Symmetry, Integrability and Geometry: Methods and Applications On Lie Algebroids and Poisson Algebras

We introduce and study a class of Lie algebroids associated to faithful modules which is motivated by the notion of cotangent Lie algebroids of Poisson manifolds. We also give a classification of transitive Lie algebroids and describe Poisson algebras by using the notions of algebroid and Lie connections.

Moreover, the bracket (1) of two closed forms is again closed. In fact, the properties above characterize the Lie algebroid structure on the cotangent bundle that comes from a Poisson bracket on the base manifold (see [11,7]). More precisely, given a Lie algebroid ( The aim of the present work is to put the study of cotangent Lie algebroids of Poisson manifolds in an algebraic framework. In the first part (Section 3), we study and characterize a class of Lie algebroids with properties similar to (a), (b), which we call Lie algebroids of Poisson type. To this end, we will work in a slightly more general context than that of vector bundles over manifolds, using the notion of Lie-Rinehart algebra (see [11,12]), also called a Lie pseudoalgebra (see [17,20,21,22] and, particularly, [16] for many interesting remarks on the evolution of these notions and as a general reference), although we will continue using the denomination "Lie algebroid" for them (as they are the algebraic version of the geometric Lie algebroids). The dictionary in Example 1 can be used at any time to obtain the corresponding expressions for geometric Lie algebroids. We give several examples illustrating the different situations that can appear.
In the second part (Sections 4 and 5), we deepen in the relationship between transitive algebroids and Poisson structures for a certain class of spaces, those of the form Der(A) ⊕ V , where A is a commutative algebra and V an A-module. We describe parametrizations of the transitive Lie algebroids on Der(A) ⊕ V following the techniques exposed in [24,23], which are based on the use of a connection on a Lie algebroid. For completeness, we include a subsection in the preliminaries devoted to the topic of connections in an algebraic setting. Once the parametrization is given, we apply it to prove that a transitive algebroid endowed with a connection is isomorphic to one of the form Der(A) ⊕ V . Finally, we obtain new classes of Poisson algebras on A ⊕ V starting from Poisson algebras on A.

Preliminaries
Throughout the document, unless otherwise explicitly stated, A denotes an associative, commutative algebra with identity element 1 A , over a commutative ring R with identity element 1 R .

Derivations and connections in commutative algebras
In subsequent sections, we will need to introduce connections on an algebroid. In our algebraic setting, the most appropriate notion of connection is Koszul's one, which is given in terms of derivations. Definition 1. A derivation of the algebra A over R is a map X ∈ Hom R (A, A) satisfying the Leibniz rule The set of derivations of A over R is denoted Der R (A) or simply Der(A) when there is no risk of confusion about the ring R. The set Der(A) has an R-Lie algebra structure when endowed with the commutator of endomorphisms, given by [ Note also that, if A as an R-module is faithful, then for every X ∈ Der(A) we have X(1 A ) = 0 and indeed X(r) = 0 for every r ∈ R viewed as a subalgebra of A.
Not every A-module M admits a connection in this sense, but it is easy to see that any free A-module does. Of course, arbitrary A-modules do not need to be free. So, in order to obtain a big enough class of modules for which we can guarantee the existence of a Koszul connection, we will make a brief digression on modules of differentials and Connes connections (see [4]).
Let Ω 1 (A) be the A-module defined by the kernel of the multiplication A ⊗ R A → A. Define the map d : A → Ω 1 (A) by da = 1 ⊗ a − a ⊗ 1, which is a derivation of A over R with values into Ω 1 (A). It is clear from the definition that Ω 1 (A) = Span A {df : f ∈ A}: Since the elements of Ω 1 (A) lie in the kernel of the multiplication map, if a j ⊗ b j ∈ Ω 1 (A), then a j b j = 0 and therefore In fact, Ω 1 (A) is the submodule of C 1 (Der(A), A) (the 1-component of the differential algebra C(Der(A), A) of Chevalley-Eilenberg cochains of the Lie algebra Der(A) with values in the Der(A)-module A) generated by the elements df , f ∈ A (see [6]). Note, in particular, that this implies Ω 1 (A) ⊂ Der * (A).

