Free Courant and derived Leibniz pseudoalgebras

We introduce the category of generalized Courant pseudoalgebras and show that it admits a free object on any anchored module over `functions'. The free generalized Courant pseudoalgebra is built from two components: the generalized Courant pseudoalgebra associated to a symmetric Leibniz pseudoalgebra and the free symmetric Leibniz pseudoalgebra on an anchored module. Our construction is thus based on the new concept of symmetric Leibniz algebroid. We compare this subclass of Leibniz algebroids with the subclass made of Loday algebroids, which were introduced in [12] as geometric replacements of standard Leibniz algebroids. Eventually, we apply our algebro-categorical machinery to associate a differential graded Lie algebra to any symmetric Leibniz pseudoalgebra, such that the Leibniz bracket of the latter coincides with the derived bracket of the former.


1.
Introduction. The skew-symmetric non-Jacobi Courant bracket [6] on sections of T M := T M ⊕ T * M was originally introduced by Courant to formulate the integrability condition defining a Dirac structure. However its nature became clear only due to the observation by Liu, Weinstein and Xu [30] that T M endowed with the Courant bracket plays the role of a 'double' object, in the sense of Drinfeld [10], for a pair of Lie algebroids over M . Whereas any Lie bialgebra has a double which is a Lie algebra, the double of a Lie bialgebroid is not a Lie algebroid, but a Courant algebroid -a generalization of T M equipped with the Courant bracket. There is another way of viewing Courant algebroids as a generalization of Lie algebroids. This requires a change in the definition of the Courant bracket and the use of an analog of the non-antisymmetric Dorfman bracket [8]. The traditional Courant bracket then becomes the skew-symmetrization of the new one [33]. This change replaces one defect with another: a version of the Jacobi identity is satisfied, while the bracket is no longer skew-symmetric. Such algebraic structures have been introduced by Loday [28] under the name of Leibniz algebras. Canonical examples of Leibniz algebras arise often as derived brackets introduced by Kosmann-Schwarzbach [23,24]. Since Leibniz brackets appear naturally in Geometry and Physics in the form of 'algebroid brackets', i.e., brackets on sections of vector bundles, there were a number of attempts to formalize the concept of Leibniz algebroid [17,16,13,20,18,19,27,31,35,38]. Note also that a Leibniz algebroid is the horizontal categorification of a Leibniz algebra; vertical categorification leads to Leibniz n-algebras and Leibniz n-algebroids [1,22,26,9,4].
It is important to observe that, despite the sheaf-theoretic nature of classical Differential Geometry, most textbooks present it in terms of global sections and morphisms between them. This global viewpoint is possible, since all morphisms between modules of sections are local -and even differential -operators in all their arguments (indeed, locality allows one to localize the operators in a way that they commute with restrictions). It follows that a map that is not a differential operator in one of its arguments is not a true geometric concept [5,Appendix 3]. However, none of the aforementioned definitions of Leibniz algebroids requires any differentiability condition for the first argument of the bracket and thus none of these concepts is geometric. In [12], the authors propose -under the name of Loday algebroid -a new notion of Leibniz algebroid, which is geometric and includes the vast majority of Leibniz brackets that can be found in the literature, in particular Courant brackets.
In the present article, we introduce the category of generalized Courant pseudoalgebras and show that it admits a free object on any anchored module over 'functions'. Our construction is based on another new concept: symmetric Leibniz algebroids and pseudoalgebras. We compare this subclass of Leibniz algebroids with the subclass made of Loday algebroids. The prototypical example of a generalized Courant pseudoalgebra is the one associated to an arbitrary symmetric Leibniz pseudoalgebra. Both notions are fundamental ingredients of the free generalized Courant pseudoalgebra. As an application, we show how the associated generalized Courant pseudoalgebra allows one to prove that any symmetric Leibniz pseudoalgebra bracket can be represented as a universal derived bracket. This formal-noncommutative-geometry-related problem, see Section 7, was one of the motivations for the present work. The possibility to encode numerous types of (homotopy) algebras in a (co)homological vector field of a possibly formal noncommutative manifold [11], is an example that emphasizes the importance of free algebras, see Equation (58). The prominence of the latter was recognized already in the late fifties, in particular by the Polish mathematical school. Also free Lie pseudoalgebras (free Lie-Rinehart algebras) were yet successfully used [21]. The construction of the free Leibniz and Courant pseudoalgebras was a second inducement.
