Cohomology and deformations of Courant pairs

In this note we define a notion of Courant pair as a Courant algebra over the Lie algebra of linear derivations on an associative algebra. We study formal deformations of Courant pairs by constructing a cohomology bicomplex with coefficients in a module from the cochain complexes defining Hochschild cohomology and Leibniz cohomology.


Introduction
The aim of this article is to define a notion of Courant pair as a Courant algebra over the Lie algebra Der(A) of linear derivations on an associative algebra A and describe a one parameter formal deformation theory of Courant pairs. We construct a cohomology bicomplex using the Hochschild cohomology for associative algebras and cohomology of Leibniz algebras respectively.
The notion of Courant algebras was defined by H. Bursztyn and coauthors ( [5] ) in the context of reduction of Courant algebroids and generalized complex structures. It is used to interpret the moment map in symplectic geometry as an object which controls an extended part of the action of Courant algebras on Courant algebroids. By definition, a Courant algebra over a Lie algebra g is a Leibniz algebra L equipped with a Leibniz algebra homomorphism to the Lie algebra g. One special case of this type of algebras is an exact Courant algebra where the Leibniz algebra homomorphism is a surjective map onto g and the kernel of the map is an abelian Lie algebra ( [5], [27]). In fact, any Leibniz algebra L can be viewed as an exact Courant algebra over the associated Lie algebra L Lie obtained by taking the quotient linear space L/L ann , where L ann is also called the Leibniz kernel of L is the module generated by elements of the form [x, x] for x ∈ L. The notion of Leibniz algebras were introduced by A. Bloh [4] under the name D-algebras and it is rediscovered by J.-L. Loday in connection with cyclic homology and Hochschild homology of matrix algebras ( [22], [24]). The (co) homology with coefficients in a representation associated to Leibniz algebras has been developed in [23] . Any Lie algebra is also a Leibniz algebra, as in the presence of skew symmetry of the Lie bracket the Jacobi identity for the bracket is equivalent to the Leibniz identity. It is also important to note that a Courant algebras can be viewed as a model to get Leibniz algebras in the form of hemisemidirect products ( as introduced in [19] ) by starting with a Lie algebra g and a g-modules. One of the Lie algebras mostly encountered in geometry or in mathematical physics is the Lie algebra χ(M ) of smooth vector fields on a smooth manifold M . This Lie algebra (which is also a Leibniz algebra ) can also be described as the space of linear derivations on the associative algebra C ∞ (M ) of smooth functions on the given manifold. Here, we are interested in finding Leibniz algebras as a canonical extension of the Lie algebra Der(A) of linear derivations on an arbitrary associative algebra A both over the same coefficient ring K. This leads us to consider a notion of Courant pair denoted simply by (A, L). The deformation of algebraic and analytic structures are an important aspects if one studies their properties. They characterize the local behaviour in a small neighbourhood in the variety of a given type of objects.
A Courant pair (A, L) consists of an associative algebra A and a Leibniz algebra L over the same coefficient field K, equipped with a Leibniz algebra homomorphism µ : L −→ g, where g denotes the Lie algebra of K-linear derivations on A under the commutator bracket. If we replace the Leibniz algebra in the definition by a Lie algebra then what we obtain is a Leibniz pair (A, L) introduced by M. Flato et al. in [10]. The notion of Leibniz pair was defined as a generalization of the Poisson algebras and a formal deformation theory is studied in [10]. Later on it is found as a special case of a homotopy algebra introduced by H. Kajiura and J. Stashef in [18]. They referred this as an open closed homotopy algebra (OCHA). A Lie algebroid or its algebraic counterpart, called a Lie-Rinehart algebra ( [29]) is also a special type of Leibniz pair by considering associative algebras also to be a commutative algebra.
A Courant pair appears naturally when one consider a Lie algebroid ( [26] ) or more generally a Leibniz algebroid ( [16] ) ( or Loday algebroid, Courant algebroid ( [17], [16], [5], [19], [30] ) ) over a smooth base manifold. In all these cases the underline Lie algebra of the Courant algebras are considered to be the Lie algebra of vector fields or space of linear derivations of the associative algebra of functions on the manifold. More examples can be found if one allow Leibniz algebras in place of Lie algebra L in the definition of a Leibniz pairs (A, L). Here, we consider this more general case of Leibniz pair in the name of Courant pair which will form a category containing the category of Leibniz pairs as a full subcategory. The cohomology space is different if one consider a Leibniz pair as an object in the larger category of Courant pairs. The paper is organized as follows: In section 2, we recall the Hochschild cohomology of an associative algebra with coefficient in a bimodule, a Leibniz algebra cohomology complex with coefficient in a representation and the notion of Courant algebras with examples. Section 3 contains definition and various natural examples of the notion of Courant pairs. We discuss about representations or modules over a Courant pair as a suitable semi-direct products in the category of associative and Leibniz algebras respectively with corresponding modules. In section 4, we present the construction of a deformation bicomplex for a Courant pair (A, L) from Hochschild cohomology complex of the associative algebra A and Leibniz cohomology complex for the Leibniz algebra L. In section 5 we define one parameter formal deformation of a Courant pair, define the notion of equivalence of deformations, and study rigidity. In the last section we discuss about the obstruction cochains that arises in extending deformation of finite order with given infinitesimal. We mentioned examples to show that one can find many more deformations of a Leibniz pair when it is deformed as a Courant pair by using the new bicomplex cohomology.
One of the main results in any deformation theory is to prove that obstruction cochains are cocycles. In our context, this consists of three parts; one arising from deformations of the considered associative algebra, one from the deformation of the Leibniz algebra and the other from deformation of the Leibniz homomorphism of a Courant pair. In [11], M. Gerstenhaber showed that the obstruction cochains arising from deformations of associative algebras are cocycles and in [3] D. Balavoine showed the obstruction cochains arising from deformations of Leibniz algebras are cocycles. The other part is done by a direct computation and given in an appendix.
Throughout we will consider vector spaces over a field K of characteristics zero and all maps are K-linear unless otherwise it is specified .

