ( v τ (1) , · · · , v τ ( k − ) , g l ( v τ ( k ) , · · · , v τ ( p ) )] g , (cid:1) . For any σ ∈ S ( l , p − l ) and l + ≤ j ≤ p , we define τ = τ σ, j ∈ S ( l , , p − l − by τ ( i ) = σ ( i ) , ≤ i ≤ l ; σ ( j ) , i = l + σ ( i − , l + ≤ i ≤ j ; σ ( i ) , j + ≤ i ≤ p . Thus we have ε ( τ ; v , · · · , v p ) = ε ( σ ; v , · · · , v p )( − v σ ( j ) ( v σ ( l + + ··· + v σ ( j − ) . Then X σ ∈ S ( l , p − l ) ε ( σ ) p X j = l + ( − ¯ β (cid:16) f k − (cid:0) ρ ( g l ( v σ (1) , · · · , v σ ( l ) )) v σ ( j ) , v σ ( l + , · · · , ˆ v σ ( j ) , · · · , v σ ( p ) (cid:1) , (cid:17) OMOTOPY O -OPERATORS AND HOMOTOPY POST-LIE ALGEBRAS 11 = X τ ∈ S ( l , , p − l − ε ( τ ) (cid:16) f k − (cid:0) ρ ( g l ( v τ (1) , · · · , v τ ( l ) )) v τ ( l + , v τ ( l + , · · · , v τ ( p ) (cid:1) , (cid:17) . Therefore, we obtain([ µ g + ρ, f ] kNR ◦ g l ) (cid:16) ( x , v ) , · · · , ( x p , v p ) (cid:17) = X σ ∈ S ( k − , l ) ε ( σ )( − n ( v σ (1) + ··· + v σ ( k − ) (cid:0) [ f k − ( v σ (1) , · · · , v σ ( k − ) , g l ( v σ ( k ) , · · · , v σ ( p ) )] g , (cid:1) − ( − m X σ ∈ S ( l , , p − l − ε ( σ ) (cid:16) f k − (cid:0) ρ ( g l ( v σ (1) , · · · , v σ ( l ) )) v σ ( l + , v σ ( l + , · · · , v σ ( p ) (cid:1) , (cid:17) . On the other hand, (cid:16) g l ◦ [ µ g + ρ, f ] kNR (cid:17)(cid:16) ( x , v ) , · · · , ( x p , v p ) (cid:17) = X σ ∈ S ( k , n − k ) ε ( σ ) g l (cid:16) [ µ g + ρ, f ] kNR (cid:0) ( x σ (1) , v σ (1) ) , · · · , ( x σ ( k ) , v σ ( k ) ) (cid:1) , ( x σ ( k + , v σ ( k + ) , · · · , ( x σ ( p ) , v σ ( p ) ) (cid:17) = X σ ∈ S ( k , p − k ) ε ( σ ) k X i = ( − α ′ (cid:16) g l (cid:0) ρ ( f k − ( v σ (1) , · · · , ˆ v σ ( i ) , · · · , v σ ( k ) )) v σ ( i ) , v σ ( k + · · · , v σ ( p ) (cid:1) , (cid:17) , where α ′ = v σ ( i ) ( v σ ( i + + · · · + v σ ( k ) ). For any σ ∈ S ( k , p − k ) and 1 ≤ i ≤ k , we define τ = τ σ, i ∈ S ( k − , , p − k ) by τ ( j ) = σ ( j ) , ≤ j ≤ i − σ ( j + , i ≤ j ≤ k − σ ( i ) , j = k ; σ ( j ) , k + ≤ j ≤ p . Thus we have ε ( τ ; v , · · · , v p ) = ε ( σ ; v , · · · , v p )( − v σ ( i ) ( v σ ( i + + ··· + v σ ( k ) ) . Then we have (cid:16) g l ◦ [ µ g + ρ, f ] kNR (cid:17)(cid:16) ( x , v ) , · · · , ( x p , v p ) (cid:17) = X σ ∈ S ( k − , , p − k ) ε ( σ ) (cid:16) g l (cid:0) ρ ( f k − ( v σ (1) , · · · , v σ ( k − )) v σ ( k ) , v σ ( k + · · · , v σ ( p ) (cid:1) , (cid:17) . By (13), we obtain that the derived bracket ~ · , · (cid:127) is closed on sC ∗ ( h , g ), and given by (12). There-fore, ( sC ∗ ( h , g ) , ~ · , · (cid:127) ) is a gLa.Moreover, by Im ρ ⊂ Der ( h ), we have [ µ g + ρ, λµ h ] NR = . We define a linear map d = : [ λµ h , · ] NR on the graded space C ∗ ( g ⊕ h , g ⊕ h ). For all g ∈ Hom n ( S ( h ) , g ), we have(d g ) p (cid:16) ( x , v ) , · · · , ( x p , v p ) (cid:17) = [ λµ h , g ] pNR (cid:16) ( x , v ) , · · · , ( x p , v p ) (cid:17) = (cid:16) λµ h ◦ g p − − ( − n g p − ◦ λµ h (cid:17)(cid:16) ( x , v ) , · · · , ( x p , v p ) (cid:17) = X ≤ i < j ≤ p ( − n + + v i ( v + ··· + v i − ) + v j ( v + ··· + v j − ) + v i v j (cid:16) g p − ( λ [ v i , v j ] h , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v p ) , (cid:17) . Thus d is closed on the subspace sC ∗ ( h , g ), and is given by (11). By [ λµ h , λµ h ] NR =
0, we ob-tain that d = . Moreover, by [ µ g + ρ, λµ h ] NR =
0, we deduce that d is a derivation of thegLa ( sC ∗ ( h , g ) , ~ · , · (cid:127) ). Therefore, ( sC ∗ ( h , g ) , ~ · , · (cid:127) , d) is a dgLa. (cid:3) Homotopy O -operators of weight λ can be characterized as Maurer-Cartan elements of theabove dgLa. Note that an element T = P + ∞ i = T i ∈ Hom( S ( h ) , g ) is of degree 0 if and only if thecorresponding element T ∈ s Hom( S ( h ) , g ) is of degree 1. Theorem 2.16.
Let ρ be an action of a sgLa ( g , [ · , · ] g ) on a sgLa ( h , [ · , · ] h ) . A degree elementT = P + ∞ i = T i ∈ Hom( S ( h ) , g ) is a homotopy O -operator of weight λ on g with respect to the action ρ if and only if T = P + ∞ i = T i is a Maurer-Cartan element of the dgLa( sC ∗ ( h , g ) , ~ · , · (cid:127) , d) , i.e. d T + ~ T , T (cid:127) = . Proof.
For a degree 0 element T = P + ∞ i = T i of the graded vector space C ∗ ( h , g ), we writed T + ~ T , T (cid:127) = X i (d T + ~ T , T (cid:127) ) i with (d T + ~ T , T (cid:127) ) i ∈ Hom( S i ( h ) , g ) . By Theorem 2.15, we have(d T + ~ T , T (cid:127) ) p ( v , · · · , v p ) = X ≤ i < j ≤ p ( − + v i ( v + ··· + v i − ) + v j ( v + ··· + v j − ) + v i v j T p − ( λ [ v i , v j ] h , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v p ) − X k + l = p + X σ ∈ S ( l , , p − l − ε ( σ ) T k − (cid:16) ρ (cid:0) T l ( v σ (1) , · · · , v σ ( l ) ) (cid:1) v σ ( l + , v σ ( l + , · · · , v σ ( p ) (cid:17) − X k + l = p + X σ ∈ S ( k − , , p − k ) ε ( σ ) T l (cid:16) ρ (cid:0) T k − ( v σ (1) , · · · , v σ ( k − ) (cid:1) v σ ( k ) , v σ ( k + , · · · , v σ ( p ) (cid:17) + X k + l = p + X σ ∈ S ( k − , l ) ε ( σ )[ T k − ( v σ (1) , · · · , v σ ( k − ) , T l ( v σ ( k ) , · · · , v σ ( p ) )] g = − X ≤ i < j ≤ p ( − v i ( v + ··· + v i − ) + v j ( v + ··· + v j − ) + v i v j T p − ( λ [ v i , v j ] h , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v p ) − X k + l = p + X σ ∈ S ( l , , p − l − ε ( σ ) T k − (cid:16) ρ (cid:0) T l ( v σ (1) , · · · , v σ ( l ) ) (cid:1) v σ ( l + , v σ ( l + , · · · , v σ ( p ) (cid:17) + X k + l = p + X σ ∈ S ( k − , l ) ε ( σ )[ T k − ( v σ (1) , · · · , v σ ( k − ) , T l ( v σ ( k ) , · · · , v σ ( p ) )] g . Thus, T = P + ∞ i = T i ∈ Hom( S ( h ) , g ) is a homotopy O -operator of weight λ on g with respect to theaction ρ if and only if T = P + ∞ i = T i is a Maurer-Cartan element of the dgLa ( sC ∗ ( h , g ) , ~ · , · (cid:127) , d). (cid:3) We note that a Lie algebra is a sgLa concentrated at degree −
1. Moreover, a Lie algebraaction is the same as an action of the sgLa on a graded vector space concentrated at degree − Corollary 2.17.
