Deformations and homotopy theory of relative Rota-Baxter Lie algebras
aa r X i v : . [ m a t h . QA ] A ug DEFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIEALGEBRAS
ANDREY LAZAREV, YUNHE SHENG, AND RONG TANGA bstract . We determine the L ∞ -algebra that controls deformations of a relative Rota-Baxter Liealgebra and show that it is an extension of the dg Lie algebra controlling deformations of theunderlying LieRep pair by the dg Lie algebra controlling deformations of the relative Rota-Baxteroperator. Consequently, we define the cohomology of relative Rota-Baxter Lie algebras and relateit to their infinitesimal deformations. A large class of relative Rota-Baxter Lie algebras is obtainedfrom triangular Lie bialgebras and we construct a map between the corresponding deformationcomplexes. Next, the notion of a homotopy relative Rota-Baxter Lie algebra is introduced. Weshow that a class of homotopy relative Rota-Baxter Lie algebras is intimately related to pre-Lie ∞ -algebras . C ontents
1. Introduction 21.1. Rota-Baxter operators 21.2. Deformations 21.3. Cohomology theories 31.4. Homotopy invariant construction of Rota-Baxter Lie algebras 31.5. Outline of the paper 31.6. Notation and conventions 42. Maurer-Cartan characterizations of
LieRep pairs and relative Rota-Baxter operators 42.1. Bidegrees and the Nijenhuis-Richardson bracket 42.2. MC characterization, deformations and cohomology of LieRep pairs 52.3. MC characterization, deformations and cohomologies of relative Rota-Baxteroperators 63. Maurer-Cartan characterization and deformations of relative Rota-Baxter Lie algebras 73.1. L ∞ -algebras and higher derived brackets 83.2. The L ∞ -algebra that controls deformations of relative Rota-Baxter Lie algebras 94. Cohomology and infinitesimal deformations of relative Rota-Baxter Lie algebras 134.1. Cohomology of relative Rota-Baxter Lie algebras 134.2. Infinitesimal deformations of relative Rota-Baxter Lie algebras 154.3. Cohomology of Rota-Baxter Lie algebras 174.4. Cohomology and infinitesimal deformations of triangular Lie bialgebras 185. Homotopy relative Rota-Baxter Lie algebras 225.1. Homotopy relative Rota-Baxter operators on L ∞ -algebras 225.2. Strict homotopy relative Rota-Baxter operators on L ∞ -algebras and pre-Lie ∞ -algebras 26References 29 Key words and phrases.
Cohomology, deformation, L ∞ -algebra, MC element, Rota-Baxter algebra, r -matrix,triangular Lie bialgebra.
1. I ntroduction
In this paper we initiate the study of deformations and cohomology of relative Rota-Baxter Liealgebras and their homotopy versions.1.1.
Rota-Baxter operators.
The concept of Rota-Baxter operators on associative algebras wasintroduced by G. Baxter [6] in his study of fluctuation theory in probability. Recently it has foundmany applications, including Connes-Kreimer’s [12] algebraic approach to the renormalizationin perturbative quantum field theory. Rota-Baxter operators lead to the splitting of operads [3,45], and are closely related to noncommutative symmetric functions and Hopf algebras [16, 27,55]. Recently the relationship between Rota-Baxter operators and double Poisson algebras werestudied in [23]. In the Lie algebra context, a Rota-Baxter operator was introduced independentlyin the 1980s as the operator form of the classical Yang-Baxter equation that plays importantroles in many subfields of mathematics and mathematical physics such as integrable systems andquantum groups [10, 47]. For further details on Rota-Baxter operators, see [25, 26].To better understand the classical Yang-Baxter equation and related integrable systems, themore general notion of an O -operator (later also called a relative Rota-Baxter operator or a gen-eralized Rota-Baxter operator) on a Lie algebra was introduced by Kupershmidt [33]; this notioncan be traced back to Bordemann [7]. Relative Rota-Baxter operators provide solutions of theclassical Yang-Baxter equation in the semidirect product Lie algebra and give rise to pre-Liealgebras [2].1.2. Deformations.
The concept of a formal deformation of an algebraic structure began withthe seminal work of Gerstenhaber [20, 21] for associative algebras. Nijenhuis and Richardsonextended this study to Lie algebras [43, 44]. More generally, deformation theory for algebrasover quadratic operads was developed by Balavoine [4]. For more general operads we refer thereader to [31, 37, 40], and the references therein.There is a well known slogan, often attributed to Deligne, Drinfeld and Kontsevich: everyreasonable deformation theory is controlled by a di ff erential graded (dg) Lie algebra, determinedup to quasi-isomorphism . This slogan has been made into a rigorous theorem by Lurie andPridham, cf. [38, 46], and a recent simple treatment in [24].It is also meaningful to deform maps compatible with given algebraic structures. Recently,the deformation theory of morphisms was developed in [8, 18, 19], the deformation theory of O -operators was developed in [53] and the deformation theory of diagrams of algebras was studiedin [5, 17] using the minimal model of operads and the method of derived brackets [32, 39, 54].Sometimes a dg Lie algebra up to quasi-isomorphism controlling a deformation theory man-ifests itself naturally as an L ∞ -algebra . This often happens when one tries to deform severalalgebraic structures as well as a compatibility relation between them, such as diagrams of al-gebras mentioned above. We will see that this also happens in the study of deformations of arelative Rota-Baxter Lie algebra, which consists of a Lie algebra, its representation and a relativeRota-Baxter operator (see Definition 2.10 below). We apply Voronov’s higher derived bracketsconstruction [54] to construct the L ∞ -algebra that characterizes relative Rota-Baxter Lie alge-bras as Maurer-Cartan ( MC ) elements in it. This leads, by a well-known procedure of twisting,to an L ∞ -algebra controlling deformations of relative Rota-Baxter Lie algebras. Moreover, weshow that this L ∞ -algebra is an extension of the dg Lie algebra that controls deformations of LieRep pairs (a
LieRep pair consists of a Lie algebra and a representation) given in [1] by the dgLie algebra that controls deformations of relative Rota-Baxter operators given in [53].
EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 3
Cohomology theories.
A classical approach for studying a mathematical structure is asso-ciating invariants to it. Prominent among these are cohomological invariants, or simply cohomol-ogy, of various types of algebras. Cohomology controls deformations and extension problems ofthe corresponding algebraic structures. Cohomology theories of various kinds of algebras havebeen developed and studied in [11, 20, 29, 30]. More recently these classical constructions havebeen extended to strong homotopy (or infinity) versions of the algebras, cf. for example [28].In the present paper we study the cohomology theory for relative Rota-Baxter Lie algebras. Arelative Rota-Baxter Lie algebra consists of a Lie algebra, its representation and an operator on ittogether with appropriate compatibility conditions. Constructing the corresponding cohomologytheory is not straightforward due to the complexity of these data. We solve this problem byconstructing a deformation complex for a relative
Rota-Baxter Lie algebra and endowing it withan L ∞ -structure . Infinitesimal deformations of relative Rota-Baxter Lie algebras are classifiedby the second cohomology group. Moreover, we show that there is a long exact sequence ofcohomology groups linking the cohomology of LieRep pairs introduced in [1], the cohomologyof O -operators introduced in [53] and the cohomology of relative Rota-Baxter Lie algebras.The above general framework has two important special cases: Rota-Baxter Lie algebras andtriangular Lie bialgebras and we introduce the corresponding cohomology theories for these ob-jects. We also show that infinitesimal deformations of Rota-Baxter Lie algebras and triangularLie bialgebras are classified by the corresponding second cohomology groups.1.4. Homotopy invariant construction of Rota-Baxter Lie algebras.
Homotopy invariant al-gebraic structures play a prominent role in modern mathematical physics [52]. Historically, thefirst such structure was that of an A ∞ -algebra introduced by Stashe ff in his study of based loopspaces [49]. Relevant later developments include the work of Lada and Stashe ff [34, 51] about L ∞ -algebras in mathematical physics and the work of Chapoton and Livernet [9] about pre-Lie ∞ -algebras. Strong homotopy (or infinity-) versions of a large class of algebraic structures werestudied in the context of operads in [37, 41].Dotsenko and Khoroshkin studied the homotopy of Rota-Baxter operators on associative alge-bras in [15], and noted that “in general compact formulas are yet to be found”. For Rota-BaxterLie algebras, one encounters a similarly challenging situation. In this paper, we use the approachof L ∞ -algebras and their MC elements to formulate the notion of a (strong) homotopy versionof a relative Rota-Baxter Lie algebra, which consists of an L ∞ -algebra, its representation and ahomotopy relative Rota-Baxter operator. We show that strict homotopy relative Rota-Baxter oper-ators give rise to pre-Lie ∞ -algebras, and conversely the identity map is a strict homotopy relativeRota-Baxter operator on the subadjacent L ∞ -algebra of a pre-Lie ∞ -algebra.1.5. Outline of the paper.
In Section 2, we briefly recall the deformation theory and the co-homology of
LieRep pairs and relative Rota-Baxter operators. In Section 3, we establish thedeformation theory of relative Rota-Baxter Lie algebras. In Section 4, we introduce the corre-sponding cohomology theory and explain how it is related to infinitesimal deformations of rela-tive Rota-Baxter Lie algebras in the usual way. In Section 4.3, we study the cohomology theory ofRota-Baxter Lie algebras. In Section 4.4, we explain how the cohomology theories of triangularLie bialgebras and of relative Rota-Baxter Lie algebras are related. In Section 5, we introduce thenotion of a homotopy relative Rota-Baxter operator and characterize it as an MC element in a cer-tain L ∞ -algebra. Finally, we exhibit a close relationship between homotopy relative Rota-BaxterLie algebras of a certain kind and pre-Lie ∞ -algebras. ANDREY LAZAREV, YUNHE SHENG, AND RONG TANG
Notation and conventions.
Throughout this paper, we work with a coe ffi cient field K whichis of characteristic 0, and R is a pro-Artinian K -algebra, that is a projective limit of local Artinian K -algebras.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 . That is, S ( V ) : = T ( V ) / I , where T ( V ) is the tensor algebra and I is the 2-sided ideal of T ( V ) generated by all homogeneous elements of the form x ⊗ y − ( − xy y ⊗ x . We will write S ( V ) = ⊕ + ∞ i = S i ( V ). Moreover, we denote the reduced symmetric algebra by ¯ 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 degrees of v i . For a permutation σ ∈ S n and v , · · · , v n ∈ V , the Koszul sign ε ( σ ) = ε ( σ ; v , · · · , v n ) ∈ {− , } is defined by v ⊙ · · · ⊙ v n = ε ( σ ; v , · · · , v n ) v σ (1) ⊙ · · · ⊙ v σ ( n ) . The desuspension operator s − changes the grading of V according to the rule ( s − V ) i : = V i + .The degree − s − : V → s − V is defined by sending v ∈ V to its copy s − v ∈ s − V .A degree 1 element θ ∈ g is called an MC element of a di ff erential graded Lie algebra( ⊕ k ∈ Z g k , [ · , · ] , d ) if it satisfies the MC equation : d θ + [ θ, θ ] = .
2. M aurer -C artan characterizations of LieRep pairs and relative R ota -B axter operators Bidegrees and the Nijenhuis-Richardson bracket.
Let g be a vector space. For all n ≥ C n ( g , g ) : = Hom( ∧ n + g , g ) . Let g and g be two vector spaces and elements in g will bedenoted by x , y , z , x i and elements in g will be denoted by u , v , w , v i . For a multilinear map f : ∧ k g ⊗ ∧ l g → g , we define ˆ f ∈ C k + l − (cid:0) g ⊕ g , g ⊕ g (cid:1) byˆ f (cid:0) ( x , v ) , · · · , ( x k + l , v k + l ) (cid:1) : = X τ ∈ S ( k , l ) ( − τ (cid:16) f ( x τ (1) , · · · , x τ ( k ) , v τ ( k + , · · · , v τ ( k + l ) ) , (cid:17) . Similarly, for f : ∧ k g ⊗ ∧ l g → g , we define ˆ f ∈ C k + l − (cid:0) g ⊕ g , g ⊕ g (cid:1) byˆ f (cid:0) ( x , v ) , · · · , ( x k + l , v k + l ) (cid:1) : = X τ ∈ S ( k , l ) ( − τ (cid:16) , f ( x τ (1) , · · · , x τ ( k ) , v τ ( k + , · · · , v τ ( k + l ) ) (cid:17) . The linear map ˆ f is called a lift of f . We define g k , l : = ∧ k g ⊗ ∧ l g . The vector space ∧ n ( g ⊕ g )is isomorphic to the direct sum of g k , l , k + l = n . Definition 2.1.
A linear map f ∈ Hom (cid:0) ∧ k + l + ( g ⊕ g ) , g ⊕ g (cid:1) has a bidegree k | l , which is denotedby || f || = k | l , if f satisfies the following two conditions:(i) If X ∈ g k + , l , then f ( X ) ∈ g and if X ∈ g k , l + , then f ( X ) ∈ g ;(ii) In all the other cases f ( X ) = . We denote the set of homogeneous linear maps of bidegree k | l by C k | l ( g ⊕ g , g ⊕ g ).It is clear that this gives a well-defined bigrading on the vector space Hom (cid:0) ∧ k + l + ( g ⊕ g ) , g ⊕ g (cid:1) . We have k + l ≥ , k , l ≥ − k + l + ≥ k + , l + ≥ g be a vector space. We consider the graded vector space C ∗ ( g , g ) = ⊕ + ∞ n = C n ( g , g ) = ⊕ + ∞ n = Hom( ∧ n + g , g ) . Then C ∗ ( g , g ) equipped with the Nijenhuis-Richardson bracket [ P , Q ] NR = P ¯ ◦ Q − ( − pq Q ¯ ◦ P , ∀ P ∈ C p ( g , g ) , Q ∈ C q ( g , g ) , (1) EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 5 is a graded Lie algebra, where P ¯ ◦ Q ∈ C p + q ( g , g ) is defined by( P ¯ ◦ Q )( x , · · · , x p + q + ) = X σ ∈ S ( q + , p ) ( − σ P ( Q ( x σ (1) , · · · , x σ ( q + ) , x σ ( q + , · · · , x σ ( p + q + ) . (2) Remark 2.2.
In fact, the Nijenhuis-Richardson bracket is the commutator of coderivations on thecofree conilpotent cocommutative coalgebra ¯ S c ( s − g ). See [43, 50] for more details.The following lemmas are very important in our later study. Lemma 2.3.
The Nijenhuis-Richardson bracket on C ∗ ( g ⊕ g , g ⊕ g ) is compatible with thebigrading. More precisely, if || f || = k f | l f , || g || = k g | l g , then [ f , g ] NR has bidegree ( k f + k g ) | ( l f + l g ) . Proof.
It follows from direct computation. (cid:3)
Remark 2.4.
In our later study, the subspaces C k | ( g ⊕ g , g ⊕ g ) and C − | l ( g ⊕ g , g ⊕ g ) willbe frequently used. By the above lift map, we have the following isomorphisms: C k | ( g ⊕ g , g ⊕ g ) (cid:27) Hom( ∧ k + g , g ) ⊕ Hom( ∧ k g ⊗ g , g ) , (3) C − | l ( g ⊕ g , g ⊕ g ) (cid:27) Hom( ∧ l g , g ) . (4) Lemma 2.5. If || f || = ( − | k and || g || = ( − | l, then [ f , g ] NR = . Consequently, ⊕ + ∞ l = C − | l ( g ⊕ g , g ⊕ g ) is an abelian subalgebra of the graded Lie algebra ( C ∗ ( g ⊕ g , g ⊕ g ) , [ · , · ] NR ) Proof.
