Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures

Given a Lie algebroid with a representation, we construct a graded Lie algebra whose Maurer-Cartan elements characterize relative Rota-Baxter operators on Lie algebroids. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extendability of order deform...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2022
Main Authors:	Liu, Meijun, Liu, Jiefeng, Sheng, Yunhe
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2022
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Cite this:	Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures. Meijun Liu, Jiefeng Liu and Yunhe Sheng. SIGMA 18 (2022), 054, 26 pages

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author	Liu, Meijun Liu, Jiefeng Sheng, Yunhe
author_facet	Liu, Meijun Liu, Jiefeng Sheng, Yunhe
citation_txt	Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures. Meijun Liu, Jiefeng Liu and Yunhe Sheng. SIGMA 18 (2022), 054, 26 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	Given a Lie algebroid with a representation, we construct a graded Lie algebra whose Maurer-Cartan elements characterize relative Rota-Baxter operators on Lie algebroids. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extendability of order deformations to order + 1 deformations of relative Rota-Baxter operators in terms of this cohomology theory. We also construct a graded Lie algebra on the space of multi-derivations of a vector bundle whose Maurer-Cartan elements characterize left-symmetric algebroids. We show that there is a homomorphism from the controlling graded Lie algebra of relative Rota-Baxter operators on Lie algebroids to the controlling graded Lie algebra of left-symmetric algebroids. Consequently, there is a natural homomorphism from the cohomology groups of a relative Rota-Baxter operator to the deformation cohomology groups of the associated left-symmetric algebroid. As applications, we give the controlling graded Lie algebra and the cohomology theory of Koszul-Vinberg structures on left-symmetric algebroids.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 054, 26 pages Deformations and Cohomologies of Relative Rota–Baxter Operators on Lie Algebroids and Koszul–Vinberg Structures Meijun LIU a, Jiefeng LIU a and Yunhe SHENG b a) School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, China E-mail: liumj281@nenu.edu.cn, liujf534@nenu.edu.cn b) Department of Mathematics, Jilin University, Changchun 130012, Jilin, China E-mail: shengyh@jlu.edu.cn Received February 02, 2022, in final form July 07, 2022; Published online July 13, 2022 https://doi.org/10.3842/SIGMA.2022.054 Abstract. Given a Lie algebroid with a representation, we construct a graded Lie algebra whose Maurer–Cartan elements characterize relative Rota–Baxter operators on Lie alge- broids. We give the cohomology of relative Rota–Baxter operators and study infinitesimal deformations and extendability of order n deformations to order n+1 deformations of rela- tive Rota–Baxter operators in terms of this cohomology theory. We also construct a graded Lie algebra on the space of multi-derivations of a vector bundle whose Maurer–Cartan ele- ments characterize left-symmetric algebroids. We show that there is a homomorphism from the controlling graded Lie algebra of relative Rota–Baxter operators on Lie algebroids to the controlling graded Lie algebra of left-symmetric algebroids. Consequently, there is a natural homomorphism from the cohomology groups of a relative Rota–Baxter operator to the de- formation cohomology groups of the associated left-symmetric algebroid. As applications, we give the controlling graded Lie algebra and the cohomology theory of Koszul–Vinberg structures on left-symmetric algebroids. Key words: cohomology; deformation; Lie algebroid; Rota–Baxter operator; Koszul–Vinberg structure; left-symmetric algebroid 2020 Mathematics Subject Classification: 53D17; 53C25; 58A12; 17B70 1 Introduction In this paper we use Maurer–Cartan elements to study deformations and cohomologies of relative Rota–Baxter operators on Lie algebroids. Applications are given to study deformations and cohomologies of Koszul–Vinberg structures on left-symmetric algebroids. 1.1 Relative Rota–Baxter operators on Lie algebroids and Koszul–Vinberg structures The concept of Rota–Baxter operators on associative algebras was introduced by G. Baxter [3] and G.-C. Rota [40, 41] in the 1960s. It also plays an important role in the Connes–Kreimer’s algebraic approach [8] to the renormalization in perturbative quantum field theory. In [23], Ku- pershmidt introduced the notion of a relative Rota–Baxter operator (also called an O-operator) on a Lie algebra in order to better understand the relationship between the classical Yang–Baxter equation and the related integrable systems. In addition, the defining relationship of a relative Rota–Baxter operator was also called the Schouten curvature in [22]. See [6, 11, 17, 18, 19, 39, 45] for more details on relative Rota–Baxter operators and their applications. mailto:liumj281@nenu.edu.cn mailto:liujf534@nenu.edu.cn mailto:shengyh@jlu.edu.cn https://doi.org/10.3842/SIGMA.2022.054 2 M. Liu, J. Liu and Y. Sheng The notion of a Lie algebroid was introduced by Pradines in 1967, which is a generalization of Lie algebras and tangent bundles. See [30] for the general theory about Lie algebroids. Relative Rota–Baxter operators (also called O-operators) on Lie algebroids were introduced in [28] as a method to construct left-symmetric algebroids. The notion of a left-symmetric algebroid is a geometric generalization of a left-symmetric algebra (also called pre-Lie algebras, see the survey article [5] for more details). See [4, 28, 33, 34] for more details and applications of left-symmetric algebroids. In [26], motivated by the theory of Lie bialgebroids [31], the notion of a left-symmetric bialgebroid was introduced as a geometric generalization of a left-symmetric bialgebra [2]. The double of a left-symmetric bialgebroid is not a left-symmetric algebroid anymore, but a pre- symplectic algebroid [27]. This result is parallel to the fact that the double of a Lie bialgebroid is a Courant algebroid [29]. As a Poisson structure π on a manifold gives rise to a Lie bialgebroid, a Koszul–Vinberg structure H on a flat manifold gives rise to a left-symmetric bialgebroid. In particular, if the Koszul–Vinberg structure H is nondegenerate, the inverse of H is a pseudo- Hessian structure [42, 43] on a flat manifold. Therefore, Koszul–Vinberg structures and pseudo- Hessian structures are respectively symmetric analogues of Poisson structures and symplectic structures. See [1, 4, 47] for recent studies on Koszul–Vinberg structures. 1.2 Deformations and cohomologies The theory of deformation plays a prominent role in mathematics and physics. The idea of treating deformation as a tool to study the algebraic structures was introduced by Gerstenhaber in his work on associative algebras [15, 16] and then was extended to Lie algebras by Nijenhuis and Richardson [36, 38]. One remarkable result in Poisson geometry is that M. Kontsevich [21] proved that every Poisson manifold has a deformation quantization. There is a well known slogan, often attributed to Deligne, Drinfeld and Kontsevich: every reasonable deformation theory is controlled by a differential graded Lie algebra, determined up to quasi-isomorphisms. A suitable deformation theory of an algebraic structure can be summarized as the following general principle: on the one hand, for a given object with an algebraic structure, there should exist a differential graded Lie algebra whose Maurer–Cartan elements characterize deformations of this object. On the other hand, there should exist a suitable cohomology so that the infinites- imal of a formal deformation can be identified with a cohomology class, and then a theory of the obstruction to the integration of an infinitesimal deformation can be developed using this coho- mology theory. It is well-known that deformations of Poisson structures are controlled by the differential graded Lie algebra constructed by the Schouten–Nijenhuis bracket of multi-vector fields. Infinitesimal deformations and extendibility of order n deformations of a Poisson structure are characterized in terms of the Poisson cohomology [20, 25]. There also exists a differential graded Lie algebra and a deformation cohomology given by M. Crainic and I. Moerdijk in [9] on the space of multi-derivations which controls deformations of Lie algebroids. See [13, 14] for more details on simultaneous deformations of algebras and morphisms and their applications in Poisson geometry. 