Let ρ : g → Der ( h ) be an action of a Lie algebra g on a Lie algebra h . Then alinear map T : h → g is an O -operator of weight λ on g with respect to the action ρ if and onlyif T is a Maurer-Cartan element of the dgLa ( ⊕ dim( h ) n = Hom( ∧ n h , g ) , ~ · , · (cid:127) , d) , where the di ff erential d : Hom( ∧ n h , g ) → Hom( ∧ n + h , g ) is given by (d g )( v , · · · , v n + ) = X ≤ i < j ≤ n + ( − n + i + j − g ( λ [ v i , v j ] h , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v n + ) , for all g ∈ C n ( h , g ) and v , · · · , v n + ∈ h , and the graded Lie bracket ~ · , · (cid:127) : Hom( ∧ n h , g ) × Hom( ∧ m h , g ) −→ Hom( ∧ m + n h , g ) OMOTOPY O -OPERATORS AND HOMOTOPY POST-LIE ALGEBRAS 13 is given by (cid:18) g , g (cid:19) ( v , · · · , v m + n ): = − X σ ∈ S ( m , , n − ( − σ g (cid:16) ρ (cid:0) g ( v σ (1) , · · · , v σ ( m ) ) (cid:1) v σ ( m + , v σ ( m + , · · · , v σ ( m + n ) (cid:17) + ( − mn X σ ∈ S ( n , , m − ( − σ g (cid:16) ρ (cid:0) g ( v σ (1) , · · · , v σ ( n ) ) (cid:1) v σ ( n + , v σ ( n + , · · · , v σ ( m + n ) (cid:17) − ( − mn X σ ∈ S ( n , m ) ( − σ [ g ( v σ (1) , · · · , v σ ( n ) ) , g ( v σ ( n + , · · · , v σ ( m + n ) )] g for all g ∈ Hom( ∧ n h , g ) , g ∈ Hom( ∧ m h , g ) and v , · · · , v m + n ∈ h . If the Lie algebra h is abelian in the above corollary, we recover the gLa that controls thedeformations of O -operators of weight 0 given in [45, Proposition 2.3].3. O perator homotopy post -L ie algebras In this section, we first recall the notion of a post-Lie algebra, and then give the definition of anoperator homotopy post-Lie algebra as a variation of a homotopy post-Lie algebra. We constructa 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 algebrasnaturally arise from homotopy O -operators of weight 1. Definition 3.1. ([48]) A post-Lie algebra ( g , [ · , · ] g , ⊲ ) consists of a Lie algebra ( g , [ · , · ] g ) and abinary product ⊲ : g ⊗ g → g such that x ⊲ [ y , z ] g = [ x ⊲ y , z ] g + [ y , x ⊲ z ] g , (14) [ x , y ] g ⊲ z = a ⊲ ( x , y , z ) − a ⊲ ( y , x , z ) , (15)here a ⊲ ( x , y , z ) : = x ⊲ ( y ⊲ z ) − ( x ⊲ y ) ⊲ z and x , y , z ∈ g . 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 apre-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]. Proposition 3.3.
Let ( g , [ · , · ] g , ⊲ ) be a post-Lie algebra. Then the bracket [ · , · ] C defined by [ x , y ] C : = x ⊲ y − y ⊲ x + [ x , y ] g , ∀ x , y ∈ g , (16) is a Lie bracket. We denote this Lie algebra by g C and call it the sub-adjacent Lie algebra of ( g , [ · , · ] g , ⊲ ). Definition 3.4. An operator homotopy post-Lie algebra is a sgLa ( g , [ · , · ] g ) equipped with acollection ( k ≥
1) of linear maps θ k : ⊗ k g → g of degree 1 satisfying, for every collection ofhomogeneous elements x , · · · , x n , x n + ∈ g ,(i) ( graded symmetry ) for every σ ∈ S n − , n ≥ θ n ( x σ (1) , · · · , x σ ( n − , x n ) = ε ( σ ) θ n ( x , · · · , x n − , x n ) , (17) (ii) ( graded derivation ) for all n ≥ θ n ( x , · · · , x n − , [ x n , x n + ] g ) = ( − x + ··· + x n − + [ θ n ( x , · · · , x n − , x n ) , x n + ] g + ( − ( x + ··· + x n − + x n + [ x n , θ n ( x , · · · , x n − , x n + )] g , (18)(iii) for all n ≥ X i + j = n + i ≥ , j ≥ X σ ∈ S ( i − , , j − ε ( σ ) θ j ( θ i ( v σ (1) , · · · , v σ ( i − , v σ ( i ) ) , v σ ( i + , · · · , v σ ( n − , v n ) + X i + j = n + i ≥ , j ≥ X σ ∈ S ( j − , i − ( − α ε ( σ ) θ j ( v σ (1) , · · · , v σ ( j − , θ i ( v σ ( j ) , · · · , v σ ( n − , v n ))(19) = X ≤ i < j ≤ n − ( − β θ n − ([ x i , x j ] g , x , · · · , ˆ x i , · · · , ˆ x j , · · · , x n ) , where β = x i ( x + · · · + x i − ) + x j ( x + · · · + x j − ) + x i x j + α = x σ (1) + x σ (2) + · · · + x σ ( j − .The notion of a pre-Lie ∞ algebra was introduced in [10]. See [32] for more applications of pre-Lie ∞ algebras in geometry. Recall that a pre-Lie ∞ algebra is a graded vector space V equippedwith a collection of linear maps θ k : ⊗ k V → V , k ≥ , of degree 1 with the property that, for anyhomogeneous elements v , · · · , v n ∈ V , we have(i) (graded symmetry) for every σ ∈ S n − , θ n ( v σ (1) , · · · , v σ ( n − , v n ) = ε ( σ ) θ n ( v , · · · , v n − , v n ) , (ii) for all n ≥ X i + j = n + i ≥ , j ≥ X σ ∈ S ( i − , , j − ε ( σ ) θ j ( θ i ( v σ (1) , · · · , v σ ( i − , v σ ( i ) ) , v σ ( i + , · · · , v σ ( n − , v n ) + X i + j = n + i ≥ , j ≥ X σ ∈ S ( j − , i − ( − α ε ( σ ) θ j ( v σ (1) , · · · , v σ ( j − , θ i ( v σ ( j ) , · · · , v σ ( n − , v n )) = , where α = v σ (1) + v σ (2) + · · · + v σ ( j − .It is straightforward to see that an operator homotopy post-Lie algebra ( g , [ · , · ] g , { θ k } ∞ k = ) reducesto a pre-Lie ∞ algebra when ( g , [ · , · ] g ) is an abelian sgLa. Remark 3.5.
The explicit definition of a homotopy post-Lie algebra is still not known though itsoperad should be the minimal model of the post-Lie operad by general construction as noted in theintroduction [27]. Since a Rota-Baxter Lie algebra of weight 1 gives a post-Lie algebra, we expectthat a Rota-Baxter homotopy Lie algebra of weight 1 (a homotopy Lie algebra with a Rota-Baxteroperator of weight 1) induces a homotopy post-Lie algebra. Based on preliminary computationsin this regards, there should be more terms in the right hand side of (19) in the definition ofa homotopy post-Lie algebra, suggesting that the homotopy post-Lie algebra is di ff erent fromthe operator homotopy post-Lie algebra just defined. So we have chosen a di ff erent name fordistinction.Now we construct the dgLa that characterizes operator homotopy post-Lie algebras as Maurer-Cartan elements. Let ( h , [ · , · ] h ) be a sgLa. Denote by¯ C n ( h , h ) = Hom n ( S ( h ) , s − Der ( h )) , ¯ C ∗ ( h , h ) : = ⊕ n ∈ Z ¯ C n ( h , h ) . OMOTOPY O -OPERATORS AND HOMOTOPY POST-LIE ALGEBRAS 15 • Define a graded linear map ∂ : ¯ C n ( h , h ) → ¯ C n + ( h , h ) by( ∂β ) p ( v , · · · , v p ) = X ≤ i < j ≤ s ( − n + + v i ( v + ··· + v i − ) + v j ( v + ··· + v j − ) + v i v j β p − ([ v i , v j ] h , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v p ) , • Define a graded bracket operation[ · , · ] c : ¯ C m ( h , h ) × ¯ C n ( h , h ) −→ ¯ C m + n + ( h , h )by[ α, β ] cp ( v , · · · , v p ) = − X k + l = p + X σ ∈ S ( l , , p − l − ε ( σ ) α k − (cid:16) s (cid:0) β l ( v σ (1) , · · · , v σ ( l ) ) (cid:1) v σ ( l + , v σ ( l + , · · · , v σ ( p ) (cid:17) + ( − ( m + n + X k + l = p + X σ ∈ S ( k − , , p − k ) ε ( σ ) β l (cid:16) s (cid:0) α k − ( v σ (1) , · · · , v σ ( k − ) (cid:1) v σ ( k ) , v σ ( k + , · · · , v σ ( p ) (cid:17) − X k + l = p + X σ ∈ S ( k − , l ) ( − ( n + v σ (1) + ··· + v σ ( k − ) ε ( σ ) s − [ s α k − ( v σ (1) , · · · , v σ ( k − ) , s β l ( v σ ( k ) , · · · , v σ ( p ) )]for all α = P i α i ∈ ¯ C m ( h , h ) , β = P i β i ∈ ¯ C n ( h , h ) with α i , β i ∈ Hom( S i ( h ) , s − Der ( h ))and v , · · · , v p ∈ h . Here we write ∂β = P i ( ∂β ) i with ( ∂β ) i ∈ Hom( S i ( h ) , s − Der ( h )) and[ α, β ] c = P i [ α, β ] ci with [ α, β ] ci ∈ Hom( S i ( h ) , s − Der ( h )). Theorem 3.6.
