Deformations and homotopy theory of relative Rota-Baxter Lie algebras

We determine the \emph{$L_\infty$-algebra} that controls deformations of a relative Rota-Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying LieRep pair by the dg Lie algebra controlling deformations of the relative Rota-Baxter operator. Consequently, we define the {\em cohomology} of relative Rota-Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota-Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a \emph{homotopy} relative Rota-Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota-Baxter Lie algebras is intimately related to \emph{pre-Lie$_\infty$-algebras}.

In this paper we initiate the study of deformations and cohomology of relative Rota-Baxter Lie algebras and their homotopy versions.
1.1. Rota-Baxter operators. The concept of Rota-Baxter operators on associative algebras was introduced by G. Baxter [6] in his study of fluctuation theory in probability. Recently it has found many applications, including Connes-Kreimer's [12] algebraic approach to the renormalization in 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 were studied in [23]. In the Lie algebra context, a Rota-Baxter operator was introduced independently in the 1980s as the operator form of the classical Yang-Baxter equation that plays important roles in many subfields of mathematics and mathematical physics such as integrable systems and quantum 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, the more general notion of an O-operator (later also called a relative Rota-Baxter operator or a generalized Rota-Baxter operator) on a Lie algebra was introduced by Kupershmidt [33]; this notion can be traced back to Bordemann [7]. Relative Rota-Baxter operators provide solutions of the classical Yang-Baxter equation in the semidirect product Lie algebra and give rise to pre-Lie algebras [2].

1.2.
Deformations. The concept of a formal deformation of an algebraic structure began with the seminal work of Gerstenhaber [20,21] for associative algebras. Nijenhuis and Richardson extended this study to Lie algebras [43,44]. More generally, deformation theory for algebras over quadratic operads was developed by Balavoine [4]. For more general operads we refer the reader to [31,37,40], and the references therein.
There is a well known slogan, often attributed to Deligne, Drinfeld and Kontsevich: every reasonable deformation theory is controlled by a differential graded (dg) Lie algebra, determined up to quasi-isomorphism. This slogan has been made into a rigorous theorem by Lurie and Pridham, 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 Ooperators was developed in [53] and the deformation theory of diagrams of algebras was studied in [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 manifests itself naturally as an L ∞ -algebra. This often happens when one tries to deform several algebraic structures as well as a compatibility relation between them, such as diagrams of algebras mentioned above. We will see that this also happens in the study of deformations of a relative Rota-Baxter Lie algebra, which consists of a Lie algebra, its representation and a relative Rota-Baxter operator (see Definition 2.10 below). We apply Voronov's higher derived brackets construction [54] to construct the L ∞ -algebra that characterizes relative Rota-Baxter Lie algebras 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, we show 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 dg Lie algebra that controls deformations of relative Rota-Baxter operators given in [53]. 1.3. Cohomology theories. A classical approach for studying a mathematical structure is associating invariants to it. Prominent among these are cohomological invariants, or simply cohomology, of various types of algebras. Cohomology controls deformations and extension problems of the corresponding algebraic structures. Cohomology theories of various kinds of algebras have been developed and studied in [11,20,29,30]. More recently these classical constructions have been 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. A relative Rota-Baxter Lie algebra consists of a Lie algebra, its representation and an operator on it together with appropriate compatibility conditions. Constructing the corresponding cohomology theory is not straightforward due to the complexity of these data. We solve this problem by constructing a deformation complex for a relative Rota-Baxter Lie algebra and endowing it with an L ∞ -structure. Infinitesimal deformations of relative Rota-Baxter Lie algebras are classified by the second cohomology group. Moreover, we show that there is a long exact sequence of cohomology groups linking the cohomology of LieRep pairs introduced in [1], the cohomology of 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 and triangular Lie bialgebras and we introduce the corresponding cohomology theories for these objects. We also show that infinitesimal deformations of Rota-Baxter Lie algebras and triangular Lie bialgebras are classified by the corresponding second cohomology groups.