1.3 Summary of the results and outline of the paper Since Koszul–Vinberg structures are symmetric analogues of Poisson structures, while there is a full developed deformation and cohomology theories for Poisson structures, it is natural to develop the deformation and cohomology theories for Koszul–Vinberg structures. Note that a Koszul–Vinberg structure on a left-symmetric algebroid is a relative Rota–Baxter operator on its sub-adjacent Lie algebroid with respect to a certain representation (Proposition 6.5). Thus we develop the deformation and cohomology theories for relative Rota–Baxter operators on Lie algebroids first. Inspired by the construction of the differential graded Lie algebra controlling Deformations and Cohomologies of Relative Rota–Baxter Operators on Lie Algebroids 3 deformations of a relative Rota–Baxter operator on a Lie algebra in [44], we construct a suitable differential graded Lie algebra that controls deformations of relative Rota–Baxter operators on Lie algebroids. See [10, 44] for more details on cohomologies and deformations of relative Rota– Baxter operators on Lie algebras and associative algebras. Following the idea of M. Crainic and I. Moerdijk in [9], we also construct a differential graded Lie algebra that controls defor- mations of a left-symmetric algebroid. There is a natural homomorphism from the controlling algebra of relative Rota–Baxter operators to the controlling algebra of left-symmetric algebroids. Using the controlling algebra of relative Rota–Baxter operators on Lie algebroids, we construct a differential graded Lie algebra whose Maurer–Cartan elements are Koszul–Vinberg structures. Consequently, we establish a cohomology theory for Koszul–Vinberg structures. We hope that our study on Koszul–Vinberg structures will draw more attention to the geometry of Koszul– Vinberg structures. The paper is organized as follows. In Section 2, first we construct a differential graded Lie algebra that controls deformations of relative Rota–Baxter operators on Lie algebroids. Then we give the cohomology theories of relative Rota–Baxter operators on Lie algebroids induced by this differential graded Lie algebra. In Section 3, we give the cohomology of Rota–Baxter operators on Lie algebroids and analyze the cohomology of the Rota–Baxter operator on an action Lie algebroid. In Section 4, first we show that infinitesimal deformations of a relative Rota–Baxter operator are classified by the first cohomology group. Then for an order n deformation, we define its obstruction class, which is a cohomology class in the second cohomology group, and show that an order n deformation of a relative Rota–Baxter operator is extendable if and only if its obstruction class is trivial. In Section 5, we construct a graded Lie algebra whose Maurer– Cartan elements are precisely left-symmetric algebroids. The deformation cohomology of left- symmetric algebroids can be given directly using this graded Lie algebra. We show that there is a homomorphism from the controlling graded Lie algebra of relative Rota–Baxter operators on Lie algebroids to the controlling graded Lie algebra of left-symmetric algebroids. Consequently, there is a natural homomorphism from the cohomology groups of a relative Rota–Baxter operator to the deformation cohomology groups of the associated left-symmetric algebroid. In Section 6, we give the deformation and cohomology theories of Koszul–Vinberg structures on left-symmetric algebroids as applications of the above general framework. 1.4 Conventions and notations We will adopt the following notations and conventions throughout the paper. Let i, j be positive integers. A permutation σ of {1, 2, . . . , i+ j} is called an (i; j)-unshuffle if σ(1) < · · · < σ(i) and σ(i+ 1) < · · · < σ(i+ j). The set of all (i; j)-unshuffle will be denoted by S(i;j). The notion of an (i1, . . . , ik)-unshuffle and the set S(i1,...,ik) are defined analogously. 2 Maurer–Cartan characterizations and cohomologies of relative Rota–Baxter operators on Lie algebroids 2.1 The controlling algebra of relative Rota–Baxter operators on Lie algebroids In this subsection, given a Lie algebroid with a representation we construct a graded Lie algebra whose Maurer–Cartan elements characterize relative Rota–Baxter operators on Lie algebroids. Consequently, we obtain the differential graded Lie algebra that controls deformations of a rela- tive Rota–Baxter operator. Definition 2.1. A Lie algebroid structure on a vector bundle A −→ M is a pair that consists of a Lie algebra structure [·, ·]A on the section space Γ(A) and a bundle map aA : A −→ TM , 4 M. Liu, J. Liu and Y. Sheng called the anchor, such that the following relation is satisfied: [x, fy]A = f [x, y]A + aA(x)(f)y, ∀f ∈ C∞(M), x, y ∈ Γ(A). When the image of aA is of constant rank, we call A a regular Lie algebroid. For a vector bundle E −→ M , we denote by D(E) the gauge Lie algebroid of the frame bundle F(E), which is also called the covariant differential operator bundle of E. See [30] for more details on the gauge Lie algebroid. Let (A, [·, ·]A, aA) and (B, [·, ·]B, aB) be two Lie algebroids (with the same base), a base- preserving homomorphism from A to B is a bundle map φ : A −→ B such that aB ◦ φ = aA, φ[x, y]A = [φ(x), φ(y)]B, ∀x, y ∈ Γ(A). Recall that a representation of a Lie algebroid A on a vector bundle E is a base-preserving morphism ρ form A to the Lie algebroid D(E). Denote a representation by (E; ρ). The dual representation of a Lie algebroid A on E∗ is the bundle map ρ∗ : A −→ D(E∗) given by ⟨ρ∗(x)(ξ), u⟩ = aA(x)⟨ξ, u⟩ − ⟨ξ, ρ(x)(u)⟩, ∀x ∈ Γ(A), ξ ∈ Γ(E∗), u ∈ Γ(E). Given a representation (E; ρ), the cohomology ofA with coefficients in E is the cohomology of the cochain complex (⊕+∞ k=0C k(A, E), ∂ρ), where Ck(A, E) = Γ(Hom(∧kA, E)) and the coboundary operator ∂ρ : C k(A, E) → Ck+1(A, E) is defined by ∂ρϖ(x1, . . . , xk+1) = k+1∑ i=1 (−1)i+1ρ(xi)ϖ(x1, . . . , x̂i, . . . , xk+1) + ∑ i<j (−1)i+jϖ([xi, xj ]A, x1, . . . , x̂i, . . . , x̂j , . . . , xk+1), for ϖ ∈ Ck(A, E) and x1, . . . , xk+1 ∈ Γ(A). Definition 2.2. A LieRep pair is a pair of a Lie algebroid (A, [·, ·]A, aA) and a representation ρ of A on a vector bundle E. We denote a LieRep pair by (A, [·, ·]A, aA; ρ), or simply by (A; ρ). Definition 2.3 ([28]). Let (A, [·, ·]A, aA; ρ) be a LieRep pair. A bundle map T : E −→ A is called a relative Rota–Baxter operator on a LieRep pair (A, [·, ·]A, aA; ρ) if [T (u), T (v)]A = T (ρ(T (u))(v)− ρ(T (v))(u)), ∀u, v ∈ Γ(E). Definition 2.4. Let (g = ⊕k∈Zgk, [·, ·], d) be a differential graded Lie algebra. An element θ ∈ g1 is called a Maurer–Cartan element of g if it satisfies dθ + 1 2 [θ, θ] = 0. In particular, a Maurer–Cartan element of a graded Lie algebra (g = ⊕k∈Zgk, [·, ·]) is an element θ ∈ g1 satisfying [θ, θ] = 0. Let (A, [·, ·]A, aA; ρ) be a LieRep pair. Consider the graded vector space C∗(E,A) = ⊕k≥0Ck(E,A), where Ck(E,A) := Γ ( Hom ( ∧kE,A )) . Now we give the controlling algebra of relative Rota–Baxter operators on Lie algebroids, which is the main tool in the following study. Deformations and Cohomologies of Relative Rota–Baxter Operators on Lie Algebroids 5 Theorem 2.5. For P ∈ Cm(E,A) and Q ∈ Cn(E,A), we define a bracket operation [[P,Q]] (u1, u2, . . . , um+n) = ∑ σ∈S(m,1,n−1) (−1)σP (ρ(Q(uσ(1), . . . , uσ(m)))uσ(m+1), uσ(m+2), . . . , uσ(m+n)) (2.1) − (−1)mn ∑ σ∈S(n,1,m−1) (−1)σQ(ρ(P (uσ(1), . . . , uσ(n)))uσ(n+1), uσ(n+2), . . . , uσ(m+n)) + (−1)mn ∑ σ∈S(n,m) (−1)σ[P (uσ(1), uσ(2), . . . , uσ(n)), Q(uσ(n+1), uσ(n+2), . . . , uσ(m+n))]A, where u1, u2, . . . , um+n ∈ Γ(E). Then (C∗(E,A), [[·, ·]]) is a graded Lie algebra and its Maurer– Cartan elements are precisely relative Rota–Baxter operators on (A; ρ). Proof. It is straightforward to check that [[·, ·]] is skew-symmetric in all arguments and function linear. Thus [[P,Q]] ∈ Cm+n(E,A) for all P ∈ Cm(E,A) and Q ∈ Cn(E,A), which implies that [[·, ·]] is well defined. It was shown in [44] that the bracket [[·, ·]] provides a graded Lie algebra structure on the graded vector space ⊕k≥0HomR ( ∧kΓ(E),Γ(A) ) . Thus (C∗(E,A), [[·, ·]]) is a graded Lie algebra. Let T : E → A be a bundle map. By a direct calculation, we have [[T, T ]] (u1, u2) = 2(T (ρ(Tu1)u2)− T (ρ(Tu2)u1)− [Tu1, Tu2]A), ∀u1, u2 ∈ Γ(E). Thus T is a Maurer–Cartan element of the graded Lie algebra (C∗(E,A), [[·, ·]]) if and only if T is a relative Rota–Baxter operator on the LieRep pair (A, [·, ·]A, aA; ρ). ■ Let T : E −→ A be a relative Rota–Baxter operator on the LieRep pair (A, [·, ·]A, aA; ρ). By Theorem 2.5, T is a Maurer–Cartan element of the graded Lie algebra (C∗(E,A), [[·, ·]]). Note that d̃T := [[T, ·]] is a graded derivation on the graded Lie algebra (C∗(E,A), [[·, ·]]) satisfying d̃2T = 0. Therefore, (C∗(E,A), [[·, ·]] , d̃T ) is a differential graded Lie algebra. Theorem 2.6. Let (A, [·, ·]A, aA; ρ) be a LieRep pair and T : E −→ A a relative Rota–Baxter operator. Then for a bundle map T ′ : E −→ A, T +T ′ is still a relative Rota–Baxter operator on the LieRep pair (A, [·, ·]A, aA; ρ) if and only if T ′ is a Maurer–Cartan element of the differential graded Lie algebra ( C∗(E,A), [[·, ·]] , d̃T ) . Proof. Assume that T + T ′ is a relative Rota–Baxter operator on the LieRep pair (A, [·, ·]A, aA; ρ). By the fact that T is a relative Rota–Baxter operator, we have d̃TT ′ + 1 2 [[ T ′, T ′]] = [[T, T ′]]+ 1 2 [[ T ′, T ′]] = 1 2 [[ T + T ′, T + T ′]] = 0. Thus T ′ is a Maurer–Cartan element of the differential graded Lie algebra (C∗(E,A), [[·, ·]] , d̃T ). The converse can be proved similarly. We omit the details. ■ 2.2 Cohomologies of relative Rota–Baxter operators on Lie algebroids In this subsection, we give a cohomology theory of relative Rota–Baxter operators on Lie alge- broids, which will be used to study formal deformations of relative Rota–Baxter operators. Let T : E −→ A be a relative Rota–Baxter operator on a LieRep pair (A, [·, ·]A, aA; ρ). Define dT : Ck(E,A) → Ck+1(E,A) by dTP = (−1)kd̃TP = (−1)k [[T, P ]] , ∀P ∈ Ck(E,A). Since d̃T ◦ d̃T = 0, we have dT ◦ dT = 0. Thus (C∗(E,A) = ⊕k≥0Ck(E,A), dT ) is a cochain complex. Note the sign in the differential dT is motivated by Theorem 2.12 below. 6 M. Liu, J. Liu and Y. Sheng Definition 2.7. The cochain complex (C∗(E,A) = ⊕k≥0Ck(E,A), dT ) is called the cohomology complex of the relative Rota–Baxter operator T on the LieRep pair (A, [·, ·]A, aA; ρ). The corre- sponding k-th cohomology group, denoted by Hk T (E,A), is called the k-th cohomology group for the relative Rota–Baxter operator T . We give the coboundary operator dT explicitly. Proposition 2.8. For P ∈ Ck(E,A) and u1, . . . , uk+1 ∈ Γ(E), we have dTP (u1, u2, . . . , uk+1) = k+1∑ i=1 (−1)i+1[Tui, P (u1, u2, . . . , ûi, . . . , uk+1)]A + k+1∑ i=1 (−1)i+1Tρ(P (u1, u2, . . . , ûi, . . . , uk+1))(ui) + ∑ 1≤i<j≤k+1 (−1)i+jP (ρ(Tui)(uj)− ρ(Tuj)(ui), u1, . . . , ûi, . . . , ûj , . . . , uk+1). (2.2) Proof. It follows from a direct calculation. ■ It is obvious that P ∈ C1(E,A) is closed if and only if [Tu, P (v)]A − [Tv, P (u)]A − T (ρ(P (u))(v)− ρ(P (v))(u))− P (ρ(Tu)(v)− ρ(Tv)(u)) = 0, where u, v ∈ Γ(E). In the sequel, we give an alternative characterization of dT using the cohomology of Lie algebroids. First we recall a useful fact. Lemma 2.9 ([28]). Let T : E −→ A be a relative Rota–Baxter operator on a LieRep pair (A, [·, ·]A, aA; ρ). Then (E, [·, ·]T , aT = aA ◦ T ) is a Lie algebroid, where the bracket [·, ·]T is given by [u, v]T = ρ(T (u))v − ρ(T (v))u, ∀u, v ∈ Γ(E). Furthermore, T is a Lie algebroid homomorphism from (E, [·, ·]T , aT ) to (A, [·, ·]A, aA). Moreover, the Lie algebroid (E, [·, ·]T , aT ) represents on the vector bundle A. Lemma 2.10. Let T : E −→ A be a relative Rota–Baxter operator on a LieRep pair (A; ρ). Define ϱ : E → D(A) by ϱ(u)(x) := [Tu, x]A + Tρ(x)(u), x ∈ Γ(A), u ∈ Γ(E). Then ϱ is a representation of the Lie algebroid (E, [·, ·]T , aT = aA ◦ T ) on the vector bundle A. Proof. By a direct calculation, we have ϱ(fu)(x) = [T (fu), x]A + Tρ(x)(fu) = f [Tu, x]A − aA(x)(f)(Tu) + fTρ(x)(u) + TaA(x)(f)u = fϱ(u)(x) and ϱ(u)(fx) = [Tu, fx]A + Tρ(fx)(u) = f [Tu, x]A + aA(Tu)(f)(x) + fTρ(x)(u) = fϱ(u)(x) + aT (u)(f)(x). It is straightforward to check that ϱ[u, v]T = ϱ(u)ϱ(v)− ϱ(v)ϱ(u). Thus ϱ is a representation of the Lie algebroid (E, [·, ·]T , aT ) on A. ■ Deformations and Cohomologies of Relative Rota–Baxter Operators on Lie Algebroids 7 Remark 2.11. Let T : E −→ A be a relative Rota–Baxter operator on a LieRep pair (A; ρ). It is straightforward to check that (A, E, ρ, ϱ) is a matched pair of Lie algebroids, where the Lie algebroid structure on E is the Lie algebroid (E, [·, ·]T , aT ). See [32] for more details on matched pairs of Lie algebroids. Theorem 2.12. Let T : E −→ A be a relative Rota–Baxter operator on a LieRep pair (A, [·, ·]A, aA; ρ). Then the coboundary operator of the relative Rota–Baxter operator T is exactly the coboundary operator of the Lie algebroid (E, [·, ·]T , aT ) with coefficients in the representation (A; ϱ), that is, dT = ∂ϱ. Proof. By Proposition 2.8, for any P ∈ Ck(E,A) and u1, . . . , uk+1 ∈ Γ(E), we have dTP (u1, u2, . . . , uk+1) = k+1∑ i=1 (−1)i+1[Tui, P (u1, u2, . . . , ûi, . . . , uk+1)]A + k+1∑ i=1 (−1)i+1Tρ(P (u1, u2, . . . , ûi, . . . , uk+1))(ui) + ∑ 1≤i<j≤k+1 (−1)i+jP (ρ(Tui)(uj)− ρ(Tuj)(ui), u1, . . . , ûi, . . . , ûj , . . . , uk+1), = k+1∑ i=1 (−1)i+1ϱ(ui)P (u1, . . . , ûi, . . . , uk+1) + ∑ i<j (−1)i+jP ([ui, uj ]T , u1, . . . , ûi, . . . , ûj , . . . , uk+1) = ∂ϱP (u1, u2, . . . , uk+1). The conclusion follows. ■ 3 Cohomologies of Rota–Baxter operators on Lie algebroids In this section, first we give the cohomologies of Rota–Baxter operators on Lie algebroids with the help of the general framework of the cohomologies of relative Rota–Baxter operators. Then we study the cohomologies of the Rota–Baxter operator arising from an action of a Rota–Baxter Lie algebra on a manifold. Now we recall the notion of a Rota–Baxter operator on a Lie algebroid given in [19]. Definition 3.1. A Rota–Baxter operator on a regular Lie algebroid (A, [·, ·]A, aA) is a bundle map R : ker(aA) → A such that [R(x), R(y)]A = R([R(x), y]A + [x,R(y)]A), ∀x, y ∈ Γ(ker(aA)). For any x ∈ Γ(A), we define Lx : Γ(A) −→ Γ(A) by Lx(y) = [x, y]A for y ∈ Γ(A). Then L gives a representation of the Lie algebroid A on ker(aA). Thus a Rota–Baxter operator on a regular Lie algebroid (A, [·, ·]A, aA) is a relative Rota–Baxter operator on the LieRep pair (A, [·, ·]A, aA;L). A Rota–Baxter operator on a Lie algebra (g, [·, ·]g) is a linear map B : g → g such that [B(u),B(v)]g = B([B(u), v]g + [u,B(v)]g), ∀u, v ∈ g. The pair (g,B) is called a Rota–Baxter Lie algebra. 8 M. Liu, J. Liu and Y. Sheng Remark 3.2. Since a vector space is a vector bundle over a point, a Lie algebra is naturally a Lie algebroid with the anchor being zero. It is not hard to see that a Rota–Baxter operator on a Lie algebroid reduces to a Rota–Baxter operator on a Lie algebra when the underlying Lie algebroid reduces to a Lie algebra. By Theorem 2.5, we have Corollary 3.3. Let (A, [·, ·]A, aA) be a regular Lie algebroid. Then (i) ( ⊕dim(ker(aA)) k=0 Γ ( Hom ( ∧kker(aA),A )) , [[·, ·]] ) is a graded Lie algebra, where the graded Lie bracket [[·, ·]] is given by (2.1). (ii) R is a Rota–Baxter operator on the regular Lie algebroid A if and only if R is a Maurer– Cartan element of ( ⊕dim(ker(aA)) k=0 Γ ( Hom ( ∧kker(aA),A )) , [[·, ·]] ) . By Lemmas 2.9 and 2.10, we have Corollary 3.4. Let R : ker(aA) −→ A be a Rota–Baxter operator on a regular Lie algebroid (A, [·, ·]A, aA). Then (ker(aA), [·, ·]R, aR = aA ◦R) is a Lie algebroid, where the bracket [·, ·]R is given by [u, v]R := [Ru, v]A + [u,Rv]A, ∀u, v ∈ Γ(ker(aA)). Furthermore, ϱ : ker(aA) → Der(A) defined by ϱ(u)y := [R(u), y]A −R[u, y]A, ∀y ∈ Γ(A) gives a representation of the Lie algebroid (ker(aA), [·, ·]R, aR) on the vector bundle A. As a special case of Definition 2.7, we have Definition 3.5. Let R be a Rota–Baxter operator on a regular Lie algebroid (A, [·, ·]A, aA). The cohomology of the cochain complex ( ⊕kCk(ker(aA),A), dR ) , where the coboundary operator dR : Ck(ker(aA),A) → Ck+1(ker(aA),A) is given by (2.2) with T = R and ρ = L, is called the cohomology of the Rota–Baxter operator R. The corresponding k-th cohomology group, which we denote by Hk R(ker(aA),A), is called the k-th cohomology group for the Rota–Baxter operator R. At the end of this section, we analyze the cohomology of the Rota–Baxter operator arising from an action of a Rota–Baxter Lie algebra on a manifold. Let (g, [·, ·]g) be a Lie algebra. An action of g on a manifold M is a homomorphism of Lie algebras ϕ : (g, [·, ·]g) → (X(M), [·, ·]X(M)). For a Rota–Baxter operator B on a Lie algebra (g, [·, ·]g), the bracket [u, v]B = [B(u), v]g + [u,B(v)]g, ∀u, v ∈ g defines another Lie algebra structure on g. Recall from [19] that an action of a Rota– Baxter Lie algebra (g,B) on a manifold M is a homomorphism of Lie algebras ϕ : (g, [·, ·]B) → (X(M), [·, ·]X(M)). Let ϕ : (g, [·, ·]B) → (X(M), [·, ·]X(M)) be an action of the Rota–Baxter Lie algebra (g,B) on M . Consider the direct sum bundle A := (M × g) ⊕ TM . Then Γ(A) = (C∞(M)⊗g)⊕X(M). There is naturally a Lie algebroid structure on A whose anchor aA is the projection to TM and whose bracket is determined by [fu+X, gv + Y ]A := fg[u, v]g +X(g)v − Y (f)u+ [X,Y ]X(M), Deformations and Cohomologies of Relative Rota–Baxter Operators on Lie Algebroids 9 for all X,Y ∈ X(M), u, v ∈ g, f, g ∈ C∞(M). Consider the bundle map R : ker(aA) = M × g → (M × g)⊕ TM defined by R(m,u) := (m,B(u), ϕ(u)(m)), ∀m ∈ M, u ∈ g. (3.1) It was proved in [19] that the bundle map R defined by (3.1) is a Rota–Baxter operator on the Lie algebroid ((M × g)⊕ TM, [·, ·]A, aA). In the following, we establish the relations among the cohomology of the Rota–Baxter operator R on the Lie algebroid ((M × g) ⊕ TM, [·, ·]A, aA), the cohomology of the Rota– Baxter operator B on the Lie algebra g and the cohomology of the Lie algebra homomorphism ϕ : (g, [·, ·]B) → X(M). The cohomology of a Rota–Baxter operator B on a Lie algebra g is the cohomology of the cochain complex ( ⊕+∞ k=0C k(g, g), dB ) , where Ck(g, g) = Hom ( ∧kg, g ) and the coboundary operator dB : Hom ( ∧kg, g ) → Hom ( ∧k+1g, g ) is given by dBf(u1, . . . , uk+1) := k+1∑ i=1 (−1)i+1[B(ui), f(u1, . . . , ûi, . . . , uk+1)]g + k+1∑ i=1 (−1)i+1B[f(u1, . . . , ûi, . . . , uk+1), ui]g + ∑ 1≤i<j≤k+1 (−1)i+jf([B(ui), uj ]g − [B(uj), ui]g, u1, . . . , ûi, . . . , ûj , . . . , uk+1). See [44] for more details about the cohomology of Rota–Baxter operators on Lie algebras. Let ϕ : (g, [·, ·]B) → (X(M), [·, ·]X(M)) be an action of the Rota–Baxter Lie algebra (g,B) on a manifold M . The cohomology of the Lie algebra homomorphism ϕ is the cohomology of the cochain complex ( ⊕+∞ k=0C k ϕ(g,X(M)),dϕ ) , where Ck ϕ(g,X(M)) = HomR ( ∧kg,X(M) ) and the coboundary operator dϕ : C k ϕ(g,X(M)) → Ck+1 ϕ (g,X(M)) is given by dϕP (u1, . . . , uk+1) = k+1∑ i=1 (−1)i+1[ϕ(ui), P (u1, . . . , ûi, . . . , uk+1)]X(M) + ∑ i<j (−1)i+jP ([ui, uj ]B, u1, . . . , ûi, . . . , ûj , . . . , uk+1), for P ∈ Ck(g,X(M)) and u1, . . . , uk+1 ∈ g. The corresponding k-th cohomology group, which we denote by Hk ϕ(g,X(M)), is called the k-th cohomology group for the action of the Rota–Baxter Lie algebra (g,B) on M . See [12, 37] for more details on cohomology and deformations of Lie algebra homomorphisms. The cochain complex associated to the Rota–Baxter operator R on the Lie algebroid ((M×g) ⊕ TM, [·, ·]A, aA) defined by (3.1) is given by Ck(M × g, (M × g)⊕ TM) = Γ ( Hom ( ∧k(M × g), (M × g)⊕ TM )) , k ≥ 0. For any k ≥ 0, define a linear map Ξ: Ck(g, g)⊕ Ck ϕ(g,X(M)) → Ck(M × g, (M × g)⊕ TM) by Ξ(P1, P2)(m, (u1, . . . , uk)) := (m,P1(u1, . . . , uk), P2(u1, . . . , uk)(m)), (3.2) for all m ∈ M and u1, . . . , uk ∈ g. 10 M. Liu, J. Liu and Y. Sheng Proposition 3.6. Let ϕ : (g, [·, ·]B) → (X(M), [·, ·]X(M)) be an action of the Rota–Baxter Lie algebra (g,B) on a manifold M . Then Ξ defined by (3.2), is a homomorphism of cochain com- plexes from ( ⊕k≥0 ( Ck(g, g)⊕ Ck ϕ(g,X(M)) ) , (dB, dϕ) ) to ( ⊕k≥0Ck(M × g, (M × g)⊕ TM),dR ) , that is, Ξ ◦ (dB,dϕ) = dR ◦ Ξ. Consequently, Ξ induces a homomorphism Ξ∗ : Hk B(g, g)⊕Hk ϕ(g,X(M)) → Hk R(M × g, (M × g)⊕ TM), k ≥ 0, between the corresponding cohomology groups. Proof. For any P1 ∈ Ck(g, g), P2 ∈ Ck ϕ(g,X(M)) and u1, . . . , uk+1 ∈ g, we have dRΞ(P1, P2)(1⊗ u1, . . . , 1⊗ uk+1) = k+1∑ i=1 (−1)i+1[R(1⊗ ui), (P1(u1, u2, . . . , ûi, . . . , uk+1), P2(u1, . . . , ûi, . . . , uk+1))]A + k+1∑ i=1 (−1)i+1R[(P1(u1, u2, . . . , ûi, . . . , uk+1), P2(u1, . . . , ûi, . . . , uk+1)), ui]A + ∑ 1≤i<j≤k+1 (−1)i+j(P1([ui, uj ]B, u1, . . . , ûi, . . . , ûj , . . . , uk+1), P2([ui, uj ]B, u1, . . . , ûi, . . . , ûj , . . . , uk+1)) = ( k+1∑ i=1 (−1)i+1[B(ui), P1(u1, u2, . . . , ûi, . . . , uk+1)]g + k+1∑ i=1 (−1)i+1B[P1(u1, u2, . . . , ûi, . . . , uk+1), ui]g + ∑ 1≤i<j≤k+1 (−1)i+jP1([ui, uj ]B, u1, . . . , ûi, . . . , ûj , . . . , uk+1) + k+1∑ i=1 (−1)i+1[ϕ(ui), P2(u1, u2, . . . , ûi, . . . , uk+1)]X(M) + ∑ 1≤i<j≤k+1 (−1)i+jP2([ui, uj ]B, u1, . . . , ûi, . . . , ûj , . . . , uk+1) ) = (dBP1(u1, . . . , uk+1),dϕP2(u1, . . . , uk+1)) = Ξ(dBP1,dϕP2)(1⊗ u1, . . . , 1⊗ uk+1), which implies that Ξ ◦ (dB, dϕ) = dR ◦ Ξ. ■ 4 Formal deformations of relative Rota–Baxter operators on Lie algebroids In this section, we use the cohomology of relative Rota–Baxter operators on Lie algebroids to study infinitesimal deformations and extendibility of order n deformations of relative Rota– Baxter operators. Let (A, [·, ·]A, aA) be a Lie algebroid. The Lie algebroid structure on A can be extended to a Lie algebroid structure on A ⊗ R[[t]] by replacing R-linearity of the bracket and anchor by R[[t]]-linearity and we denote it by (A ⊗ R[[t]], [·, ·]A, aA). For any representation (E; ρ) of the Lie algebroid A, there is also a natural representation of A ⊗ R[[t]] on E ⊗ R[[t]] induced by ρ and we also denote it by ρ. Deformations and Cohomologies of Relative Rota–Baxter Operators on Lie Algebroids 11 Definition 4.1. A formal deformation of a relative Rota–Baxter operator T : E −→ A on a LieRep pair (A, [·, ·]A, aA; ρ) is a formal power series Tt = +∞∑ i=0 Titi ∈ Hom(E,A)[[t]] such that Tt is a relative Rota–Baxter operator on the LieRep pair (A ⊗ R[[t]], [·, ·]A, aA; ρ) and T0 = T . By a direct calculation, we see that Tt is a relative Rota–Baxter operator on the LieRep pair (A⊗ R[[t]], [·, ·]A, aA; ρ) if and only if∑ i+j=k ( [Ti(u), Tj(v)]A − Tj(ρ(Ti(u))(v)− ρ(Ti(v))(u)) ) = 0, ∀k ≥ 0, u, v ∈ Γ(E). (4.1) Definition 4.2. Let (A, [·, ·]A, aA; ρ) be a LieRep pair and T : E −→ A a relative Rota–Baxter operator. If T(n) = ∑n i=0 Titi with T0 = T , Ti ∈ Hom(E,A), i = 1, . . . , n is a relative Rota– Baxter operator on the LieRep pair ( A⊗R[[t]]/ ( tn+1 ) , [·, ·]A, aA; ρ ) , we say that T(n) is an order n deformation of the relative Rota–Baxter operator T . Furthermore, if there exists an element Tn+1 ∈ Hom(E,A) such that T(n+1) = T(n) + tn+1Tn+1 is an order n + 1 deformation of the relative Rota–Baxter operator T , we say that T(n) is extendable. An order 1 deformation of a relative Rota–Baxter operator T on a LieRep pair (A, [·, ·]A, aA; ρ) is called an infinitesimal deformation of the relative Rota–Baxter operator T . Definition 4.3. Let T : E −→ A be a relative Rota–Baxter operator on (A, [·, ·]A, aA; ρ). Two order n deformations Tt and T ′ t of T are said to be equivalent if there is a formal series Xt =∑+∞ i=1 xit i, xi ∈ Γ(A) such that exp(adXt)Tt = T ′ t modulo tn+1, (4.2) where exp denotes the exponential series and adkXt Tt = [[ Xt, [[ Xt, . . . , [[Xt, T t]], k. . . ]]]] . An order n deformation Tt of T is called trivial if Tt is equivalent to T . Theorem 4.4. There is a one-to-one correspondence between the equivalence classes of the infinitesimal deformations of a relative Rota–Baxter operator T : E −→ A on a LieRep pair (A, [·, ·]A, aA; ρ) and the first cohomology group H1 T (E,A). Proof. For k = 1 in (4.1), we have [Tu, T1v]A−[Tv, T1u]A−T (ρ(T1u)v−ρ(T1v)u)−T1(ρ(Tu)v−ρ(Tv)u)= 0, ∀u, v ∈ Γ(A), which implies that (dTT1)(u, v) = 0, i.e., T1 is a 1-cocycle. Assume that Tt and T ′ t are equivalent infinitesimal deformations of the relative Rota–Baxter operator T . Comparing the coefficients of t on both sides of (4.2) for n = 1, we obtain (T ′ 1 − T1)(u) = [Tu, x1]A + Tρ(x1)u = dTx1(u), which implies that T ′ 1 − T1 = dTx1. Thus T1 and T ′ 1 are in the same cohomology class. The converse can be proved similarly. We omit the details. ■ 12 M. Liu, J. Liu and Y. Sheng It is routine to check that Proposition 4.5. Let T : E −→ A be a relative Rota–Baxter operator on a LieRep pair (A, [·, ·]A, aA; ρ) such that H1 T (E,A) = 0. Then all infinitesimal deformations of T are trivial. Theorem 4.6. Let T(n) = ∑n i=0 Titi be an order n deformation of a relative Rota–Baxter ope- rator T on a LieRep pair (A, [·, ·]A, aA; ρ). Define Θ = 1 2 ∑ i+j=n+1 i,j≥1 [[Ti, Tj ]] . Then the 2-cochain Θ is closed, i.e., dTΘ = 0. Furthermore, T(n) is extendable if and only the cohomology class [Θ] in H2(E,A) is trivial. Proof. By a direct calculation, we have Θ(u, v) = ∑ i+j=n+1 i,j≥1 ( Tj(ρ(Ti(u))(v) + ρ(Ti(v))(u))− [Ti(u), Tj(v)]A ) , ∀u, v ∈ Γ(E). It is not hard to check that Θ(u, fv) = Θ(fu, v) = fΘ(u, v). Thus Θ ∈ C2(E,A). The rest follows directly from the fact that this deformation problem is controlled by the differential graded Lie algebra ( C∗(E,A), [[·, ·]] , d̃T ) . See the book [24] for more details. ■ The above results on infinitesimal deformations and order n deformations of relative Rota– Baxter operators on Lie algebroids can be easily applied to Rota–Baxter operators on Lie alge- broids. We omit the details. 5 The Matsushima–Nijenhuis bracket for left-symmetric algebroids In this section, we construct a graded Lie algebra whose Maurer–Cartan elements are left- symmetric algebroids and study the relation with the controlling graded Lie algebra of relative Rota–Baxter operators on Lie algebroids. 