Let us emphasize that, whereas, as indicated above, Courant algebroids and, more generally, Loday algebroids are geometric, Leibniz algebroids, symmetric Leibniz algebroids, and generalized Courant algebroids are not. Hence, this paper should be viewed as a purely algebro-categorical work. From the categorical standpoint, most definitions are quite obvious. This holds in particular for the notions of free object and of morphism between generalized Courant pseudoalgebras -see Definitions 2.2 and 4.3.
The paper is organized as follows. In Section 2, we recall the definitions of the categories of Leibniz algebroids [20], of Leibniz pseudoalgebras -their algebraic counterpart -, and of modules over them, as well as the classical notion of Courant algebroid. We then describe, in Section 3, two intersecting subclasses of Leibniz algebroids, namely the class of Loday algebroids (Definition 3.1), which are Leibniz algebroids that admit a generalized right anchor, and the class of symmetric Leibniz algebroids (Definition 3.3), a new concept, made of Leibniz algebroids that satisfy two symmetry conditions and contain Courant algebroids as a particular example.
Examples of symmetric and nonsymmetric Leibniz and Loday algebroids and pseudoalgebras are given. In Section 4, we motivate the definition of generalized Courant pseudoalgebras (Definitions 4.1 and 4.2), which are specific symmetric Leibniz pseudoalgebras. The prototypical example of a generalized Courant pseudoalgebra is the one that is naturally associated to a symmetric Leibniz pseudoalgebra (Theorem 4.4). This associated Courant pseudoalgebra is one of the two ingredients of the free generalized Courant pseudoalgebra. Moreover, Theorem 4.4 allows to understand the origin of the definition of symmetric Leibniz pseudoalgebras. The second ingredient is the free symmetric Leibniz pseudoalgebra, which we construct in Section 5 (Theorem 5.3 and Proposition 6). In Section 6, we combine the results of Section 4 and Section 5 to build the free Courant pseudoalgebra (Theorem 6.1). Finally, we apply, in Section 7, our algebro-categorical constructions to show that any symmetric Leibniz pseudoalgebra bracket is a universal derived bracket implemented by a differential graded Lie algebra that is put up from the associated Courant pseudoalgebra.

Preliminaries.
2.1. Notation and conventions. Unless otherwise specified, manifolds are made of a finite-dimensional smooth structure on a second-countable Hausdorff space.
If [−, −] is a Leibniz bracket, we denote by − • − its symmetrization, i.e., for any elements X, Y of the Leibniz algebra.

Anchored vector bundles and anchored modules.
Definition 2.1. If M is a manifold, an anchored vector bundle over M is a vector bundle E → M with a vector bundle morphism a : E → T M , called its anchor.
If R is a commutative unital ring and A is a commutative unital R-algebra, an anchored module over (R, A) is an A-module E endowed with an anchor, i.e., an A-module morphism a : E → Der A.
Of course, here T M is the tangent bundle of M and Der A is the A-module of derivations of A. If a : E → T M is an anchor, we still denote by a : ΓE → ΓT M = Der(C ∞ (M )) the corresponding C ∞ (M )-linear map between sections. Obviously, if E is an anchored vector bundle over M with anchor a, then its space ΓE of sections is an anchored module over (R, C ∞ (M )) with anchor a.
Morphisms of anchored vector bundles (resp., anchored modules) over a fixed base M (resp., over a fixed algebra A) are defined in the obvious way, and we obtain categories AncVec(M ) and AncMod(A), respectively. The algebroids (resp., pseudoalgebras) we are going to define in this article will be anchored vector bundles (resp., anchored modules) with extra structure. They will form, together with their morphisms, categories that are concrete over AncVec(M ) and AncMod(A), i.e., admit a (faithful) forgetful functor to the latter. One of our goals is to define left adjoints to these functors, or, in other words, to define the free algebroid (resp., pseudoalgebra) of a given type on a given anchored vector bundle (resp., anchored module). More generally, Definition 2.2. Let C and D be categories, such that there exists a forgetful functor For : C → D. For any D ∈ D, the corresponding free object in C over D is an object F ∈ C equipped with a D-morphism i : D → F which is universal among all pairs of this type.
For the different types of pseudoalgebras we are going to define, we will also define modules over them, using the following general principle: if V is an R-module with extra structure, then a V -module is an R-module W such that V ⊕ W is of the same type as V and contains V as a subobject and W as an abelian ideal in an appropriate sense. Similarly, a morphism of modules from the V -module W to the V -module W will be a morphism V ⊕ W → V ⊕ W sending V to V and W to W . It is possible to make these statements precise, but we prefer to keep them heuristic here and to work out the details below in the specific cases.