With above notations, ( s ¯ C ∗ ( h , h ) , [ · , · ] c , ∂ ) is a dgLa . Moreover, its Maurer-Cartanelements are precisely the operator homotopy post-Lie algebra structures on the sgLa ( h , [ · , · ] h ) .Proof. Setting λ = , g = s − Der ( h ) and ρ = s in Theorem 2.15, we find that ( s ¯ C ∗ ( h , h ) , [ · , · ] c , ∂ )is a dgLa.Let L = P + ∞ i = L i ∈ Hom ( S ( h ) , s − Der ( h )) with L i ∈ Hom( S i ( h ) , s − Der ( h )) be a Maurer-Cartanelement of the dgLa ( s ¯ C ∗ ( h , h ) , [ · , · ] c , ∂ ). We define a collection of linear maps θ k : ⊗ k h → h ( k ≥
1) of degree 1 by θ k ( v , · · · , v k ) : = (cid:0) sL k − ( v , · · · , v k − ) (cid:1) v k , ∀ v · · · , v k ∈ h . (20)By L n − ∈ Hom ( S n − ( h ) , s − Der ( h )), we obtain (17) and (18). Moreover, write ∂ L + [ L , L ] c = P i ( ∂ L + [ L , L ] c ) i with ( ∂ L + [ L , L ] c ) i ∈ Hom( S i ( h ) , s − Der ( h )). Then for all v , · · · , v n ∈ h , bya similar computation as in the proof of Theorem 2.16, we have (cid:16) s (cid:0) ( ∂ L +
12 [ L , L ] c ) n − (cid:1) ( v , · · · , v n − ) (cid:17) v n = − X ≤ i < j ≤ n − ( − v i ( v + ··· + v i − ) + v j ( v + ··· + v j − ) + v i v j sL n − ([ v i , v j ] h , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v n − ) v n − X k + l = n X σ ∈ S ( l , , n − l − ε ( σ ) sL k − (cid:16) s (cid:0) L l ( v σ (1) , · · · , v σ ( l ) ) (cid:1) v σ ( l + , v σ ( l + , · · · , v σ ( n − (cid:17) v n − X k + l = n X σ ∈ S ( k − , l ) ( − v σ (1) + ··· + v σ ( k − ε ( σ )[ sL k − ( v σ (1) , · · · , v σ ( k − ) , sL l ( v σ ( k ) , · · · , v σ ( n − )] v n = − X ≤ i < j ≤ n − ( − v i ( v + ··· + v i − ) + v j ( v + ··· + v j − ) + v i v j θ n − ([ v i , v j ] h , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v n − , v n ) − X k + l = n X σ ∈ S ( l , , n − l − ε ( σ ) θ k (cid:0) θ l + ( v σ (1) , · · · , v σ ( l ) , v σ ( l + ) , v σ ( l + , · · · , v σ ( n − , v n (cid:1) − X k + l = n X σ ∈ S ( k − , l ) ( − v σ (1) + ··· + v σ ( k − ε ( σ ) θ k (cid:0) v σ (1) , · · · , v σ ( k − , θ l + ( v σ ( k ) , · · · , v σ ( n − , v n ) (cid:1) − X k + l = n X σ ∈ S ( k − , l ) ( − ( v σ (1) + ··· + v σ ( k − + v σ ( k ) + ··· + v σ ( n − ) ε ( σ ) θ l + (cid:0) v σ ( k ) , · · · , v σ ( n − , θ k ( v σ (1) , · · · , v σ ( k − , v n ) (cid:1) . For any σ ∈ S ( k − , l ) , we define τ ∈ S ( l , k − by τ ( i ) : = ( σ ( i + k − , ≤ i ≤ l ; σ ( i − l ) , l + ≤ i ≤ n − . Thus we have ε ( τ ; v , · · · , v n − ) = ε ( σ ; v , · · · , v n − )( − ( v σ (1) + ··· + v σ ( k − )( v σ ( k ) + ··· + v σ ( n − ) . Applying thebijection between S ( k − , l ) and S ( l , k − , we obtain − X k + l = n X σ ∈ S ( k − , l ) ( − ( v σ (1) + ··· + v σ ( k − + v σ ( k ) + ··· + v σ ( n − ) ε ( σ ) θ l + (cid:0) v σ ( k ) , · · · , v σ ( n − , θ k ( v σ (1) , · · · , v σ ( k − , v n ) (cid:1) = − X k + l = n X τ ∈ S ( l , k − ( − v τ (1) + ··· + v τ ( l ) ε ( τ ) θ l + (cid:0) v τ (1) , · · · , v τ ( l ) , θ k ( v τ ( l + , · · · , v τ ( n − , v n ) (cid:1) . Therefore, we have (cid:16) s (cid:0) ( ∂ L +
12 [ L , L ] c ) n − (cid:1) ( v , · · · , v n − ) (cid:17) v n = − X ≤ i < j ≤ n − ( − v i ( v + ··· + v i − ) + v j ( v + ··· + v j − ) + v i v j θ n − ([ v i , v j ] h , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v n − , v n ) − X k + l = n X σ ∈ S ( l , , n − l − ε ( σ ) θ k (cid:0) θ l + ( v σ (1) , · · · , v σ ( l ) , v σ ( l + ) , v σ ( l + , · · · , v σ ( n − , v n (cid:1) − X k + l = n X σ ∈ S ( k − , l ) ( − v σ (1) + ··· + v σ ( k − ε ( σ ) θ k (cid:0) v σ (1) , · · · , v σ ( k − , θ l + ( v σ ( k ) , · · · , v σ ( n − , v n ) (cid:1) , which implies (19). Thus, { θ k } ∞ k = is an operator homotopy post-Lie algebra structure on thesgLa ( h , [ · , · ] h ). (cid:3) When the sgLa h reduces to a usual Lie algebra, we characterize post-Lie algebra structures on h as Maurer-Cartan elements. See [8, 9, 15, 38] for classifications of post-Lie algebras on somespecific Lie algebras. Corollary 3.7.
Let ( h , [ · , · ] h ) be a Lie algebra. Denote by ¯ C n ( h , h ) : = Hom( ∧ n h , Der ( h )) and ¯ C ∗ ( h , h ) : = ⊕ n ≥ ¯ C n ( h , h ) . Then ( ¯ C ∗ ( h , h ) , [ · , · ] c , ∂ ) is a dgLa , where the di ff erential ∂ : ¯ C n ( h , h ) → ¯ C n + ( h , h ) is given by ( ∂α )( u , · · · , u n + ) = X ≤ i < j ≤ n + ( − n + i + j − α ([ u i , u j ] h , u , · · · , ˆ u i , · · · , ˆ u j , · · · , u n + ) , and the graded Lie bracket [ · , · ] c : ¯ C n ( h , h ) × ¯ C m ( h , h ) → ¯ C m + n ( h , h ) is given by [ α, β ] c ( u , · · · , u m + n ) = − X σ ∈ S ( m , , n − ( − σ α ( β ( u σ (1) , · · · , u σ ( m ) ) u σ ( m + , u σ ( m + , · · · , u σ ( m + n ) ) + ( − mn X σ ∈ S ( n , , m − ( − σ β ( α ( u σ (1) , · · · , u σ ( n ) ) u σ ( n + , u σ ( n + , · · · , u σ ( m + n ) ) OMOTOPY O -OPERATORS AND HOMOTOPY POST-LIE ALGEBRAS 17 − ( − mn X σ ∈ S ( n , m ) ( − σ [ α ( u σ (1) , · · · , u σ ( n ) ) , β ( u σ ( n + , · · · , u σ ( m + n ) )] , for all α ∈ Hom( ∧ n h , Der ( h )) , β ∈ Hom( ∧ m h , Der ( h )) and u , · · · , u m + n ∈ h .Moreover, L ⊲ : h → Der ( h ) defines a post-Lie algebra structure on the Lie algebra ( h , [ · , · ] h ) ifand only if L ⊲ is a Maurer-Cartan element of the dgLa ( ¯ C ∗ ( h , h ) , [ · , · ] c , ∂ ) . When the sgLa h is abelian, we characterize pre-Lie ∞ -algebras as Maurer-Cartan elements,which was originally given in [10]. Corollary 3.8.