Preliminaries
2.1. Associative algebras and its cohomology. Definition 2.1. Let A be an associative algebra over K, a bimodule M over A or, an A-bimodule is a K-module equipped with two actions (left and right) of A The actions of A and K on M are compatible, that is (λa)m = λ(am) = a(λm) for λ ∈ K, a ∈ A and m ∈ M . When A has the identity 1 we always assume that 1m = m1 = m for m ∈ M .
Given a bimodule M over A, the Hochschild cochain complex of A with coefficients in M is defined as follows. Set C n (A, M ) = Hom K (A ⊗n , M ) ; n ≥ 0 where A ⊗n = A ⊗ · · · ⊗ A (n-copies). Let δ H : C n (A, M ) → C n+1 (A, M ) be the K-linear map given by δ H f (a 1 , a 2 , . . . , a n+1 ) = a 1 .f (a 2 , . . . , a n ) + n i=1 (−1) i (a 1 , . . . , a i a i+1 , . . . , a n+1 ) + (−1) n+1 (a 1 , . . . , a n−1 , a n ).a n+1 Then δ 2 H = 0 and the complex (C * (A, M ), δ H ) is called the Hochschild complex of A with coefficients in the A-bimodule M . When M = A, where the actions are given by algebra operation in A we denote the complex C * (A, A) by C * (A). The graded space C * (A) is a differential graded Lie algebra (or DGLA in short ) (see [11]). Jacobi identity ) for x, y, z ∈ L, where |x| denotes the degree of x.
A differential graded Lie algebra is a graded Lie algebra equipped with a differential d satisfying Remark 2.3. The shifted Hochschild complex C * (A, A) = C * +1 (A, A) is a DGLA where the graded Lie bracket is also called the Gerstenhaber bracket. Let φ be a Hochschild p-cochain and θ be a Hochschild q-cochain. The Gerstenhaber bracket of φ and θ is the (p + q − 1)-cochain defined by Also, note that if α 0 is the associative multiplication, then δ H can be written in terms of the Gerstenhaber bracket and the multiplication α 0 :

2.2.
Leibniz algebra and its cohomology. Now, we recall the definition of a Leibniz algebra and describe its cohomology.
Definition 2.4. A Leibniz algebra (left Leibniz algebra) is a K-module L, equipped with a bracket operation, which is K-bilinear and satisfies the Leibniz identity, In the presence of antisymmetry of the bracket operation, the Leibniz identity is equivalent to the Jacobi identity, hence any Lie algebra is a Leibniz algebra.
Remark 2.5. For a given Leibniz algebra L as defined above, the left adjoint operation [x, −] is a derivation on L for any x ∈ L. Analogously, Leibniz algebra ( right Leibniz algebra ) can be defined by requiring that the right adjoint map [−, x] is a derivation for any x ∈ L. In that case, the Leibniz identity appeared in the above definition would be of the form: Sometime, in the literature ( e.g., [22][23][24]) right Leibniz algebra has been considered. All our Leibniz algebras in the present discussion will be left Leibniz algebras unless otherwise stated.
Example 2.6. Let (g, [−, −]) be a Lie algebra and V be a g-module with the action (x, v) → x.v. Take L = g ⊕ V with the bracket given by Then (L, [−, −] L ) is a Leibniz algebra. This bracket is called hemisemidirect product in [19].  The new bracket on L is called the derived bracket(see [32]).
Then (CL * (L; M ), δ) is a cochain complex, whose cohomology is denoted by HL * (L; M ), is called the cohomology of the Leibniz algebra L with coefficients in the representation M (see [8,23]).
In particular, L is a representation of itself with the obvious actions given by the Leibniz algebra bracket of L.