It follows from Lemma 2.3. (cid:3) MC characterization, deformations and cohomology of LieRep pairs.
Let g be a vectorspace. For µ ∈ C ( g , g ) = Hom( ∧ g , g ), we have[ µ, µ ] NR ( x , y , z ) = µ ¯ ◦ µ )( x , y , z ) = (cid:16) µ ( µ ( x , y ) , z ) + µ ( µ ( y , z ) , x ) + µ ( µ ( z , x ) , y ) (cid:17) . Thus, µ defines a Lie algebra structure on g if and only if [ µ, µ ] NR = C ( g ; g ) to be 0, and define the set of n -cochains C n Lie ( g ; g ) to be C n Lie ( g ; g ) : = Hom( ∧ n g , g ) = C n − ( g , g ) , n ≥ . The
Chevalley-Eilenberg coboundary operator d CE of the Lie algebra g with coe ffi cients in theadjoint representation is defined byd CE f = ( − n − [ µ, f ] NR , ∀ f ∈ C n Lie ( g ; g ) . (5)The resulting cohomology is denoted by H ∗ Lie ( g ; g ). Definition 2.6. A LieRep pair consists of a Lie algebra ( g , [ · , · ] g ) and a representation ρ : g −→ gl ( V ) of g on a vector space V .Usually we will also use µ to indicate the Lie bracket [ · , · ] g , and denote a LieRep pair by( g , µ ; ρ ).Note that µ + ρ ∈ C | ( g ⊕ V , g ⊕ V ). Moreover, the fact that µ is a Lie bracket and ρ is arepresentation is equivalent to that [ µ + ρ, µ + ρ ] NR = . Next, the following result holds:
Proposition 2.7. ([1])
Let g and V be two vector spaces. Then (cid:0) ⊕ + ∞ k = C k | ( g ⊕ V , g ⊕ V ) , [ · , · ] NR (cid:1) isa graded Lie algebra. Its MC elements are precisely LieRep pairs. (cid:3)
ANDREY LAZAREV, YUNHE SHENG, AND RONG TANG
Let ( g , µ ; ρ ) be a LieRep pair. By Proposition 2.7, π = µ + ρ is an MC element of the graded Liealgebra (cid:0) ⊕ + ∞ k = C k | ( g ⊕ V , g ⊕ V ) , [ · , · ] NR (cid:1) . It follows from the graded Jacobi identity that d π : = [ π, · ] NR is a graded derivation of the graded Lie algebra (cid:0) ⊕ + ∞ k = C k | ( g ⊕ V , g ⊕ V ) , [ · , · ] NR (cid:1) satisfying d π = Theorem 2.8. ([1])
Let ( g , µ ; ρ ) be a LieRep pair. Then (cid:0) ⊕ + ∞ k = C k | ( g ⊕ V , g ⊕ V ) , [ · , · ] NR , d π (cid:1) is adg Lie algebra.Furthermore, ( g , µ + µ ′ ; ρ + ρ ′ ) is also a LieRep pair for µ ′ ∈ Hom( ∧ g , g ) and ρ ′ ∈ Hom( g , gl ( V )) if and only if µ ′ + ρ ′ is an MC element of the dg Lie algebra (cid:0) ⊕ + ∞ k = C k | ( g ⊕ V , g ⊕ V ) , [ · , · ] NR , d π (cid:1) . (cid:3) Let ( g , µ ; ρ ) be a LieRep pair. Define the set of 0-cochains C ( g , ρ ) to be 0. For n ≥
1, wedefine the set of n -cochains C n ( g , ρ ) to be C n ( g , ρ ) : = C ( n − | ( g ⊕ V , g ⊕ V ) = Hom( ∧ n g , g ) ⊕ Hom( ∧ n − g ⊗ V , V ) . Define the coboundary operator ∂ : C n ( g , ρ ) → C n + ( g , ρ ) by(6) ∂ f : = ( − n − [ µ + ρ, f ] NR . By Proposition 2.7, we deduce that ∂ ◦ ∂ =
0. Thus we obtain the complex ( ⊕ + ∞ n = C n ( g , ρ ) , ∂ ). Definition 2.9. ([1]) The cohomology of the cochain complex ( ⊕ + ∞ n = C n ( g , ρ ) , ∂ ) is called the coho-mology of the LieRep pair ( g , µ ; ρ ). The resulting n -th cohomology group is denoted by H n ( g , ρ ).Now we give the precise formula for ∂ . For any n -cochain f ∈ C n ( g , ρ ), by (3), we will write f = ( f g , f V ), where f g ∈ Hom( ∧ n g , g ) and f V ∈ Hom( ∧ n − g ⊗ V , V ). Then we have ∂ f = (cid:16) ( ∂ f ) g , ( ∂ f ) V (cid:17) , (7)where ( ∂ f ) g = d CE f g and ( ∂ f ) V is given by( ∂ f ) V ( x , · · · , x n , v ) = X ≤ i < j ≤ n ( − i + j f V ([ x i , x j ] g , x , · · · , ˆ x i , · · · , ˆ x j , · · · , x n , v ) + ( − n − ρ ( f g ( x , · · · , x n )) v (8) + n X i = ( − i + (cid:16) ρ ( x i ) f V ( x , · · · , ˆ x i , · · · , x n , v ) − f V (cid:0) x , · · · , ˆ x i , · · · , x n , ρ ( x i ) v (cid:1)(cid:17) . MC characterization, deformations and cohomologies of relative Rota-Baxter opera-tors. We now recall the notion of a relative Rota-Baxter operator. Let ( g , [ · , · ] g ) be a Lie algebraand ρ : g −→ gl ( V ) a representation of g on a vector space V . Definition 2.10. (i) A linear operator T : g −→ g is called a Rota-Baxter operator if(9) [ T ( x ) , T ( y )] g = T (cid:0) [ T ( x ) , y ] g + [ x , T ( y )] g (cid:1) , ∀ x , y ∈ g . Moreover, a Lie algebra ( g , [ · , · ] g ) with a Rota-Baxter operator T is called a Rota-BaxterLie algebra . We denote it by ( g , [ · , · ] g , T ).(ii) A relative Rota-Baxter Lie algebra is a triple (( g , [ · , · ] g ) , ρ, T ), where ( g , [ · , · ] g ) is a Liealgebra, ρ : g −→ gl ( V ) is a representation of g on a vector space V and T : V −→ g is a relative Rota-Baxter operator , i.e.(10) [ T u , T v ] g = T (cid:0) ρ ( T u )( v ) − ρ ( T v )( u ) (cid:1) , ∀ u , v ∈ V . EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 7
Note that a Rota-Baxter operator on a Lie algebra is a relative Rota-Baxter operator with respectto the adjoint representation.
Definition 2.11. (i) Let ( g , [ · , · ] g , T ) and ( g ′ , {· , ·} g ′ , T ′ ) be Rota-Baxter Lie algebras. A linearmap φ : g ′ → g is called a homomorphism of Rota-Baxter Lie algebras if φ is a Liealgebra homomorphism and φ ◦ T ′ = T ◦ φ. (ii) Let (( g , [ · , · ] g ) , ρ, T ) and (( g ′ , {· , ·} g ′ ) , ρ ′ , T ′ ) be two relative Rota-Baxter Lie algebras. A homomorphism from (( g ′ , {· , ·} g ′ ) , ρ ′ , T ′ ) to (( g , [ · , · ] g ) , ρ, T ) consists of a Lie algebra ho-momorphism φ : g ′ −→ g and a linear map ϕ : V ′ −→ V such that T ◦ ϕ = φ ◦ T ′ , (11) ϕρ ′ ( x )( u ) = ρ ( φ ( x ))( ϕ ( u )) , ∀ x ∈ g ′ , u ∈ V ′ . (12) In particular, if φ and ϕ are invertible, then ( φ, ϕ ) is called an isomorphism .Define a skew-symmetric bracket operation on the graded vector space ⊕ + ∞ k = Hom( ∧ k V , g ) by (cid:18) θ, φ (cid:19) : = ( − n − [[ µ + ρ, θ ] NR , φ ] NR , ∀ θ ∈ Hom( ∧ n V , g ) , φ ∈ Hom( ∧ m V , g ) . Proposition 2.12. ([53])
With the above notation, ( ⊕ + ∞ k = Hom( ∧ k V , g ) , ~ · , · (cid:127) ) is a graded Lie alge-bra. Its MC elements are precisely relative Rota-Baxter operators on ( g , [ · , · ] g ) with respect to therepresentation ( V ; ρ ) . (cid:3) Let T : V −→ g be a relative Rota-Baxter operator. By Proposition 2.12, T is an MC elementof the graded Lie algebra ( ⊕ + ∞ k = Hom( ∧ k V , g ) , ~ · , · (cid:127) ). It follows from graded Jacobi identity that d T : = ~ T , · (cid:127) is a graded derivation on the graded Lie algebra ( ⊕ + ∞ k = Hom( ∧ k V , g ) , ~ · , · (cid:127) ) satisfying d T =
0. Therefore we have
Theorem 2.13. ([53])
With the above notation, ( ⊕ + ∞ k = Hom( ∧ k V , g ) , ~ · , · (cid:127) , d T ) is a dg Lie algebra.Furthermore, T + T ′ is still a relative Rota-Baxter operator on the Lie algebra ( g , [ · , · ] g ) withrespect to the representation ( V ; ρ ) for T ′ : V −→ g if and only if T ′ is an MC element of the dgLie algebra ( ⊕ + ∞ k = Hom( ∧ k V , g ) , ~ · , · (cid:127) , d T ) . (cid:3) Now we define the cohomology governing deformations of a relative Rota-Baxter operator T : V → g . The spaces of 0-cochains C ( T ) and of 1-cochains C ( T ) are set to be 0. For n ≥ n -cochains C n ( T ) as C n ( T ) = Hom( ∧ n − V , g ).Define the coboundary operator δ : C n ( T ) → C n + ( T ) by(13) δθ = ( − n − ~ T , θ (cid:127) = ( − n − [[ µ + ρ, T ] NR , θ ] NR , ∀ θ ∈ Hom( ∧ n − V , g ) . By Proposition 2.12, ( ⊕ + ∞ n = C n ( T ) , δ ) is a cochain complex. Definition 2.14. ([53]) The cohomology of the cochain complex ( ⊕ + ∞ n = C n ( T ) , δ ) is called the co-homology of the relative Rota-Baxter operator T : V → g . The corresponding n -th cohomologygroup is denoted by H n ( T ).See [53] for explicit formulas of the coboundary operator δ.
3. M aurer -C artan characterization and deformations of relative R ota -B axter L ie algebras In this section, we apply Voronov’s higher derived brackets to construct the L ∞ -algebra thatcharacterizes relative Rota-Baxter Lie algebras as MC elements. Consequently, we obtain the L ∞ -algebra that controls deformations of a relative Rota-Baxter Lie algebra. ANDREY LAZAREV, YUNHE SHENG, AND RONG TANG L ∞ -algebras and higher derived brackets. The notion of an L ∞ -algebra was introducedby Stashe ff in [51]. See [34, 35] for more details. Definition 3.1. An L ∞ -algebra is a Z -graded vector space g = ⊕ k ∈ Z g k equipped with a collection( k ≥
1) of linear maps l k : ⊗ k g → g of degree 1 with the property that, for any homogeneouselements x , · · · , x n ∈ g , we have(i) (graded symmetry) for every σ ∈ S n , l n ( x σ (1) , · · · , x σ ( n − , x σ ( n ) ) = ε ( σ ) l n ( x , · · · , x n − , x n ) , (ii) (generalized Jacobi identity) for all n ≥ n X i = X σ ∈ S ( i , n − i ) ε ( σ ) l n − i + ( l i ( x σ (1) , · · · , x σ ( i ) ) , x σ ( i + , · · · , x σ ( n ) ) = . There is a canonical way to view a di ff erential graded Lie algebra as an L ∞ -algebra. Lemma 3.2.
Let ( g , [ · , · ] g , d ) be a dg Lie algebra. Then ( s − g , { l i } + ∞ i = ) is an L ∞ -algebra, wherel ( s − x ) = s − d ( x ) , l ( s − x , s − y ) = ( − x s − [ x , y ] g , l k = , for all k ≥ , and homogeneouselements x , y ∈ g . (cid:3) Definition 3.3. A weakly filtered L ∞ -algebra is a pair ( g , F • g ), where g is an L ∞ -algebra and F • g is a descending filtration of the graded vector space g such that g = F g ⊃ · · · ⊃ F n g ⊃ · · · and(i) there exists n ≥ k ≥ n it holds that l k ( g , · · · , g ) ⊂ F k g , (ii) g is complete with respect to this filtration, i.e. there is an isomorphism of graded vectorspaces g (cid:27) lim ←−− g / F n g . Definition 3.4.
The set of MC elements , denoted by MC ( g ), of a weakly filtered L ∞ -algebra( g , F • g ) is the set of those α ∈ g satisfying the MC equation + ∞ X k = k ! l k ( α, · · · , α ) = . (14)Let α be an MC element. Define l α k : ⊗ k g → g ( k ≥
1) by l α k ( x , · · · , x k ) = + ∞ X n = n ! l k + n ( α, · · · , α | {z } n , x , · · · , x k ) . (15) Remark 3.5.
The condition of being weakly filtered ensures convergence of the series figuringin the definition of MC elements and MC twistings above. Note that the notion of a filtered L ∞ -algebra is due to Dolgushev and Rogers [13]. For our purposes the weaker notion defined abovesu ffi ces.The following result is essentially contained in [22, Section 4]; that paper works with a di ff erenttype of L ∞ algebras than weakly filtered ones, but this does not a ff ect the arguments. Theorem 3.6.