Definition 3. Let M be an A-module. A Connes connection on M is an
Remark 2. Connes' definition of a connection (see [4]) actually does not require that A be a commutative algebra. The definition goes back to a work by N. Katz [14].
Starting from a Connes connection, we can obtain a Koszul one. If X ∈ Der(A), then we define a right A-linear pairing ϕ : The Koszul connection ∇ associated to δ can be constructed as follows: for X ∈ Der(A), ∇ X ∈ Hom R (M, M ) is the map given by applying the connection δ and then contracting the Ω 1 (A) component with ϕ. Thus, if m ∈ M is such that δ(m) = a j db j ⊗ m j , for certain a j , b j ∈ A and m j ∈ M , we have for each f ∈ A that δ(f m) = f a j db j ⊗ m j + df ⊗ m, and A basic result obtained by J. Cuntz and D. Quillen (see [5]) is that Connes connections on an A-module M are in bijective correspondence with A-linear splittings of the natural action A ⊗ R M → M . As a consequence, M admits a Connes connection if and only if it is projective. As said earlier, we will need later on to work with (Koszul) connections, so we need conditions on A to assure their existence. From what we have seen, these connections exist on any Amodule M which is free or projective. Indeed, note that a free module is always projective, but there are projective modules which are not free. In the literature, there are several well-known conditions on A guaranteeing the projective character of M (for example, that A be semi-simple as a ring). When we talk of a connection on M , unless otherwise explicitly stated, we will mean that any one of these conditions is satisfied and that the connection is Koszul.

Lie algebroids
] is a Lie bracket on F and ρ : F → Der(A) is a morphism of A-modules, called the anchor map, such that for all f ∈ A and for all X, Y ∈ F. Remark 3. Sometimes, the condition that the anchor map be a morphism of Lie algebras is included in the definition of Lie algebroid. However, this fact is a consequence of the conditions in Definition 4, as it has been noted by J.C. Herz, Y. Kosmann-Schwarzbach, F. Magri and J. Grabowski among others (see [9,10,17,8]).
, ρ ) be Lie algebroids (over the same algebra A and the same ring R). A morphism of Lie algebroids is a morphism of A-modules φ : F → F such that for all X, Y ∈ F.
Let us consider some examples. The first is the classical one.
such that for all f ∈ C ∞ (M ) and for all X, Y ∈ Γ(E): In particular, if E = T * M , then the Lie algebroid structure is given by the bracket (1) where the anchor is the Poisson mapping P . For E = T M we have the trivial Lie algebroid, where q = Id T M . Example 2. Consider the R-algebra of dual numbers over A, with the obvious operations. Clearly, A is an A -module and we can endow it with the Lie algebra structure given by the bracket: The next example will be relevant in Section 4.
3 Lie algebroids of Poisson type Definition 6. A Poisson algebra (A, { , }) is an associative algebra A together with a Lie bracket which is also a derivation for the product in A, that is, there is an R-bilinear operation If (A, { , }) is a Poisson algebra, then we can define the adjoint map ad : for f ∈ A. Then, extending the mapping df −→ ad f by linearity, we get a morphism ρ : Sometimes (by analogy with Poisson manifolds), ad f is referred to as the Hamiltonian vector field corresponding to f ∈ A, and denoted by X f (we will use this notation and terminology later in Section 5). Also, given an α ∈ Ω 1 (A), we can define dα through the usual formula for all X, Y ∈ Der(A).