5.1 The Matsushima–Nijenhuis bracket for left-symmetric algebroids Recall that a left-symmetric algebra is a pair (g, ∗g), where g is a vector space, and ∗g : g⊗g −→ g is a bilinear multiplication satisfying that for all x, y, z ∈ g, the associator (x, y, z) := x ∗g (y ∗g z)− (x ∗g y) ∗g z is symmetric in x, y, i.e., (x, y, z) = (y, x, z), or equivalently, x ∗g (y ∗g z)− (x ∗g y) ∗g z = y ∗g (x ∗g z)− (y ∗g x) ∗g z. Deformations and Cohomologies of Relative Rota–Baxter Operators on Lie Algebroids 13 Definition 5.1 ([28, 33]). A left-symmetric algebroid structure on a vector bundle A −→ M is a pair that consists of a left-symmetric algebra structure ∗A on the section space Γ(A) and a vector bundle morphism aA : A −→ TM , called the anchor, such that for all f ∈ C∞(M) and x, y ∈ Γ(A), the following conditions are satisfied: (i) x ∗A (fy) = f(x ∗A y) + aA(x)(f)y, (ii) (fx) ∗A y = f(x ∗A y). We usually denote a left-symmetric algebroid by (A, ∗A, aA). Any left-symmetric algebra is a left-symmetric algebroid over a point. Let (A, ∗A, aA) be a left-symmetric algebroid. For any x ∈ Γ(A), we define Lx : Γ(A) −→ Γ(A) and Rx : Γ(A) −→ Γ(A) by Lxy = x ∗A y, Rxy = y ∗A x, ∀y ∈ Γ(A). Condition (i) in the above definition means that Lx ∈ D(A). Condition (ii) means that the map x 7−→ Lx is C∞(M)-linear. Thus, L : A −→ D(A) is a bundle map. With the same notations, there are two maps Lx, Rx : Γ(A∗) −→ Γ(A∗) given by ⟨Lxξ, y⟩ = aA(x)⟨ξ, y⟩ − ⟨ξ, Lxy⟩, ⟨Rxξ, y⟩ = −⟨ξ,Rxy⟩, ∀x, y ∈ Γ(A), ξ ∈ Γ(A∗). (5.1) Proposition 5.2 ([28]). Let (A, ∗A, aA) be a left-symmetric algebroid. Define a skew-symmetric bilinear bracket operation [·, ·]A on Γ(A) by [x, y]A = x ∗A y − y ∗A x, ∀x, y ∈ Γ(A). Then, (A, [·, ·]A, aA) is a Lie algebroid, denoted by Ac, called the sub-adjacent Lie algebroid of (A, ∗A, aA). Furthermore, L : A −→ D(A) gives a representation of the Lie algebroid Ac. A connection ∇ on M is said to be flat if the torsion tensor and the curvature tensor of ∇ vanish identically. A manifold M endowed with a flat connection ∇ is called a flat manifold. Example 5.3. Let (M,∇) be a flat manifold. Then (TM,∇, Id) is a left-symmetric algebroid whose sub-adjacent Lie algebroid is exactly the tangent Lie algebroid. We denote this left- symmetric algebroid by T∇M . Definition 5.4. Let E be a vector bundle over M , a multiderivation of degree n is a multilinear map D ∈ Hom(ΛnΓ(E) ⊗ Γ(E),Γ(E)), such that for all f ∈ C∞(M) and ui ∈ Γ(E), i = 1, 2, . . . , n+ 1, the following conditions are satisfied: D(u1, . . . , fui, . . . , un, un+1) = fD(u1, . . . , ui, . . . , un+1), i = 1, . . . , n, D(u1, . . . , un, fun+1) = fD(u1, . . . , un, un+1) + σD(u1, . . . , un)(f)un+1, where σD ∈ Γ(Hom(ΛnΓ(E), TM)) is called the symbol. We will denote by Dern(E) the space of multiderivations of n, n ≥ 0. We denote by Der∗(E) = ⊕mDerm(E) the space of multiderivations on a vector bundle E. Remark 5.5. The terminology “multiderivation” is usually referred to skew-symmetric ope- rators, like in Crainic–Moerdijk’s deformation complex of a Lie algebroid given in [9]. For convenience, we also use the terminology “multiderivation” for the above case. Note that this kind of operators also appeared under the name Der-valued forms in [46]. 14 M. Liu, J. Liu and Y. Sheng Theorem 5.6. For D1 ∈ Derm(E) and D2 ∈ Dern(E), we define the Matsushima–Nijenhuis bracket [·, ·]MN : Derm(E)×Dern(E) → Derm+n(E) by [D1, D2]MN = D1 ◦D2 − (−1)mnD2 ◦D1, where (D1 ◦D2)(u1, u2, . . . , um+n+1) = ∑ σ∈S(m,1,n−1) (−1)σD1(D2(uσ(1), . . . , uσ(m+1)), uσ(m+2), . . . , uσ(m+n), um+n+1) + (−1)mn ∑ σ∈S(n,m) (−1)σD1(uσ(1), . . . , uσ(n), D2(uσ(n+1), uσ(n+2), . . . , uσ(m+n), um+n+1)). Then (Der∗(E), [·, ·]MN) is a graded Lie algebra. Furthermore, π ∈ Der1(E) defines a left-symmetric algebroid structure on E if and only if [π, π]MN = 0, that is, π is a Maurer–Cartan element of the graded Lie algebra (Der∗(E), [·, ·]MN). Proof. First, we show that the space of multiderivations is closed under the Matsushima– Nijenhuis bracket. For D1 ∈ Derm(E) and D2 ∈ Dern(E), by a direct calculation, we have [D1, D2]MN(fu1, u2, . . . , um+n+1) = fD1 ◦D2(u1, u2, . . . , um+n+1)− (−1)mnfD2 ◦D1(u1, u2, . . . , um+n+1) + ∑ σ∈S(m−1,1,n−1) (−1)σσD2(uσ(2), . . . , uσ(m+1))(f)D1(u1, uσ(m+2), . . . , uσ(m+n), um+n+1) + (−1)mn ∑ σ∈S(n−1,1,m−1) (−1)σσD1(uσ(2), . . . , uσ(n+1))(f) ×D2(u1, uσ(n+2), . . . , uσ(m+n), um+n+1) − (−1)mn ∑ σ∈S(n−1,1,m−1) (−1)σσD1(uσ(2), . . . , uσ(n+1))(f) ×D2(u1, uσ(n+2), . . . , uσ(m+n), um+n+1) − ∑ σ∈S(m−1,1,n−1) (−1)σσD2(uσ(2), . . . , uσ(m+1))(f)D1(u1, uσ(m+2), . . . , uσ(m+n), um+n+1) = f [D1, D2]MN(u1, u2, . . . , um+n+1), which implies that [D1, D2]MN(fu1, u2, . . . , um+n+1) = f [D1, D2]MN(u1, u2, . . . , um+n+1). It is straightforward to check that [D1, D2]MN is skew-symmetric with respect to its first m+ n arguments. Thus [D1, D2]MN is C∞(M)-linear with respect to its first m+ n arguments. By a direct calculation, we have [D1, D2]MN(u1, u2, . . . , fum+n+1) = f [D1, D2]MN(u1, u2, . . . , um+n+1) + σ[D1,D2]MN (u1, u2, . . . , um+n)(f)um+n+1, where the symbol σ[D1,D2]MN is given by σ[D1,D2]MN (u1, u2, . . . , um+n)(f) Deformations and Cohomologies of Relative Rota–Baxter Operators on Lie Algebroids 15 = ∑ σ∈S(m,1,n−1) (−1)σσD1(D2(uσ(1), . . . , uσ(m+1)), uσ(m+2), . . . , uσ(m+n)))(f) + ∑ σ∈S(n,1,m−1) (−1)σσD2(D1(uσ(1), . . . , uσ(n+1)), uσ(n+2), . . . , uσ(m+n))(f) + (−1)mn ∑ σ∈S(m,n) (−1)σσD1(uσ(1), . . . , uσ(n))(σD2(uσ(n+1), . . . , uσ(n+m)))(f) + ∑ σ∈S(m,n) (−1)σσD2(uσ(1), . . . , uσ(m))(σD1(uσ(m+1), . . . , uσ(m+n)))(f). Thus [D1, D2]MN ∈ Derm+n(E). It was shown in [7, 35] that the Matsushima–Nijenhuis bracket provides a graded Lie algebra structure on the graded vector space ⊕n≥1HomR ( Λn−1Γ(E)⊗Γ(E),Γ(E) ) . We have shown that Der∗(E) is closed under the Matsushima–Nijenhuis bracket. Thus (Der∗(E), [·, ·]MN) is a graded Lie algebra. For π ∈ Der1(E), we have π(fu1, u2) = fπ(u1, u2), π(u1, fu2) = fπ(u1, u2) + σπ(u1)(f)u2, ∀u1, u2 ∈ Γ(E). Furthermore, by a direct calculation, we have [π, π]MN(u1, u2, u3) = 2(π(π(u1, u2), u3)− π(π(u2, u1), u3)− π(u1, π(u2, u3)) + π(u2, (u1, u3))). Thus (E, π, σπ) is a left-symmetric algebroid if and only if [π, π]MN = 0. ■ Remark 5.7. The cohomology of left-symmetric algebras first appeared in the unpublished paper of Y. Matsushima. Then A. Nijenhuis constructed a graded Lie bracket in [35], which produces the cohomology theory for left-symmetric algebras. Thus the aforementioned graded Lie bracket is usually called the Matsushima–Nijenhuis bracket. Let (E, π, σπ) be a left-symmetric algebroid. By Theorem 5.6, we have [π, π]MN = 0. Because of the graded Jacobi identity, we get a coboundary operator ddef : Dern−1(E) → Dern(E) defi- ned by ddef(D) = (−1)n−1[π,D]MN, ∀D ∈ Dern−1(E). Proposition 5.8. For all D ∈ Dern−1(E), we have ddefD(u1, u2, . . . , un+1) = n∑ i=1 (−1)i+1π(ui, D(u1, u2, . . . , ûi, . . . , un+1)) + n∑ i=1 (−1)i+1π(D(u1, u2, . . . , ûi, . . . , un, ui), un+1) − n∑ i=1 (−1)i+1D(u1, u2, . . . , ûi, . . . , un, π(ui, un+1)) + ∑ 1≤i<j≤n (−1)i+jD(π(ui, uj)− π(uj , ui), u1, . . . , ûi, . . . , ûj , . . . , un+1) (5.2) 16 M. Liu, J. Liu and Y. Sheng for all ui ∈ Γ(E), i = 1, 2, . . . , n+ 1 and σddefD is given by σddefD(u1, u2, . . . , un) = n∑ i=1 (−1)i+1[σπ(ui), σD(u1, u2, . . . , ûi, . . . , un)]X(M) + ∑ 1≤i<j≤n (−1)i+jσD(π(ui, uj)− π(uj , ui), u1, . . . , ûi, . . . , ûj , . . . , un) + n∑ i=1 (−1)i+1σπ(D(u1, u2, . . . , ûi, . . . , un, ui)). Proof. It follows from straightforward verification. ■ Definition 5.9. The cochain complex ( Der∗(E) = ⊕ n≥0Dern(E), ddef ) is called the deforma- tion complex of the left-symmetric algebroid E. The corresponding k-th cohomology group, which we denote by Hk def(E), is called the k-th deformation cohomology group. Remark 5.10. The coboundary operator ddef given by (5.2) is exactly the coboundary oper- ator given in [28] in the study of deformations of left-symmetric algebroids. Here we give this coboundary operator ddef intrinsically using the Matsushima–Nijenhuis bracket. 