With the above notation, ( g , { l α k } + ∞ k = ) is a weakly filtered L ∞ -algebra, obtained from g by twisting with the MC element α . Moreover, α + α ′ is an MC element of ( g , F • g ) if and only if α ′ is an MC element of the twisted L ∞ -algebra ( g , { l α k } + ∞ k = ) . (cid:3) One method for constructing explicit L ∞ -algebras is given by Voronov’s derived brackets [54].Let us recall this construction. EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 9
Definition 3.7. A V -data consists of a quadruple ( L , h , P , ∆ ) where • ( L , [ · , · ]) is a graded Lie algebra, • h is an abelian graded Lie subalgebra of ( L , [ · , · ]), • P : L → L is a projection, that is P ◦ P = P , whose image is h and kernel is a graded Liesubalgebra of ( L , [ · , · ]), • ∆ is an element in ker( P ) such that [ ∆ , ∆ ] = Theorem 3.8. ([54])
Let ( L , h , P , ∆ ) be a V-data. Then ( h , { l k } + ∞ k = ) is an L ∞ -algebra wherel k ( a , · · · , a k ) = P [ · · · [[ |{z} k ∆ , a ] , a ] , · · · , a k ] , for homogeneous a , · · · , a k ∈ h . (16) We call { l k } + ∞ k = the higher derived brackets of the V-data ( L , h , P , ∆ ) . (cid:3) There is also an L ∞ -algebra structure on a bigger space, which is used to study simultaneousdeformations of morphisms between Lie algebras in [5, 18, 19]. Theorem 3.9. ([54])
Let ( L , h , P , ∆ ) be a V-data. Then the graded vector space s − L ⊕ h is anL ∞ -algebra where l ( s − x , a ) = ( − s − [ ∆ , x ] , P ( x + [ ∆ , a ])) , l ( s − x , s − y ) = ( − x s − [ x , y ] , l k ( s − x , a , · · · , a k − ) = P [ · · · [[ x , a ] , a ] · · · , a k − ] , k ≥ , l k ( a , · · · , a k − , a k ) = P [ · · · [[ ∆ , a ] , a ] · · · , a k ] , k ≥ . Here a , a , · · · , a k are homogeneous elements of h and x , y are homogeneous elements of L. Allthe other L ∞ -algebra products that are not obtained from the ones written above by permutationsof arguments, will vanish. (cid:3) Remark 3.10. ([18]) Let L ′ be a graded Lie subalgebra of L that satisfies [ ∆ , L ′ ] ⊂ L ′ . Then s − L ′ ⊕ h is an L ∞ -subalgebra of the above L ∞ -algebra ( s − L ⊕ h , { l k } + ∞ k = ).3.2. The L ∞ -algebra that controls deformations of relative Rota-Baxter Lie algebras. Let g and V be two vector spaces. Then we have a graded Lie algebra ( ⊕ + ∞ n = C n ( g ⊕ V , g ⊕ V ) , [ · , · ] NR ).This graded Lie algebra gives rise to a V-data, and an L ∞ -algebra naturally. Proposition 3.11.
We have a V-data ( L , h , P , ∆ ) as follows: • the graded Lie algebra ( L , [ · , · ]) is given by (cid:0) ⊕ + ∞ n = C n ( g ⊕ V , g ⊕ V ) , [ · , · ] NR (cid:1) ; • the abelian graded Lie subalgebra h is given by (17) h : = ⊕ + ∞ n = C − | ( n + ( g ⊕ V , g ⊕ V ) = ⊕ + ∞ n = Hom( ∧ n + V , g ); • P : L → L is the projection onto the subspace h ; • ∆ = .Consequently, we obtain an L ∞ -algebra ( s − L ⊕ h , { l k } + ∞ k = ) , where l i are given byl ( s − Q , θ ) = P ( Q ) , l ( s − Q , s − Q ′ ) = ( − Q s − [ Q , Q ′ ] NR , l k ( s − Q , θ , · · · , θ k − ) = P [ · · · [ Q , θ ] NR , · · · , θ k − ] NR , for homogeneous elements θ, θ , · · · , θ k − ∈ h , homogeneous elements Q , Q ′ ∈ L and all the otherpossible combinations vanish.
Proof.
Note that h = ⊕ + ∞ n = C − | ( n + ( g ⊕ V , g ⊕ V ) = ⊕ + ∞ n = Hom( ∧ n + V , g ) . By Lemma 2.5, we deducethat h is an abelian subalgebra of ( L , [ · , · ]).Since P is the projection onto h , it is obvious that P ◦ P = P . It is also straightforward to seethat the kernel of P is a graded Lie subalgebra of ( L , [ · , · ]). Thus ( L , h , P , ∆ =
0) is a V-data.The other conclusions follows immediately from Theorem 3.9. (cid:3)
By Lemma 2.3, we obtain that(18) L ′ = ⊕ + ∞ n = C n | ( g ⊕ V , g ⊕ V ) , where C n | ( g ⊕ V , g ⊕ V ) = Hom( ∧ n + g , g ) ⊕ Hom( ∧ n g ⊗ V , V )is a graded Lie subalgebra of (cid:0) ⊕ + ∞ n = C n ( g ⊕ V , g ⊕ V ) , [ · , · ] NR (cid:1) . Corollary 3.12.
With the above notation, ( s − L ′ ⊕ h , { l i } + ∞ i = ) is an L ∞ -algebra, where l i are givenby l ( s − Q , s − Q ′ ) = ( − Q s − [ Q , Q ′ ] NR , l k ( s − Q , θ , · · · , θ k − ) = P [ · · · [ Q , θ ] NR , · · · , θ k − ] NR , for homogeneous elements θ , · · · , θ k − ∈ h , homogeneous elements Q , Q ′ ∈ L ′ , and all the otherpossible combinations vanish.Moreover, ( s − L ′ ⊕ h , { l i } + ∞ i = ) is weakly filtered with n = in the sense of Definition 3.3 with thefiltration given by F = s − L ′ ⊕ h , F = P [ s − L ′ ⊕ h , h ] NR , · · · , F k = P [ · · · [ |{z} k s − L ′ ⊕ h , h ] NR , · · · , h ] NR , · · · . Proof.
The stated formulas for the L ∞ -structure follow from Remark 3.10 and Proposition 3.11.To see that the given filtration satisfies the conditions of Definition 3.3 it su ffi ces to note that anyelement h ∈ h can be written as h = P + ∞ i = h i where h i ∈ Hom( ∧ i V , g ) and that the term h : V → g is nilpotent (even has square zero) when viewed as an endomorphism of g ⊕ V . (cid:3) Now we are ready to formulate the main result in this subsection.
Theorem 3.13.
Let g and V be two vector spaces, µ ∈ Hom( ∧ g , g ) , ρ ∈ Hom( g ⊗ V , V ) andT ∈ Hom( V , g ) . Then (( g , µ ) , ρ, T ) is a relative Rota-Baxter Lie algebra if and only if ( s − π, T ) isan MC element of the L ∞ -algebra ( s − L ′ ⊕ h , { l i } + ∞ i = ) given in Corollary 3.12, where π = µ + ρ ∈ C | ( g ⊕ V , g ⊕ V ) .Proof. Since ( s − L ′ ⊕ h ) = s − (cid:0) Hom( g ∧ g , g ) ⊕ Hom( g ⊗ V , V ) (cid:1) ⊕ Hom( V , g ) ⊂ F ( s − L ′ ⊕ h ), the MC equation is well defined. Let ( s − π, T ) be an MC element of ( s − L ′ ⊕ h , { l i } + ∞ i = ). By Lemma2.3 and Lemma 2.5, we have || [ π, T ] NR || = | , || [[ π, T ] NR , T ] NR || = − | , [[[ π, T ] NR , T ] NR , T ] NR = . Then, by Corollary 3.12, we have + ∞ X k = k ! l k (cid:16) ( s − π, T ) , · · · , ( s − π, T ) (cid:17) = l (cid:16) ( s − π, T ) , ( s − π, T ) (cid:17) + l (cid:16) ( s − π, T ) , ( s − π, T ) , ( s − π, T ) (cid:17) = (cid:16) − s −
12 [ π, π ] NR ,
12 [[ π, T ] NR , T ] NR (cid:17) = (0 , . EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 11
Thus, we obtain [ π, π ] NR = π, T ] NR , T ] NR = , which implies that ( g , µ ) is a Lie algebra,( V ; ρ ) is its representation and T is a relative Rota-Baxter operator on the Lie algebra ( g , µ ) withrespect to the representation ( V ; ρ ). (cid:3) Remark 3.14.
Since the axiom defining a relative Rota-Baxter Lie algebra is not quadratic, itcan be anticipated that the deformation complex of a Rota-Baxter Lie algebra is a fully-fledged L ∞ -algebra rather than a di ff erential graded Lie algebra.Let (( g , µ ) , ρ, T ) be a relative Rota-Baxter Lie algebra. Denote by π = µ + ρ ∈ C | ( g ⊕ V , g ⊕ V ).By Theorem 3.13, we obtain that ( s − π, T ) is an MC element of the L ∞ -algebra ( s − L ′ ⊕ h , { l i } + ∞ i = )given in Corollary 3.12. Now we are ready to give the L ∞ -algebra that controls deformations ofthe relative Rota-Baxter Lie algebra. Theorem 3.15.
With the above notation, we have the twisted L ∞ -algebra (cid:0) s − L ′ ⊕ h , { l ( s − π, T ) k } + ∞ k = (cid:1) associated to a relative Rota-Baxter Lie algebra (( g , µ ) , ρ, T ) , where π = µ + ρ .Moreover, for linear maps T ′ ∈ Hom( V , g ) , µ ′ ∈ Hom( ∧ g , g ) and ρ ′ ∈ Hom( g , gl ( V )) , the triple (( g , µ + µ ′ ) , ρ + ρ ′ , T + T ′ ) is again a relative Rota-Baxter Lie algebra if and only if (cid:0) s − ( µ ′ + ρ ′ ) , T ′ (cid:1) is an MC element of the twisted L ∞ -algebra (cid:0) s − L ′ ⊕ h , { l ( s − π, T ) k } + ∞ k = (cid:1) .Proof. If (( g , µ + µ ′ ) , ρ + ρ ′ , T + T ′ ) is a relative Rota-Baxter Lie algebra, then by Theorem 3.13, wededuce that ( s − ( µ + µ ′ + ρ + ρ ′ ) , T + T ′ ) is an MC element of the L ∞ -algebra given in Corollary 3.12.Moreover, by Theorem 3.6, we obtain that ( s − ( µ ′ + ρ ′ ) , T ′ ) is an MC element of the L ∞ -algebra (cid:0) s − L ′ ⊕ h , { l ( s − π, T ) k } + ∞ k = (cid:1) . (cid:3) Let ( g , µ ) be a Lie algebra and ( V ; ρ ) a representation of ( g , µ ). By Theorem 2.8 and Lemma3.2, we have an L ∞ -algebra structure on the graded vector space ⊕ + ∞ k = s − C k | ( g ⊕ V , g ⊕ V ).Let T : V −→ g be a relative Rota-Baxter operator on a Lie algebra ( g , µ ) with respect to arepresentation ( V ; ρ ). By Theorem 2.13 and Lemma 3.2, we have an L ∞ -algebra structure on thegraded vector space ⊕ + ∞ k = Hom( ∧ k V , g ).The above L ∞ -algebras are related as follows. Theorem 3.16.
Let (( g , µ ) , ρ, T ) be a relative Rota-Baxter Lie algebra. Then the L ∞ -algebra (cid:0) s − L ′ ⊕ h , { l ( s − π, T ) k } + ∞ k = (cid:1) is a strict extension of the L ∞ -algebra ⊕ + ∞ k = s − C k | ( g ⊕ V , g ⊕ V ) by the L ∞ -algebra ⊕ + ∞ k = Hom( ∧ k V , g ) , that is, we have the following short exact sequence of L ∞ -algebras: (19) 0 −→ ⊕ + ∞ k = Hom( ∧ k V , g ) ι −→ s − L ′ ⊕ h p −→ ⊕ + ∞ k = s − C k | ( g ⊕ V , g ⊕ V ) −→ , where ι ( θ ) = (0 , θ ) and p ( s − f , θ ) = s − f .Proof. For any ( s − f , θ ) ∈ ( s − L ′ ⊕ h ) n − , by Lemma 2.3 and Lemma 2.5, we obtain that || [ π, θ ] NR || = | ( n − , || [[ π, T ] NR , θ ] NR || = − | n , [[[ π, T ] NR , T ] NR , θ ] NR = . Moreover, for 1 ≤ k ≤ n , we have || [ · · · [[ |{z} k f , T ] NR , T ] NR , · · · , T ] NR || = ( n − − k ) | k , and for n + ≤ k , we have [ · · · [[ |{z} k f , T ] NR , T ] NR , · · · , T ] NR = . Therefore, we have l ( s − π, T )1 ( s − f , θ ) = + ∞ X k = k ! l k + (cid:0) ( s − π, T ) , · · · , ( s − π, T ) | {z } k , ( s − f , θ ) (cid:1) = l ( s − π, s − f ) + l ( s − π, T , θ ) + n ! l n + ( f , T , · · · , T | {z } n ) = (cid:0) − s − [ π, f ] NR , [[ π, T ] NR , θ ] NR + n ! [ · · · [[ |{z} n f , T ] NR , T ] NR , · · · , T ] NR (cid:1) . (20)For any ( s − f , θ ) ∈ ( s − L ′ ⊕ h ) n − , ( s − f , θ ) ∈ ( s − L ′ ⊕ h ) n − , we have || f || = ( n − | , || θ || = − | ( n − , || f || = ( n − | , || θ || = − | ( n − . By Lemma 2.3, || [[ π, θ ] NR , θ ] NR || = − | ( n + n − , || [ f , θ ] NR || = ( n − | ( n − , || [ f , θ ] NR || = ( n − | ( n − . By Lemma 2.5, for 1 ≤ k , we have[[[ · · · [[ π, T ] NR , T ] NR · · · , T | {z } k ] NR , θ ] NR , θ ] NR = . By Lemma 2.3, for 1 ≤ k ≤ n −
1, we obtain that || [[ · · · [[ f , T ] NR , T ] NR · · · , T | {z } k ] NR , θ ] NR || = ( n − k − | ( n + k − . By Lemma 2.5, for n ≤ k , we have [[ · · · [[ f , T ] NR , T ] NR · · · , T | {z } k ] NR , θ ] NR = . By Lemma 2.3, for1 ≤ k ≤ n −
1, we obtain that || [[ · · · [[ f , T ] NR , T ] NR · · · , T | {z } k ] NR , θ ] NR || = ( n − k − | ( n + k − . ByLemma 2.5, for n ≤ k , we have [[ · · · [[ f , T ] NR , T ] NR · · · , T | {z } k ] NR , θ ] NR = . Therefore, we have l ( s − π, T )2 (cid:0) ( s − f , θ ) , ( s − f , θ ) (cid:1) = + ∞ X k = k ! l k + (cid:0) ( s − π, T ) , · · · , ( s − π, T ) | {z } k , ( s − f , θ ) , ( s − f , θ ) (cid:1) = l ( s − f , s − f ) + l ( s − π, θ , θ ) + n − l n + ( f , T , · · · , T | {z } n − , θ ) + ( − n n n − l n + ( f , T , · · · , T | {z } n − , θ ) = (cid:16) ( − n − s − [ f , f ] NR , [[ π, θ ] NR , θ ] NR + n − · · · [[ f , T ] NR , T ] NR · · · , T | {z } n − ] NR , θ ] NR + ( − n n n − · · · [[ f , T ] NR , T ] NR · · · , T | {z } n − ] NR , θ ] NR (cid:17) . Similarly, for m ≥
3, ( s − f i , θ i ) ∈ ( s − L ′ ⊕ h ) n i − , 1 ≤ i ≤ m , we have l ( s − π, T ) m (cid:0) ( s − f , θ ) , · · · , ( s − f m , θ m ) (cid:1) = (cid:16) , m X i = ( − α n i + − m )! [ · · · [ f i , T ] NR , · · · , T | {z } n i + − m ] NR , θ ] NR , · · · , θ i − ] NR , θ i + ] NR , · · · , θ m ] NR (cid:17) , where α = n i ( n + · · · + n i − ). Thus, ι and p are strict morphisms between L ∞ -algebras and satisfy p ◦ ι = . (cid:3) EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 13
4. C ohomology and infinitesimal deformations of relative R ota -B axter L ie algebras In this section, (( g , µ ) , ρ, T ) is a relative Rota-Baxter Lie algebra, i.e. ρ : g → gl ( V ) is arepresentation of the Lie algebra ( g , µ ) and T : V → g is a relative Rota-Baxter operator. Wedefine the cohomology of relative Rota-Baxter Lie algebras and show that the two-dimensionalcohomology groups classify infinitesimal deformations. We also establish a relationship betweenthe cohomology of relative Rota-Baxter Lie algebras and the cohomology of triangular Lie bial-gebras.4.1. Cohomology of relative Rota-Baxter Lie algebras.