1.4. Homotopy invariant construction of Rota-Baxter Lie algebras. Homotopy invariant algebraic structures play a prominent role in modern mathematical physics [52]. Historically, the first such structure was that of an A ∞ -algebra introduced by Stasheff in his study of based loop spaces [49]. Relevant later developments include the work of Lada and Stasheff [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 were studied in the context of operads in [37,41].
Dotsenko and Khoroshkin studied the homotopy of Rota-Baxter operators on associative algebras in [15], and noted that "in general compact formulas are yet to be found". For Rota-Baxter Lie algebras, one encounters a similarly challenging situation. In this paper, we use the approach of L ∞ -algebras and their MC elements to formulate the notion of a (strong) homotopy version of a relative Rota-Baxter Lie algebra, which consists of an L ∞ -algebra, its representation and a homotopy relative Rota-Baxter operator. We show that strict homotopy relative Rota-Baxter operators give rise to pre-Lie ∞ -algebras, and conversely the identity map is a strict homotopy relative Rota-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 cohomology of LieRep pairs and relative Rota-Baxter operators. In Section 3, we establish the deformation theory of relative Rota-Baxter Lie algebras. In Section 4, we introduce the corresponding cohomology theory and explain how it is related to infinitesimal deformations of relative Rota-Baxter Lie algebras in the usual way. In Section 4.3, we study the cohomology theory of Rota-Baxter Lie algebras. In Section 4.4, we explain how the cohomology theories of triangular Lie bialgebras and of relative Rota-Baxter Lie algebras are related. In Section 5, we introduce the notion of a homotopy relative Rota-Baxter operator and characterize it as an MC element in a certain L ∞ -algebra. Finally, we exhibit a close relationship between homotopy relative Rota-Baxter Lie algebras of a certain kind and pre-Lie ∞ -algebras. 1.6. Notation and conventions. Throughout this paper, we work with a coefficient field K which is of characteristic 0, and R is a pro-Artinian K-algebra, that is a projective limit of local Artinian K-algebras.
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 − (−1) xy y ⊗ x. We will write . Moreover, we denote the reduced symmetric algebra byS(V) : . Denote the product of homogeneous elements v 1 , · · · , v n ∈ V in S n (V) by v 1 ⊙ · · · ⊙ v n . The degree of v 1 ⊙ · · · ⊙ v n is by definition the sum of the degrees of v i . For a permutation σ ∈ S n and v 1 , · · · , v n ∈ V, the Koszul sign The desuspension operator s −1 changes the grading of V according to the rule (s A degree 1 element θ ∈ g 1 is called an MC element of a differential graded Lie algebra 2. Maurer-Cartan characterizations of LieRep pairs and relative Rota-Baxter operators 2.1. Bidegrees and the Nijenhuis-Richardson bracket. Let g be a vector space. For all n ≥ 0, set C n (g, g) := Hom(∧ n+1 g, g). Let g 1 and g 2 be two vector spaces and elements in g 1 will be denoted by x, y, z, x i and elements in g 2 will be denoted by u, v, w, v i . For a multilinear map f : ∧ k g 1 ⊗ ∧ l g 2 → g 1 , we definef ∈ C k+l−1 g 1 ⊕ g 2 , g 1 ⊕ g 2 bŷ Similarly, for f : ∧ k g 1 ⊗ ∧ l g 2 → g 2 , we definef ∈ C k+l−1 g 1 ⊕ g 2 , g 1 ⊕ g 2 bŷ The linear mapf is called a lift of f . We define g k,l := ∧ k g 1 ⊗ ∧ l g 2 . The vector space ∧ n (g 1 ⊕ g 2 ) is isomorphic to the direct sum of g k,l , k + l = n.
A linear map f ∈ Hom ∧ k+l+1 (g 1 ⊕g 2 ), g 1 ⊕g 2 has a bidegree k|l, which is denoted by || f || = k|l, if f satisfies the following two conditions: (i) If X ∈ g k+1,l , then f (X) ∈ g 1 and if X ∈ g k,l+1 , then f (X) ∈ g 2 ; (ii) In all the other cases f (X) = 0. We denote the set of homogeneous linear maps of bidegree k|l by C k|l (g 1 ⊕ g 2 , g 1 ⊕ g 2 ).
The following lemmas are very important in our later study.