2.3.
Courant algebras over a Lie algebra. Now, we recall the definition of Courant algebras from [5]. Let g be a Lie algebra over K.
Definition 2.10. A Courant algebra over a Lie algebra g is a Leibniz algebra L equipped with a homomorphism of Leibniz algebras π : L → g. We will simply denote it as π : L −→ g is a Courant algebra.
Example 2.11. A Courant algebroid over a smooth manifold M gives an example of a Courant algebra over the Lie algebra of vector fields g = Γ(T M ) by taking a as the Leibniz algebra structure on the space of sections of the underlying vector bundle of the Courant algebroid. From [30] it follows that any Courant algebra is actually an example of a 2-term L ∞ algebra [2,21].
12. An exact Courant algebra over a Lie algebra g is a Courant algebra π : L −→ g for which π is a surjective linear map and For an exact Courant algebra π : L −→ g, there are two canonical actions of g on h: for any a such that π(a) = g. Thus the K-module h equipped with these two actions denoted by the same bracket notation [−, −], is a representation of g ( viewed as a Leibniz algebra ).
The next example will give a natural exact Courant algebra associated with any representation of the Lie algebra g. Example 2.13. As in Example 2.6, g be a Lie algebra acting on the vector space V . Then L = g⊕V becomes a Courant algebra over g with the Leibniz algebra homomorphism given by projection on g and the Leibniz bracket on L given by where g.h denotes the action of the Lie algebra g on V .
In [27] it is shown that exact Courant algebras over a Lie algebra g can be characterised via Leibniz 2-cocycles, and the automorphism group of a given exact Courant algebra is in a one-toone correspondence with first Leibniz cohomology space of g. Exact Courant algebras also appeared in the general study of Leibniz algebra extension and a discussion of some unified product for Leibniz algebras is in [1,28].
Example 2.14. If we consider a Leibniz representation h of the Lie algebra g, then L = g ⊕ h becomes a Courant algebra over g via the bracket where the actions (left and right) of g on h are denoted by the same bracket [−, −].

Courant pairs
In this section we introduce the notion of a Courant pair and discuss about various natural examples . We define representation or modules of a Courant pair as a suitable semi-direct product in the category of associative and Leibniz algebras respectively with corresponding modules.
Let A be an associative algebra over K and by g, we denote the Lie algebra (Der K (A), [−, −] c ) of linear derivations of A with commutator bracket [−, −] c . Now we define a Courant pair as a Courant algebra over the Lie algebra g which is canonically associated to a given associative algebra A.  Every Leibniz pair defined in [10] is an example of a Courant pair. In fact, if (A, L) is a Leibniz pair then by definition, A is an associative algebra and L is a Lie algebra over some common coefficient ring K, connected by a Lie algebra morphism µ : L → g. And since the category of Lie algebras is a subcategory of the Leibniz algebra category, it follows that a Leibniz pair is also a Courant pair. In particular, Remark 3.4. In particular, for A to be the space of smooth functions on a smooth manifold this type of examples of Courant pair appears as twisted Dirac structure of order zero in [6].
In the sequel, one can find Courant pairs associated to an algebra by considering modules and flat connections on the module of the algebra. Let A be an associative algebra and V be an unitary A-module. A derivation law, or Koszul connection on V is an A-linear mapping ∇ :  [26]). Now, ΓE is a Lie algebra and ΓF is a ΓE-module (it follows from flatness of ∇). Therefore, by taking hemisemidirect product of ΓE and ΓF , we have a Leibniz algebra bracket on ΓE ⊕ ΓF given by for X, Y ∈ ΓE and α, β ∈ ΓF . Consider A = C ∞ (M ), and L = ΓE ⊕ ΓF , then (A, L) is a Courant pair with the Leibniz algebra morphism: where µ = ρ • π 1 , and π 1 is the projection map onto ΓE.     (1)μ(x, 0)(a, 0) = µ(x)(a) for any x ∈ L, a ∈ A.
Remark 3.11. Every Courant pair (A, L) is a module over itself. This fact will be useful in the later section while considering deformations. We may also canonically define morphisms of Courant pairs and hence morphisms of modules of a given Courant pair (A, L). Note 3.12. It is important to notice here that for each α ∈ P the map A −→ M defined by sending a toμ(0, α)(a, 0) is a derivation of A into M . This yields a map φ : P → Der(A, M ), which we will use in the double complex defined later.