2.3. Leibniz algebroids. In this paper, we consider left Leibniz brackets, i.e., bilinear brackets that satisfy the left Jacobi identity Alternatively, one could work with right Leibniz brackets, which are defined similarly, except that one requires the right Jacobi identity We first recall the definition of a Leibniz algebroid given in [20]. Note that this notion of Leibniz algebroid does not impose any differentiability requirement on the first argument of the bracket and is thus not a geometric concept.
for any f ∈ C ∞ (M ) and X, Y ∈ ΓE.
It is easily checked that the Leibniz rule (4) and the Jacobi identity imply that a is a Leibniz algebra morphism: where the RHS bracket is the Lie bracket on ΓT M . We will essentially deal with the algebraic counterpart of Leibniz algebroids: Definition 2.4. Let R be commutative unital ring and let A be a commutative unital R-algebra. A Leibniz pseudoalgebra (or Leibniz-Rinehart algebra) over (R, A) is an anchored module (E, a) over (R, A) endowed with a Leibniz R-algebra structure [−, −], such that, for all f ∈ A and X, Y ∈ E, where the RHS is the commutator.
If the A-module E is faithful, the last requirement is again a consequence of the Leibniz rule and the Jacobi identity.
The space of sections of a Leibniz algebroid over M is obviously a Leibniz pseudoalgebra over (R, C ∞ (M )).
Of course, if, in Definitions 2.3 and 2.4, the Leibniz bracket is antisymmetric, we get a Lie algebroid and a Lie pseudoalgebra, respectively.
Leibniz algebroids over M and Leibniz pseudoalgebras over (R, A) are the objects of categories LeiOid M and LeiPsAlg (R, A). The morphisms of these categories are defined as follows.
Definition 2.6. Let (E 1 , [−, −] 1 , a 1 ) and (E 2 , [−, −] 2 , a 2 ) be two Leibniz pseudoalgebras over the same pair (R, A). A Leibniz pseudoalgebra morphism between them is an A-module morphism φ : We now define (bi)modules over Leibniz algebroids and pseudoalgebras. Recall first the definition of a (bi)module over a Leibniz R-algebra (V, [−, −]). By the general heuristic described above, this is an R-module W with a Leibniz R-algebra structure [−, −] on V ⊕ W containing V as a subalgebra and W as an abelian ideal. Therefore, this bracket has to be the original bracket on V × V , and 0 on W × W , so it is determined by the values of [x, w] ∈ W and [w, x] ∈ W , where x ∈ V and w ∈ W . Setting µ l (x)(w) = [x, w] and µ r (x)(w) = [w, x], we recover the usual notion: A (bi)module over a Leibniz R-algebra (V, [−, −]) is an R-module W together with a left and a right action µ l ∈ Hom R (V ⊗ R W, W ) and µ r ∈ Hom R (W ⊗ R V, W ), which satisfy the following requirements for all x, y ∈ V.
In particular, let ∇ be a representation of (V, [−, −]) on W , i.e., a Leibniz Ralgebra morphism V → End R W . Then µ = ∇ and µ r = −∇ is a (bi)module structure over V on W .
Definition 2.7. Let (E 1 , [−, −], a) be a Leibniz algebroid over M . A module over the Leibniz algebroid E 1 is a C ∞ (M )-module E 2 , which is a module (µ , µ r ) over the Leibniz R-algebra ΓE 1 whose left action satisfies the Leibniz rule for any f ∈ C ∞ (M ), X ∈ ΓE 1 , and Y ∈ E 2 .
In this definition, the C ∞ (M )-module E 2 is not required to define a locally free sheaf of modules over the function sheaf C ∞ M of M , i.e., it is not required to be a module of sections of a vector bundle. Moreover, just as for the Leibniz bracket on ΓE 1 , we do not impose any differentiability condition on µ r . Similarly, , a) be a Leibniz pseudoalgebra over (R, A). A module over the Leibniz pseudoalgebra E 1 is an A-module E 2 (hence an R-module), which is a module over the Leibniz R-algebra E 1 whose left action µ satisfies the Leibniz rule for any f ∈ A, X ∈ E 1 , and Y ∈ E 2 .
for any X, Y ∈ ΓE.