5.2 Relations between the graded Lie algebra (C∗(E,A), [[·, ·]]) and (Der∗(E), [·, ·]MN) Let (A, [·, ·]A, aA; ρ) be a LieRep pair. We define a bundle map Φ: Ck(E,A) → HomR(Λ kΓ(E)⊗ Γ(E),Γ(E)) as follows: for P ∈ Ck(E,A), Φ(P )(u1, . . . , uk, uk+1) = ρ(P (u1, . . . , uk))uk+1, ∀u1, . . . , uk+1 ∈ Γ(E). (5.3) Lemma 5.11. With the above notations, Φ(P ) ∈ Derk(E) and σΦ(P ) = aA ◦ P. Proof. By the properties of the representation ρ, we have Φ(P )(fu1, . . . , uk, uk+1) = ρ(P (fu1, . . . , uk))uk+1 = fρ(P (u1, . . . , uk))uk+1 = fΦ(P )(u1, . . . , uk, uk+1). Since Φ(P ) is skew-symmetric with respect to its first k arguments, Φ(P ) is C∞(M)-linear with respect to its first k arguments. Similarly, by a direct calculation, we have Φ(P )(u1, . . . , uk, fuk+1) = ρ(P (u1, . . . , uk))(fuk+1) = fρ(P (u1, . . . , uk))(uk+1) + aA(P (u1, . . . , uk))(f)uk+1 = fΦ(P )(fu1, . . . , uk, uk+1) + σΦ(P )(u1, . . . , uk)(f)uk+1. Thus Φ(P ) ∈ Derk(E). ■ Recall from Theorems 2.5 and 5.6 that (C∗(E,A), [[·, ·]]) and (Der∗(E), [·, ·]MN) are graded Lie algebras whose Maurer–Cartan elements are relative Rota–Baxter operators and left-symmetric algebroids respectively. Theorem 5.12. Let (A, [·, ·]A, aA; ρ) be a LieRep pair. Then Φ given by (5.3) is a homomor- phism of graded Lie algebras from (C∗(E,A), [[·, ·]]) to (Der∗(E), [·, ·]MN). Deformations and Cohomologies of Relative Rota–Baxter Operators on Lie Algebroids 17 Proof. On the one hand, for P ∈ Cn(E,A), Q ∈ Cm(E,A), we have Φ([[P,Q]])(u1, u2, . . . , um+n+1) = ρ([[P,Q]] (u1, . . . , um+n))um+n+1 = ∑ S(m,1,n−1) (−1)σρ(P (ρ(Q(uσ(1), . . . , uσ(m)))uσ(m+1), uσ(m+2), . . . , uσ(m+n)))um+n+1 −(−1)mn ∑ S(n,1,m−1) (−1)σρ(Q(ρ(P (uσ(1), . . . , uσ(n)))uσ(n+1), uσ(n+2), . . . , uσ(m+n)))um+n+1 +(−1)mn ∑ S(n,m) (−1)σρ([P (uσ(1), . . . , uσ(n)), Q(uσ(n+1), uσ(n+2), . . . , uσ(m+n))]A)um+n+1 = ∑ S(m,1,n−1) (−1)σρ(P (ρ(Q(uσ(1), . . . , uσ(m)))uσ(m+1), uσ(m+2), . . . , uσ(m+n)))um+n+1 −(−1)mn ∑ S(n,1,m−1) (−1)σρ(Q(ρ(P (uσ(1), . . . , uσ(n)))uσ(n+1), uσ(n+2), . . . , uσ(m+n)))um+n+1 +(−1)mn ∑ S(n,m) (−1)σρ(P (uσ(1), . . . , uσ(n)))ρ(Q(uσ(n+1), uσ(n+2), . . . , uσ(m+n)))um+n+1 +(−1)mn ∑ S(n,m) (−1)σρ(Q(uσ(n+1), uσ(n+2), . . . , uσ(m+n)))ρ(P (uσ(1), . . . , uσ(n)))um+n+1. On the other hand, we have (Φ(P ) ◦ Φ(Q))(u1, u2, . . . , um+n+1) = ∑ S(m,1,n−1) (−1)σρ(P (ρ(Q(uσ(1), . . . , uσ(m)))uσ(m+1), uσ(m+2), . . . , uσ(m+n)))um+n+1 + (−1)mn ∑ S(n,m) (−1)σρ(P (uσ(1), . . . , uσ(n)))ρ(Q(uσ(n+1), uσ(n+2), . . . , uσ(m+n)))um+n+1, and −(−1)mn(Φ(Q) ◦ Φ(P ))(u1, u2, . . . , um+n+1) = ∑ S(n,1,m−1) (−1)σρ(Q(ρ(P (uσ(1), . . . , uσ(n)))uσ(n+1), uσ(n+2), . . . , uσ(m+n)))um+n+1 − (−1)mn ∑ S(n,m) (−1)σρ(Q(uσ(n+1), uσ(n+2), . . . , uσ(m+n)))ρ(P (uσ(1), . . . , uσ(n)))um+n+1. Thus, we have Φ([[P,Q]]) = Φ(P ) ◦ Φ(Q)− (−1)mnΦ(Q) ◦ Φ(P ) = [Φ(P ),Φ(Q)]MN, that is, Φ is a homomorphism from (C∗(E,A), [[·, ·]]) to (Der∗(E), [·, ·]MN). ■ The following conclusion has been proved in [28] by a direct calculation. We give an intrinsic proof. Corollary 5.13. Let T : E −→ A be a relative Rota–Baxter operator on a LieRep pair (A, [·, ·]A, aA; ρ). Then (E, ∗T , aT = aA ◦ T ) is a left-symmetric algebroid, where ∗T is given by u ∗T v = ρ(Tu)(v), ∀u, v ∈ Γ(E). 18 M. Liu, J. Liu and Y. Sheng Proof. Since T is a relative Rota–Baxter operator on a LieRep pair (A, [·, ·]A, aA; ρ), by Theo- rem 2.5, we have [[T, T ]] = 0. By Theorem 5.12, we have [Φ(T ),Φ(T )]MN = 0. By Theorem 5.6, Φ(T ) provides a left-symmetric algebroid structure on E. Note that u ∗T v = Φ(T )(u, v) = ρ(Tu)(v). Thus (E, ∗T , aT = aA ◦ T ) is a left-symmetric algebroid. ■ Theorem 5.14. Let T be a relative Rota–Baxter operator on a LieRep pair (A; ρ). Then Φ given by (5.3) is a homomorphism from the cochain complex (C∗(E,A),dT ) to (Der∗(E),ddef), that is, ddef ◦Φ = Φ◦dT . Consequently, Φ induces a homomorphism Φ∗ : Hk(E,A) → Hk def(E) from the cohomology groups of the relative Rota–Baxter operator T to the deformation cohomology groups of the induced left-symmetric algebroid (E, ∗T , aT ). Proof. By Theorem 5.12, we have Φ([[P,Q]]) = [Φ(P ),Φ(Q)]MN. Note that the left-symmetric algebroid structure on E is given by Φ(T ). For P ∈ Ck(E,A), we have ddefΦ(P ) = (−1)k[Φ(T ),Φ(P )]MN = Φ ( (−1)k [[T, P ]] ) = Φ(dTP ), which implies that ddef ◦ Φ = Φ ◦ dT . The rest is direct. ■ At the end of this section, we show that a formal deformation of a relative Rota–Baxter operator induces a formal deformation of the associated left-symmetric algebroid. Recall that a formal deformation of a left-symmetric algebroid (A, ∗A, aA) is a left-symmetric algebroid (A⊗ R[[t]], ∗t, at) with power series ∗t = +∞∑ i=0 µit i ∈ Der1(A)[[t]], at = +∞∑ i=0 ait i ∈ Hom(A, TM)[[t]], such that (A⊗ R[[t]], ∗t, at)t=0 = (A, ∗A, aA). Proposition 5.15. Let Tt be a formal deformation of the relative Rota–Baxter operator T : E −→ A on a LieRep pair (A, [·, ·]A, aA; ρ). Then (E⊗R[[t]], ∗t, at = aA◦Tt) is a formal deformation of the left-symmetric algebroid (E, ∗T , aT ) associated to the relative Rota–Baxter operator T , where u ∗t v = ρ(Tt(u))v, ∀u, v ∈ Γ(E). Proof. Since Tt is a formal deformation of the relative Rota–Baxter operator T , by Corol- lary 5.13, (E⊗R[[t]], ∗t, at = aA◦Tt) is a left-symmetric algebroid. Note that (E⊗R[[t]], ∗t, at)t=0 = (E, ∗T , aT ). Thus (E ⊗ R[[t]], ∗t, at) is a formal deformation of the left-symmetric algebroid (E, ∗T , aT ). ■ Deformations and Cohomologies of Relative Rota–Baxter Operators on Lie Algebroids 19 6 Maurer–Cartan characterizations and cohomology of Koszul–Vinberg structures on left-symmetric algebroids In this section, we apply the controlling graded Lie algebra associated to relative Rota–Baxter operators to construct a graded Lie algebra whose Maurer–Cartan elements are precisely Koszul– Vinberg structures. Then we use this graded Lie algebra to study deformations of Koszul– Vinberg structures. 6.1 Maurer–Cartan characterizations of Koszul–Vinberg structures Let us first recall the cochain complex of a left-symmetric algebroid with coefficients in the trivial representation. See [28] for the general theory of cohomology of left-symmetric algebroids. Let (A, ∗A, aA) be a left-symmetric algebroid. The set of n-cochains is given by Cn(A) = Γ ( ∧n−1A∗ ⊗A∗), n ≥ 1. For all φ ∈ Cn(A) and xi ∈ Γ(A), i = 1, . . . , n+ 1, the coboundary operator δA is given by δAφ(x1, . . . , xn+1) = n∑ i=1 (−1)i+1aA(xi)φ(x1, . . . , x̂i, . . . , xn+1) − n∑ i=1 (−1)i+1φ(x1, . . . , x̂i, . . . , xn, xi ∗A xn+1) + ∑ 1≤i<j≤n (−1)i+jφ([xi, xj ]A, x1, . . . , x̂i, . . . , x̂j , . . . , xn+1). (6.1) Let (A, ∗A, aA) be a left-symmetric algebroid. Define Sym2(A) = {H ∈ A⊗A\|H(α, β) = H(β, α), ∀α, β ∈ Γ(A∗)}. For any H ∈ Sym2(A), the bundle map H♯ : A∗ −→ A is given by H♯(α)(β) = H(α, β). In [26], the authors introduced [H,H] ∈ Γ ( ∧2A⊗A ) as follows [H,H](α1, α2, α3) = aA(H ♯(α1))⟨H♯(α2), α3⟩ − aA(H ♯(α2))⟨H♯(α1), α3⟩ + ⟨α1, H ♯(α2) ∗A H♯(α3)⟩ − ⟨α2, H ♯(α1) ∗A H♯(α3)⟩ − ⟨α3, [H ♯(α1), H ♯(α2)]A⟩, (6.2) for all α1, α2, α3 ∈ Γ(A∗). Suppose that H♯ : A∗ −→ A is nondegenerate. Then (H♯)−1 : A −→ A∗ is also a symmetric bundle map, which gives rise to an element, denoted byH−1, in Sym2(A∗). Proposition 6.1 ([26]). Let (A, ∗A, aA) be a left-symmetric algebroid and H ∈ Sym2(A). If H is nondegenerate, then [H,H] = 0 if and only if δA(H −1) = 0, i.e. H−1 is a 2-cocycle on the left-symmetric algebroid A. Recall that a pseudo-Hessian metric g is a pseudo-Riemannian metric g on a flat manifold (M,∇) such that g can be locally expressed by gij = ∂2φ ∂xi∂xj , where φ ∈ C∞(M) and { x1, . . . , xn } is an affine coordinate system with respect to ∇. Then the pair (∇, g) is called a pseudo-Hessian structure on M . A manifold M with a pseudo-Hessian structure (∇, g) is called a pseudo- Hessian manifold. See [43] for more details about pseudo-Hessian manifolds. Let (M,∇) be a flat manifold and g a pseudo-Riemannian metric on M . Then (M,∇, g) is a pseudo-Hessian manifold if and only if δT∇Mg = 0, where δT∇M is the coboundary operator given by (6.1) associated to the left-symmetric algebroid T∇M given in Example 5.3. Now we give the main structure studied in this section. 