Proof. It follows from direct computation.
Thus, µ defines a Lie algebra structure on g if and only if [µ, µ] NR = 0.
Define the set of 0-cochains C 0 Lie (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−1 (g, g), n ≥ 1. The Chevalley-Eilenberg coboundary operator d CE of the Lie algebra g with coefficients in the adjoint representation is defined by 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 a Rota-Baxter operator on a Lie algebra is a relative Rota-Baxter operator with respect to the adjoint representation.
Now we define the cohomology governing deformations of a relative Rota-Baxter operator T : V → g. The spaces of 0-cochains C 0 (T ) and of 1-cochains C 1 (T ) are set to be 0. For n ≥ 2, define the vector space of n-cochains C n (T ) as C n (T ) = Hom(∧ n−1 V, g).
See [53] for explicit formulas of the coboundary operator δ.

Maurer-Cartan characterization and deformations of relative Rota-Baxter Lie algebras
In this section, we apply Voronov's higher derived brackets to construct the L ∞ -algebra that characterizes relative Rota-Baxter Lie algebras as MC elements. Consequently, we obtain the L ∞ -algebra that controls deformations of a relative Rota-Baxter Lie algebra.
3.1. L ∞ -algebras and higher derived brackets. The notion of an L ∞ -algebra was introduced by Stasheff 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 homogeneous elements x 1 , · · · , x n ∈ g, we have (i) (graded symmetry) for every σ ∈ S n , There is a canonical way to view a differential graded Lie algebra as an L ∞ -algebra.
where g is an L ∞ -algebra and F • g is a descending filtration of the graded vector space g such that g = F 1 g ⊃ · · · ⊃ F n g ⊃ · · · and (i) there exists n ≥ 1 such that for all 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 vector spaces g 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 0 satisfying the MC equation Let α be an MC element. Define l α k : Remark 3.5. The condition of being weakly filtered ensures convergence of the series figuring in 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 above suffices.
The following result is essentially contained in [22,Section 4]; that paper works with a different type of L ∞ algebras than weakly filtered ones, but this does not affect the arguments.
Theorem 3.6. With the above notation, (g, {l α k } +∞ k=1 ) 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=1 ). One method for constructing explicit L ∞ -algebras is given by Voronov's derived brackets [54]. Let us recall this construction.
We call {l k } +∞ k=1 the higher derived brackets of the V-data (L, h, P, ∆). There is also an L ∞ -algebra structure on a bigger space, which is used to study simultaneous deformations of morphisms between Lie algebras in [5,18,19].
Here a, a 1 , · · · , a k are homogeneous elements of h and x, y are homogeneous elements of L. All the other L ∞ -algebra products that are not obtained from the ones written above by permutations of arguments, will vanish.
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=0 C n (g ⊕ V, g ⊕ V), [·, ·] NR ). This graded Lie algebra gives rise to a V-data, and an L ∞ -algebra naturally.
Since P is the projection onto h, it is obvious that P • P = P. It is also straightforward to see that 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.
By Lemma 2.3, we obtain that is an L ∞ -algebra, where l i are given by for homogeneous elements θ 1 , · · · , θ k−1 ∈ h, homogeneous elements Q, Q ′ ∈ L ′ , and all the other possible combinations vanish. Moreover is weakly filtered with n = 3 in the sense of Definition 3.3 with the filtration given by 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 suffices to note that any element h ∈ h can be written as h = +∞ i=1 h i where h i ∈ Hom(∧ i V, g) and that the term h 1 : V → g is nilpotent (even has square zero) when viewed as an endomorphism of g ⊕ V.
Now we are ready to formulate the main result in this subsection.
Remark 3.14. Since the axiom defining a relative Rota-Baxter Lie algebra is not quadratic, it can be anticipated that the deformation complex of a Rota-Baxter Lie algebra is a fully-fledged L ∞ -algebra rather than a differential graded Lie algebra.
Let ((g, µ), ρ, T ) be a relative Rota-Baxter Lie algebra. Denote by ) given in Corollary 3.12. Now we are ready to give the L ∞ -algebra that controls deformations of the relative Rota-Baxter Lie algebra.