The nondegeneracy of the scalar product allows us to identify E with its dual E * , and we will use this identification implicitly in the following. Note that (11) is equivalent to where Y • Z denotes the symmetrized bracket. Similarly, (12) easily implies the invariance of the scalar product, which in turn shows that a is the anchor of the left adjoint map: Hence, a Courant algebroid is a particular Leibniz algebroid. When defining a derivation D : we get out of (13) that The fact that (17) is a consequence of the 'invariance' condition (13) and the nondegeneracy of the scalar product, will be of importance later on. Let us moreover stress that (17) implies a differentiability condition for the first argument of the Leibniz bracket: It is now clear that As already indicated above, we view the conditions (19) and (20), as well as their consequence as the invariance properties of the scalar product. We will come back to this idea in Subsection 4.2.

Subclasses of Leibniz algebroids.
3.1. Loday algebroids. In [12], the authors observe that a Leibniz algebroid -in the sense of the present paper -is not a proper geometric concept. They suggest a new notion of Leibniz algebroid, called Loday algebroid, which has a right anchor satisfying a condition analogous to (18), and is therefore geometric. Moreover, they show that almost all 'Leibniz algebroids' met in the literature are Loday algebroids in their sense.
for any X, Y ∈ ΓE and f ∈ C ∞ (M ).
Let us mention that the right anchor D can be viewed as a bundle map D : For instance, it is clear from what was said above that, in the case of Courant algebroids, the derivation D is given by The algebraic version of Loday algebroids is defined as follows: 3.2. Symmetric Leibniz algebroids. We now introduce another subclass of Leibniz algebroids, symmetric Leibniz algebroids, which also contains Courant algebroids as a particular example. Let us briefly mention the origins of the next definition. The problem, when searching for a free (generalized) Courant algebroid (or, better, pseudoalgebra), or when trying to represent a Leibniz algebroid (pseudoalgebra) bracket by a derived bracket, is the absence of a differentiability condition on the first argument of the involved bracket. It turns out that both issues can be reduced to the two fundamental symmetry conditions (24) and (25).
The definition can be equivalently formulated as follows: Proof. It suffices to show that, for a Leibniz algebroid that satisfies the first condition, the conditions (25) and (27) are equivalent. Note first that the Jacobi identity implies that [X, The first condition (26) means that the symmetrized bracket is C ∞ (M )-linear between the two variables X, Y . The second condition (27) is a C ∞ (M )-linearity condition in a combination of symmetrized products.
Proof. It follows from the differentiability properties (4) and (23), and from the antisymmetry a[X, Y ] = −a[Y, X] of the bracket of vector fields, that the C ∞ (M )linearity conditions (24) and (25) are equivalent to (28) and (29).  (28) and (29) are direct consequences of the symmetry and the invariance properties of the scalar product. For definitions concerning the other examples, we refer the reader to [12], Section 5.
We examine the first example.
be the Lie algebroid differential, and denote by the Lie algebroid Lie derivative, where i X is the interior product. There is a Leibniz bracket on sections of the vector bundle E ⊕ ∧E * . Indeed, set, for any X, Y ∈ ΓE and any ω, η ∈ Γ(∧E * ), This is a Loday algebroid bracket with left anchor a(X + ω) = a E (X) and right anchor see [12], Section 5. If this Loday algebroid E ⊕ ∧E * is symmetric, Condition (28) is satisfied in particular for 0-forms, i.e., we have for any f, g, h ∈ C ∞ (M ) and any X, Y ∈ ΓE. If we choose f = 0 and g = 1, we find that a E = 0, so that the considered Lie algebroid E is a Lie algebra bundle (28) and (29) The 'Courant pseudoalgebra' 'morphism' f 1 : F → E 0 can easily be handled, hence, we do not insist on it here. On the other hand, the searched universal 'scalar product' (−|−) on F should of course satisfy a condition of the type for any m, m ∈ F. The RHS of this condition is visibly defined on F F, where is the symmetric tensor product over A, but it does not provide a 'universal product' (−|−). The way out is to compose the map f 2 : F F → C ∞ (M ), defined by with the 'universal scalar product' (−|−) : F × F → F F, given by The 'Courant pseudoalgebra' 'morphism' (f 1 , f 2 ) then respects the 'metrics' (−|−) and (−|−) 0 . To make sure that the 'universal scalar product' satisfies the compatibility condition (21), it is natural to replace the 'product' (32) by the 'product' (we use the same notation as before) defined by In view of the invariance conditions (19) and (20) for the free generalized Courant pseudoalgebra, as well as the definition of 'generalized'. When trying to prove that the actions µ and µ r are well-defined on the symmetric tensor product, we discover the first symmetry condition (24) for F. When attempting to show that they are well-defined on the quotient R(F), one finds the second symmetry condition (25) for F. It then suffices to force these properties in F, i.e., to pass again to the quotient.