Deformation complex of a Courant pair:
In the present section, we introduce the deformation complex of a Courant pair. We shall observe subsequently that the second and the third cohomologies associated to this cohomology complex encode all the information about deformations. Suppose (M,P) is a module of the Courant pair (A, L). So M is an A-bimodule and P is a Leibniz-L-module. We need to consider the Hochschild complex C * (A, M ) and Leibniz algebra complex C * (L, P ) in the sequel. Here we will denote the tensor modules ⊗ p A simply by A p . Let In the horizontal direction, we have for all p and q the Leibniz algebra coboundary δ L : C p,q (A, L) −→ C p,q+1 (A, L). Since all the vertical coboundaries are Lie module homomorphisms, it follows that they commute with all the horizontal ones. Therefore, we can define a total cochain complex as follows: For n ≥ 0.  On the other hand if we take (R, χ(R n )) as a Courant pair and (R, R) as a module over it, then HL * (R, χ(R n ); R, R) ∼ = HL * (χ(R n ), R).
We know that the Leibniz algebra cohomology space HL * (χ(R n ), R) and the Lie algebra cohomology space H * (χ(R n ), R) are different as there are new generators for the dual Leibniz algebra structure over the Leibniz cohomology space (see [7], [25]).
This shows that, while considering a Leibniz pair (R, χ(R n )) as an object in the category of Courant pairs, we have a somewhat different cohomology space.

Deformation of Courant pairs:
In this section, we study formal deformation of Courant pairs. All the basic notions of deformation theory of algebraic structures are originally due to M. Gerstenhaber ( [11], [12], [13], [14] ).
Here we briefly describe the analogous concepts related to deformation of a Courant pair. This can be viewed as a generalization of the deformation of Leibniz pairs (in [10]) to a larger category by allowing more deformations.
Let (A, L) be a Courant pair, then using remark (3.2), it can be viewed as a triplet (α, µ, λ). Recall that a formal 1-parameter family of deformations of an associative algebra A is an associative algebra bracket α t on the K Definition 5.1. A deformation of a Courant pair (A, L) whose structure maps are given by the triplet (α, µ, λ) is defined as a tuple of the form (α t , µ t , λ t ), where α t is a deformation of α, i.e., an associative K
By above definition we have also the following equations, using the fact that α t and λ t are deformations of the associative multiplication α on A and the Leibniz bracket λ on L, respectively: λ t (x, λ t (y, z)) = λ t (λ t (x, y), z) + λ t (y, λ t (x, z)).
such thatα Definition 5.6. Any deformation of Courant pair equivalent to the deformation (α, µ, λ) is said to be a trivial deformation.
Theorem 5.7. The cohomology class of the infinitesimal of the deformation (α t , µ t , λ t ) of Courant pair is determined by the equivalence class of (α t , µ t , λ t ).
Proof. Let (Φ t , Ψ t ) : (α t , µ t , λ t ) → (α t ,μ t ,λ t ) be an equivalence of these two deformations. Then expanding and comparing coefficients of t in the equations (9), (10), (11), we getα Definition 5.8. A Courant pair is said to be rigid if every deformation of the Courant pair is trivial.
Theorem 5.9. A non trivial deformation of a Courant pair is equivalent to a deformation whose n-infinitesimal is not a coboundary for some n ≥ 1.