We define the cohomology of a rela-tive Rota-Baxter Lie algebra using the twisted L ∞ -algebra given in Theorem 3.15.By Theorem 3.15, (cid:0) s − L ′ ⊕ h , { l ( s − π, T ) k } + ∞ k = (cid:1) is an L ∞ -algebra, where π = µ + ρ , h and L ′ are givenby (17) and (18) respectively. In particular, we have Lemma 4.1. (cid:0) s − L ′ ⊕ h , l ( s − π, T )1 (cid:1) is a complex, i.e. l ( s − π, T )1 ◦ l ( s − π, T )1 = . Proof.
Since (cid:0) s − L ′ ⊕ h , { l ( s − π, T ) k } + ∞ k = (cid:1) is an L ∞ -algebra, we have l ( s − π, T )1 ◦ l ( s − π, T )1 = . (cid:3) Define the set of 0-cochains C ( g , ρ, T ) to be 0, and define the set of 1-cochains C ( g , ρ, T ) tobe gl ( g ) ⊕ gl ( V ). For n ≥
2, define the space of n -cochains C n ( g , ρ, T ) by C n ( g , ρ, T ) : = C n ( g , ρ ) ⊕ C n ( T ) = C ( n − | ( g ⊕ V , g ⊕ V ) ⊕ C − | ( n − ( g ⊕ V , g ⊕ V ) = (cid:16) Hom( ∧ n g , g ) ⊕ Hom( ∧ n − g ⊗ V , V ) (cid:17) ⊕ Hom( ∧ n − V , g ) . Define the coboundary operator D : C n ( g , ρ, T ) → C n + ( g , ρ, T ) by(21) D ( f , θ ) = ( − n − (cid:0) − [ π, f ] NR , [[ π, T ] NR , θ ] NR + n ! [ · · · [[ |{z} n f , T ] NR , T ] NR , · · · , T ] NR (cid:1) , where f ∈ Hom( ∧ n g , g ) ⊕ Hom( ∧ n − g ⊗ V , V ) and θ ∈ Hom( ∧ n − V , g ) . Theorem 4.2.
With the above notation, ( ⊕ + ∞ n = C n ( g , ρ, T ) , D ) is a cochain complex, i.e. D ◦ D = . Proof.
For any ( f , θ ) ∈ C n ( g , ρ, T ), we have ( s − f , θ ) ∈ ( s − L ′ ⊕ h ) n − . By (21), we deduce that D ( f , θ ) = ( − n − l ( s − π, T )1 ( s − f , θ ) . By Lemma 4.1, we obtain that ( ⊕ + ∞ n = C n ( g , ρ, T ) , D ) is a cochain complex. (cid:3) Definition 4.3.
The cohomology of the cochain complex ( ⊕ + ∞ n = C n ( g , ρ, T ) , D ) is called the co-homology of the relative Rota-Baxter Lie algebra (( g , µ ) , ρ, T ). We denote its n -th cohomologygroup by H n ( g , ρ, T )Define a linear operator h T : C n ( g , ρ ) → C n + ( T ) by h T f : = ( − n − n ! [ · · · [[ |{z} n f , T ] NR , T ] NR , · · · , T ] NR . (22)By (21) and (22), the coboundary operator can be written as(23) D ( f , θ ) = ( ∂ f , δθ + h T f ) , where ∂ is given by (6), and δ is given by (13). More precisely,( δθ )( v , · · · , v n ) = n X i = ( − i + [ T v i , θ ( v , · · · , ˆ v i , · · · , v n )] g + n X i = ( − i + T ρ ( θ ( v , · · · , ˆ v i , · · · , v n ))( v i )(24) + X ≤ i < j ≤ n ( − i + j θ ( ρ ( T v i )( v j ) − ρ ( T v j )( v i ) , v , · · · , ˆ v i , · · · , ˆ v j , · · · , v n ) . Now we give the formulas for h T in terms of multilinear maps. Lemma 4.4.
The operator h T : Hom( ∧ n g , g ) ⊕ Hom( ∧ n − g ⊗ V , V ) → Hom( ∧ n V , g ) is given by ( h T f )( v , · · · , v n ) = ( − n f g ( T v , · · · , T v n ) + n X i = ( − i + T f V (cid:0) T v , · · · , T v i − , T v i + , · · · , T v n , v i (cid:1) , (25) where f = ( f g , f V ) , and f g ∈ Hom( ∧ n g , g ) , f V ∈ Hom( ∧ n − g ⊗ V , V ) and v , · · · , v n ∈ V . Proof.
By Remark 2.2, it is convenient to view the elements of ⊕ + ∞ n = C n ( g ⊕ V ; g ⊕ V ) as coderiva-tions of ¯ S c (cid:0) s − ( g ⊕ V ) (cid:1) . The coderivations corresponding to f and T will be denoted by ¯ f and ¯ T respectively. Then, by induction, we have[ · · · [[ |{z} n f , T ] NR , T ] NR , · · · , T ] NR (cid:0) ( x , v ) , · · · , ( x n , v n ) (cid:1) = n X i = ( − i ni !(cid:0) ¯ T ◦ · · · ◦ ¯ T | {z } i ◦ ( ¯ f g + ¯ f V ) ◦ ¯ T · · · ◦ ¯ T | {z } n − i (cid:1)(cid:0) ( x , v ) , · · · , ( x n , v n ) (cid:1) = (cid:0) n ! f g ( T v , · · · , T v n ) , (cid:1) + (cid:16) ( − n n X k = ( − n − i ( n − T f V (cid:0) T v , · · · , T v i − , T v i + , · · · , T v n , v i (cid:1) , (cid:17) = (cid:0) n ! f g ( T v , · · · , T v n ) , (cid:1) + (cid:16) n ! n X k = ( − n − i + T f V (cid:0) T v , · · · , T v i − , T v i + , · · · , T v n , v i (cid:1) , (cid:17) , which implies that (25) holds. (cid:3) The formula of the coboundary operator D can be well-explained by the following diagram: · · · −→ C n ( g , ρ ) h T ' ' ❖❖❖❖❖❖❖❖❖❖❖ ∂ / / C n + ( g , ρ ) h T ( ( PPPPPPPPPPPP ∂ / / C n + ( g , ρ ) −→ · · ·· · · −→ C n ( T ) δ / / C n + ( T ) δ / / C n + ( T ) −→ · · · . Theorem 4.5.
Let (( g , µ ) , ρ, T ) be a relative Rota-Baxter Lie algebra. Then there is a short exactsequence of the cochain complexes: −→ ( ⊕ + ∞ n = C n ( T ) , δ ) ι −→ ( ⊕ + ∞ n = C n ( g , ρ, T ) , D ) p −→ ( ⊕ + ∞ n = C n ( g , ρ ) , ∂ ) −→ , where ι and p are the inclusion map and the projection map.Consequently, there is a long exact sequence of the cohomology groups: · · · −→ H n ( T ) H n ( ι ) −→ H n ( g , ρ, T ) H n ( p ) −→ H n ( g , ρ ) c n −→ H n + ( T ) −→ · · · , where the connecting map c n is defined by c n ([ α ]) = [ h T α ] , for all [ α ] ∈ H n ( g , ρ ) . EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 15
Proof.
By (23), we have the short exact sequence of chain complexes which induces a long exactsequence of cohomology groups. Also by (23), c n is given by c n ([ α ]) = [ h T α ] . (cid:3) Infinitesimal deformations of relative Rota-Baxter Lie algebras.
In this subsection, weintroduce the notion of R -deformations of relative Rota-Baxter Lie algebras, where R is a localpro-Artinian K -algebra. Since R is the projective limit of local Artinian K -algebras, R is equippedwith an augmentation ǫ : R → K . See [14, 31] for more details about R -deformation theory ofalgebraic structures. Then we restrict our study to infinitesimal deformations, i.e. R = K [ t ] / ( t ),using the cohomology theory introduced in Section 4.1.Replacing the K -vector spaces and K -linear maps by R -modules and R -linear maps in Defi-nition 2.10 and Definition 2.11, it is straightforward to obtain the definitions of R -relative Rota-Baxter Lie algebras and homomorphisms between them.Any relative Rota-Baxter Lie algebra (( g , [ · , · ] g ) , ρ, T ) can be viewed as an R -relative Rota-Baxter Lie algebra with the help of the augmentation map ǫ . More precisely, the R -modulestructure on g and V are given by r · x : = ǫ ( r ) x , r · u : = ǫ ( r ) u , ∀ r ∈ R , x ∈ g , u ∈ V . Definition 4.6. An R -deformation of a relative Rota-Baxter Lie algebra (( g , [ · , · ] g ) , ρ, T ) consistsof an R -Lie algebra structure [ · , · ] R on the tensor product R ⊗ K g , an R -Lie algebra homomor-phism ρ R : R ⊗ K g → gl R ( R ⊗ K V ) and an R -linear map T R : R ⊗ K V → R ⊗ K g , which isa relative Rota-Baxter operator such that ( ǫ ⊗ K Id g , ǫ ⊗ K Id V ) is an R -relative Rota-Baxter Liealgebra homomorphism from (( R ⊗ K g , [ · , · ] R ) , ρ R , T R ) to (( g , [ · , · ] g ) , ρ, T ).Thereafter, we denote an R -deformation of (( g , [ · , · ] g ) , ρ, T ) by a triple (( R ⊗ K g , [ · , · ] R ) , ρ R , T R ).Next we discuss equivalences between R -deformations. Definition 4.7.
Let (( R ⊗ K g , [ · , · ] R ) , ρ R , T R ) and (( R ⊗ K g , [ · , · ] ′ R ) , ρ ′ R , T ′ R ) be two R -deformationsof a relative Rota-Baxter Lie algebra (( g , [ · , · ] g ) , ρ, T ). We call them equivalent if there existsan R -relative Rota-Baxter Lie algebra isomorphism ( φ, ϕ ) : (( R ⊗ K g , [ · , · ] ′ R ) , ρ ′ R , T ′ R ) → (( R ⊗ K g , [ · , · ] R ) , ρ R , T R ) such that( ǫ ⊗ K Id g , ǫ ⊗ K Id V ) = ( ǫ ⊗ K Id g , ǫ ⊗ K Id V ) ◦ ( φ, ϕ ) . (26) Definition 4.8. A K [ t ] / ( t )-deformation of the relative Rota-Baxter Lie algebra (( g , [ · , · ] g ) , ρ, T )is call an infinitesimal deformation .Let R = K [ t ] / ( t ) and (( R ⊗ K g , [ · , · ] R ) , ρ R , T R ) be an infinitesimal deformation of (( g , [ · , · ] g ) , ρ, T ).Since (( R ⊗ K g , [ · , · ] R ) , ρ R , T R ) is an R -relative Rota-Baxter Lie algebra, there exist ω , ω ∈ Hom( g ∧ g , g ), ̺ , ̺ ∈ gl ( V ) and T , T ∈ Hom( V , g ) such that[ · , · ] R = ω + t ω , ρ R = ̺ + t ̺ , T R = T + t T . (27)Since ( ǫ ⊗ K Id g , ǫ ⊗ K Id V ) is an R -relative Rota-Baxter Lie algebra homomorphism from (( R ⊗ K g , [ · , · ] R ) , ρ R , T R ) to (( g , [ · , · ] g ) , ρ, T ), we deduce that ω = [ · , · ] g , ̺ = ρ, T = T . Therefore, an infinitesimal deformation of (( g , [ · , · ] g ) , ρ, T ) is determined by the triple ( ω , ̺ , T ).Now we analyze the conditions on ( ω , ̺ , T ). First by the fact that ( R ⊗ K g , [ · , · ] g + t ω ) is an R -Lie algebra, we get d CE ω = . (28) Then since ( R ⊗ K V ; ρ + t ̺ ) is a representation of ( R ⊗ K g , [ · , · ] g + t ω ), we obtain ρ ( ω ( x , y )) + ̺ ([ x , y ] g ) = [ ρ ( x ) , ̺ ( y )] + [ ̺ ( x ) , ρ ( y )] . (29)Finally by the fact that T + t T is an R -linear relative Rota-Baxter operator on the R -Lie algebra( R ⊗ K g , [ · , · ] g + t ω ) with respect to the representation ( R ⊗ K V ; ρ + t ̺ ), we obtain[ T u , T v ] g + [ T u , T v ] g + ω ( T u , T v ) = T (cid:16) ρ ( T u ) v − ρ ( T v ) u + ̺ ( T u ) v − ̺ ( T v ) u (cid:17) (30) + T (cid:16) ρ ( T u ) v − ρ ( T v ) u (cid:17) . Proposition 4.9.
The triple ( ω , ̺ , T ) determines an infinitesimal deformation of the relativeRota-Baxter Lie algebra (( g , [ · , · ] g ) , ρ, T ) if and only if ( ω , ̺ , T ) is a -cocycle of the relativeRota-Baxter Lie algebra (( g , [ · , · ] g ) , ρ, T ) .Proof. By (28), (29) and (30), ( ω , ̺ , T ) is a 2-cocycle if and only if ( ω , ̺ , T ) determines aninfinitesimal deformation of the relative Rota-Baxter Lie algebra (( g , [ · , · ] g ) , ρ, T ). (cid:3) If two infinitesimal deformations determined by ( ω , ̺ , T ) and ( ω ′ , ̺ ′ , T ′ ) are equivalent,then there exists an R -relative Rota-Baxter Lie algebra isomorphism ( φ, ϕ ) from (( R ⊗ K g , [ · , · ] g + t ω ′ ) , ρ + t ̺ ′ , T + t T ′ ) to (( R ⊗ K g , [ · , · ] g + t ω ) , ρ + t ̺ , T + t T ). By (26), we deduce that φ = Id g + tN , ϕ = Id V + tS , where N ∈ gl ( g ) , S ∈ gl ( V ) . (31)Since Id g + tN is an isomorphism from ( R ⊗ K g , [ · , · ] g + t ω ′ ) to ( R ⊗ K g , [ · , · ] g + t ω ), we get(32) ω ′ − ω = d CE N . By the equality (Id V + tS )( ρ + t ̺ ′ )( y ) u = ( ρ + t ̺ ) (cid:0) (Id g + tN ) y )(Id V + tS ) u , we deduce that(33) ̺ ′ ( y ) u − ̺ ( y ) u = ρ ( Ny ) u + ρ ( y ) S u − S ρ ( y ) u , ∀ y ∈ g , u ∈ V . By the equality (Id g + tN ) ◦ ( T + t T ′ ) = ( T + t T ) ◦ (Id V + tS ), we obtain(34) T ′ − T = − N ◦ T + T ◦ S . Theorem 4.10.
There is a one-to-one correspondence between equivalence classes of infinites-imal deformations of the relative Rota-Baxter Lie algebra (( g , [ · , · ] g ) , ρ, T ) and the second coho-mology group H ( g , ρ, T ) .Proof. By (32), (33) and (34), we deduce that( ω ′ , ̺ ′ , T ′ ) − ( ω , ̺ , T ) = D ( N , S ) , which implies that ( ω , ̺ , T ) and ( ω ′ , ̺ ′ , T ′ ) are in the same cohomology class if and only ifthe corresponding infinitesimal deformations of (( g , [ · , · ] g ) , ρ, T ) are equivalent. (cid:3) Remark 4.11.