hold true. We refer to such a tuple as a generalized pre-Courant pseudoalgebra.  It is clear that, for any X ∈ E 1 , we have (X|−) ∈ Hom A (E 1 , E 2 ). By nondegenerate scalar product, we mean here that any ∆ ∈ Hom A (E 1 , E 2 ) reads ∆ = (X|−), and that the map Y → (Y |−) is injective, so that X is unique. Indeed, in the aforementioned geometric case, nondegeneracy in each fiber, see Definition 2.10, implies these requirements.
Proof. We use the above notation; in particular f ∈ A and X, Y, Z ∈ E 1 . In view of the invariance relations and Leibniz rule for the left action, we get On the other hand, the Leibniz rule for the bracket [−, −] gives The properties of the anchor and the scalar product imply that D is a derivation It now follows from (38), (39), (40), and nondegeneracy that i.e., that (E 1 , [−, −], a) is a Loday pseudoalgebra.
Furthermore, the latter is a symmetric Leibniz pseudoalgebra if and only if the conditions (28) and (29) are satisfied -the proof in the geometric situation remains valid in the present algebraic case -. It is easily seen that these requirements are fulfilled due to the invariance relation (37).
A priori it might seem natural to define generalized Courant pseudoalgebras as generalized pre-Courant pseudoalgebras endowed with a nondegenerate scalar product (−|−). In view of the candidate (34), (33), we choose however the following more general (see Proposition 4) definition: Whereas Courant algebroids are Leibniz algebroids endowed with a scalar product, such that some invariance conditions are satisfied, generalized Courant pseudoalgebras are symmetric Leibniz pseudoalgebras that are endowed with a 'scalar product' (a symmetric A-bilinear map) valued in a 'representation' (a module over the Leibniz pseudoalgebra) and satisfy similar invariance conditions. In other words, the representation (C ∞ (M ), a, −a) is replaced by a 'representation' (E 2 , µ , µ r ), the C ∞ (M )-valued scalar product is replaced by an E 2 -valued 'scalar product', and the symmetry of the Leibniz pseudoalgebra substitutes for the nondegeneracy of the classical scalar product.   4.2. Generalized Courant pseudoalgebra associated to a symmetric Leibniz pseudoalgebra. The next theorem describes this generalized Courant pseudoalgebra, which is actually the prototypical example. It is also the motivation for the introduction of symmetric Leibniz algebroids. Moreover, it will turn out that the generalized Courant pseudoalgebra associated to a symmetric Leibniz pseudoalgebra is one of the two components of the free Courant pseudoalgebra -see Subsection 4.1, introductory remark. Denote by the symmetric tensor product over A, take the subset of the A-module E 2 , and let Inv be the A-submodule of E 2 generated by Inv.

The quotient R(E) = E 2 / Inv
is an E-module with actionsμ andμ r induced by and These data, together with the universal scalar product Remark 3. The associated generalized Courant pseudoalgebra is a very natural construction over a symmetric Leibniz pseudoalgebra, whose scalar product is the universal scalar product given by the symmetric tensor product and whose actions are the 'invariant' Courant actions.  Proof. (i) We first show that µ (X) and µ r (X) are well-defined on E 2 (note that we do of course not intend to show that they are A-linear on E 2 ; indeed, they are visibly only R-linear). Since the RHSs of (43) and (44) are symmetric in Y 1 , Y 2 , it suffices to prove that they respect the 'defining relations' of the tensor product over

A. The only nonobvious condition is that
(ii) It remains to check the 'Leibniz morphism conditions' (6), (7), and (8), as well as the Leibniz rule (9). The Leibniz rule is clear from (i). The morphism conditions are also straightforwardly checked. To verify for instance note first that the right adjoint action [−, X] on a symmetrized product vanishes: We now get Hence, the result. and which are similar to (19) and (20). Since (21) does however not hold in general, we consider the quotient A-module Proof. It suffices to show that the actions descend to the quotient; indeed, the induced maps then inherit the required properties.