20 M. Liu, J. Liu and Y. Sheng Definition 6.2. Let (A, ∗A, aA) be a left-symmetric algebroid. (i) If H ∈ Sym2(A) satisfies [H,H] = 0, then H is called a Koszul–Vinberg structure on the left-symmetric algebroid A; (ii) If B ∈ Sym2(A∗) is nondegenerate and satisfies δAB = 0, then B is called a pseudo- Hessian structure on the left-symmetric algebroid A. Let (A, ∗A, aA) be a left-symmetric algebroid, and H ∈ Sym2(A). Define α ∗H♯ β = LH♯(α)β −RH♯(β)α− dA(H(α, β)), ∀α, β ∈ Γ(A∗), (6.3) where L is the Lie derivation of the sub-adjacent Lie algebroid Ac, R and dA are given by ⟨Rxα, y⟩ = −⟨α, y ∗A x⟩, dAf(x) = aA(x)f, ∀x, y ∈ Γ(A), f ∈ C∞(M). Theorem 6.3 ([26]). If H is a Koszul–Vinberg structure on a left-symmetric algebroid (A, ∗A, aA), then (A∗, ∗H♯ , aH♯ = aA ◦ H♯) is a left-symmetric algebroid, and H♯ is a left- symmetric algebroid homomorphism from (A∗, ∗H♯ , aH♯) to (A, ∗A, aA). The sub-adjacent Lie algebroid of the left-symmetric algebroid (A∗, ∗H♯ , aH♯) is (A∗, [·, ·]H♯ , aH♯), where [·, ·]H♯ is given by [α, β]H♯ = LH♯(α)β − LH♯(β)α, ∀α, β ∈ Γ(A∗), (6.4) where L is given by (5.1). Proposition 6.4 ([26]). With the above notations, for all α, β ∈ Γ(A∗), we have H♯([α, β]H♯)− [H♯(α), H♯(β)]A = [H,H](α, β, ·). Note that L : A −→ D(A∗) is a representation of the sub-adjacent Lie algebroid Ac on the dual bundle A∗. Thus, by Proposition 6.4, we have Proposition 6.5. H is a Koszul–Vinberg structure on a left-symmetric algebroid (A, ∗A, aA) if and only if H♯ : A∗ −→ A is a relative Rota–Baxter operator on the LieRep pair (Ac;L). By Theorem 2.5 and Proposition 6.5, we have Lemma 6.6. Let (A, ∗A, aA) be a left-symmetric algebroid and H ∈ Sym2(A). (i) ( C∗(A∗,A) := ⊕k≥0Γ ( Hom ( ∧kA∗,A )) , [[·, ·]] ) is a graded Lie algebra, where the bracket [[·, ·]] is given by (2.1), in which ρ = L is given by (5.1). (ii) H is a Koszul–Vinberg structure on the left-symmetric algebroid if and only if H♯ is a Maurer–Cartan element of the graded Lie algebra (C∗(A∗,A), [[·, ·]]). For k ≥ 0, define Ψ: Γ ( ∧kA⊗A ) −→ Ck(A∗,A) by ⟨Ψ(φ)(α1, . . . , αk), αk+1⟩ = ⟨φ, α1 ∧ · · · ∧ αk ⊗ αk+1⟩, ∀α1, . . . , αk+1 ∈ Γ(A∗), (6.5) and Υ: Ck(A∗,A) −→ Γ ( ∧kA⊗A ) by ⟨Υ(P ), α1 ∧ · · · ∧ αk ⊗ αk+1⟩ = ⟨P (α1, . . . , αk), αk+1⟩, ∀α1, . . . , αk+1 ∈ Γ(A∗). Obviously we have Ψ ◦Υ = Id, Υ ◦Ψ = Id. By Lemma 6.6, we have Deformations and Cohomologies of Relative Rota–Baxter Operators on Lie Algebroids 21 Theorem 6.7. Let (A, ∗A, aA) be a left-symmetric algebroid. Then, there is a graded Lie bracket [[·, ·]]KV : Γ ( ∧kA⊗A ) × Γ ( ∧lA⊗A ) −→ Γ ( ∧k+lA⊗A ) on the graded vector space C∗ KV(A∗) := ⊕k≥1C k KV(A∗) with Ck KV(A∗) := Γ ( ∧k−1A⊗A ) given by [[φ, ϕ]]KV := Υ [[Ψ(φ),Ψ(ϕ)]] , ∀φ ∈ Γ ( ∧kA⊗A ) , ϕ ∈ Γ ( ∧lA⊗A ) . Furthermore, H ∈ Sym2(A) is a Koszul–Vinberg structure on the left-symmetric algebroid A if and only if H is a Maurer–Cartan element of the graded Lie algebra (C∗ KV(A∗), [[·, ·]]KV). More precisely, we have [[H,H]]KV (α1, α2, α3) = 2[H,H](α1, α2, α3), ∀α1, α2, α3 ∈ Γ(A∗), where [H,H] is given by (6.2). Remark 6.8. We characterize a Koszul–Vinberg structure on a left-symmetric algebroid A as a Maurer–Cartan element of the graded Lie algebra (C∗ KV(A∗), [[·, ·]]KV). This is parallel to the fact that a Poisson structure is a Maurer–Cartan element of the graded Lie algebra given by the Schouten–Nijenhuis bracket of multi-vector fields. 6.2 Cohomologies and deformations of Koszul–Vinberg structures Let H ∈ Sym2(A) be a Koszul–Vinberg structure on a left-symmetric algebroid (A, ∗A, aA). Define δA∗ : Ck KV(A∗) −→ Ck+1 KV (A∗) by δA∗φ = (−1)k−1 [[H,φ]]KV , ∀φ ∈ Ck KV(A∗). By the graded Jacobi identity, we have δA∗ ◦ δA∗ = 0. Thus (C∗ KV(A∗), δA∗) is a cochain comp- lex. Denote by Hk KV(A∗) the k-th cohomology group, called the k-th cohomology group of the Koszul–Vinberg structure H. Furthermore, we have Proposition 6.9. For φ ∈ Ck KV(A∗), we have δA∗φ(α1, . . . , αk+1) = k∑ i=1 (−1)i+1aH♯(αi)φ(α1, . . . , α̂i, . . . , αk+1) − k∑ i=1 (−1)i+1φ(α1, . . . , α̂i, . . . , αk, αi ∗H♯ αk+1) + ∑ 1≤i<j≤k (−1)i+jφ([αi, αj ]H♯ , α1, . . . , α̂i, . . . , α̂j , . . . , αk+1), where α1, . . . , αk+1 ∈ Γ(A∗), ∗H♯ is given by (6.3) and [·, ·]H♯ is given by (6.4). Proof. It follows by a direct calculation. ■ Remark 6.10. Note that this coboundary operator δA∗ is just the coboundary operator given by (6.1) associated to the left-symmetric algebroid (A∗, ∗H♯ , aH♯) in Theorem 6.3. By Corollary 5.13 and Proposition 6.5, we have Proposition 6.11. Let H be a Koszul–Vinberg structure on a left-symmetric algebroid (A, ∗A, aA). Then (A∗, ·H♯ , aH♯ = aA ◦H♯) is a left-symmetric algebroid, where ·H♯ is given by α ·H♯ β = LH♯(α)β, ∀α, β ∈ Γ(A∗). 22 M. Liu, J. Liu and Y. Sheng Remark 6.12. The left-symmetric algebroids (A∗, ·H♯ , aH♯) and (A∗, ∗H♯ , aH♯) have the same sub-adjacent Lie algebroid (A∗, [·, ·]H♯ , aH♯). By Lemma 2.10, we have Proposition 6.13. Let H be a Koszul–Vinberg structure on a left-symmetric algebroid (A, ∗A, aA). Then ϱ : A∗ −→ D(A), ϱ(α)(x) = [H♯(α), x]A +H♯(Lxα), ∀x ∈ Γ(A), α ∈ Γ(A∗) (6.6) is a representation of the sub-adjacent Lie algebroid (A∗, [·, ·]H♯ , aH♯) on the vector bundle A. Remark 6.14. The representation ϱ given by (6.6) is exactly the dual representation of the left multiplication operation of the left-symmetric algebroid (A∗, ∗H♯ , aH♯). More precisely, let us denote by L : A∗ −→ D(A∗) the left multiplication operation of the left-symmetric algebroid (A∗, ∗H♯ , aH♯), then we have ⟨Lαx, β⟩ = aH♯(α)⟨x, β⟩ − ⟨x, α ∗H♯ β⟩ = aH♯(α)⟨x, β⟩ − ⟨x,LH♯(α)β −RH♯(β)α− dA(H(α, β))⟩ = aH♯(α)⟨x, β⟩−aA(H ♯(α))⟨x, β⟩+[H♯(α), x]A−⟨α, x ∗A H♯(β)⟩+aA(x)H(α, β) = ⟨[H♯(α), x]A +H♯(Lxα), β⟩ = ⟨ϱ(α)(x), β⟩. Thus we have Lαx = ϱ(α)(x). Let H be a Koszul–Vinberg structure on a left-symmetric algebroid (A, ∗A, aA). By Theo- rem 2.12, for P ∈ Ck(A∗,A) and α1, . . . , αk+1 ∈ Γ(A∗), the coboundary operator dH♯ : Ck(A∗,A) −→ Ck+1(A∗,A) of the relative Rota–Baxter operator H♯ is given by dH♯P (α1, . . . , αk+1) = k+1∑ i=1 (−1)i+1[H♯(αi), P (α1, α2, . . . , α̂i, . . . , αk+1)]A + k+1∑ i=1 (−1)i+1H♯ ( LP (α1,α2,...,α̂i,...,αk+1)αi ) + ∑ 1≤i<j≤k+1 (−1)i+jP ([αi, αj ]H♯ , α1, . . . , α̂i, . . . , α̂j , . . . , αk+1). Denote byHk(A∗,A) the k-th cohomology group, called the k-th cohomology group of the relative Rota–Baxter operator H♯. Proposition 6.15. With the above notations, the map Ψ defined by (6.5) is a cochain iso- morphism between cochain complexes (C∗ KV(A∗), δA∗) and (C∗(A∗,A), dH♯), i.e., we have the following commutative diagram: · · · −→ Ck+1 KV (A∗) Ψ �� δA∗ // Ck+2 KV (A∗) Ψ �� // · · · · · · −→ Ck(A∗,A) d H♯ // Ck+1(A∗,A) // · · · . Consequently, Ψ induces an isomorphism map Ψ∗ between the corresponding cohomology groups. Deformations and Cohomologies of Relative Rota–Baxter Operators on Lie Algebroids 23 Proof. It is straightforward to see that Ψ is a graded Lie algebra isomorphism between the graded Lie algebra (C∗(A∗,A), [[·, ·]]) and (C∗ KV(A∗), [[·, ·]]KV). Thus for any P ∈ Ck+1 KV (A∗), we have Ψ(δA∗P ) = Ψ((−1)k [[H,P ]]KV) = (−1)k [[Ψ(H),Ψ(P )]] = dH♯Ψ(P ), which implies that dH♯◦Ψ = Ψ◦δA∗ , i.e., the map Ψ is a cochain map between cochain complexes (C∗ KV(A∗), δA∗) and (C∗(A∗,A),dH♯). Consequently, for any k ≥ 0, Ψ induces an isomorphism between the corresponding cohomology groups. ■ Now we introduce a new cochain complex, whose cohomology groups control deformations of Koszul–Vinberg structures. Let H be a Koszul–Vinberg structure on a left-symmetric algebroid (A, ∗A, aA). For all α1, α2, α3 ∈ Γ(A∗), define C̃1 KV(A∗) = { x ∈ C1 KV(A∗) \| H(Rxα1, α2) = H(α1, Rxα2) } , C̃2 KV(A∗) = { φ ∈ C2 KV(A∗) \| φ(α1, α2) = φ(α2, α1) } , C̃3 KV(A∗) = { φ ∈ C3 KV(A∗) \| φ(α1, α2, α3) + c.p. = 0 } , C̃k KV(A∗) = Ck KV(A∗), k ≥ 4. It is straightforward to verify that the cochain complex ( C̃∗ KV(A∗), δA∗ ) is a subcomplex of the cochain complex (C∗ KV(A∗), δA∗). Denote by H̃k KV(A∗) the k-th cohomology group. Definition 6.16. Let H be a Koszul–Vinberg structure on a left-symmetric algebroid (A, ∗A, aA). A formal deformation of the Koszul–Vinberg structure H is a formal power se- ries Ht = +∞∑ i=0 Hit i ∈ Sym2(A)[[t]] such that Ht is a Koszul–Vinberg structure on the left-symmetric algebroid (A⊗ R[[t]], ∗A, aA) and H0 = H. Note that Ht is a formal deformation of the Koszul–Vinberg structure H if and only if H♯ t is a formal deformation of the relative Rota–Baxter operator H♯ on the LieRep pair (Ac;L). Definition 6.17. Let H be a Koszul–Vinberg structure on a left-symmetric algebroid (A, ∗A, aA). If H(n) = ∑n i=0Hit i with H0 = H, Hi ∈ Sym2(A), i = 1, . . . , n is a Koszul– Vinberg structure on the left-symmetric algebroid ( A⊗R[[t]]/ ( tn+1 ) , ∗A, aA ) , we say that H(n) is an order n deformation of the Koszul–Vinberg structure H. Furthermore, if there exists an element Hn+1 ∈ Sym2(A) such that H(n+1) = H(n)+ tn+1Hn+1 is an order n deformation of the Koszul–Vinberg structure H, we say that H(n) is extendable. We call an order 1 deformation of the Koszul–Vinberg structure H on a left-symmetric algebroid (A, ∗A, aA) an infinitesimal deformation of the Koszul–Vinberg structure H. It is not hard to check that H(n) is an order n deformation of the Koszul–Vinberg structure H if and only if H♯ (n) is an order n deformation of the relative Rota–Baxter operator H♯ on the LieRep pair (Ac;L). Definition 6.18. Let H be a Koszul–Vinberg structure on a left-symmetric algebroid (A, ∗A, aA). Two order n deformations Ht and H ′ t of H are said to be equivalent if there exists a formal series Xt = ∑+∞ i=1 xit i, xi ∈ Γ(A) such that exp(adXt)Ht = H ′ t modulo tn+1, 24 M. Liu, J. Liu and Y. Sheng where exp denotes the exponential series and adkXt Ht = [[ Xt, [[ Xt, . . . , [[Xt, Ht]]KV, k. . . ]] KV ]] KV . An order n deformation Ht of H is called trivial if Ht is equivalent to H. Proposition 6.19. Let H be a Koszul–Vinberg