Proof . All the required properties can be checked directly from the definition of the bracket (3).
To verify the Jacobi identity, it is easier to consider a system {df i } i∈I of generators of Ω 1 (A), and to write the elements α ∈ Ω 1 (A) as α = g i df i for some g i ∈ A.  (2)) and the bracket (1): for all α, β ∈ Ω 1 (M ). The proof is based on the fact that there exists a finite subset This, in turn, is a consequence of Whitney's embedding theorem and the fact that the sheaf of germs of smooth functions is soft (see [2]).
Theorem 1 tells us that given a Poisson algebra we have a Lie algebroid canonically associated to it. We are interested now in the reciprocal: When does a Lie algebroid (Ω 1 (A), [[·, ·]], ρ) determine a Poisson structure on A?.
Given a Lie algebroid (Ω 1 (A), [[·, ·]], ρ), for any f, g ∈ A we can define The bracket { , } defined in this way is clearly R-bilinear and satisfies the Leibniz rule Thus, in order to get a Poisson structure on A we only need to take care of the skew-symmetry and the Jacobi identity. for all α, β ∈ Ω 1 (A).
If we assume that the anchor map is skew-symmetric, then the new operation defined by (4) is also skew-symmetric. Now, let us turn our attention to the Jacobi identity.
, ρ) be a Lie algebroid, and define Q ∈ Λ 2 (Der(A)) by Q(df, dg) = dg(ρ(df )) for f, g ∈ A. The following conditions are equivalent: Here, [·, ·] SN denotes the Schouten-Nijenhuis bracket on multiderivations [15]. Another important issue for us is to determine the form of the bracket of a Lie algebroid of Poisson type. We know that the classical example of Poisson manifolds leads to brackets of the type (1). The following result characterizes the class of such algebroids.
Proof . Note first that, if α = df and β = dg for some f, g, ∈ A, then Since every element of Ω 1 (A) is a linear combination of elements of the form df i (f i ∈ A), it is enough to prove the statement for α = f 1 dg 1 and β = f 2 dg 2 , which can be done by a direct computation.
Interchanging the roles of f and h, we have These relations imply that df , dh and C(df, dh) are linearly dependent for arbitrary f and h.
In particular, if df and dh are linearly independent, then C(df, dh) = 0, and hence for any α, β ∈ Ω(M ). Such graded Poisson brackets were called differential in [3].
Let us summarize these results.
if and only if: (a) ρ is skew-symmetric (b) One of the following conditions holds: Under these conditions, the Lie bracket [[·, ·]] is reconstructed from ρ by the formula Of course, the basic example of this situation is the cotangent Lie algebroid of a symplectic manifold.
Note that the matrix representation of ρ relative to the given basis in Der(A) and Ω 1 (A) is Therefore, ρ is skew-symmetric and of rank 2 (ρ is not injective). A long but straightforward computation shows that (Ω 1 (A), ][·, ·]], ρ) is a Lie algebroid.
Let us show that this Lie algebroid is of Poisson type by checking the property [[df, dg]] = d{f, g}. We have for p, q ∈ A: The dx 1 factor in the expansion of this expression (the other cases are similar) is On the other hand, the Lie algebroid bracket is For k = 1, we compute the coefficient of dx 1 : which is the same as above. Thus, (Ω 1 (A), [[·, ·]], ρ) is a Lie algebroid of Poisson type and its bracket is just given by formula (3).