(i) Left action. Let I(X, Y, Z), or just I, be any element in Inv ⊂ E 2 , and let f ∈ A and W ∈ E. Since The latter actually holds true: (ii) Right action. In view of the annihilation of symmetrized products by right adjoint actions and due to the symmetry condition (25), It follows from (47) and denote by J 1 ( resp., J 2 ) the A-module generated by J 1 ( resp., J 2 ). The A-module (J 1 + J 2 ) := J 1 + J 2 is an ideal of the Leibniz pseudoalgebra E, so that the quotient E/(J 1 + J 2 ) inherits a symmetric Leibniz pseudoalgebra structure.
Proof. The left adjoint action [W, −], W ∈ E, satisfies the Leibniz rule with respect to the Leibniz bracket (X, and similarly for J 2 .
As for the right action [−, W ], W ∈ E, recall that it vanishes on every symmetrized bracket. Since the first term of an element of J 2 is symmetric as well, the sets J 1 and J 2 vanish under the right action.
For any f ∈ A, W ∈ E, and Q ∈ J 1 , we have now Hence, [f W, Q] ∈ J 2 , and [Q, f W ] = 0. Therefore, Equations (51), (52), (53), and (54) imply that the A-submodule J 1 + J 2 ⊂ E is a Leibniz R-algebra ideal. To see that J 1 + J 2 is a Leibniz pseudoalgebra ideal, it now suffices to recall that a is A-linear and that any symmetrized bracket belongs to ker a.
We write for short C := (Sym)LeiPsAlg (R, A) and D := AncMod(A). Therefore, for any M ∈ D, we can define the free (symmetric) Leibniz pseudoalgebra over M, see Definition 2.2. It is made of an object F(S) M ∈ C and a D-morphism i : M → F(S) M, such that, for any object E ∈ C and any D-morphism φ : We first recall the construction of the free Leibniz algebra over an R-module [29]. Let V be an R-module and let T V = k≥1 V ⊗ R k be the reduced tensor R-module For instance, The next theorem has been conjectured at the beginning of Subsection 4.1. Proof. We denote by F n M = M ⊗ R n (resp., F n M = 1≤k≤n M ⊗ R k ) the grading (resp., the filtration) of F M.
(i) Module structure. Equation (56) provides a well-defined A-module structure on F n M, if we are given a well-defined A-module structure on F n−1 M, n ≥ 2. Since the RHS of (56) is R-multilinear, the 'action' is well-defined from F n M into F n M. We extend it by linearity to F n M = F n M ⊕ F n−1 M. It is now straightforwardly checked that this extension satisfies all the A-module requirements, except, maybe, the condition f (gµ) = (f g)µ, where f, g ∈ A and µ ∈ F n M. As for the latter, note first that, if f, g ∈ A, m, m i ∈ M, and m = m 2 ⊗ . . . ⊗ m n , we have The A-module structures on the filters F n M, n ≥ 1, naturally induce an A-module structure on F M.
(ii) Universal anchor map. Since F M is the free Leibniz algebra over M, the map a : M → Der A factors through the inclusion M → F M: The Leibniz algebra morphism F a is actually A-linear. Indeed, in view of the decomposition f m = i≤n−1 m i , where m i ∈ F i M is a decomposed tensor, we obtain first m ⊗ f m = [m, f m], where the notation is the same as above. Since, by induction, F a is A-linear on F n−1 M, we then have (iii) Leibniz pseudoalgebra conditions. To see that (F M, [−, −] Lei , F a) (in the following we omit subscript Lei) is a Leibniz pseudoalgebra, it now suffices to check that (4) is satisfied. We proceed by induction and assume that, for any f ∈ A, m ∈ F n−1 M and m ∈ F M, n ≥ 2, the bracket [m, f m ] satisfies Condition (4). Indeed, for n = 2, we have It is easily seen that (4) is then also satisfied in F n M:  Proof. We characterize the classes and the mentioned induced data by the symbol 'tilde'. It has already been said that (FS M, [−, −] Lei , (F a) ) is a symmetric Leibniz pseudoalgebra (as usual we will omit Lei). On the other hand, it is clear from the definition of (F a) thatĩ : M m →m ∈ FS M is an anchored A-module map.