Obstruction Cocycles:
In this section, we discuss about the problem of realising a 2-cocycle in C 2 tot as the infinitesimal of a deformation of the Courant pair. To this end first we define an obstruction cochain and then proceed by detecting any obstructions to extend a given deformation modulo t n to a deformation modulo t n+1 , n ≥ 1. Let N be a positive integer. Definition 6.1. A deformation of Courant pair of order N is a (α t , µ t , λ t ) such that α t = N i=0 α i t i and λ t = N i=0 λ i t i are deformations modulo t n+1 of A and L respectively, that is α t and λ t satisfy (8), (7) respectively for 0 ≤ i ≤ N, and µ t = N i=0 µ i t i satisfies equations (5) and (6) If there exists a 2-cochain (α N +1 , µ N +1 , λ N +1 ) ∈ C 2 tot , such that the triple (α t ,μ t ,λ t ) withμ t = µ t + µ N +1 t N +1 ,α t = α t + α N +1 t N +1 , andλ t = λ t + λ N +1 t N +1 is a deformation of Courant pair of order (N + 1), then we say that (α t , µ t , λ t ) extends to a deformation of Courant pair of order (N + 1). Definition 6.2. Let (α t , µ t , λ t ) be a deformation of Courant pair of order N . Consider the cochains Θ(A), Θ 1 , Θ 2 , Θ(L), where j (a, b), c); (x, a), b); µ j (y, a)) − µ i (y, µ j (x, a)); (x, y), z).
tot is called the obstruction cochain for extending the deformation of Courant pair (α t , µ t , λ t ) of order N to a deformation of order N + 1.
Proof. Θ(A) is the obstruction cochain for extending the N -th order deformation of the associative algebra A to a deformation of order N + 1, So, it is a 3-cocycle in the Hochschild complex for associative algebra, i.e. (16) δ H Θ(A) = 0. Now, Θ(L) is the obstruction cochain for extending the N -th order deformation of the Leibniz algebra L to a deformation of order N + 1. So, it is a 3-cocycle in the deformation complex for a Leibniz algebra, i.e. (17) δ L Θ(L) = 0.
Moreover, these deformations are obtained only when we consider the Leibniz pair (C ∞ (R 3 ), L) as a Courant pair.
Appendix A. Obstruction Cocycle: In this appendix, we present an explicit computation involved (in the proof of Theorem 6.3 ) to show that an obstruction cochain arising in the extension of finite order deformation is a 3-cocycle.
Here we follow the notations uded in the previous in the sections.
Lemma A.1. The cochains Θ(A) ∈ Hom(A 3 , A) and Θ 1 ∈ Hom(L, C 2 (A, A)) satisfy the identity, Proof. We want to show that for x ∈ L and a, b, c ∈ A we have From equation (29), for each n < N + 1 we write as By using the DGLA structure on the Hochschild cochain space ( see Remark 2.3 ) Further, α t is a deformation of α so we have the following equation using obstruction cocycles for associative algebras. Therefore, Also using equation (31), Finally, using expressions in equations (35), (34), and graded Jacobi identity for the Gerstenhaber bracket (see Remark 2.3), we have the required identity Lemma A.2. The cochains Θ 1 ∈ Hom(L, C 2 (A, A)) and Θ 2 ∈ Hom(L 2 , C 1 (A, A)) satisfy the identity, δ H Θ 2 + δ L Θ 1 = 0. for x, y, z ∈ L and a ∈ A. First, we compute δ L Θ 2 (x, y, z)(a). From definition of δ L we have In equation (47), using the action of L on C p (A, A) (recall from Section 4) and the Gerstenhaber bracket for Hochschild cochains, the first term in the right hand side can be written as: Similarly, we write the expressions for [y, Θ 2 (x, z)] and [z, Θ 2 (x, y)] respectively. Further, in equation (47), the last three terms in the right hand side can be expressed as follows.
We proceed by rewritting the equation (22) as follows: (49) f Replacing y by λ j (y, z), we can write the first term in the right hand side of the equation (48) as follows. ]. (50) Using equation (49), the last term in the right hand side of equation (50) ]. Similarly, we can write the second and third term in the right hand side of the equation (48). Now, substituting the final expression of δ v Θ(L)(x, y, z)(a) and δ L Θ 2 (x, y, z)(a) in the left hand side of the equation (46) and by considering equation (23) we get the result.

Conclusions:
The deformation of a given structure characterize the local behaviour in the variety of a given type of objects. In order to study the deformation one needs a suitable notion of module or representation and then construct a deformation complex. In this work we develop a formal deformation of Courant pairs which includes the classical cases of Lie algebras and Leibniz pairs. Moreover, There are plenty of examples of the Courant pairs including the Leibniz algebras and Leibniz algebroids appear in algebra, geometry and in Mathematical physics. A Courant pair (A, L) is a Leibniz pseudoalgebra over (K, A) if L is also an A-module and the structure map is an A-module homomorphism such that following condition is satisfied: [x, f y] = f [x, y] + µ(x)(f )y where x, y ∈ L and f ∈ A. So it is also natural to expect a deformation theory for Leibniz-Rinehart algebras by deducing the required deformation cohomology. As in the classical cases of algebras over quadratic operads (e.g., Ass, Lie, Leib,...etc. ), one may look for a differential graded Lie algebra structure on the cohomology space of a Courant pair. Also, a Courant pair can be described as a non skew-symmetric version of OCHA and subsequently as an algebra over an operad. These questions are planed to be addressed in a separate discussion.