One can study deformations of relative Rota-Baxter Lie algebras over more gen-eral bases such as R = K [ t ] / ( t n ), R = K [[ t ]] = lim ←−− n K [ t ] / ( t n ) or indeed over di ff erential gradedlocal pro-Artinian rings. EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 17
Cohomology of Rota-Baxter Lie algebras.
In this subsection, we define the cohomologyof Rota-Baxter Lie algebras with the help of the general framework of the cohomology of relativeRota-Baxter Lie algebras.Let ( g , [ · , · ] g , T ) be a Rota-Baxter Lie algebra. We define the set of 0-cochains C ( g , T ) to be0, and define the set of 1-cochains C ( g , T ) to be C ( g , T ) : = Hom( g , g ). For n ≥
2, define thespace of n -cochains C n RB ( g , T ) by C n RB ( g , T ) : = C n Lie ( g ; g ) ⊕ C n ( T ) = Hom( ∧ n g , g ) ⊕ Hom( ∧ n − g , g ) . Define the embedding i : C n RB ( g , T ) → C n ( g , ad , T ) by i ( f , θ ) = ( f , f , θ ) , ∀ f ∈ Hom( ∧ n g , g ) , θ ∈ Hom( ∧ n − g , g ) . Denote by Im n ( i ) = i ( C n RB ( g , T )). Then we have Proposition 4.12.
With the above notation, ( ⊕ + ∞ n = Im n ( i ) , D ) is a subcomplex of the cochain com-plex ( ⊕ + ∞ n = C n ( g , ad , T ) , D ) associated to the relative Rota-Baxter Lie algebra (( g , [ · , · ] g ) , ad , T ) .Proof. Let ( f , f , θ ) ∈ Im n ( i ). By the definition of D , we have D ( f , f , θ ) = (( ∂ ( f , f )) g , ( ∂ ( f , f )) V , δθ + h T ( f , f )) . By (7), we have ( ∂ ( f , f )) g = d CE f . And for any x , · · · , x n + ∈ g , we have( ∂ ( f , f )) V ( x , · · · , x n + ) = X ≤ i < j ≤ n ( − i + j f ([ x i , x j ] g , x , · · · , ˆ x i , · · · , ˆ x j , · · · , x n , x n + ) + ( − n − [ f ( x , · · · , x n ) , x n + ] g + n X i = ( − i + (cid:16) [ x i , f ( x , · · · , ˆ x i , · · · , x n , x n + )] g − f (cid:0) x , · · · , ˆ x i , · · · , x n , [ x i , x n + ] g (cid:1)(cid:17) = (d CE f )( x , · · · , x n + ) . Thus, we obtain D ( f , f , θ ) = (d CE f , d CE f , δθ + h T ( f , f )) = i (d CE f , δθ + h T ( f , f )), which impliesthat ( ⊕ n Im n ( i ) , D ) is a subcomplex. (cid:3) We define the projection p : Im n ( i ) → C n RB ( g , T ) by p ( f , f , θ ) = ( f , θ ) , ∀ f ∈ Hom( ∧ n g , g ) , θ ∈ Hom( ∧ n − g , g ) . Then for n ≥ , we define D RB : C n RB ( g , T ) → C n + ( g , T ) by D RB = p ◦ D ◦ i . More precisely, D RB ( f , θ ) = (cid:16) d CE f , δθ + Ω f (cid:17) , ∀ f ∈ Hom( ∧ n g , g ) , θ ∈ Hom( ∧ n − g , g ) , (35)where δ is given by (24) and Ω : Hom( ∧ n g , g ) → Hom( ∧ n g , g ) is defined by( Ω f )( x , · · · , x n ) = ( − n (cid:16) f ( T x , · · · , T x n ) − n X i = T f ( T x , · · · , T x i − , x i , T x i + , · · · , T x n ) (cid:17) . Theorem 4.13.
The map D RB is a coboundary operator, i.e. D RB ◦ D RB = .Proof. By Proposition 4.12 and i ◦ p = Id, we have D RB ◦ D RB = p ◦ D ◦ i ◦ p ◦ D ◦ i = p ◦ D ◦ D ◦ i = , which finishes the proof. (cid:3) Definition 4.14.
Let ( g , [ · , · ] g , T ) be a Rota-Baxter Lie algebra. The cohomology of the cochaincomplex ( ⊕ + ∞ n = C n RB ( g , T ) , D RB ) is taken to be the cohomology of the Rota-Baxter Lie algebra ( g , [ · , · ] g , T ). Denote the n -th cohomology group by H n RB ( g , T ) . Theorem 4.15.
There is a short exact sequence of the cochain complexes: −→ ( ⊕ + ∞ n = C n ( T ) , δ ) ι −→ ( ⊕ + ∞ n = C n RB ( g , T ) , D RB ) p −→ ( ⊕ + ∞ n = C n Lie ( g ; g ) , d CE ) −→ , where ι ( θ ) = (0 , θ ) and p ( f , θ ) = f for all f ∈ Hom( ∧ n g , g ) and θ ∈ Hom( ∧ n − g , g ) .Consequently, there is a long exact sequence of the cohomology groups: · · · −→ H n ( T ) H n ( ι ) −→ H n RB ( g , T ) H n ( p ) −→ H n Lie ( g , g ) c n −→ H n + ( T ) −→ · · · , where the connecting map c n is defined by c n ([ α ]) = [ Ω α ] , for all [ α ] ∈ H n Lie ( g , g ) . Proof.
By (35), we have the short exact sequence of cochain complexes which induces a longexact sequence of cohomology groups. (cid:3)
Remark 4.16.
The approach used to define D RB , can be also used to obtain the L ∞ -algebra struc-ture { l k } + ∞ k = on ⊕ n C n RB ( g , T ) controlling deformations of Rota-Baxter Lie algebras. By Theorem3.15, we have the L ∞ -algebra ( ⊕ n C n ( g , ad , T ) , { l k } + ∞ k = ) which controls deformations of the relativeRota-Baxter Lie algebra ( g , ad , T ). Define l k by l k ( X , · · · , X k ) : = p l k ( i ( X ) , · · · , i ( X k )) , for all homogeneous elements X i ∈ ⊕ n C n RB ( g , T ) . Then ( ⊕ n C n RB ( g , T ) , { l k } + ∞ k = ) is an L ∞ -algebraembedded into ( ⊕ n C n ( g , ad , T ) , { l k } + ∞ k = ) as an L ∞ -subalgebra. Remark 4.17.
Similarly to the study of infinitesimal deformations of relative Rota-Baxter Liealgebras, we can show that infinitesimal deformations of Rota-Baxter Lie algebras are classifiedby the second cohomology group H ( g , R ) . We omit the details.4.4.
Cohomology and infinitesimal deformations of triangular Lie bialgebras.
In this sub-section, all vector spaces are assumed to be finite-dimensional. First we define the cohomologyof triangular Lie bialgebras with the help of the general cohomological framework for relativeRota-Baxter Lie algebras. Then we establish the standard classification result for infinitesimaldeformations of triangular Lie bialgebras using this cohomology theory.Recall that a Lie bialgebra is a vector space g equipped with a Lie algebra structure [ · , · ] g : ∧ g −→ g and a Lie coalgebra structure δ : g −→ ∧ g such that δ is a 1-cocycle on g withcoe ffi cients in ∧ g . The Lie bracket [ · , · ] g in a Lie algebra g naturally extends to the Schouten-Nijenhuis bracket [ · , · ] SN on ∧ • g = ⊕ k ≥ ∧ k + g . More precisely, we have[ x ∧ · · · ∧ x p , y ∧ · · · ∧ y q ] SN = X ≤ i ≤ p ≤ j ≤ q ( − i + j [ x i , y j ] g ∧ x ∧ · · · ˆ x i · · · ∧ x p ∧ y ∧ · · · ˆ y j · · · ∧ y q . An element r ∈ ∧ g is called a skew-symmetric r-matrix [47] if it satisfies the classical Yang-Baxter equation [ r , r ] SN =
0. It is well known [33] that r satisfies the classical Yang-Baxterequation if and only if r ♯ is a relative Rota-Baxter operator on g with respect to the coadjointrepresentation, where r ♯ : g ∗ → g is defined by h r ♯ ( ξ ) , η i = h r , ξ ∧ η i for all ξ, η ∈ g ∗ .Let r be a skew-symmetric r -matrix. Define δ r : g −→ ∧ g by δ r ( x ) = [ x , r ] SN , for all x ∈ g . Then ( g , [ · , · ] g , δ r ) is a Lie bialgebra, which is called a triangular Lie bialgebra . From now on, wewill denote a triangular Lie bialgebra by ( g , [ · , · ] g , r ). EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 19
Definition 4.18.
Let ( g , [ · , · ] g , r ) and ( g , {· , ·} g , r ) be triangular Lie bialgebras. A linear map φ : g → g is called a homomorphism from ( g , {· , ·} g , r ) to ( g , [ · , · ] g , r ) if φ is a Lie algebrahomomorphism and ( φ ⊗ φ )( r ) = r . If φ is invertible, then φ is called an isomorphism betweentriangular Lie bialgebras.The above definition is consistent with the equivalence between r -matrices given in [10].Let g be a Lie algebra and r ∈ ∧ g a skew-symmetric r -matrix. Define the set of 0-cochainsand 1-cochains to be zero and define the set of k -cochains to be ∧ k g . Define d r : ∧ k g → ∧ k + g by(36) d r χ = [ r , χ ] SN , ∀ χ ∈ ∧ k g . Then d r = . Denote by H k ( r ) the corresponding k -th cohomology group, called the k-th coho-mology group of the skew-symmetric r-matrix r .For any k ≥
1, define Ψ : ∧ k + g −→ Hom( ∧ k g ∗ , g ) by(37) h Ψ ( χ )( ξ , · · · , ξ k ) , ξ k + i = h χ, ξ ∧ · · · ∧ ξ k ∧ ξ k + i , ∀ χ ∈ ∧ k + g , ξ , · · · , ξ k + ∈ g ∗ . By [53, Theorem 7.7], we have(38) Ψ (d r χ ) = δ ( Ψ ( χ )) , ∀ χ ∈ ∧ k g . Thus (Im( Ψ ) , δ ) is a subcomplex of the cochain complex ( ⊕ k C k ( r ♯ ) , δ ) associated to the relativeRota-Baxter operator r ♯ , where Im( Ψ ) : = ⊕ k { Ψ ( χ ) |∀ χ ∈ ∧ k g } and δ is the coboundary operatorgiven by (13) for the relative Rota-Baxter operator r ♯ .In the following, we define the cohomology of a triangular Lie bialgebra ( g , [ · , · ] g , r ). Wedefine the set of 0-cochains C ( g , r ) to be 0, and define the set of 1-cochains to be C ( g , r ) : = Hom( g , g ). For n ≥
2, define the space of n -cochains C n TLB ( g , r ) by C n TLB ( g , r ) : = Hom( ∧ n g , g ) ⊕ ∧ n g . Define the embedding i : C n TLB ( g , r ) → C n ( g , ad ∗ , r ♯ ) = Hom( ∧ n g , g ) ⊕ Hom( ∧ n − g ⊗ g ∗ , g ∗ ) ⊕ Hom( ∧ n − g ∗ , g ) by i ( f , χ ) = ( f , f ⋆ , Ψ ( χ )) , ∀ f ∈ Hom( ∧ n g , g ) , χ ∈ ∧ n g , where f ⋆ ∈ Hom( ∧ n − g ⊗ g ∗ , g ∗ ) is defined by h f ⋆ ( x , · · · , x n − , ξ ) , x n i = −h ξ, f ( x , · · · , x n − , x n ) i . (39)Denote by Im n ( i ) the image of i , i.e. Im n ( i ) : = { i ( f , χ ) |∀ ( f , χ ) ∈ C n TLB ( g , r ) } . Proposition 4.19.
With the above notation, ( ⊕ n Im n ( i ) , D ) is a subcomplex of the cochain complex ( C n ( g , ad ∗ , r ♯ ) , D ) associated to the relative Rota-Baxter Lie algebra (( g , [ · , · ] g ) , ad ∗ , r ♯ ) .Proof. Let ( f , f ⋆ , Ψ ( χ )) ∈ Im n ( i ). By the definition of D , we have D ( f , f ⋆ , Ψ ( χ )) = (( ∂ ( f , f ⋆ )) g , ( ∂ ( f , f ⋆ )) g ∗ , δ Ψ ( χ ) + h r ♯ ( f , f ⋆ )) . By (7), we have ( ∂ ( f , f ⋆ )) g = d CE f . By (8), we can deduce that ( ∂ ( f , f ⋆ )) g ∗ = (d CE f ) ⋆ . By (24)and (25), we can deduce that δ Ψ ( χ ) + h r ♯ ( f , f ⋆ ) ∈ Im( Ψ ) . Thus, we obtain D (( f , f ⋆ , Ψ ( χ ))) = (d CE f , (d CE f ) ⋆ , δ Ψ ( χ ) + h r ♯ ( f , f ⋆ )) = i (cid:16) d CE f , Ψ − (cid:0) δ Ψ ( χ ) + h r ♯ ( f , f ⋆ ) (cid:1)(cid:17) , which implies that ( ⊕ n Im n ( i ) , D ) is a subcomplex. (cid:3) Define the projection p : Im n ( i ) → C n TLB ( g , r ) by p ( f , f ⋆ , θ ) = ( f , θ ♭ ) , ∀ f ∈ Hom( ∧ n g , g ) , θ ∈ { Ψ ( χ ) |∀ χ ∈ ∧ n g } , where θ ♭ ∈ ∧ n g is defined by h θ ♭ , ξ ∧ · · · ∧ ξ n i = h θ ( ξ , · · · , ξ n − ) , ξ n i . Define the coboundaryoperator D TLB : C n TLB ( g , r ) → C n + ( g , r ) for a triangular Lie bialgebra by D TLB = p ◦ D ◦ i . Theorem 4.20.
The map D TLB is a coboundary operator, i.e. D TLB ◦ D
TLB = .Proof. Since i ◦ p = Id when restricting on the image of i , we have D TLB ◦ D
TLB = p ◦ D ◦ i ◦ p ◦ D ◦ i = p ◦ D ◦ D ◦ i = , which finishes the proof. (cid:3) Definition 4.21.
Let ( g , [ · , · ] g , r ) be a triangular Lie algebra. The cohomology of the cochain com-plex ( ⊕ + ∞ n = C n TLB ( g , r ) , D TLB ) is called the cohomology of the triangular Lie bialgebra ( g , [ · , · ] g , r ).Denote the n -th cohomology group by H n TLB ( g , r ).Now we give the precise formula for the coboundary operator D TLB . By the definition of i , p , D and (38), we have D TLB ( f , χ ) = (cid:16) d CE f , Θ f + d r χ (cid:17) , ∀ f ∈ Hom( ∧ n g , g ) , χ ∈ ∧ n g , (40)where d r is given by (36) and Θ : Hom( ∧ n g , g ) → ∧ n + g is defined by Θ f = Ψ − ( h r ♯ ( f , f ⋆ )) . The precise formula of Θ is given as follows. Lemma 4.22.