As for freeness, let (E, [−, −] , a ) be a symmetric Leibniz pseudoalgebra. Any anchored module map φ : M → E uniquely extends to a Leibniz pseudoalgebra map F φ : F M → E. To see that F φ descends to FS M, it suffices to show that it vanishes on J 1 and J 2 . Observe first that, for any µ, ν ∈ F M, we have It follows now from the A-linearity of F φ and the symmetry of E that F φ annihilates J 1 and J 2 . It is also straightforwardly checked that the induced map (F φ) : FS M → E is a map of Leibniz pseudoalgebras such that (F φ) ĩ = φ. As for uniqueness of this extension, note that any Leibniz pseudoalgebra morphism FS φ : FS M → E that extends φ, implements a Leibniz pseudoalgebra morphism (FS φ)¯: F M µ → FS φ(μ) ∈ E that extends φ; hence, (FS φ)¯= F φ and FS φ = (F φ) .
6. Free Courant pseudoalgebra. There is a forgetful functor For : CrtPsAlg → AncMod between the categories of generalized Courant (R, A)-pseudoalgebras and anchored A-modules. Proof. Let C and φ be as in the statement of the theorem. Due to freeness of FS M, the anchored module map φ uniquely factors through FS M, thus leading to a unique Leibniz pseudoalgebra morphism φ 1 : FS M → E 1 such that φ 1ĩ = φ.
As for φ 2 , note that the A-linear map descends to the quotient R(FS M). Indeed, if I(μ,ν,τ ) ∈ Inv (in the following we omit the symbol 'tilde'), we have since C is a generalized Courant pseudoalgebra. The resulting A-linear map R(FS M) → E 2 will still be denoted by φ 2 . Since it is clear that the requirement (41), i.e., is satisfied. It now suffices to check that the conditions (10), i.e., hold as well. Let us detail the last case. Letμ,ν,τ ∈ FS M (and omit again the 'tilde'). Sinceμ the application of φ 2 leads to The latter coincides with the value of µ r (φ 2 × φ 1 ) on the arguments (ν τ ) and µ. Finally, uniqueness of φ 1 was already mentioned, and uniqueness of φ 2 is a consequence of (41) and (57). where All the ingredients of this free object are implicit: the free Leibniz algebra bracket, the A-module structure, and the anchor on T E are defined inductively, whereas the module FS E and its 'representation' module are quotients by the abstract symmetry conditions (24) and (25) and by the abstract invariance condition (42), respectively. It follows that the associated and the free generalized Courant pseudoalgebras are important rather by their existence than by their description in concrete situations -see Section 7. Moreover, the free generalized Courant bracket is not geometric, in the sense that it is not Loday. This can be quite easily checked by an argument to absurdity.
7. Symmetric Leibniz pseudoalgebra bracket as derived bracket. Many algebraic and algebro-geometric concepts can be encoded in a (co)homological vector field of a possibly formal and noncommutative manifold. For instance, if P is a quadratic Koszul operad, a P ∞ −structure on a finitedimensional graded vector space V over a field K of characteristic zero, is essentially a sequence n of n-ary brackets of degree 2 − n on V , which satisfy a sequence R n of defining relations, n ∈ {1, 2, . . .}. These structures are 1:1 with cohomological vector fields δ ∈ Der 1 (F gr P ! (sV * )) (58) of the 'manifold' with function algebra F gr P ! (sV * ) -the free graded algebra over the Koszul dual operad P ! of P on the suspended linear dual sV * of V .
In the case P = Lie, the latter is the graded symmetric tensor algebra sV * without unit. The n-ary brackets n of the Lie infinity structure on V are obtained, up to (de)suspension, as the transposes of the projections to n sV * of the restriction of δ to sV * . T. Voronov uses an alternative method and constructs a Lie infinity structure on sV , starting, in the main, from a cohomological vector field δ of a Lie superalgebra, and using higher derived brackets [37].
The higher derived brackets modus operandi goes through in the geometric situation of Lie n-algebroids, n ≥ 1, [4], in particular, as well-known, for Lie algebroids.
Another geometric context, where this technique can be applied, is the case of Loday algebroids: if E denotes a vector bundle, there is a 1:1 correspondence between Loday algebroid structures on E and equivalence classes of cohomological vector fields δ ∈ Der 1 (D • (E), ) [12]. The latter is nonobvious and far from the known solution in the purely algebraic case P = Lei of Leibniz infinity or just Leibniz structures: we have to consider specific derivations of specific multidifferential operators D • (E), as well as the symmetrization of the half-shuffle or Zinbiel multiplication. In particular, the Loday algebroid bracket is the derived bracket induced by the graded Lie algebra Der(D • (E), ) and its interior derivation implemented by the cohomological field δ.