Theorem 3.15. With the above notation, we have the twisted L ∞ -algebra Moreover, for linear maps T ′ ∈ Hom(V, g), µ ′ ∈ Hom(∧ 2 g, g) and ρ ′ ∈ Hom(g, gl(V)), the triple is a relative Rota-Baxter Lie algebra, then by Theorem 3.13, we deduce that (s −1 (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 ( be a Lie algebra and (V; ρ) a representation of (g, µ). By Theorem 2.8 and Lemma 3.2, we have an L ∞ -algebra structure on the graded vector space . Let T : V −→ g be a relative Rota-Baxter operator on a Lie algebra (g, µ) with respect to a representation (V; ρ). By Theorem 2.13 and Lemma 3.2, we have an L ∞ -algebra structure on the graded vector space ⊕ +∞ k=1 Hom(∧ k V, g). The above L ∞ -algebras are related as follows.

Cohomology and infinitesimal deformations of relative Rota-Baxter Lie algebras
In this section, ((g, µ), ρ, T ) is a relative Rota-Baxter Lie algebra, i.e. ρ : g → gl(V) is a representation of the Lie algebra (g, µ) and T : V → g is a relative Rota-Baxter operator. We define the cohomology of relative Rota-Baxter Lie algebras and show that the two-dimensional cohomology groups classify infinitesimal deformations. We also establish a relationship between the cohomology of relative Rota-Baxter Lie algebras and the cohomology of triangular Lie bialgebras.
4.1. Cohomology of relative Rota-Baxter Lie algebras. We define the cohomology of a relative Rota-Baxter Lie algebra using the twisted L ∞ -algebra given in Theorem 3.15.
Proof. By Remark 2.2, it is convenient to view the elements of ⊕ +∞ n=0 C n (g ⊕ V; g ⊕ V) as coderivations ofS c s −1 (g ⊕ V) . The coderivations corresponding to f and T will be denoted byf andT respectively. Then, by induction, we have which implies that (25) holds.
The formula of the coboundary operator D can be well-explained by the following diagram: Theorem 4.5. Let ((g, µ), ρ, T ) be a relative Rota-Baxter Lie algebra. Then there is a short exact sequence of the cochain complexes: , ρ), ∂) −→ 0, where ι and p are the inclusion map and the projection map.
Consequently, there is a long exact sequence of the cohomology groups: where the connecting map c n is defined by c n ([α]) = [h T α], for all [α] ∈ H n (g, ρ).
Proof. By (23), we have the short exact sequence of chain complexes which induces a long exact sequence of cohomology groups. Also by (23), c n is given by c n ([α]) = [h T α].

4.2.
Infinitesimal deformations of relative Rota-Baxter Lie algebras. In this subsection, we introduce the notion of R-deformations of relative Rota-Baxter Lie algebras, where R is a local pro-Artinian K-algebra. Since R is the projective limit of local Artinian K-algebras, R is equipped with an augmentation ǫ : R → K. See [14,31] for more details about R-deformation theory of algebraic structures. Then we restrict our study to infinitesimal deformations, i.e. R = K[t]/(t 2 ), 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 Definition 2.10 and Definition 2.11, it is straightforward to obtain the definitions of R-relative Rota-Baxter Lie algebras and homomorphisms between them.

Cohomology of Rota-Baxter Lie algebras.
In this subsection, we define the cohomology of Rota-Baxter Lie algebras with the help of the general framework of the cohomology of relative Rota-Baxter Lie algebras.
Consequently, there is a long exact sequence of the cohomology groups: where the connecting map c n is defined by c n ([α]) = [Ωα], for all [α] ∈ H n Lie (g, g). Proof. By (35), we have the short exact sequence of cochain complexes which induces a long exact sequence of cohomology groups.