Let V be a Z -graded vector space. Denote ¯ C n ( V , V ) : = Hom n ( S ( V ) , gl ( V )) , ¯ C ∗ ( V , V ) : = ⊕ n ∈ Z ¯ C n ( V , V ) . Then ( ¯ C ∗ ( V , V ) , [ · , · ] c ) is a gLa , where the graded Lie bracket [ · , · ] c : ¯ C m ( V , V ) × ¯ C n ( V , V ) −→ ¯ C m + n ( V , V ) is given by [ α, β ] cp ( v , · · · , v p ) = − X k + l = p + X σ ∈ S ( l , , p − l − ε ( σ ) α k − (cid:16) β l ( v σ (1) , · · · , v σ ( l ) ) v σ ( l + , v σ ( l + , · · · , v σ ( p ) (cid:17) + ( − mn X k + l = p + X σ ∈ S ( k − , , p − k ) ε ( σ ) β l (cid:16) α k − ( v σ (1) , · · · , v σ ( k − ) v σ ( k ) , v σ ( k + , · · · , v σ ( p ) (cid:17) − X k + l = p + X σ ∈ S ( k − , l ) ( − n ( v σ (1) + ··· + v σ ( k − ) ε ( σ )[ α k − ( v σ (1) , · · · , v σ ( k − ) , β l ( v σ ( k ) , · · · , v σ ( p ) )] , for all α = P i α i ∈ ¯ C m ( V , V ) , β = P β i ∈ ¯ C n ( V , V ) with α i , β i ∈ Hom( S i ( V ) , V ) .Moreover, L = P ∞ i = L i ∈ Hom ( S ( h ) , gl ( h )) defines a pre-Lie ∞ algebra structure by θ k ( v , · · · , v k ) : = L k − ( v , · · · , v k − ) v k , ∀ v · · · , v k ∈ h , (21) on the graded vector space V if and only if L = P ∞ i = 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 charac-terize 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. Proposition 3.9. ([5])
Let T : h → g be an O -operator of weight . Then ( h , [ · , · ] h , ⊲ ) is a post-Liealgebra, where ⊲ is given by u ⊲ v = ρ ( T u ) v , ∀ u , v ∈ h . In the sequel, we generalize the above relation to homotopy O -operators of weight 1 and oper-ator homotopy post-Lie algebras.Define a graded linear map Ψ : C ∗ ( h , g ) → ¯ C ∗ ( h , h ) of degree 0 by Ψ ( f ) = s − ◦ ρ ◦ f , ∀ f ∈ Hom m ( S ( h ) , g ) . Therefore, we have Ψ ( f ) k = s − ◦ ρ ◦ f k . In the following, we set λ = Theorem 3.10.
Let ( g , [ · , · ] g ) and ( h , [ · , · ] h ) be sgLa ’s and ρ : g → Der ( h ) a sgLa action. Then Ψ is a homomorphism of dgLa ’s from ( sC ∗ ( h , g ) , ~ · , · (cid:127) , d) to ( s ¯ C ∗ ( h , h ) , [ · , · ] c , ∂ ) . Proof.
For all f = P i f i ∈ Hom m ( S ( h ) , g ) , g = P i g i ∈ Hom n ( S ( h ) , g ) with f i , g i ∈ Hom( S i ( h ) , g ),we write [ Ψ ( f ) , Ψ ( g )] c − Ψ ( (cid:18) f , g (cid:19) ) = P i (cid:16) [ Ψ ( f ) , Ψ ( g )] c − Ψ ( (cid:18) f , g (cid:19) ) (cid:17) i . Since s − ◦ ρ : g → s − Der ( h )is a sgLa homomorphism, we have (cid:16) [ Ψ ( f ) , Ψ ( g )] c − Ψ ( (cid:18) f , g (cid:19) ) (cid:17) p ( v , · · · , v p ) = − X k + l = p + X σ ∈ S ( l , , p − l − ε ( σ ) Ψ ( f ) k − (cid:16) s (cid:0) Ψ ( g ) l ( v σ (1) , · · · , v σ ( l ) ) (cid:1) v σ ( l + , v σ ( l + , · · · , v σ ( p ) (cid:17) + ( − ( m + n + X k + l = p + X σ ∈ S ( k − , , p − k ) ε ( σ ) Ψ ( g ) l (cid:16) s (cid:0) Ψ ( f ) k − ( v σ (1) , · · · , v σ ( k − ) (cid:1) v σ ( k ) , v σ ( k + , · · · , v σ ( p ) (cid:17) − X k + l = p + X σ ∈ S ( p − l , l ) ( − ( n + v σ (1) + ··· + v σ ( p − l ) ) ε ( σ ) s − [ s Ψ ( f ) k − ( v σ (1) , · · · , v σ ( p − l ) ) , s Ψ ( g ) l ( v σ ( p − l + , · · · , v σ ( p ) )] + X k + l = p + X σ ∈ S ( l , , p − l − ε ( σ ) s − ρ (cid:16) f k − (cid:16) ρ (cid:0) g l ( v σ (1) , · · · , v σ ( l ) ) (cid:1) v σ ( l + , v σ ( l + , · · · , v σ ( p ) (cid:17)(cid:17) − ( − ( m + n + X k + l = p + X σ ∈ S ( k − , , p − k ) ε ( σ ) s − ρ (cid:16) g l (cid:16) ρ (cid:0) f k − ( v σ (1) , · · · , v σ ( k − ) (cid:1) v σ ( k ) , v σ ( k + , · · · , v σ ( p ) (cid:17)(cid:17) + X k + l = p + X σ ∈ S ( p − l , l ) ( − n ( v σ (1) + ··· + v σ ( p − l ) ) + m + ε ( σ ) s − ρ [ f k − ( v σ (1) , · · · , v σ ( p − l ) ) , g l ( v σ ( p − l + , · · · , v σ ( p ) )] g = − X k + l = p + X σ ∈ S ( l , , p − l − ε ( σ ) s − ρ (cid:16) f k − (cid:16) ρ (cid:0) g l ( v σ (1) , · · · , v σ ( l ) ) (cid:1) v σ ( l + , v σ ( l + , · · · , v σ ( p ) (cid:17)(cid:17) + ( − ( m + n + X k + l = p + X σ ∈ S ( k − , , p − k ) ε ( σ ) s − ρ (cid:16) g l (cid:16) ρ ( f k − ( v σ (1) , · · · , v σ ( k − ) (cid:1) v σ ( k ) , v σ ( k + , · · · , v σ ( p ) (cid:17)(cid:17) − X k + l = p + X σ ∈ S ( p − l , l ) ( − ( n + v σ (1) + ··· + v σ ( p − l ) ) ε ( σ ) s − [ ρ (cid:0) f k − ( v σ (1) , · · · , v σ ( p − l ) ) (cid:1) , ρ (cid:0) g l ( v σ ( p − l + , · · · , v σ ( p ) ) (cid:1) ] + X k + l = p + X σ ∈ S ( l , , p − l − ε ( σ ) s − ρ (cid:16) f k − (cid:16) ρ (cid:0) g l ( v σ (1) , · · · , v σ ( l ) ) (cid:1) v σ ( l + , v σ ( l + , · · · , v σ ( p ) (cid:17)(cid:17) − ( − ( m + n + X k + l = p + X σ ∈ S ( k − , , p − k ) ε ( σ ) s − ρ (cid:16) g l (cid:16) ρ (cid:0) f k − ( v σ (1) , · · · , v σ ( k − ) (cid:1) v σ ( k ) , v σ ( k + , · · · , v σ ( p ) (cid:17)(cid:17) + X k + l = p + X σ ∈ S ( p − l , l ) ( − n ( v σ (1) + ··· + v σ ( p − l ) ) + m + ε ( σ ) s − ρ ([ f k − ( v σ (1) , · · · , v σ ( p − l ) ) , g l ( v σ ( p − l + , · · · , v σ ( p ) )] g ) = . Moreover, for all g ∈ Hom n ( S ( h ) , g ), we have (cid:16) ( ∂ ◦ Ψ − Ψ ◦ d)( g ) (cid:17) p ( v , · · · , v p ) = (cid:0) ∂ ( Ψ ( g )) (cid:1) p ( v , · · · , v p ) − (cid:0) Ψ (d g ) (cid:1) p ( v , · · · , v p ) = X ≤ i < j ≤ p ( − n + + v i ( v + ··· + v i − ) + v j ( v + ··· + v j − ) + v i v j ( Ψ ( g )) p − ([ v i , v j ] h , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v p ) − X ≤ i < j ≤ p ( − n + + v i ( v + ··· + v i − ) + v j ( v + ··· + v j − ) + v i v j s − ρ (cid:0) g p − ([ v i , v j ] h , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v p ) (cid:1) = . OMOTOPY O -OPERATORS AND HOMOTOPY POST-LIE ALGEBRAS 19 Thus, we deduce that Ψ is a homomorphism. (cid:3) Now we are ready to show that the homotopy O -operators of weight 1 induce operator homo-topy post-Lie algebras. Theorem 3.11.
Let T = P + ∞ i = T i ∈ Hom ( S ( h ) , g ) be a homotopy O -operator of weight on a sgLa ( g , [ · , · ] g ) with respect to an action ρ : g → Der ( h ) . Then ( h , [ · , · ] h , { θ k } ∞ k = ) is an operatorhomotopy post-Lie algebra, where θ k : ⊗ k h → h ( k ≥ are linear maps of degree defined by θ k ( v , · · · , v k ) : = ρ ( T k − ( v , · · · , v k − ) (cid:1) v k , ∀ v · · · , v k ∈ h . (22) Proof.
By Theorem 2.16 and Theorem 3.10, we deduce that Ψ ( T ) ia a Maurer-Cartan elementof the dgLa ( s ¯ C ∗ ( h , h ) , [ · , · ] c , ∂ ). Moreover, by Theorem 3.6, we obtain that ( h , [ · , · ] h , { θ k } ∞ k = ) is anoperator homotopy post-Lie algebra. (cid:3) Corollary 3.12.
Let T = P + ∞ i = T i ∈ Hom ( S ( V ) , g ) be a homotopy O -operator of weight on a sgLa ( g , [ · , · ] g ) with respect to an action ρ : g → gl ( V ) . Then ( V , { θ k } ∞ k = ) is a pre-Lie ∞ algebra,where θ k : ⊗ k V → V ( k ≥ are linear maps of degree defined by θ k ( v , · · · , v k ) : = ρ ( T k − ( v , · · · , v k − ) (cid:1) v k , ∀ v · · · , v k ∈ V . (23)It is straightforward to obtain the following result. Proposition 3.13.