For any f ∈ Hom( ∧ n g , g ) and ξ , · · · , ξ n + ∈ g ∗ , we have h Θ f , ξ ∧ · · · ∧ ξ n + i = n + X i = ( − i + h ξ i , f ( r ♯ ( ξ ) , · · · , r ♯ ( ξ i − ) , r ♯ ( ξ i + ) , · · · , r ♯ ( ξ n + )) i . (41) Proof.
By the definition of h r ♯ given by (25), we have h Θ f , ξ ∧ · · · ∧ ξ n + i = Ψ − ( h r ♯ ( f , f ⋆ ))( ξ , · · · , ξ n + ) = h h r ♯ ( f , f ⋆ )( ξ , · · · , ξ n ) , ξ n + i = ( − n h f ( r ♯ ( ξ ) , · · · , r ♯ ( ξ n )) , ξ n + i + n X i = ( − i + h r ♯ f ⋆ ( r ♯ ( ξ ) , · · · , r ♯ ( ξ i − ) , r ♯ ( ξ i + ) , · · · , r ♯ ( ξ n ) , ξ i ) , ξ n + i = n + X i = ( − i + h ξ i , f ( r ♯ ( ξ ) , · · · , r ♯ ( ξ i − ) , r ♯ ( ξ i + ) , · · · , r ♯ ( ξ n + )) i , which finishes the proof. (cid:3) Theorem 4.23.
Let ( g , [ · , · ] g , r ) be a triangular Lie bialgebra. Then there is a short exact sequenceof cochain complexes: −→ ( ⊕ + ∞ n = ∧ n g , d r ) ι −→ ( ⊕ + ∞ n = C n TLB ( g , r ) , D TLB ) p −→ ( ⊕ + ∞ n = C n Lie ( g ; g ) , d CE ) −→ , where ι ( χ ) = (0 , χ ) and p ( f , χ ) = f for all χ ∈ ∧ n g and f ∈ Hom( ∧ n g , g ) .Consequently, there is a long exact sequence of cohomology groups: (42) · · · −→ H n ( r ) H n ( ι ) −→ H n TLB ( g , r ) H n ( p ) −→ H n Lie ( g , g ) c n −→ H n + ( r ) −→ · · · , EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 21 where the connecting map c n is defined by c n ([ α ]) = [ Θ α ] , for all [ α ] ∈ H n Lie ( g , g ) . Proof.
By (40), we have the short exact sequence of cochain complexes which induces a longexact sequence of cohomology groups. (cid:3)
Remark 4.24.
In a forthcoming paper [36], we will use the functorial approach to give the L ∞ -algebra structure on ⊕ + ∞ n = C n TLB ( g , r ) that control deformations of triangular Lie bialgebras, andestablish the relationship with the L ∞ -algebra ( ⊕ + ∞ n = C n ( g , ad ∗ , r ♯ ) , { l k } + ∞ k = ) given by Theorem 3.15.We will now consider R -deformations and infinitesimal deformations of triangular Lie bial-gebras using the above cohomology theory, where R is a local pro-Artinian K -algebra with theaugmentation ǫ : R → K .Any triangular Lie bialgebra ( g , [ · , · ] g , r ) can be viewed as a triangular R -Lie bialgebra with thehelp of the augmentation map ǫ . Definition 4.25. An R -deformation of a triangular Lie bialgebra ( g , [ · , · ] g , r ) contains of an R -Lie algebra structure [ · , · ] R on the tensor product R ⊗ K g and a skew-symmetric r -matrix X ∈ ( R ⊗ K g ) ⊗ R ( R ⊗ K g ) (cid:27) R ⊗ K g ⊗ K g such that ǫ ⊗ K Id g is an R -Lie algebra homomorphism from( R ⊗ K g , [ · , · ] R ) to ( g , [ · , · ] g ) and ( ǫ ⊗ K Id g ⊗ K Id g )( X ) = r . Definition 4.26.
Let ( R ⊗ K g , [ · , · ] R , X ) and ( R ⊗ K g , [ · , · ] ′ R , X ′ ) be two R -deformations of a triangu-lar Lie bialgebra ( g , [ · , · ] g , r ). We call them equivalent if there exists a triangular R -Lie bialgebraisomorphism φ : ( R ⊗ K g , [ · , · ] ′ R , X ′ ) → ( R ⊗ K g , [ · , · ] R , X ) such that ǫ ⊗ K Id g = ( ǫ ⊗ K Id g ) ◦ φ. (43) Definition 4.27. A K [ t ] / ( t )-deformation of the triangular Lie bialgebra ( g , [ · , · ] g , r ) is called an infinitesimal deformation .Let R = K [ t ] / ( t ) and ( R ⊗ K g , [ · , · ] R , X ) be an infinitesimal deformation of ( g , [ · , · ] g , r ). Since( R ⊗ K g , [ · , · ] R , X ) is a triangular R -Lie bialgebra, there exist ω , ω ∈ Hom( g ∧ g , g ) and X , X ∈∧ K g such that [ · , · ] R = ω + t ω , X = X + t X . (44)Since ǫ ⊗ K Id g is an R -Lie algebra homomorphism from ( R ⊗ K g , [ · , · ] R ) to ( g , [ · , · ] g ), we deducethat ω = [ · , · ] g . By ( ǫ ⊗ K Id g ⊗ K Id g )( X ) = r , we deduce that X = r . Therefore, an infinitesimaldeformation of ( g , [ · , · ] g , r ) is determined by a pair ( ω , X ). By the fact that ( R ⊗ K g , [ · , · ] g + t ω )is an R -Lie algebra, we get d CE ω = . (45)Then by the fact that r + t X is a skew-symmetric r -matrix of the R -Lie algebra ( R ⊗ K g , [ · , · ] g + t ω ),we deduce that 2(d r X + Θ ω ) = . (46) Proposition 4.28.
The pair ( ω , X ) determines an infinitesimal deformation of the triangularLie bialgebra ( g , [ · , · ] g , r ) if and only if ( ω , X ) is a -cocycle of the triangular Lie bialgebra ( g , [ · , · ] g , r ) , i.e. D TLB ( ω , X ) = .Proof. By (45) and (46), we deduce that ( ω , X ) is a 2-cocycle if and only if ( ω , X ) determinesan infinitesimal deformation of the triangular Lie bialgebra ( g , [ · , · ] g , r ). (cid:3) If two infinitesimal deformations of a triangular Lie bialgebra ( g , [ · , · ] g , r ) corresponding to( ω , X ) and ( ω ′ , X ′ ) are equivalent, then there exists a triangular R -Lie bialgebra isomorphism φ from ( R ⊗ K g , [ · , · ] g + t ω ′ , R + t X ′ ) to ( R ⊗ K g , [ · , · ] g + t ω , R + t X ). By (43), we deduce that φ = Id g + tN , where N ∈ gl ( g ) . (47)Since Id g + tN is an isomorphism from ( R ⊗ K g , [ · , · ] g + t ω ′ ) to ( R ⊗ K g , [ · , · ] g + t ω ), we get(48) ω ′ − ω = d CE N . By the equality (cid:0) (Id g + tN ) ⊗ (Id g + tN ) (cid:1) ( r + t X ′ ) = ( r + t X ), we obtain(49) X ′ − X = − (Id g ⊗ N + N ⊗ Id g )( r ) = Θ N . Theorem 4.29.
There is a one-to-one correspondence between the space of equivalence classesof infinitesimal deformations of ( g , [ · , · ] g , r ) and the second cohomology group H ( g , r ) . Proof.
By (48) and (49), we deduce that( ω ′ , X ′ ) − ( ω , X ) = D TLB ( N ) , which implies that ( ω , X ) and ( ω ′ , X ′ ) are in the same cohomology class if and only if thecorresponding infinitesimal deformations of ( g , [ · , · ] g , r ) are equivalent. (cid:3)
5. H omotopy relative R ota -B axter L ie algebras In this section, we introduce the notion of a homotopy relative Rota-Baxter Lie algebra, whichconsists of an L ∞ -algebra, its representation and a homotopy relative Rota-Baxter operator. Wecharacterize homotopy relative Rota-Baxter operators as MC elements in a certain L ∞ -algebra.We show that strict homotopy relative Rota-Baxter operators induce pre-Lie ∞ -algebras.5.1. Homotopy relative Rota-Baxter operators on L ∞ -algebras. Denote by Hom n ( ¯ S ( V ) , V )the space of degree n linear maps from the graded vector space ¯ S ( V ) = ⊕ + ∞ i = S i ( V ) to the Z -gradedvector space V . Obviously, an element f ∈ Hom n ( ¯ S ( V ) , V ) is the sum of f i : S i ( V ) → V . We willwrite 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 ) . As the gradedversion of the Nijenhuis-Richardson bracket given in [43, 44], the graded Nijenhuis-Richardsonbracket [ · , · ] 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 ) , (50)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) , (51)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 − ) . (52)The following result is well-known and, in fact, can be taken as a definition of an L ∞ -algebra. Theorem 5.1.
With the above notation, ( C ∗ ( V , V ) , [ · , · ] NR ) is a graded Lie algebra. Its MC ele-ments P + ∞ k = l k are the L ∞ -algebra structures on V. (cid:3) EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 23
Definition 5.2. ([35]) A representation of an L ∞ -algebra ( g , { l k } + ∞ k = ) on a graded vector space V consists of linear maps ρ k : S k − ( g ) ⊗ V → V , k ≥
1, of degree 1 with the property that, for anyhomogeneous elements x , · · · , x n − ∈ g , v ∈ V , we have n − X i = X σ ∈ S ( i , n − i − ε ( σ ) ρ n − i + ( l i ( x σ (1) , · · · , x σ ( i ) ) , x σ ( i + , · · · , x σ ( n − , v )(53) + n X i = X σ ∈ S ( n − i , i − ε ( σ )( − x σ (1) + ··· + x σ ( n − i ) ρ n − i + ( x σ (1) , · · · , x σ ( n − i ) , ρ i ( x σ ( n − i + , · · · , x σ ( n − , v )) = . Let ( V , { ρ k } + ∞ k = ) be a representation of an L ∞ -algebra ( g , { l k } + ∞ k = ). There is an L ∞ -algebra structureon the direct sum g ⊕ V given by l k (cid:0) ( x , v ) , · · · , ( x k , v k ) (cid:1) : = (cid:0) l k ( x , · · · , x k ) , k X i = ( − x i ( x i + + ··· + x k ) ρ k ( x , · · · , x i − , x i + , · · · , x k , v i ) (cid:1) . This L ∞ -algebra is called the semidirect product of the L ∞ -algebra ( g , { l k } + ∞ k = ) and ( V , { ρ k } + ∞ k = ), anddenoted by g ⋉ ρ V .Now we are ready to define our main object of study in this section. Definition 5.3.
Let ( V , { ρ k } + ∞ k = ) be a representation of an L ∞ -algebra ( g , { l k } + ∞ k = ). A degree 0 el-ement T = P + ∞ k = T k ∈ Hom( ¯ S ( V ) , g ) with T k ∈ Hom( S k ( V ) , g ) is called a homotopy relativeRota-Baxter operator on an L ∞ -algebra ( g , { l k } + ∞ k = ) with respect to the representation ( V , { ρ k } + ∞ k = ) ifthe following equalities hold for all p ≥ v , · · · , v p ∈ V , X k + ··· + km = t ≤ t ≤ p − X σ ∈ S ( k , ··· , km , , p − − t ) ε ( σ ) m ! · T p − t (cid:16) ρ m + (cid:16) T k (cid:0) v σ (1) , · · · , v σ ( k ) (cid:1) , · · · , T k m (cid:0) v σ ( k + ··· + k m − + , · · · , v σ ( t ) (cid:1) , v σ ( t + (cid:17) , v σ ( t + , · · · , v σ ( p ) (cid:17) = X k + ··· + k n = p X σ ∈ S ( k , ··· , kn ) ε ( σ ) n ! l n (cid:16) T k (cid:0) v σ (1) , · · · , v σ ( k ) (cid:1) , · · · , T k n (cid:0) v σ ( k + ··· + k n − + , · · · , v σ ( p ) (cid:1)(cid:17) . A homotopy relative Rota-Baxter operator on an L ∞ -algebra is a generalization of an O -operator on a Lie 2-algebra introduced in [48]. Definition 5.4.
Let ( g , { l k } + ∞ k = ) be an L ∞ -algebra. A degree 0 element T = P + ∞ k = T k ∈ Hom( ¯ S ( g ) , g )with T k ∈ Hom( S k ( g ) , g ) is called a homotopy Rota-Baxter operator on an L ∞ -algebra ( g , { l k } + ∞ k = )if the following equalities hold for all p ≥ x , · · · , x p ∈ g , X k + ··· + km = t ≤ t ≤ p − X σ ∈ S ( k , ··· , km , , p − − t ) ε ( σ ) m ! · T p − t (cid:16) l m + (cid:16) T k (cid:0) x σ (1) , · · · , x σ ( k ) (cid:1) , · · · , T k m (cid:0) x σ ( k + ··· + k m − + , · · · , x σ ( t ) (cid:1) , x σ ( t + (cid:17) , x σ ( t + , · · · , x σ ( p ) (cid:17) = X k + ··· + k n = p X σ ∈ S ( k , ··· , kn ) ε ( σ ) n ! l n (cid:16) T k (cid:0) x σ (1) , · · · , x σ ( k ) (cid:1) , · · · , T k n (cid:0) x σ ( k + ··· + k n − + , · · · , x σ ( p ) (cid:1)(cid:17) . Remark 5.5.
A homotopy Rota-Baxter operator T = P + ∞ k = T k ∈ Hom( ¯ S ( g ) , g ) on an L ∞ -algebra( g , { l k } + ∞ k = ) is a homotopy relative Rota-Baxter operator with respect to the adjoint representation. If moreover the L ∞ -algebra reduces to a Lie algebra ( g , [ · , · ] g ), then the resulting linear operator T : g −→ g is a Rota-Baxter operator . Definition 5.6. (i) An L ∞ -algebra ( g , { l k } + ∞ k = ) with a homotopy Rota-Baxter operator T = P + ∞ k = T k ∈ Hom( ¯ S ( g ) , g ) is called a homotopy Rota-Baxter Lie algebra . We denote itby (cid:0) g , { l k } + ∞ k = , { T k } + ∞ k = (cid:1) .(ii) A homotopy relative Rota-Baxter Lie algebra is a triple (cid:0) ( g , { l k } + ∞ k = ) , { ρ k } + ∞ k = , { T k } + ∞ k = (cid:1) , where( g , { l k } + ∞ k = ) is an L ∞ -algebra, ( V , { ρ k } + ∞ k = ) is a representation of g on a graded vector space V and T = P + ∞ k = T k ∈ Hom( ¯ S ( V ) , g ) is a homotopy relative Rota-Baxter operator.A representation of an L ∞ -algebra will give rise to a V-data as well as an L ∞ -algebra thatcharacterize homotopy relative Rota-Baxter operators as MC elements. Proposition 5.7.