Remark 4.16. The approach used to define D RB , can be also used to obtain the L ∞ -algebra structure {l k } +∞ k=1 on ⊕ n C n RB (g, T ) controlling deformations of Rota-Baxter Lie algebras. By Theorem 3.15, we have the L ∞ -algebra (⊕ n C n (g, ad, T ), {l k } +∞ k=1 ) which controls deformations of the relative Rota-Baxter Lie algebra (g, ad, T ). Define l k by l k (X 1 , · · · , X k ) := pl k (i(X 1 ), · · · , i(X k )), for all homogeneous elements X i ∈ ⊕ n C n RB (g, T ). Then (⊕ n C n  R). We omit the details. 4.4. Cohomology and infinitesimal deformations of triangular Lie bialgebras. In this subsection, all vector spaces are assumed to be finite-dimensional. First we define the cohomology of triangular Lie bialgebras with the help of the general cohomological framework for relative Rota-Baxter Lie algebras. Then we establish the standard classification result for infinitesimal deformations 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 : ∧ 2 g −→ g and a Lie coalgebra structure δ : g −→ ∧ 2 g such that δ is a 1-cocycle on g with coefficients in ∧ 2 g. The Lie bracket [·, ·] g in a Lie algebra g naturally extends to the Schouten-Nijenhuis bracket [·, ·] SN on ∧ • g = ⊕ k≥0 ∧ k+1 g. More precisely, we have An element r ∈ ∧ 2 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-Baxter equation if and only if r ♯ is a relative Rota-Baxter operator on g with respect to the coadjoint representation, where r ♯ : g * → g is defined by r ♯ (ξ), η = r, ξ ∧ η for all ξ, η ∈ g * .
The above definition is consistent with the equivalence between r-matrices given in [10]. Let g be a Lie algebra and r ∈ ∧ 2 g a skew-symmetric r-matrix. Define the set of 0-cochains and 1-cochains to be zero and define the set of k-cochains to be ∧ k g. Define d r : ∧ k g → ∧ k+1 g by (36) d r χ = [r, χ] SN , ∀χ ∈ ∧ k g.
Then d 2 r = 0. Denote by H k (r) the corresponding k-th cohomology group, called the k-th cohomology group of the skew-symmetric r-matrix r.
Definition 4.21. Let (g, [·, ·] g , r) be a triangular Lie algebra. The cohomology of the cochain complex (⊕ +∞ n=0 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 where d r is given by (36) and Θ : Hom(∧ n g, g) → ∧ n+1 g is defined by The precise formula of Θ is given as follows.
Consequently, there is a long exact sequence of cohomology groups: where the connecting map c n is defined by c n ([α]) = [Θα], for all [α] ∈ H n Lie (g, g). Proof. By (40), we have the short exact sequence of cochain complexes which induces a long exact sequence of cohomology groups.
Remark 4.24. In a forthcoming paper [36], we will use the functorial approach to give the L ∞algebra structure on ⊕ +∞ n=0 C n TLB (g, r) that control deformations of triangular Lie bialgebras, and establish the relationship with the L ∞ -algebra (⊕ +∞ n=0 C n (g, ad * , r ♯ ), {l k } +∞ k=1 ) given by Theorem 3.15. We will now consider R-deformations and infinitesimal deformations of triangular Lie bialgebras using the above cohomology theory, where R is a local pro-Artinian K-algebra with the augmentation ǫ : R → K.
Any triangular Lie bialgebra (g, [·, ·] g , r) can be viewed as a triangular R-Lie bialgebra with the help 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) 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.

Homotopy relative Rota-Baxter Lie algebras
In this section, we introduce the notion of a homotopy relative Rota-Baxter Lie algebra, which consists of an L ∞ -algebra, its representation and a homotopy relative Rota-Baxter operator. We characterize 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. As the graded version of the Nijenhuis-Richardson bracket given in [43,44], the graded Nijenhuis-Richardson bracket [·, ·] NR on the graded vector space C * (V, V) is given by: The following result is well-known and, in fact, can be taken as a definition of an L ∞ -algebra.
Definition 5.2. ( [35]) A representation of an L ∞ -algebra (g, {l k } +∞ k=1 ) on a graded vector space V consists of linear maps ρ k : S k−1 (g) ⊗ V → V, k ≥ 1, of degree 1 with the property that, for any homogeneous elements x 1 , · · · , x n−1 ∈ g, v ∈ V, we have There is an L ∞ -algebra structure on the direct sum g ⊕ V given by This L ∞ -algebra is called the semidirect product of the L ∞ -algebra (g, {l k } +∞ k=1 ) and (V, {ρ k } +∞ k=1 ), and denoted by g ⋉ ρ V. Now we are ready to define our main object of study in this section.