Let ( g , [ · , · ] g , { θ k } ∞ k = ) be an operator homotopy post-Lie algebra. Then ( g , { l k } ∞ k = ) is an L ∞ -algebra, where l = θ , l is defined by (24) l ( x , y ) = θ ( x , y ) + ( − xy θ ( y , x ) + [ x , y ] g , and for k ≥ , l k is defined by (25) l k ( x , · · · , x k ) = k X i = ( − x i ( x i + + ··· + x k ) θ k ( x , · · · , ˆ x i , · · · , x k , x i ) . Definition 3.14. ([21, 31]) Let ( V , { l k } ∞ k = ) be an L ∞ -algebra and ( V ′ , { l ′ m } ∞ m = ) an L ∞ -algebra inwhich l ′ m = m ≥ m =
2. A curved L ∞ -algebra homomorphism from ( V , { l k } ∞ k = )to ( V ′ , l ′ ) consists of a collection of degree 0 graded multilinear maps f k : V ⊗ k → V ′ , k ≥ f n ( v σ (1) , · · · , v σ ( n ) ) = ε ( σ ) f n ( v , · · · , v n ) , for any n ≥ v , · · · , v n ∈ V , and n X i = X σ ∈ S ( i , n − i ) ε ( σ ) f n − i + ( l i ( v σ (1) , · · · , v σ ( i ) ) , v σ ( i + , · · · , v σ ( n ) ) = n X i = X σ ∈ S ( i , n − i ) ε ( σ ) l ′ ( f i ( v σ (1) , · · · , v σ ( i ) ) , f n − i ( v σ ( i + , · · · , v σ ( n ) )) . Then combining Theorem 3.11 and Theorem 3.13, we obtain
Theorem 3.15.
Let T = P + ∞ i = T i ∈ Hom( S ( h ) , g ) be a homotopy O -operator of weight on a sgLa ( g , [ · , · ] g ) with respect to the action ρ : g → Der ( h ) . Then T is a curved L ∞ -algebrahomomorphism from the L ∞ -algebra ( h , { l k } ∞ k = ) to ( g , [ · , · ] g ) .Proof. For all v , · · · , v n ∈ h , we have n X i = X σ ∈ S ( i , n − i ) ε ( σ ) T n − i + ( l i ( v σ (1) , · · · , v σ ( i ) ) , v σ ( i + , · · · , v σ ( n ) ) = X σ ∈ S (1 , n − ε ( σ ) T n (cid:16) ρ ( T ) v σ (1) , v σ (2) , · · · , v σ ( n ) (cid:17) + X σ ∈ S (2 , n − ε ( σ ) T n − ( ρ ( T ( v σ (1) )) v σ (2) + ( − v σ (1) v σ (2) ρ ( T ( v σ (2) )) v σ (1) + [ v σ (1) , v σ (2) ] g , · · · , v σ ( n ) ) + n X i = X σ ∈ S ( i , n − i ) i X l = ε ( σ )( − v σ ( l ) ( v σ (1 + + ··· + v σ ( i ) ) T n − i + (cid:16) ρ ( T i − ( v σ (1) , · · · , ˆ v σ ( l ) , · · · , v σ ( i ) ) (cid:1) ˆ v σ ( l ) , v σ ( i + , · · · , v σ ( n ) (cid:17) = X ≤ i < j ≤ n ( − v i ( v + ··· + v i − ) + v j ( v + ··· + v j − ) + v i v j T n − ([ v i , v j ] h , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v n ) n X i = X σ ∈ S ( i − , , n − i ) ε ( σ ) T n − i + (cid:16) ρ ( T i − ( v σ (1) , · · · , v σ ( i − ) (cid:1) v σ ( i ) , v σ ( i + , · · · , v σ ( n ) (cid:17) (9) = n X i = X σ ∈ S ( i , n − i ) ε ( σ )[ T i ( v σ (1) , · · · , v σ ( i ) ) , T n − i ( v σ ( i + , · · · , v σ ( n ) )] g , which implies that T is a curved L ∞ -algebra homomorphism. (cid:3) Similarly, the above result also holds for homotopy O -operators of weight 0.4. C lassification of term skeletal operator homotopy post -L ie algebras In general, it is expected that the 2-term homotopy of an algebra structure is equivalent tothe categorification of this algebraic structure, and the 2-term homotopy algebras are quasi-isomorphic to the 2-term skeletal homotopy algebras, which are classified by the third coho-mological group. Baez and Crans [3] accomplished these for Lie algebras. In this spirit, weclassify 2-term skeletal operator homotopy post-Lie algebras by the third cohomology group of apost-Lie algebra. For this purpose, we first define representations of post-Lie algebras and thendevelop the corresponding cohomology theory.4.1.
Representations of post-Lie algebras.
Here we introduce the notion of a representation ofa post-Lie algebra ( g , [ · , · ] g , ⊲ ) on a vector space V . We show that there is naturally an inducedrepresentation of the subadjacent Lie algebra g C on Der ( g , V ). This fact plays a crucial role in ourstudy of cohomology groups of post-Lie algebras in the next subsection. Definition 4.1. A representation of a post-Lie algebra ( g , [ · , · ] g , ⊲ ) on a vector space V is atriple ( ρ, µ, ν ), where ρ : g −→ gl ( V ) is a representation of the Lie algebra ( g , [ · , · ] g ) on V , and µ, ν : g −→ gl ( V ) are linear maps satisfying ρ ( x ⊲ y ) = µ ( x ) ◦ ρ ( y ) − ρ ( y ) ◦ µ ( x ) , (26) ν ([ x , y ] g ) = ρ ( x ) ◦ ν ( y ) − ρ ( y ) ◦ ν ( x ) , (27) µ ([ x , y ] g ) = µ ( x ) ◦ µ ( y ) − µ ( x ⊲ y ) − µ ( y ) ◦ µ ( x ) + µ ( y ⊲ x ) , (28) ν ( y ) ◦ ρ ( x ) = µ ( x ) ◦ ν ( y ) − ν ( y ) ◦ µ ( x ) − ν ( x ⊲ y ) + ν ( y ) ◦ ν ( x ) , ∀ x , y ∈ g . (29)Let ( g , [ · , · ] g , ⊲ ) be a post-Lie algebra and ( V ; ρ, µ, ν ) a representation of ( g , [ · , · ] g , ⊲ ). By Propo-sition 3.3 and (28), we deduce that ( V ; µ ) is a representation of the sub-adjacent Lie algebra( g , [ · , · ] C ). It is obvious that ( g ; ad , L ⊲ , R ⊲ ) is a representation of a post-Lie algebra on itself,which is called the regular representation . OMOTOPY O -OPERATORS AND HOMOTOPY POST-LIE ALGEBRAS 21 Let ( V ; ρ, µ, ν ) be a representation of a post-Lie algebra ( g , [ · , · ] g , ⊲ ). Define a Lie bracket[ · , · ] ρ : ⊗ ( g ⊕ V ) → g ⊕ V by[ x + v , x + v ] ρ : = [ x , x ] g + ρ ( x ) v − ρ ( x ) v , (30)and a bilinear operation ⊲ µ,ν : ⊗ ( g ⊕ V ) → g ⊕ V by( x + v ) ⊲ µ,ν ( x + v ) : = x ⊲ x + µ ( x ) v + ν ( x ) v . (31)By straightforward computations, we have Theorem 4.2.
With the above notations, ( g ⊕ V , [ · , · ] ρ , ⊲ µ,ν ) is a post-Lie algebra. This post-Lie algebra is called the semidirect product of the post-Lie algebra ( g , [ · , · ] g , ⊲ ) andthe representation ( V ; ρ, µ, ν ), and denoted by g ⋉ ρ,µ,ν V . Proposition 4.3.
Let ( V ; ρ, µ, ν ) be a representation of a post-Lie algebra ( g , [ · , · ] g , ⊲ ) . Then ( V ; ρ + µ − ν ) is a representation of the sub-adjacent Lie algebra ( g , [ · , · ] C ) .Proof. By Theorem 4.2, we have the semidirect product post-Lie algebra g ⋉ ρ,µ,ν V . Consideringits sub-adjacent Lie algebra structure [ · , · ] C , we have[( x + v ) , ( x + v )] C = ( x + v ) ⊲ µ,ν ( x + v ) − ( x + v ) ⊲ µ,ν ( x + v ) + [ x + v , x + v ] ρ = x ⊲ x + µ ( x ) v + ν ( x ) v − x ⊲ x − µ ( x ) v − ν ( x ) v + [ x , x ] g + ρ ( x ) v − ρ ( x ) v = [ x , x ] C + ( ρ + µ − ν )( x ) v − ( ρ + µ − ν )( x ) v . (32)Thus, ( V ; ρ + µ − ν ) is a representation of the sub-adjacent Lie algebra ( g , [ · , · ] C ). (cid:3) If ( ρ, µ, ν ) = ( ad , L ⊲ , R ⊲ ) is the regular representation of a post-Lie algebra ( g , [ · , · ] g , ⊲ ), then ad + L ⊲ − R ⊲ is the adjoint representation of the sub-adjacent Lie algebra ( g , [ · , · ] C ). Corollary 4.4.