structure on a left-symmetric algebroid (A, ∗A, aA) and Ht, H ′ t ∈ Sym2(A)[[t]]. Two order n deformations Ht and H ′ t of the Koszul– Vinberg structure H are equivalent if and only if the two order n deformations H♯ t and (H ′)♯t of the relative Rota–Baxter operator H♯ on the LieRep pair (Ac;L) are equivalent. Proof. It follows from that Ψ defined by (6.5) is a graded Lie algebra isomorphism between the graded Lie algebra (C∗(A∗,A), [[·, ·]]) and (C∗ KV(A∗), [[·, ·]]KV). ■ Proposition 6.20. Let H be a Koszul–Vinberg structure on a left-symmetric algebroid (A, ∗A, aA). Then there is a one-to-one correspondence between equivalence classes of infinitesi- mal deformations of the Koszul–Vinberg structure H and the second cohomology group H̃2 KV(A∗). Proof. Assume that Ht and H ′ t are equivalent infinitesimal deformations of the Koszul–Vinberg structure H. By Theorem 4.4 and Proposition 6.19, there exists an element x ∈ Γ(A) such that H′ 1 −H1 = δA∗x. Since H′ 1 and H1 are symmetric, for all α1, α2 ∈ Γ(A∗), we have δA∗x(α1, α2) = δA∗x(α2, α1), which implies that H(Rxα1, α2) = H(α1, Rxα2), i.e., x ∈ C̃1 KV(A∗). Thus H′ 1 and H1 are in the same cohomology class of H̃2 KV(A∗). The converse can be proved similarly. We omit the details. ■ Similarly to Proposition 4.5, we have Proposition 6.21. Let H be a Koszul–Vinberg structure on a left-symmetric algebroid (A, ∗A, aA) such that H̃2 KV(A∗) = 0. Then all infinitesimal deformations of the Koszul–Vinberg structure H are trivial. Theorem 6.22. Let H be a Koszul–Vinberg structure on a left-symmetric algebroid (A, ∗A, aA). Let H(n) = ∑n i=0Hit i be an order n deformation of H. Define Θ = 1 2 ∑ i+j=n+1 i,j≥1 [[Hi,Hj ]]KV . (6.7) Then the 3-cochain Θ is closed, i.e., δA∗Θ = 0. Furthermore, H(n) is extendable if and only if the cohomology class [Θ] in H̃3 KV(A∗) is trivial. Proof. For any α1, α2, α3 ∈ Γ(A∗) and i, j ≥ 1, we have [[Hi,Hj ]]KV (α1, α2, α3) = aA(H♯ i(α1))⟨H♯ j(α2), α3⟩+ aA(H♯ j(α1))⟨H♯ i(α2), α3⟩ − aA(H♯ i(α2))⟨H♯ j(α1), α3⟩ − aA(H♯ j(α2))⟨H♯ i(α1), α3⟩ + ⟨α1,H♯ i(α2) ∗A H♯ j(α3)⟩+ ⟨α1,H♯ j(α2) ∗A H♯ i(α3)⟩ − ⟨α2,H♯ i(α1) ∗A H♯ j(α3)⟩ − ⟨α2,H♯ j(α1) ∗A H♯ i(α3)⟩ − ⟨α3, [H♯ i(α1),H♯ j(α2)]A⟩ − ⟨α3, [H♯ j(α1),H♯ i(α2)]A⟩. Deformations and Cohomologies of Relative Rota–Baxter Operators on Lie Algebroids 25 It is straightforward to check that [[Hi,Hj ]]KV (α1, α2, α3) + [[Hi,Hj ]]KV (α3, α1, α2) + [[Hi,Hj ]]KV (α2, α3, α1) = 0, which implies that Θ defined by (6.7) is in C̃3 KV(A∗). 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Phys. 17 (2020), 2050199, 28 pages, arXiv:2004.01774. https://doi.org/10.1016/j.aim.2021.107834 https://arxiv.org/abs/2009.03492 https://doi.org/10.1515/crll.1990.408.57 https://arxiv.org/abs/1303.3903 https://doi.org/10.1023/B:MATH.0000027508.00421.bf https://arxiv.org/abs/q-alg/9709040 https://doi.org/10.2991/jnmp.1999.6.4.5 https://arxiv.org/abs/math.QA/9910188 https://doi.org/10.1007/978-3-642-31090-4 https://doi.org/10.4310/jdg/1214433987 https://doi.org/10.1016/j.difgeo.2018.04.003 https://doi.org/10.1016/j.difgeo.2018.04.003 https://arxiv.org/abs/1705.08299 https://doi.org/10.1007/s11005-017-0973-8 https://arxiv.org/abs/1604.00146 https://doi.org/10.1002/mana.201300339 https://arxiv.org/abs/1312.6526 https://doi.org/10.4310/jdg/1214459842 https://arxiv.org/abs/dg-ga/9508013 https://doi.org/10.1017/CBO9781107325883 https://doi.org/10.1215/S0012-7094-94-07318-3 https://doi.org/10.1017/S0017089500032055 https://doi.org/10.4064/bc54-0-7 https://doi.org/10.4064/bc54-0-7 https://doi.org/10.1142/S0129167X0500320X https://doi.org/10.1090/S0002-9904-1966-11401-5 https://doi.org/10.1090/S0002-9904-1966-11401-5 https://doi.org/10.1090/S0002-9904-1967-11703-8 https://doi.org/10.1090/S0002-9904-1967-11703-8 https://doi.org/10.1016/0021-8693(68)90004-5 https://doi.org/10.1007/s10485-016-9431-5 https://arxiv.org/abs/1306.3046 https://doi.org/10.1090/S0002-9904-1969-12156-7 https://doi.org/10.1090/S0002-9904-1969-12158-0 https://doi.org/10.5802/aif.794 https://doi.org/10.1142/9789812707536 https://doi.org/10.1007/s00220-019-03286-x https://doi.org/10.1007/s00220-019-03286-x https://arxiv.org/abs/1803.09287 https://doi.org/10.4171/jncg/59 https://arxiv.org/abs/0710.4309 https://doi.org/10.1017/S0305004114000541 https://arxiv.org/abs/1304.4353 https://doi.org/10.1142/S0219887820501996 https://arxiv.org/abs/2004.01774 1 Introduction 1.1 Relative Rota–Baxter operators on Lie algebroids and Koszul–Vinberg structures 1.2 Deformations and cohomologies 1.3 Summary of the results and outline of the paper 1.4 Conventions and notations 2 Maurer–Cartan characterizations and cohomologies of relative Rota–Baxter operators on Lie algebroids 2.1 The controlling algebra of relative Rota–Baxter operators on Lie algebroids 2.2 Cohomologies of relative Rota–Baxter operators on Lie algebroids 3 Cohomologies of Rota–Baxter operators on Lie algebroids 4 Formal deformations of relative Rota–Baxter operators on Lie algebroids 5 The Matsushima–Nijenhuis bracket for left-symmetric algebroids 5.1 The Matsushima–Nijenhuis bracket for left-symmetric algebroids 5.2 Relations between the graded Lie algebra (C^(E,A),[[cdot,cdot]]) and (Der^(E),[cdot,cdot]_{MN}) 6 Maurer–Cartan characterizations and cohomology of Koszul–Vinberg structures on left-symmetric algebroids 6.1 Maurer–Cartan characterizations of Koszul–Vinberg structures 6.2 Cohomologies and deformations of Koszul–Vinberg structures References
id	nasplib_isofts_kiev_ua-123456789-211733
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-15T07:47:38Z
publishDate	2022
publisher	Інститут математики НАН України
record_format	dspace
spelling	Liu, Meijun Liu, Jiefeng Sheng, Yunhe 2026-01-09T12:55:05Z 2022 Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures. Meijun Liu, Jiefeng Liu and Yunhe Sheng. SIGMA 18 (2022), 054, 26 pages 1815-0659 2020 Mathematics Subject Classification: 53D17; 53C25; 58A12; 17B70 arXiv:2108.08906 https://nasplib.isofts.kiev.ua/handle/123456789/211733 https://doi.org/10.3842/SIGMA.2022.054 Given a Lie algebroid with a representation, we construct a graded Lie algebra whose Maurer-Cartan elements characterize relative Rota-Baxter operators on Lie algebroids. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extendability of order deformations to order + 1 deformations of relative Rota-Baxter operators in terms of this cohomology theory. We also construct a graded Lie algebra on the space of multi-derivations of a vector bundle whose Maurer-Cartan elements characterize left-symmetric algebroids. We show that there is a homomorphism from the controlling graded Lie algebra of relative Rota-Baxter operators on Lie algebroids to the controlling graded Lie algebra of left-symmetric algebroids. Consequently, there is a natural homomorphism from the cohomology groups of a relative Rota-Baxter operator to the deformation cohomology groups of the associated left-symmetric algebroid. As applications, we give the controlling graded Lie algebra and the cohomology theory of Koszul-Vinberg structures on left-symmetric algebroids. This research was supported by the National Key Research and Development Program of China (2021YFA1002000), the National Natural Science Foundation of China (11901501, 11922110), the China Postdoctoral Science Foundation (2021M700750), and the Fundamental Research Funds for the Central Universities (2412022QD033). We give our warmest thanks to the referees for their very useful comments that improve the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures Article published earlier
spellingShingle	Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures Liu, Meijun Liu, Jiefeng Sheng, Yunhe
title	Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures
title_full	Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures
title_fullStr	Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures
title_full_unstemmed	Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures
title_short	Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures
title_sort	deformations and cohomologies of relative rota-baxter operators on lie algebroids and koszul-vinberg structures
url	https://nasplib.isofts.kiev.ua/handle/123456789/211733
work_keys_str_mv	AT liumeijun deformationsandcohomologiesofrelativerotabaxteroperatorsonliealgebroidsandkoszulvinbergstructures AT liujiefeng deformationsandcohomologiesofrelativerotabaxteroperatorsonliealgebroidsandkoszulvinbergstructures AT shengyunhe deformationsandcohomologiesofrelativerotabaxteroperatorsonliealgebroidsandkoszulvinbergstructures

Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures

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