A homotopy relative Rota-Baxter operator on an L ∞ -algebra is a generalization of an Ooperator on a Lie 2-algebra introduced in [48].
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.
(i) An L ∞ -algebra (g, {l k } +∞ k=1 ) with a homotopy Rota-Baxter operator T = +∞ k=1 T k ∈ Hom(S(g), g) is called a homotopy Rota-Baxter Lie algebra. We denote it by g, is a representation of g on a graded vector space V and T = +∞ k=1 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 that characterize homotopy relative Rota-Baxter operators as MC elements.
. Then the following quadruple forms a V-data: • the graded Lie algebra (L, [·, ·]) is given by • 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; 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 ImP = h is an abelian graded Lie subalgebra and ker P is a graded Lie subalgebra. Since ∆ = +∞ k=1 (l k + ρ k ) is the semidirect product L ∞ -algebra structure on g ⊕ V, we have [∆, ∆] NR = 0 and P(∆) = 0. Thus (L, h, P, ∆) is a V-data. Hence by Theorem 3.8, we obtain the higher derived brackets {l k } +∞ k=1 on the abelian graded Lie subalgebra h.
Remark 5.9. In fact, the above argument shows that h, F • (h) is a filtered L ∞ -algebra in the sense of [13].
Thus, (56) holds if and only if T = +∞ k=1 T k ∈ Hom(S(V), g) is a homotopy relative Rota-Baxter operator on (g, {l k } +∞ k=1 ) with respect to the representation (V, {ρ k } +∞ k=1 ). At the end of this section, we show that a homotopy relative Rota-Baxter operator corresponding to a representation V naturally gives rise to a L ∞ structure on V.
(iii) By the definition of a homotopy relative Rota-Baxter operator and (57), we deduce that T is an L ∞ -algebra homomorphism.
. A degree 0 element T ∈ Hom(V, g) is called a strict homotopy relative Rota-Baxter operator on an L ∞algebra (g, {l k } +∞ k=1 ) with respect to the representation (V, {ρ k } +∞ k=1 ) if the following equalities hold for all p ≥ 1 and all homogeneous elements v 1 , · · · , v p ∈ V, Remark 5.13. A strict homotopy relative Rota-Baxter operator is just a homotopy relative Rota-Baxter operator T = +∞ i=1 T i ∈ Hom(S(V), g), in which T i = 0 for all i ≥ 2. Let V be a graded vector space. Denote by Hom n (S(V) ⊗ V, V) the space of degree n linear maps 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 : As the graded version of the Matsushima-Nijenhuis bracket given in [9], the graded Matsushima-Nijenhuis bracket [·, ·] MN on the graded vector space C * (V, V) is given by: where α = n(v σ(1) + v σ(2) + · · · + v σ(i−1) ). Then the graded vector space C * (V, V) equipped with the graded 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 of pre-Lie ∞ -algebras in geometry.
where Φ( f k ) is given by Proof. It follows from a direct but tedious computation. We omit details.
In the classical case, the symmetrization of a pre-Lie algebra gives rise to a Lie algebra. The following result generalizes this construction to pre-Lie ∞ -algebras and L ∞ -algebras.
Proof. It follows from Theorem 5.1, Theorem 5.14 and Theorem 5.15.
Thus, we deduce that (g, {L k } +∞ k=1 ) 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 respect to (g, {L k } +∞ k=1 ).
Thus, (V, {θ k } +∞ k=1 ) is a pre-Lie ∞ -algebra. Corollary 5.19. With the above conditions, the linear map T is a strict L ∞ -algebra homomorphism from the sub-adjacent L ∞ -algebra V C to the initial L ∞ -algebra (g, {l k } +∞ k=1 ). Proof. It follows from Theorem 5.18 and Corollary 5.16.
At the end of this section, we give the necessary and sufficient conditions on an L ∞ -algebra admitting a compatible pre-Lie ∞ -algebra.