Let ( V ; ρ, µ, ν ) be a representation of a post-Lie algebra ( g , [ · , · ] g , ⊲ ) . Then thesemidirect product post-Lie algebras g ⋉ ρ,µ,ν V and g ⋉ ,ρ + µ − ν, V given by the representations ( V ; ρ, µ, ν ) and ( V ; 0 , ρ + µ − ν, respectively have the same sub-adjacent Lie algebra g C ⋉ ρ + µ − ν Vgiven by (32) , which is the semidirect product of the Lie algebra ( g , [ · , · ] C ) and its representation ( V ; ρ + µ − ν ) . Let ( V ; ρ, µ, ν ) be a representation of a post-Lie algebra ( g , [ · , · ] g , ⊲ ). We set Der ( g , V ) : = n f ∈ Hom( g , V ) (cid:12)(cid:12)(cid:12) f ([ x , y ] g ) = ρ ( x ) f ( y ) − ρ ( y ) f ( x ) , ∀ x , y ∈ g o (33)and define ˆ ρ : g → Hom (cid:16)
Der ( g , V ) , Hom( g , V ) (cid:17) by (cid:16) ˆ ρ ( x )( f ) (cid:17) y : = µ ( x ) f ( y ) + ν ( y ) f ( x ) − f ( x ⊲ y ) , ∀ x , y ∈ g , f ∈ Der ( g , V ) . (34)By a straightforward computation, we deduce Lemma 4.5.
For all x ∈ g , we have ˆ ρ ( x ) ∈ gl ( Der ( g , V )) . Theorem 4.6.
Let ( V ; ρ, µ, ν ) be a representation of a post-Lie algebra ( g , [ · , · ] g , ⊲ ) . Then ( Der ( g , V ); ˆ ρ ) is a representation of the sub-adjacent Lie algebra ( g , [ · , · ] C ) , where ˆ ρ is given by (34) . Proof.
By (34), for all x , y , z ∈ g and f ∈ Der ( g , V ), we have (cid:16)(cid:0) [ ˆ ρ ( x ) , ˆ ρ ( y )] − ˆ ρ ([ x , y ] C ) (cid:1) ( f ) (cid:17) z = ˆ ρ ( x )( ˆ ρ ( y )( f )) z − ˆ ρ ( y )( ˆ ρ ( x )( f )) z − ˆ ρ ([ x , y ] C )( f ) z = µ ( x )( ˆ ρ ( y )( f )) z + ν ( z )( ˆ ρ ( y )( f )) x − ( ˆ ρ ( y )( f ))( x ⊲ z ) − µ ( y )( ˆ ρ ( x )( f )) z − ν ( z )( ˆ ρ ( x )( f )) y + ( ˆ ρ ( x )( f ))( y ⊲ z ) − µ ([ x , y ] C ) f ( z ) − ν ( z ) f ([ x , y ] C ) + f ([ x , y ] C ⊲ z ) = µ ( x ) (cid:16) µ ( y ) f ( z ) + ν ( z ) f ( y ) − f ( y ⊲ z ) (cid:17) + ν ( z ) (cid:16) µ ( y ) f ( x ) + ν ( x ) f ( y ) − f ( y ⊲ x ) (cid:17) − (cid:16) µ ( y ) f ( x ⊲ z ) + ν ( x ⊲ z ) f ( y ) − f ( y ⊲ ( x ⊲ z )) (cid:17) − µ ( y ) (cid:16) µ ( x ) f ( z ) + ν ( z ) f ( x ) − f ( x ⊲ z ) (cid:17) − ν ( z ) (cid:16) µ ( x ) f ( y ) + ν ( y ) f ( x ) − f ( x ⊲ y ) (cid:17) + (cid:16) µ ( x ) f ( y ⊲ z ) + ν ( y ⊲ z ) f ( x ) − f ( x ⊲ ( y ⊲ z )) (cid:17) − µ (cid:16) x ⊲ y − y ⊲ x + [ x , y ] g (cid:17) f ( z ) − ν ( z ) f ( x ⊲ y − y ⊲ x + [ x , y ] g ) + f (cid:16) ( x ⊲ y − y ⊲ x + [ x , y ] g ) ⊲ z (cid:17) (15) , (28) = µ ( x ) ν ( z ) f ( y ) + ν ( z ) µ ( y ) f ( x ) + ν ( z ) ν ( x ) f ( y ) − ν ( x ⊲ z ) f ( y ) − µ ( y ) ν ( z ) f ( x ) − ν ( z ) µ ( x ) f ( y ) − ν ( z ) ν ( y ) f ( x ) + ν ( y ⊲ z ) f ( x ) − ν ( z ) f ([ x , y ] g ) (33) = µ ( x ) ν ( z ) f ( y ) + ν ( z ) µ ( y ) f ( x ) | {z } + ν ( z ) ν ( x ) f ( y ) − ν ( x ⊲ z ) f ( y ) − µ ( y ) ν ( z ) f ( x ) | {z } − ν ( z ) µ ( x ) f ( y ) − ν ( z ) ν ( y ) f ( x ) + ν ( y ⊲ z ) f ( x ) | {z } − ν ( z ) ρ ( x ) f ( y ) + ν ( z ) ρ ( y ) f ( x ) | {z } (29) = . Thus, (
Der ( g , V ); ˆ ρ ) is a representation of the sub-adjacent Lie algebra ( g , [ · , · ] C ). (cid:3) Cohomology groups of post-Lie algebras.
In this subsection, we define the cohomologygroups of a post-Lie algebra with coe ffi cients in an arbitrary representation. Furthermore, weestablish precise relationship between the cohomology groups of a post-Lie algebra and those ofits subadjacent Lie algebra.Let ( V ; ρ, µ, ν ) be a representation of a post-Lie algebra ( g , [ · , · ] g , ⊲ ). We have the natural iso-morphism Φ : Hom( ∧ n − g ⊗ g , V ) −→ Hom( ∧ n − g , Hom( g , V ))defined by(35) (cid:16) Φ ( ω )( x , · · · , x n − ) (cid:17) x n : = ω ( x , · · · , x n − , x n ) , ∀ x , · · · , x n − , x n ∈ g . Define the set of 0-cochains to be 0. For n ≥
1, we define the set of n -cochains C n Der ( g , V ) by(36) C n Der ( g , V ) : = n f ∈ Hom( ∧ n − g ⊗ g , V ) (cid:12)(cid:12)(cid:12)(cid:12) Φ ( f )( x , · · · , x n − ) ∈ Der ( g , V ) , ∀ x , · · · , x n − ∈ g o . For all f ∈ C n Der ( g , V ) , x , · · · , x n + ∈ g , define the operator δ : C n Der ( g , V ) −→ Hom( ∧ n g ⊗ g , V )by ( δ f )( x , · · · , x n + ) = n X i = ( − i + µ ( x i ) f ( x , · · · , ˆ x i , · · · , x n , x n + ) + n X i = ( − i + ν ( x n + ) f ( x , · · · , ˆ x i , · · · , x n , x i ) − n X i = ( − i + f ( x , · · · , ˆ x i , · · · , x n , x i ⊲ x n + )(37) OMOTOPY O -OPERATORS AND HOMOTOPY POST-LIE ALGEBRAS 23 + X ≤ i < j ≤ n ( − i + j f ( x i ⊲ x j − x j ⊲ x i + [ x i , x j ] g , x , · · · , ˆ x i , · · · , ˆ x j , · · · , x n , x n + ) . Proposition 4.7.
For all f ∈ C n Der ( g , V ) , we have δ f ∈ C n + Der ( g , V ) .Proof. By the definition of C n + Der ( g , V ), we just need to prove that Φ ( δ f )( x , · · · , x n ) is in Der ( g , V )for any x , · · · , x n ∈ g . For all x , y ∈ g , we have Φ ( δ f )( x , · · · , x n )([ x , y ] g ) (35) = ( δ f )( x , · · · , x n , [ x , y ] g ) (37) = n X i = ( − i + µ ( x i ) f ( x , · · · , ˆ x i , · · · , x n , [ x , y ] g ) + n X i = ( − i + ν ([ x , y ] g ) f ( x , · · · , ˆ x i , · · · , x n , x i ) − n X i = ( − i + f ( x , · · · , ˆ x i , · · · , x n , x i ⊲ [ x , y ] g ) + X ≤ i < j ≤ n ( − i + j f ( x i ⊲ x j − x j ⊲ x i + [ x i , x j ] g , x , · · · , ˆ x i , · · · , ˆ x j , · · · , x n , [ x , y ] g ) (14) , (27) = n X i = ( − i + µ ( x i ) (cid:16) ρ ( x ) f ( x , · · · , ˆ x i , · · · , x n , y ) − ρ ( y ) f ( x , · · · , ˆ x i , · · · , x n , x ) (cid:17) + n X i = ( − i + (cid:16) ρ ( x ) ν ( y ) − ρ ( y ) ν ( x ) (cid:17) f ( x , · · · , ˆ x i , · · · , x n , x i ) − n X i = ( − i + ρ ( x i ⊲ x ) f ( x , · · · , ˆ x i , · · · , x n , y ) + n X i = ( − i + ρ ( y ) f ( x , · · · , ˆ x i , · · · , x n , x i ⊲ x ) − n X i = ( − i + ρ ( x ) f ( x , · · · , ˆ x i , · · · , x n , x i ⊲ y ) + n X i = ( − i + ρ ( x i ⊲ y ) f ( x , · · · , ˆ x i , · · · , x n , x ) + X ≤ i < j ≤ n ( − i + j ρ ( x ) f ( x i ⊲ x j − x j ⊲ x i + [ x i , x j ] g , x , · · · , ˆ x i , · · · , ˆ x j , · · · , x n , y ) − X ≤ i < j ≤ n ( − i + j ρ ( y ) f ( x i ⊲ x j − x j ⊲ x i + [ x i , x j ] g , x , · · · , ˆ x i , · · · , ˆ x j , · · · , x n , x ) (26) = n X i = ( − i + (cid:16) ρ ( x ) ν ( y ) − ρ ( y ) ν ( x ) (cid:17) f ( x , · · · , ˆ x i , · · · , x n , x i ) + n X i = ( − i + ρ ( x ) (cid:16) µ ( x i ) f ( x , · · · , ˆ x i , · · · , x n , y ) (cid:17) + n X i = ( − i + ρ ( y ) f ( x , · · · , ˆ x i , · · · , x n , x i ⊲ x ) − n X i = ( − i + ρ ( x ) f ( x , · · · , ˆ x i , · · · , x n , x i ⊲ y ) − n X i = ( − i + ρ ( y ) (cid:16) µ ( x i ) f ( x , · · · , ˆ x i , · · · , x n , x ) (cid:17) + X ≤ i < j ≤ n ( − i + j ρ ( x ) f ( x i ⊲ x j − x j ⊲ x i + [ x i , x j ] g , x , · · · , ˆ x i , · · · , ˆ x j , · · · , x n , y ) − X ≤ i < j ≤ n ( − i + j ρ ( y ) f ( x i ⊲ x j − x j ⊲ x i + [ x i , x j ] g , x , · · · , ˆ x i , · · · , ˆ x j , · · · , x n , x ) = ρ ( x ) (cid:16) Φ ( δ f )( x , · · · , x n ) y (cid:17) − ρ ( y ) (cid:16) Φ ( δ f )( x , · · · , x n ) x (cid:17) . Thus we deduce that Φ ( δ f )( x , · · · , x n ) is in Der ( g , V ). (cid:3) To prove that the operator δ is indeed a coboundary operator, i.e. δ ◦ δ = , we need somepreparations. Proposition 4.8.