Let ( g , { l k } + ∞ k = ) be an L ∞ -algebra and ( V , { ρ k } + ∞ k = ) a representation of ( g , { l k } + ∞ k = ) .Then the following quadruple forms a V-data: • the graded Lie algebra ( L , [ · , · ]) is given by ( C ∗ ( g ⊕ V , g ⊕ V ) , [ · , · ] NR ) ; • the abelian graded Lie subalgebra h is given by h : = ⊕ n ∈ Z Hom n ( ¯ S ( V ) , g ); • P : L → L is the projection onto the subspace h ; • ∆ = P + ∞ k = ( l k + ρ k ) .Consequently, ( h , { l k } + ∞ k = ) is an L ∞ -algebra, where l k is given by (16) .Proof. By Theorem 5.1, we obtain that ( C ∗ ( g ⊕ V , g ⊕ V ) , [ · , · ] NR ) is a graded Lie algebra. Moreover,by (52) we deduce that Im P = h is an abelian graded Lie subalgebra and ker P is a graded Liesubalgebra. Since ∆ = P + ∞ k = ( l k + ρ k ) is the semidirect product L ∞ -algebra structure on g ⊕ V , wehave [ ∆ , ∆ ] NR = P ( ∆ ) =
0. Thus ( L , h , P , ∆ ) is a V-data. Hence by Theorem 3.8, we obtainthe higher derived brackets { l k } + ∞ k = on the abelian graded Lie subalgebra h . (cid:3) Moreover, for all n ≥
1, we set F n ( h ) = Π + ∞ i = n Hom( S i ( V ) , g ) . (54) Lemma 5.8.
With above notation, ( h , { l k } + ∞ k = ) is a weakly filtered L ∞ -algebra.Proof. By (54), we have h = F ( h ) ⊃ · · · ⊃ F n ( h ) ⊃ · · · . Moreover, by Lemma 2.3, we have l k ( F n ( h ) , F n ( h ) , · · · , F n k ( h )) ⊂ F n + n + ··· + n k ( h ) ⊂ F k ( h ) . (55)Thus, we deduce that (cid:0) h , F • ( h ) (cid:1) is a weakly filtered L ∞ -algebra with n = (cid:3) Remark 5.9.
In fact, the above argument shows that (cid:0) h , F • ( h ) (cid:1) is a filtered L ∞ -algebra in the senseof [13]. Theorem 5.10.
With the above notation, a degree element T = P + ∞ k = T k ∈ Hom( ¯ S ( V ) , g ) is a ho-motopy relative Rota-Baxter operator on ( g , { l k } + ∞ k = ) with respect to the representation ( V , { ρ k } + ∞ k = ) if and only if T = P + ∞ k = T k is an MC element of the L ∞ -algebra ( h , { l k } + ∞ k = ) .Proof. By Remark 2.2, we will view the elements of C ∗ ( g ⊕ V , g ⊕ V ) as coderivations of ¯ S c ( g ⊕ V ).Moreover, we view ⊕ n ∈ Z Hom n ( ¯ S ( V ) , g ) as an abelian graded Lie subalgebra of the graded Liealgebra Coder ( ¯ S c ( g ⊕ V )) and we denote by ¯ P the projection onto this Lie subalgebra. Thecoderivations of ¯ S c ( g ⊕ V ) corresponding to P + ∞ k = l k , P + ∞ k = ρ k and P + ∞ k = T k will be denoted by ¯ l , ¯ ρ EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 25 and ¯ T respectively. Then T = P + ∞ k = T k is an MC element of the L ∞ -algebra ( h , { l k } + ∞ k = ) if and onlyif(56) ¯ P + ∞ X n = n ! [ · · · [[ |{z} n ¯ l + ¯ ρ, ¯ T ] , ¯ T ] , · · · , ¯ T ] = . In fact, we have[ · · · [[ |{z} n ¯ l + ¯ ρ, ¯ T ] , ¯ T ] , · · · , ¯ T ] = n X i = ( − i ni !(cid:0) ¯ T ◦ · · · ◦ ¯ T | {z } i ◦ (¯ l + ¯ ρ ) ◦ ¯ T ◦ · · · ◦ ¯ T | {z } n − i (cid:1) . We denote by pr g the natural projections from ¯ S ( g ⊕ V ) to g . Thus, for all v , · · · , v p ∈ V , we have (cid:0) pr g ◦ [ · · · [[ |{z} n ¯ l + ¯ ρ, ¯ T ] , ¯ T ] , · · · , ¯ T ] (cid:1) ( v , · · · , v p ) = (cid:0) pr g ◦ ¯ l ◦ ¯ T · · · ◦ ¯ T | {z } n (cid:1) ( v , · · · , v p ) − n (cid:0) pr g ◦ ¯ T ◦ ¯ ρ ◦ ¯ T · · · ◦ ¯ T | {z } n − (cid:1) ( v , · · · , v p ) . By (52), we obtain that (cid:0) pr g ◦ ¯ l ◦ ¯ T · · · ◦ ¯ T | {z } n (cid:1) ( v , · · · , v p ) = X k + ··· + k n = p X σ ∈ S ( k , ··· , kn ) ε ( σ ) l n (cid:16) T k (cid:0) v σ (1) , · · · , v σ ( k ) (cid:1) , · · · , T k n (cid:0) v σ ( k + ··· + k n − + , · · · , v σ ( p ) (cid:1)(cid:17) and n (cid:0) pr g ◦ ¯ T ◦ ¯ ρ ◦ ¯ T · · · ◦ ¯ T | {z } n − (cid:1) ( v , · · · , v p ) = n X k + ··· + kn − = t ≤ t ≤ p − X τ ∈ S ( k , ··· , kn − , p − t ) ε ( τ ) · (cid:16) pr g ◦ ¯ T ◦ ¯ ρ (cid:17)(cid:16) T k (cid:0) v τ (1) , · · · , v τ ( k ) (cid:1) , · · · , T k n − (cid:0) v τ ( k + ··· + k n − + , · · · , v τ ( t ) (cid:1) , v τ ( t ) + , · · · , v τ ( p ) (cid:17) = n X k + ··· + kn − = t ≤ t ≤ p − X σ ∈ S ( k , ··· , kn − , , p − − t ) ε ( σ ) · T p − t (cid:16) ρ n (cid:16) T k (cid:0) v σ (1) , · · · , v σ ( k ) (cid:1) , · · · , T k n − (cid:0) v σ ( k + ··· + k n − + , · · · , v σ ( t ) (cid:1) , v σ ( t + (cid:17) , v σ ( t + , · · · , v σ ( p ) (cid:17) . Thus, (56) holds if and only if T = P + ∞ k = T k ∈ Hom( ¯ S ( V ) , g ) is a homotopy relative Rota-Baxteroperator on ( g , { l k } + ∞ k = ) with respect to the representation ( V , { ρ k } + ∞ k = ). (cid:3) At the end of this section, we show that a homotopy relative Rota-Baxter operator correspond-ing to a representation V naturally gives rise to a L ∞ structure on V . Proposition 5.11.
Let T = P + ∞ k = T k ∈ Hom( ¯ S ( V ) , g ) be a homotopy relative Rota-Baxter operatoron ( g , { l k } + ∞ k = ) with respect to the representation ( V , { ρ k } + ∞ k = ) . (i) e [ · , T ] NR (cid:16) P + ∞ k = ( l k + ρ k ) (cid:17) is an MC element of the graded Lie algebra ( C ∗ ( g ⊕ V , g ⊕ V ) , [ · , · ] NR ) ; (ii) there is an L ∞ -algebra structure on V given by l t + ( v , · · · , v t + ) = X k + ··· + k m = t X σ ∈ S ( k , ··· , km , ε ( σ ) m ! ρ m + (cid:16) T k (cid:0) v σ (1) , · · · , v σ ( k ) (cid:1) , · · · , T k m (cid:0) v σ ( k + ··· + k m − + , · · · , v σ ( t ) (cid:1) , v σ ( t + (cid:17) ;(57) (iii) T is an L ∞ -algebra homomorphism from the L ∞ -algebra ( V , { l k } + ∞ k = ) to ( g , { l k } + ∞ k = ) .Proof. (i) For any X ∈ Hom( S i ( g ) ⊗ S j ( V ) , g ) and Y ∈ Hom( S i ( g ) ⊗ S j ( V ) , V ), we have[ · · · [[ |{z} i + X , T ] NR , T ] NR , · · · , T ] NR = , [ · · · [[ |{z} i + Y , T ] NR , T ] NR , · · · , T ] NR = . Thus, [ · , T ] NR is a locally nilpotent derivation of ( C ∗ ( g ⊕ V , g ⊕ V ) , [ · , · ] NR ). Since e [ · , T ] NR is anautomorphism of ( C ∗ ( g ⊕ V , g ⊕ V ) , [ · , · ] NR ), we have[ e [ · , T ] NR (cid:16) + ∞ X k = ( l k + ρ k ) (cid:17) , e [ · , T ] NR (cid:16) + ∞ X k = ( l k + ρ k ) (cid:17) ] NR = e [ · , T ] NR [ + ∞ X k = ( l k + ρ k ) , + ∞ X k = ( l k + ρ k )] NR = , which implies that e [ · , T ] NR (cid:16) P + ∞ k = ( l k + ρ k ) (cid:17) is an MC element.(ii) By (56), e [ · , T ] NR (cid:16) P + ∞ k = ( l k + ρ k ) (cid:17) | V is a sub L ∞ -algebra structure on V . It is straightforward todeduce that the L ∞ -algebra structure on V is just given by (57).(iii) By the definition of a homotopy relative Rota-Baxter operator and (57), we deduce that T is an L ∞ -algebra homomorphism. (cid:3) Strict homotopy relative Rota-Baxter operators on L ∞ -algebras and pre-Lie ∞ -algebras.Definition 5.12. Let ( V , { ρ k } + ∞ k = ) be a representation of an L ∞ -algebra ( g , { l k } + ∞ k = ). A degree 0element T ∈ Hom( V , g ) is called a strict homotopy relative Rota-Baxter operator on an L ∞ -algebra ( g , { l k } + ∞ k = ) with respect to the representation ( V , { ρ k } + ∞ k = ) if the following equalities holdfor all p ≥ v , · · · , v p ∈ V , l p (cid:0) T v , · · · , T v p (cid:1) = p X i = ( − ( v i + + ··· + v p ) v i T ρ p ( T v , · · · , T v i − , T v i + , · · · , T v p , v i ) . (58) Remark 5.13.
A strict homotopy relative Rota-Baxter operator is just a homotopy relative Rota-Baxter operator T = P + ∞ i = T i ∈ Hom( ¯ S ( V ) , g ), in which T i = i ≥ V be a graded vector space. Denote by Hom n ( S ( V ) ⊗ V , V ) the space of degree n linearmaps from the graded vector space S ( V ) ⊗ V to the graded vector space V . Obviously, an element f ∈ Hom n ( S ( V ) ⊗ V , V ) is the sum of f i : S i − ( V ) ⊗ V → V . We will write f = P + ∞ i = f i .Set C n ( V , V ) : = Hom n ( S ( V ) ⊗ V , V ) and C ∗ ( V , V ) : = ⊕ n ∈ Z C n ( V , V ) . As the graded version of theMatsushima-Nijenhuis bracket given in [9], the graded Matsushima-Nijenhuis bracket [ · , · ] MN onthe graded vector space C ∗ ( V , V ) is given by:[ f , g ] MN : = f ⋄ g − ( − mn g ⋄ f , ∀ f = + ∞ X i = f i ∈ C m ( V , V ) , g = + ∞ X j = g j ∈ C n ( V , V ) , (59)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) , (60)while f i ⋄ g j ∈ Hom( S i + j − ( V ) ⊗ V , V ) is defined by( f i ⋄ g j )( v , · · · , v i + j − ) EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 27 = X σ ∈ S ( j − , , i − ε ( σ ) f i ( g j ( v σ (1) , · · · , v σ ( j − , v σ ( j ) ) , v σ ( j + , · · · , v σ ( i + j − , v i + j − )(61) + X σ ∈ S ( i − , j − ( − α ε ( σ ) f i ( v σ (1) , · · · , v σ ( i − , g j ( v σ ( i ) , · · · , v σ ( i + j − , v i + j − )) , where α = n ( v σ (1) + v σ (2) + · · · + v σ ( i − ). Then the graded vector space C ∗ ( V , V ) equipped with thegraded Matsushima-Nijenhuis bracket [ · , · ] MN is a graded Lie algebra.The notion of a pre-Lie ∞ -algebra was introduced in [9]. See [42] for more applications ofpre-Lie ∞ -algebras in geometry. Theorem 5.14. ([9])
Let P + ∞ k = θ k be a degree linear map from the graded vector space S ( V ) ⊗ Vto the graded vector space V. Then ( V , { θ k } + ∞ k = ) is a pre-Lie ∞ -algebra if and only if P + ∞ k = θ k is an MC element of the graded Lie algebra ( C ∗ ( V , V ) , [ · , · ] MN ) . (cid:3) Now we show that there is a close relationship between the graded Lie algebra ( C ∗ ( V , V ) , [ · , · ] MN )and ( C ∗ ( V , V ) , [ · , · ] NR ). Define a graded linear map Φ : C ∗ ( V , V ) → C ∗ ( V , V ) of degree 0 by Φ ( f ) = + ∞ X k = Φ ( f ) k = Φ ( f k ) , ∀ f = + ∞ X k = f k ∈ Hom m ( S ( V ) ⊗ V , V ) , where Φ ( f k ) is given by Φ ( f k )( v , · · · , v k ) = X σ ∈ S ( k − , ε ( σ ) f k ( v σ (1) , · · · , v σ ( k ) ) = k X i = ( − v i ( v i + + ··· + v k ) f k ( v , · · · , ˆ v i , · · · , v k , v i ) . Theorem 5.15. Φ is a homomorphism from the graded Lie algebra ( C ∗ ( V , V ) , [ · , · ] MN ) to thegraded Lie algebra ( C ∗ ( V , V ) , [ · , · ] NR ) .Proof. It follows from a direct but tedious computation. We omit details. (cid:3)
In the classical case, the symmetrization of a pre-Lie algebra gives rise to a Lie algebra. Thefollowing result generalizes this construction to pre-Lie ∞ -algebras and L ∞ -algebras. Corollary 5.16.
Let ( g , { θ k } + ∞ k = ) be a pre-Lie ∞ -algebra and we define l k byl k ( x , · · · , x k ) = Φ ( θ k )( x , · · · , x k ) = k X i = ( − x i ( x i + + ··· + x k ) θ k ( x , · · · , ˆ x i , · · · , x k , x i ) . (62) Then ( g , { l k } + ∞ k = ) is an L ∞ -algebra. We denote this L ∞ -algebra by g C and call it the sub-adjacent L ∞ -algebra of ( g , { θ k } + ∞ k = ) . Moreover, ( g , { θ k } + ∞ k = ) is called the compatible pre-Lie ∞ -algebra struc-ture on the L ∞ -algebra g C .Proof. It follows from Theorem 5.1, Theorem 5.14 and Theorem 5.15. (cid:3)
Let ( g , { θ k } + ∞ k = ) be a pre-Lie ∞ -algebra. For all k ≥
1, we define L k : S k − ( g ) ⊗ g → g by L k ( x , · · · , x k − , x k ) = θ k ( x , · · · , x k − , x k ) . (63) Proposition 5.17.
With the above notation, ( g , { L k } + ∞ k = ) is a representation of the sub-adjacentL ∞ -algebra g C . Moreover, the identity map Id : g → g is a strict homotopy relative Rota-Baxteroperator on the L ∞ -algebra g C with respect to the representation ( g , { L k } + ∞ k = ) . Proof.