Remark 5.
The anchor map of the cotangent Lie algebroid is not injective in general. It is only true in the symplectic case.
Finally, let us consider an example of Lie algebroid of Poisson type whose bracket does not have the form (3).

Transitive Lie algebroids
To motivate the definition of transitive Lie algebroids let us consider for a moment the geometric example of a Lie algebroid (E, [[·, ·]], q), where E → M is a vector bundle over a manifold M (recall Example 1). If the anchor map q : Γ(E) → Γ(T M ) is an epimorphism, the algebroid (E, [[·, ·]], q) is said to be transitive. In this case, it is possible to construct the so-called Atiyah sequence of the algebroid, which is the short exact sequence Thus, the existence of a section for the anchor q (equivalently, a linear connection on E) implies that, locally, E = T M ⊕ g. Note also that the fibre of the bundle g over the point x ∈ M , g x is a Lie algebra (called the isotropy Lie algebra of the algebroid E at x ∈ M ) with the bracket given, for α, β ∈ g x , by where X, Y ∈ Γ(E) are any sections such that X(x) = α and Y (x) = β.
There is a more general notion, the extension of a Lie-Rinehart algebra, that generalizes the transitivity condition for a geometric Lie algebroid (see [13]).

Transitive algebroids induced by connections
Let V be a unitary A-module and ∇ a connection on V .
Definition 11. If V is endowed with a Lie algebra structure [·, ·] V , then a connection ∇ is said to be a Lie connection if for all v 1 , v 2 ∈ V and for all X ∈ Der(A).
We have the following result, which gives us a description of transitive Lie algebroids in terms of Lie connections (see, also [20,21]).
. Let ∇ be a Lie connection on V . If there exists a 2-form B ∈ Ω 2 (A; V ) with values in V , such that, for any X 1 , X 2 , X 3 ∈ Der(A) and v ∈ V the following conditions hold: , ρ) is a transitive Lie algebroid with anchor map ρ = pr 1 and bracket Moreover, for X, Y ∈ Der(A). Here ι 1 : Der(A) → Der(A) ⊕ V and ι 2 : V → Der(A) ⊕ V are the inclusion maps.
Proof . It is clear that Der(A) ⊕ V is an A-module. We must show that the bracket defined by (5) is Lie. The skew-symmetry and the R-bilinearity are immediate, while the Jacobi identity and the Leibniz rule are checked by lengthy but straightforward computations. Since ρ = pr 1 is clearly A-linear, we have that Der(A) ⊕ V is a Lie algebroid. The transitivity is obvious in view of the sequence Moreover, a direct computation shows that Remark 7. The conditions (a) and (b) in this theorem have the following interpretation. Condition (a) states that the curvature of the connection ∇ is given by the composition C ∇ = ad R • B, where ad R : V → V is the right adjoint with respect to the bracket on V . On the other hand, (b) expresses the Bianchi identity for a connection with torsion (see also [20,21]).

Parametrization of transitive algebroids on DerA ⊕ V
The following result states that the converse of Theorem 4 is also true.
is an A-linear Lie bracket.
(ii) The mapping ∇ : Der(A) → Hom(V, V ) given by for v ∈ V and X ∈ Der(A), is a Lie connection on V .  Proof . The proof is a lengthy but straightforward computation developing the definitions.
As a consequence of Theorems 4 and 5, we have the following.

Algebroid connections
In this subsection, we first generalize Theorem 5 to the case of a transitive Lie algebroid which is not necessarily of the form Der(A)⊕V , but we require that it be endowed with a Lie algebroid connection (see [19,21]), considered as a section of the anchor map. Then, in Theorem 6 we construct a Lie algebroid structure on Der(A) ⊕ V which is isomorphic to the given algebroid. As stated in Section 2.1, we will assume some condition guaranteeing the existence of a connection on (F, [[·, ·]], ρ), to be precise, we will assume that F is free. It is known that the existence of a section γ of ρ in the short exact sequence be a transitive Lie algebroid and γ : Der(A) → F a Lie algebroid connection for F. Then: is an A-linear Lie bracket.  Proof . First, note that the exactness of (6) allows us to identify V with its image under ι in F, and that Im ι = ker ρ in F. Then, since ρ is a morphism of Lie algebras, we have Thus, the bracket [·, ·] V is well-defined. By a similar argument it can be proved that ∇ and B are well-defined. The proof of the remaining properties follows the same guidelines that those in Theorem 5. Proof . We know that (Der(A) ⊕ V, ·, · , pr 1 ) is a Lie algebroid by Theorem 4. Moreover, φ : Der(A) ⊕ V −→ F, is an A-module isomorphism such that (ρ • φ)(X, v) = ρ(γ(X) + ι(v)) = X = pr 1 (X, v) and, by a straightforward computation, φ( (X 1 , v 1 ), (X 2 , v 2 ) ) = [[φ(X 1 , v 1 ), φ(X 2 , v 2 )]].

Poisson algebras on A ⊕ V
Recall that, if (A, { , }) is a Poisson algebra, then we define for every f ∈ A the Hamiltonian derivation X f ∈ Der(A) as the adjoint map X f = {f, ·}. The following standard properties will be used below: (ii) the mapping f → X f is a Lie algebra morphism, i.e., X {f 1 ,f 2 } = [X f 1 , X f 2 ]. Now, on A ⊕ V we can define a product given by for f 1 , f 2 ∈ A and v 1 , v 2 ∈ V . This makes A ⊕ V a commutative ring. Under certain conditions, this ring is also a Poisson algebra, as shown by the following result.
Proof . The Jacobi identity and the Leibniz rule are proved by direct computations, making use of the A-linearity of the bracket [ , ] V .  Proof . By Theorem 6, we know that (F, [[·, ·]], ρ) is isomorphic to (Der(A) ⊕ V, ·, · , pr 1 ). The statement then follows from Theorem 7.