Let ( V ; ρ, µ, ν ) be a representation of a post-Lie algebra ( g , [ · , · ] g , ⊲ ) . Then wehave Φ ◦ δ = d ˆ ρ ◦ Φ , where d ˆ ρ is the coboundary operator of the sub-adjacent Lie algebra g C withcoe ffi cients in the representation ( Der ( g , V ); ˆ ρ ) given in Theorem 4.6 and δ is defined by (37) .Proof. For all f ∈ C n Der ( g , V ) and x , · · · , x n + ∈ g , we have (cid:16) d ˆ ρ ( Φ ( f ))( x , · · · , x n ) (cid:17) x n + = n X i = ( − i + (cid:16) ˆ ρ ( x i ) Φ ( f )( x , · · · , ˆ x i , · · · , x n ) (cid:17) x n + + X ≤ i < j ≤ n ( − i + j (cid:16) Φ ( f )([ x i , x j ] C , x , · · · , ˆ x i , · · · , ˆ x j , · · · , x n ) (cid:17) x n + = n X i = ( − i + µ ( x i ) Φ ( f )( x , · · · , ˆ x i , · · · , x n ) x n + + n X i = ( − i + ν ( x n + ) Φ ( f )( x , · · · , ˆ x i , · · · , x n ) x i − n X i = ( − i + Φ ( f )( x , · · · , ˆ x i , · · · , x n )( x i ⊲ x n + ) + X ≤ i < j ≤ n ( − i + j (cid:16) Φ ( f )( x i ⊲ x j − x j ⊲ x i + [ x i , x j ] g , x , · · · , ˆ x i , · · · , ˆ x j , · · · , x n ) (cid:17) x n + = n X i = ( − i + µ ( x i ) f ( x , · · · , ˆ x i , · · · , x n , x n + ) + n X i = ( − i + ν ( x n + ) f ( x , · · · , ˆ x i , · · · , x n , x i ) − n X i = ( − i + f ( x , · · · , ˆ x i , · · · , x n , x i ⊲ x n + ) + X ≤ i < j ≤ n ( − i + j f ( x i ⊲ x j − x j ⊲ x i + [ x i , x j ] g , x , · · · , ˆ x i , · · · , ˆ x j , · · · , x n , x n + ) (37) = ( δ f )( x , · · · , x n + ) (35) = (cid:16) Φ ( δ f )( x , · · · , x n ) (cid:17) x n + , which implies that d ˆ ρ ◦ Φ = Φ ◦ δ . (cid:3) Theorem 4.9.
The operator δ : C n Der ( g , V ) −→ C n + Der ( g , V ) defined by (37) satisfies δ ◦ δ = .Proof. By Proposition 4.8, we have δ = Φ − ◦ d ˆ ρ ◦ Φ . Thus, by the fact that d ˆ ρ ◦ d ˆ ρ =
0, we obtain δ ◦ δ = Φ − ◦ d ˆ ρ ◦ d ˆ ρ ◦ Φ = . (cid:3) Let ( V ; ρ, µ, ν ) be a representation of a post-Lie algebra ( g , [ · , · ] g , ⊲ ). Denote by C ∗ Der ( g , V ) : = L n ≥ C n Der ( g , V ). Then we have the cochain complex ( C ∗ Der ( g , V ) , δ ). Denote the set of closed n -cochains by Z n ( g , V ) and the set of exact n -cochains by B n ( g , V ). We denote by H n ( g , V ) = Z n ( g , V ) / B n ( g , V ), and call them the cohomology groups of the post-Lie algebra ( g , [ · , · ] g , ⊲ )with coe ffi cients in the representation ( V ; ρ, µ, ν ). OMOTOPY O -OPERATORS AND HOMOTOPY POST-LIE ALGEBRAS 25 It is obvious that f ∈ C Der ( g , V ) is closed if and only if f ∈ Der ( g , V ) and ν ( y ) f ( x ) + µ ( x ) f ( y ) − f ( x ⊲ y ) = , ∀ x , y ∈ g . Also f ∈ C Der ( g , V ) is closed if and only if Φ ( f ) ∈ Hom( g , Der ( g , V )) and ν ( x ) f ( x , x ) + µ ( x ) f ( x , x ) − f ( x , x ⊲ x ) − ν ( x ) f ( x , x ) − µ ( x ) f ( x , x ) + f ( x , x ⊲ x ) − f ( x ⊲ x , x ) + f ( x ⊲ x , x ) − f ([ x , x ] g , x ) = , ∀ x , x , x ∈ g . There is a close relationship between the cohomology groups of post-Lie algebras and those ofthe corresponding sub-adjacent Lie algebras.
Theorem 4.10.
Let ( V ; ρ, µ, ν ) be a representation of a post-Lie algebra ( g , [ · , · ] g , ⊲ ) . Thenthe cohomology group H n ( g , V ) of the post-Lie algebra ( g , [ · , · ] g , ⊲ ) and the cohomology groupH n − ( g C , Der ( g , V )) of the subadjacent Lie algebra g C are isomorphic for all n ≥ .Proof. By Proposition 4.8, we deduce that Φ is an isomorphism from the cochain complex (cid:0) C ∗ Der ( g , V ) , δ (cid:1) to the cochain complex (cid:0) C ∗− ( g , Der ( g , V )) , d ˆ ρ (cid:1) . Thus, Φ induces an isomorphism Φ ∗ from H ∗ ( g , V ) to H ∗− ( g C , Der ( g , V )). (cid:3) In the above theorem, if [ · , · ] g and ρ are zero, then the post-Lie algebra is a pre-Lie algebra andwe 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 cohomologygroup H n ( g , V ) of the pre-Lie algebra ( g , ⊲ ) and the cohomology group H n − ( g C , Hom( g , V )) ofthe subadjacent Lie algebra g C are isomorphic for all n ≥ . Classification of 2-term skeletal operator homotopy post-Lie algebras.
In this subsec-tion, we first give an equivalent definition of an operator homotopy post-Lie algebra and thenclassify 2-term skeletal operator homotopy post-Lie algebras using the third cohomology groupgiven in Section 4.2For all i ≥
1, let Θ i : ∧ i − g ⊗ g → g be a graded linear map of degree 2 − i . Define D ( Θ i ) : ⊙ i − s − g ⊗ s − g → s − g by D ( Θ i ) = ( − i ( i − s − ◦ Θ i ◦ s ⊗ i , which is a graded linear map of degree 1. This is illustrated by the following commutative dia-gram: ∧ i − g ⊗ g Θ i −−−−−→ g ⊗ i s − y y s − ⊙ i − s − g ⊗ s − g D ( Θ i ) −−−−−→ s − g Using this process, we can give an equivalent definition of an operator homotopy post-Liealgebra as follows.