In the present paper, we investigate which Leibniz algebroid or Leibniz pseudoalgebra brackets can be viewed as derived brackets. The difficulty, in the passage from algebras to algebroids or pseudoalgebras, is the replacement of scalars (in a field or ring) by functions (in an algebra over this ring). Additional obstruction comes -in the Leibniz pseudoalgebra setting -from the absence of a differentiability condition on the first argument of the bracket. This is one of the origins of the subclass of symmetric Leibniz pseudoalgebras. We will show that each symmetric Leibniz pseudoalgebra bracket can be universally represented by a derived bracket.
Let (E, [−, −], a) be a symmetric Leibniz pseudoalgebra (over (R, A)) and let be the associated generalized Courant pseudoalgebra. Recall that R(E) is the universal representation A-module E 2 / Inv , where the denominator is the Asubmodule induced by We denote by E Lie the Lie R-algebra E/I obtained as the quotient of the Leibniz We denote its degree by | − | , its elements of degree 1 (resp., degree 0, degree 2) by X, Y, . . . (resp., byX,Ȳ , . . . , by (X Y ) , . . . ), and its elements of arbitrary degree by a, b, . . . We will endow this graded module with a differential graded Lie R-algebra As {−, −} must be of degree 0 and graded antisymmetric, it is naturel to set (s is understood, wherever possible): whereas the brackets of pairs of elements of degrees (1, 2), (2, 1), and (2, 2) are of degree > 2 and must therefore vanish.
Of course, we have to check well-definedness. For the first bracket, of degree (0, 0) elements, remember that [T • U, Y ] = 0 (even without passing to E Lie -in view of the Jacobi identity) and note that Well-definedness is now also clear for degree (0, 1) and (1, 0) elements. As for degree (0, 2) and (2, 0) elements, we already proved above thatμ (X) is well-defined on the quotient R(E), for any X ∈ E. Concerning the argumentX, it suffices to observe thatμ Regarding the graded Jacobi identity, it is easily seen that, if it holds for three elements of some given degrees, it also holds for elements whose degrees are any permutation of the initial ones. Therefore, it suffices to check the identity for the degrees (0, 0, 0), (0, 0, 1), (0, 0, 2), and (0, 1, 1). Indeed, all other cases are permutations or their sum of degrees > 2. In the four relevant cases, the graded Jacobi identity is a direct consequence of the definitions.
If the abovementioned degree −1 derivation δ does exist, it sends X ∈ E to δX ∈ E Lie . A naturel choice is δX :=X .
Further, since δ(X Y ) = δ{X, Y }, the graded derivation property shows that the latter is equal to When adopting this definition we have to check that δ is well-defined. Well-definedness on E 2 is a direct consequence of the first symmetry condition, whereas well-definedness on the quotient R(E) requires that for any f ∈ A , X, Y, Z ∈ E . However, the LHS of the latter reads where the sum of the first three terms vanishes due to the second symmetry condition, while the last term is of the form 'right adjoint action on a symmetrized product' and thus vanishes -as recalled above. Finally, the map δ is a well-defined degree −1 map on the R-module D • (E), which is visibly R-linear and of square 0. It now suffices to prove that δ is a graded derivation for {−, −}. If the graded derivation property holds for degree (i, j) elements, it is also valid for degree (j, i) elements. Hence, we only examine the degrees (0, 0), (0, 1), (0, 2), (1, 1), and (1, 2). These verifications are straightforward, at least if one remembers the second symmetry condition (in fact, here we do not even need the symmetry assumption: we use the second symmetry condition with f = 1 ∈ A, and in this case it is satisfied in any Leibniz algebra).
Eventually , whose bracket is of degree 0 and whose differential has degree −1.
In this definition, all algebras are over R. In the following, we consider representations of the Leibniz algebra (E, [−, −]) (concentrated in degree 0), and we restrict ourselves to representations ξ, such that ξ(E) (resp., In other words, all brackets of the type { {ξ(X), { {ξ(Y ), ξ(Z)} } } } vanish. We refer to such representations as A-linear nilpotent representations. The above suspension, for instance, is A-linear and nilpotent.
In fact, the suspension is the best possible representation: Hence, the uniqueness of Ξ . We adopt these definitions. Clearly, the map Ξ is of degree 0, but we must verify that it is well-defined. In the case ofX, we get It remains to prove that Ξ respects the brackets and the differentials, and actually induces a degree 0 morphism of graded Leibniz algebras.