For all x , · · · , x n ∈ g , by the definition of pre-Lie ∞ -algebras, we have n − X i = X σ ∈ S ( i , n − i − ε ( σ ) L n − i + ( l i ( x σ (1) , · · · , x σ ( i ) ) , x σ ( i + , · · · , x σ ( n − , x n ) + n X i = X σ ∈ S ( n − i , i − ε ( σ )( − x σ (1) + ··· + x σ ( n − i ) L n − i + ( x σ (1) , · · · , x σ ( n − i ) , L i ( x σ ( n − i + , · · · , x σ ( n − , x n )) (62) , (63) = n − X i = X τ ∈ S ( i − , , n − i − ε ( τ ) θ n − i + ( θ i ( x τ (1) , · · · , x τ ( i − , x τ ( i ) ) , x τ ( i + , · · · , x τ ( n − , x n ) + n X i = X τ ∈ S ( n − i , i − ε ( τ )( − x τ (1) + ··· + x τ ( n − i ) θ n − i + ( x τ (1) , · · · , x τ ( n − i ) , θ i ( x τ ( n − i + , · · · , x τ ( n − , x n )) = . Thus, we deduce that ( g , { L k } + ∞ k = ) is a representation of the sub-adjacent L ∞ -algebra g C . By(62), we deduce that Id is a strict homotopy relative Rota-Baxter operator on g C with respectto ( g , { L k } + ∞ k = ). (cid:3) Now we are ready to show that strict homotopy relative Rota-Baxter operators on an L ∞ -algebra( g , { l k } + ∞ k = ) induce pre-Lie ∞ -algebras. This generalizes the result given in [2]. Theorem 5.18.
Let T ∈ Hom( V , g ) be a strict homotopy relative Rota-Baxter operator on anL ∞ -algebra ( g , { l k } + ∞ k = ) with respect to the representation ( V , { ρ k } + ∞ k = ) . 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 k ( T v , · · · , T v k − , v k ) , ∀ v · · · , v k ∈ V . (64) Proof.
By the fact that ρ k is a linear map of degree 1 from graded vector space S k − ( g ) ⊗ V to V ,we deduce the graded symmetry condition of θ k . Moreover, for all v · · · , v n ∈ V , we have 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 − ( − v σ (1) + v σ (2) + ··· + v σ ( j − ε ( σ ) θ j ( v σ (1) , · · · , v σ ( j − , θ i ( v σ ( j ) , · · · , v σ ( n − , v n )) (64) = X i + j = n + i ≥ , j ≥ X σ ∈ S ( i − , , j − ε ( σ ) ρ j (cid:0) T ρ i ( T v σ (1) , · · · , T v σ ( i − , v σ ( i ) ) , T v σ ( i + , · · · , T v σ ( n − , v n (cid:1) + X i + j = n + i ≥ , j ≥ X σ ∈ S ( j − , i − ( − v σ (1) + v σ (2) + ··· + v σ ( j − ε ( σ ) ρ j (cid:0) T v σ (1) , · · · , T v σ ( j − , ρ i ( T v σ ( j ) , · · · , T v σ ( n − , v n ) (cid:1) = X i + j = n + i ≥ , j ≥ X τ ∈ S ( i , j − i X s = ( − v τ ( s ) ( v τ ( s + + ··· + v τ ( i ) ) ε ( τ ) · ρ j (cid:0) T ρ i ( T v τ (1) , · · · , ˆ T v τ ( s ) , · · · , T v τ ( i ) , v τ ( s ) ) , T v τ ( i + , · · · , T v τ ( n − , v n (cid:1) + X i + j = n + i ≥ , j ≥ X τ ∈ S ( j − , i − ( − v τ (1) + v τ (2) + ··· + v τ ( j − ε ( τ ) ρ j (cid:0) T v τ (1) , · · · , T v τ ( j − , ρ i ( T v τ ( j ) , · · · , T v τ ( n − , v n ) (cid:1) EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 29 (58) = X i + j = n + i ≥ , j ≥ X τ ∈ S ( i , j − ε ( τ ) ρ j (cid:0) l i ( T v τ (1) , · · · , T v τ ( i ) ) , T v τ ( i + , · · · , T v τ ( n − , v n (cid:1) + X i + j = n + i ≥ , j ≥ X τ ∈ S ( j − , i − ( − v τ (1) + v τ (2) + ··· + v τ ( j − ε ( τ ) ρ j (cid:0) T v τ (1) , · · · , T v τ ( j − , ρ i ( T v τ ( j ) , · · · , T v τ ( n − , v n ) (cid:1) (53) = . Thus, ( V , { θ k } + ∞ k = ) is a pre-Lie ∞ -algebra. (cid:3) Corollary 5.19.
With the above conditions, the linear map T is a strict L ∞ -algebra homomor-phism from the sub-adjacent L ∞ -algebra V C to the initial L ∞ -algebra ( g , { l k } + ∞ k = ) .Proof. It follows from Theorem 5.18 and Corollary 5.16. (cid:3)
At the end of this section, we give the necessary and su ffi cient conditions on an L ∞ -algebraadmitting a compatible pre-Lie ∞ -algebra. Proposition 5.20.
Let ( g , { l k } + ∞ k = ) be an L ∞ -algebra. Then there exists a compatible pre-Lie ∞ -algebra if and only if there exists an invertible strict homotopy relative Rota-Baxter operator on ( g , { l k } + ∞ k = ) .Proof. Let T be an invertible strict homotopy relative Rota-Baxter operator on ( g , { l k } + ∞ k = ) withrespect to a representation ( V , { ρ k } + ∞ k = ). By Theorem 5.18, ( V , { θ k } + ∞ k = ) is a pre-Lie ∞ -algebra struc-ture, where θ k is defined by (64). Since T is an invertible linear map, there is an isomorphicpre-Lie ∞ -algebra structure { Θ k } + ∞ k = on g given by Θ k ( x , · · · , x k ) : = T θ k ( T − x , · · · , T − x k − , T − x k ) = T ρ k ( x , · · · , x k − , T − x k )(65)for all x · · · , x k ∈ g . Since T is a strict homotopy relative Rota-Baxter operator, we have l k ( x , · · · , x k − , x k ) = k X i = ( − ( x i + + ··· + x k ) x i T ρ k ( x , · · · , x i − , x i + , · · · , x k , T − x i ) = k X i = ( − ( x i + + ··· + x k ) x i Θ k ( x , · · · , ˆ x i , · · · , x k , x i ) . Therefore ( g , { Θ k } + ∞ k = ) is a compatible pre-Lie ∞ -algebra of ( g , { l k } + ∞ k = ).Conversely, by Proposition 5.17, the identity map Id is a strict homotopy relative Rota-Baxteroperator on the sub-adjacent L ∞ -algebra g C with respect to the representation ( g , { L k } + ∞ k = ). (cid:3) Acknowledgements.
This research was partially supported by NSFC (11922110). R. Tang is alsoFunded by China Postdoctoral Science Foundation (2020M670833). This work was completedin part while the first author was visiting Max Planck Institute for Mathematics in Bonn and hewishes to thank this institution for excellent working conditions.R eferences [1] D. Arnal, Simultaneous deformations of a Lie algebra and its modules. Di ff erential geometry and mathematicalphysics (Liege, 1980 / Leuven, 1981), 3-15,
Math. Phys. Stud. , 3, Reidel, Dordrecht, 1983. 2, 3, 5, 6[2] C. Bai, A unified algebraic approach to the classical Yang-Baxter equation.
J. Phys. A: Math. Theor. (2007),11073-11082. 2, 28[3] C. Bai, O. Bellier, L. Guo and X. Ni, Spliting of operations, Manin products and Rota-Baxter operators. Int.Math. Res. Not. (2013), 485-524. 2 [4] D. Balavoine, Deformations of algebras over a quadratic operad. Operads: Proc. of Renaissance Conferences(Hartford, CT / Luminy, 1995),
Contemp. Math. 202
Amer. Math. Soc., Providence, RI, 1997, 207-34. 2[5] S. Barmeier and Y. Fr´egier, Deformation-obstruction theory for diagrams of algebras and applications to geom-etry. to appear in
J. Noncommut. Geom. arXiv:1806.05142. 2, 9[6] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity.
Pacific J. Math. (1960), 731-742. 2[7] M. Bordemann, Generalized Lax pairs, the modified classical Yang-Baxter equation, and a ffi ne geometry of Liegroups. Comm. Math. Phys. (1990), 201-216. 2[8] D. V. Borisov, Formal deformations of morphisms of associative algebras.
Int. Math. Res. Not. (2005), 2499-2523. 2[9] F. Chapoton and M. Livernet, Pre-Lie algebras and the rooted trees operad. Int. Math. Res. Not. (2001), 395-408. 3, 26, 27[10] V. Chari and A. Pressley, A Guide to Quantum Groups. Cambridge University Press, 1994. 2, 19[11] C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras. Trans. Amer. Math. Soc. (1948), 85-124. 3[12] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. TheHopf algebra structure of graphs and the main theorem. Comm. Math. Phys. (2000), 249-273. 2[13] V. A. Dolgushev and C. L. Rogers, A version of the Goldman-Millson Theorem for filtered L ∞ -algebras. J.Algebra (2015), 260-302. 8, 24[14] M. Doubek, M. Markl and P. Zima, Deformation theory (lecture notes).
Arch. Math. (Brno) (2007), 333-371.15[15] V. Dotsenko and A. Khoroshkin, Quillen homology for operads via Gr¨obner bases. Doc. Math. (2013),707-747. 3[16] K Ebrahimi-Fard, D. Manchon and F. Patras, A noncommutative Bohnenblust-Spitzer identity for Rota-Baxteralgebras solves Bogoliubov’s counterterm recursion. J. Noncommut. Geom. (2009), 181-222. 2[17] Y. Fr´egier, M. Markl and D. Yau, The L ∞ -deformation complex of diagrams of algebras. New York J. Math. (2009), 353-392. 2[18] Y. Fr´egier, and M. Zambon, Simultaneous deformations and Poisson geometry. Compos. Math. (2015),1763-1790. 2, 9[19] Y. Fr´egier, and M. Zambon, Simultaneous deformations of algebras and morphisms via derived brackets.
J.Pure Appl. Algebra (2015), 5344-5362. 2, 9[20] M. Gerstenhaber, The cohomology structure of an associative ring.
Ann. Math. (1963), 267-288. 2, 3[21] M. Gerstenhaber, On the deformation of rings and algebras. Ann. Math. (2) (1964), 59-103. 2[22] E. Getzler, Lie theory for nilpotent L ∞ -algebras. Ann. Math. (2) (2009), 271-301. 8[23] M. E. Goncharov and P. S. Kolesnikov, Simple finite-dimensional double algebras.
J. Algebra (2018),425-438. 2[24] A. Guan, A. Lazarev, Y. Sheng, and R. Tang, Review of deformation theory II: a homotopical approach.
Adv.Math. (China) (2020), 278-298. 2[25] L. Guo, What is a Rota-Baxter algebra? Notices of the AMS (2009), 1436-1437. 2[26] L. Guo, An introduction to Rota-Baxter algebra. Surveys of Modern Mathematics, 4. International Press,Somerville, MA; Higher Education Press, Beijing, 2012. xii +
226 pp. 2[27] L. Guo, Properties of Free Baxter Algebras.
Adv. Math. (2000), 346-374. 2[28] A. Hamilton and A. Lazarev, Cohomology theories for homotopy algebras and noncommutative geometry.
Algebr. Geom. Topol. (2009), 1503-1583. 3[29] D. K. Harrison, Commutative algebras and cohomology. Trans. Amer. Math. Soc. (1962), 191-204. 3[30] G. Hochschild, On the cohomology groups of an associative algebra.
Ann. Math. (2) (1945), 58-67. 3[31] M. Kontsevich and Y. Soibelman, Deformation theory. I [Draft], http: // / soibel / Book-vol1.ps, 2010. 2, 15[32] Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras.
Ann. Inst. Fourier (Grenoble) (1996), 1243-1274. 2[33] B. A. Kupershmidt, What a classical r -matrix really is. J. Nonlinear Math. Phys. (1999), 448-488. 2, 18[34] T. Lada and J. Stashe ff , Introduction to sh Lie algebras for physicists. Internat. J. Theoret. Phys. (1993),1087-1103. 3, 8[35] T. Lada and M. Markl, Strongly homotopy Lie algebras. Comm. Algebra (1995), 2147-2161. 8, 23 EFORMATIONS AND HOMOTOPY THEORY OF RELATIVE ROTA-BAXTER LIE ALGEBRAS 31 [36] A. Lazarev, Y. Sheng and R. Tang, Homotopy Rota-Baxter algebras, triangular L ∞ -bialgebras and higher de-rived brackets. arXiv:2008.00059. DAG X: Formal moduli problems , available at http: // / lurie / papers / DAG-X.pdf2[39] M. Markl, Intrinsic brackets and the L ∞ -deformation theory of bialgebras. J. Homotopy Relat. Struct. (2010),177-212. 2[40] M. Markl, Deformation Theory of Algebras and Their Diagrams. Regional Conference Series in Mathematics ,Number 116, American Mathematical Society (2011). 2[41] M. Markl, S. Shnider and J. D. Stashe ff , Operads in Algebra, Topology and Physics. American MathematicalSociety, Providence, RI, 2002. 3[42] S. A. Merkulov, Nijenhuis infinity and contractible di ff erential graded manifolds. Compos. Math. (2005),1238-1254. 27[43] A. Nijenhuis and R. Richardson, Cohomology and deformations in graded Lie algebras.
Bull. Amer. Math. Soc. (1966), 1-29. 2, 5, 22[44] A. Nijenhuis and R. Richardson, Commutative algebra cohomology and deformations of Lie and associativealgebras. J. Algebra (1968), 42-105. 2, 22[45] J. Pei, C. Bai and L. Guo, Splitting of Operads and Rota-Baxter Operators on Operads. Appl. Categor. Struct. (2017), 505-538. 2[46] J. P. Pridham, Unifying derived deformation theories. Adv. Math. (2010), 772-826. 2[47] M. A. Semyonov-Tian-Shansky, What is a classical R-matrix?
Funct. Anal. Appl. (1983), 259-272. 2, 18[48] Y. Sheng, Categorification of pre-Lie algebras and solutions of 2-graded classical Yang-Baxter equations. The-ory Appl. Categ. (2019), 269-294. 23[49] J. Stashe ff , Homotopy associativity of H-spaces. I, II. Trans. Amer. Math. Soc. (1963), 275-292; ibid. (1963), 293-312. 3[50] J. Stashe ff , The intrinsic bracket on the deformation complex of an associative algebra. J. Pure Appl. Algebra (1993), 231-235. 5[51] J. Stashe ff , Di ff erential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras. Quantumgroups (Leningrad, 1990) , 120-137, Lecture Notes in Math., 1510,
Springer, Berlin, ff , L -infinity and A -infinity structures. High. Struct. (2019), 292-326. 3[53] R. Tang, C. Bai, L. Guo and Y. Sheng, Deformations and their controlling cohomologies of O -operators. Comm.Math. Phys. (2019), 665-700. 2, 3, 7, 19[54] Th. Voronov, Higher derived brackets and homotopy algebras.
J. Pure Appl. Algebra (2005), 133-153. 2, 8,9[55] H. Yu, L. Guo and J.-Y. Thibon, Weak quasi-symmetric functions, Rota-Baxter algebras and Hopf algebras.
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