Definition 4.12. An operator homotopy post-Lie algebra is a graded Lie algebra ( g , [ · , · ] g )equipped with a collection of linear maps Θ k : ⊗ k g → g , k ≥ , of degree 2 − k satisfying,for any homogeneous elements x , · · · , x n , x n + ∈ g , the following conditions hold:(i) (graded antisymmetry) for every σ ∈ S n − , n ≥ Θ n ( x σ (1) , · · · , x σ ( n − , x n ) = χ ( σ ) Θ n ( x , · · · , x n − , x n ) , (38) (ii) (graded derivation) for all n ≥ Θ n ( x , · · · , x n − , [ x n , x n + ] g ) = [ Θ n ( x , · · · , x n − , x n ) , x n + ] g (39) + ( − x n ( x + ··· + x n − + n ) [ x n , Θ n ( x , · · · , x n − , x n + )] g , (iii) for all n ≥ X ≤ i < j ≤ n − ( − β Θ n − ([ x i , x j ] g , x , · · · , ˆ x i , · · · , ˆ x j , · · · , x n ) = X i + j = n + i ≥ , j ≥ X σ ∈ S ( i − , , j − ( − i ( j − χ ( σ ) Θ j ( Θ i ( x σ (1) , · · · , x σ ( i − , x σ ( i ) ) , x σ ( i + , · · · , x σ ( n − , x n )(40) + X i + j = n + i ≥ , j ≥ X σ ∈ S ( j − , i − ( − j − ( − α χ ( σ ) Θ j ( x σ (1) , · · · , x σ ( j − , Θ i ( x σ ( j ) , · · · , x σ ( n − , x n )) , where β = x i ( x + · · · + x i − ) + x j ( x + · · · + x j − ) + x i x j + i + j and α = i ( x σ (1) + x σ (2) + · · · + x σ ( j − ).By (39), for n =
1, we have Θ ([ x , x ] g ) = [ Θ ( x ) , x ] g + ( − x [ x , Θ ( x )] g . (41)By (40), for n =
1, we have Θ = , which implies that ( g , Θ ) is a complex. By (40), for n = = − Θ ( Θ ( x ) , x ) − ( − x Θ ( x , Θ ( x )) + Θ ( Θ ( x , x )) . (42)Now we will show that the corresponding cohomology space H ∗ ( g ) of the complex ( g , Θ ) en-joys a graded post-Lie algebra structure and this justifies our definition of an “operator homotopypost-Lie algebra”. Theorem 4.13.
Let ( g , [ · , · ] g , { Θ k } ∞ k = ) be an operator homotopy post-Lie algebra. Then the coho-mology space H ∗ ( g ) is a graded post-Lie algebra.Proof. For any homogeneous element x ∈ ker( Θ ), we denote by x ∈ H ∗ ( g ) its cohomologicalclass. First we define a graded bracket operation [ · , · ] on the graded vector space H ∗ ( g ) by[ ¯ x , ¯ y ] : = [ x , y ] g , ∀ ¯ x , ¯ y ∈ H ∗ ( g ) . If ¯ x = ¯ x ′ , then there exists X ∈ g such that x ′ = x + Θ ( X ). Hence by (41), we have[ ¯ x ′ , ¯ y ] = [ x ′ , y ] g = [ x + Θ ( X ) , y ] g = [ x , y ] g + Θ ([ X , y ] g ) = [ x , y ] g = [ ¯ x , ¯ y ] , which implies that [ · , · ] is well-defined. It is straightforward to obtain that ( H ∗ ( g ) , [ · , · ]) is a gradedLie algebra.Then we define a multiplication ⊲ on the graded vector space H ∗ ( g ) by¯ x ⊲ ¯ y : = Θ ( x , y ) , ∀ ¯ x , ¯ y ∈ H ∗ ( g ) . Similarly, by (42), we can deduce that ⊲ is well-defined. By (39) for n =
2, we have¯ x ⊲ [¯ y , ¯ z ] = Θ ( x , [ y , z ] g ) = [ Θ ( x , y ) , z ] g + ( − xy [ y , Θ ( x , z )] g = [ ¯ x ⊲ ¯ y , ¯ z ] + + ( − xy [¯ y , ¯ x ⊲ ¯ z ] . Similarly, by (40) for n =
3, we have[ ¯ x , ¯ y ] ⊲ ¯ z = a ⊲ ( ¯ x , ¯ y , ¯ z ) − a ⊲ (¯ y , ¯ x , ¯ z ) . Therefore, ( H ∗ ( g ) , [ · , · ] , ⊲ ) is a graded post-Lie algebra. (cid:3) By truncation, we obtain the definition of a 2-term operator homotopy post-Lie algebra.
OMOTOPY O -OPERATORS AND HOMOTOPY POST-LIE ALGEBRAS 27 Definition 4.14. A is a 2-term graded Lie algebra( g = g ⊕ g − , [ · , · ] g ) equipped with • a linear map Θ : g − → g ; • a linear map Θ : g i ⊗ g j → g i + j , − ≤ i + j ≤ • a linear map Θ : ∧ g ⊗ g → g − such that for all x , y , z , w ∈ g and a , b ∈ g − , the following equalities hold:( a ) Θ ([ x , a ] g ) = [ x , Θ ( a )] g ;( a ) [ Θ ( a ) , b ] g = [ a , Θ ( b )] g ;( b ) Θ ( x , [ y , z ] g ) = [ Θ ( x , y ) , z ] g + [ y , Θ ( x , z )] g ;( b ) Θ ( x , [ y , a ] g ) = [ Θ ( x , y ) , a ] g + [ y , Θ ( x , a )] g ;( b ) Θ ( a , [ x , y ] g ) = [ Θ ( a , x ) , y ] g + [ x , Θ ( a , y )] g ;( c ) Θ ( x , y , [ z , w ] g ) = [ Θ ( x , y , z ) , w ] g + [ z , Θ ( x , y , w )] g ;( d ) Θ Θ ( x , a ) = Θ ( x , Θ ( a ));( d ) Θ Θ ( a , x ) = Θ ( Θ ( a ) , x );( d ) Θ ( Θ ( a ) , b ) = Θ ( a , Θ ( b ));( e ) Θ ( x , Θ ( y , z )) − Θ ( Θ ( x , y ) , z ) − Θ ( y , Θ ( x , z )) +Θ ( Θ ( y , x ) , z ) − Θ ([ x , y ] g , z ) = Θ Θ ( x , y , z );( e ) Θ ( x , Θ ( y , a )) − Θ ( Θ ( x , y ) , a ) − Θ ( y , Θ ( x , a )) +Θ ( Θ ( y , x ) , a ) − Θ ([ x , y ] g , a ) = Θ ( x , y , Θ ( a ));( e ) Θ ( a , Θ ( y , z )) − Θ ( Θ ( a , y ) , z ) − Θ ( y , Θ ( a , z )) +Θ ( Θ ( y , a ) , z ) − Θ ([ a , y ] g , z ) = Θ ( Θ ( a ) , y , z );( f ) Θ ( x , Θ ( y , z , w )) − Θ ( y , Θ ( x , z , w )) +Θ ( z , Θ ( x , y , w )) +Θ ( Θ ( y , z , x ) , w ) − Θ ( Θ ( x , z , y ) , w ) +Θ ( Θ ( x , y , z ) , w ) − Θ ( Θ ( x , y ) − Θ ( y , x ) + [ x , y ] g , z , w ) − Θ ( Θ ( y , z ) − Θ ( z , y ) + [ y , z ] g , x , w ) +Θ ( Θ ( x , z ) − Θ ( z , x ) + [ x , z ] g , y , w ) − Θ ( y , z , Θ ( x , w )) +Θ ( x , z , Θ ( y , w )) − Θ ( x , y , Θ ( z , w )) = . A 2-term operator homotopy post-Lie algebra ( g = g ⊕ g − , [ · , · ] g , Θ , Θ , Θ ) is said to be skeletal if Θ = . Remark 4.15.
If the underlying graded Lie algebra ( g ⊕ g − , [ · , · ] g ) in a 2-term operator homotopypost-Lie algebra ( g = g ⊕ g − , [ · , · ] g , Θ , Θ , Θ ) is abelian, then ( g ⊕ g − , Θ , Θ , Θ ) reduces toa 2-term pre-Lie ∞ algebra or equivalently a pre-Lie 2-algebra.In [42], the author showed that skeletal pre-Lie 2-algebras are classified by the third cohomol-ogy group of pre-Lie algebras. See [3] also for more details for the classification of skeletal Lie2-algebras. Similarly, we have Theorem 4.16.
There is a one-to-one correspondence between -term skeletal operator homo-topy post-Lie algebras and triples (( h , [ · , · ] h , ⊲ ) , ( V ; ρ, µ, ν ) , ω ) , where ( h , [ · , · ] h , ⊲ ) is a post-Lie al-gebra, ( V ; ρ, µ, ν ) is a representation of the post-Lie algebra ( h , [ · , · ] h , ⊲ ) and ω ∈ Hom( ∧ h ⊗ h , V ) is a -cocycle of ( h , [ · , · ] h , ⊲ ) with coe ffi cients in ( V ; ρ, µ, ν ) .Proof. Let ( g = g ⊕ g − , [ · , · ] g , Θ , Θ , Θ ) be a 2-term skeletal operator homotopy post-Lie al-gebra, i.e. Θ =
0. Then by Condition ( e ) in Definition 4.14, we deduce that ( g , [ · , · ] g , Θ ) is apost-Lie algebra. Define linear maps ρ, µ, ν from g to gl ( g − ) by ρ ( x )( a ) : = [ x , a ] g , µ ( x )( a ) : = Θ ( x , a ) , ν ( x )( a ) : = Θ ( a , x ) , ∀ x ∈ g , a ∈ g − . Obviously, ( g − ; ρ ) is a representation of the Lie algebra ( g , [ · , · ] g ). Then by ( b ), ( b ), ( e ) and( e ) in Definition 4.14, we deduce (26)-(29) respectively. Thus, ( g − ; ρ, µ, ν ) is a representationof the post-Lie algebra ( g , [ · , · ] g , Θ ). Finally, by ( c ) and ( f ) in Definition 4.14, we deduce that Θ is a 3-cocycle of the post-Lie algebra ( g , [ · , · ] g , Θ ) with coe ffi cients in the representation( g − ; ρ, µ, ν ). The proof of the other direction is similar. So the details will be omitted. (cid:3)
Acknowledgements.
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