On a Lie Algebraic Characterization of Vector Bundles

We prove that a vector bundle $\pi : E \to M$ is characterized by the Lie algebra generated by all differential operators on $E$ which are eigenvectors of the Lie derivative in the direction of the Euler vector field. Our result is of Pursell-Shanks type but it is remarkable in the sense that it is the whole fibration that is characterized here. The proof relies on a theorem of [Lecomte P., J. Math. Pures Appl. (9) 60 (1981), 229-239] and inherits the same hypotheses. In particular, our characterization holds only for vector bundles of rank greater than 1.


Introduction
Algebraic characterization of spaces goes back to Gel'fand and Kolmogorov [1], who proved in 1939 that two compact topological spaces are homeomorphic if and only if the associative commutative algebras of continuous functions growing on them are isomorphic. Since then, similar characterizations have proved useful for defining algebraically noncommutative spaces for instance.  [8] which states that the Lie-algebra of compactly supported vector fields on a manifold characterizes the latter manifold. Similar results have been obtained for various Lie subalgebras of Vect(M ), the Lie algebra of vector fields on M .
Rather than considering Lie subalgebras of vector fields J. Grabowski and N. Poncin [4,5,6] have characterized the manifold by means of the Lie algebra of linear differential operators and the Poisson algebra of fiberwise polynomial functions on the cotangent bundle. Recently, they extended, together with Kotov, their characterizations to the super context, see [2,3].
Another way of generalizing Pursell-Shanks' result is the Lie algebraic characterization of vector bundles. Let us, for instance, point out a result of P. Lecomte [7] which shows that vector bundles are isomorphic if and only if their Lie algebras of infinitesimal automorphisms are isomorphic.
In the present paper we propose a hybrid approach in the sense that we give a characterization of vector bundles in terms of a Lie algebra which is not a Lie subalgebra of vector fields. More precisely, we consider for a vector bundle E → M , the Lie algebra D E (E) of homogeneous differential operators, i.e., the Lie algebra generated by all differential operators D : C ∞ (E) → C ∞ (E) which are eigenvectors with constant eigenvalues of L E , the Lie derivative in the direction of the Euler vector field. We prove that the Lie algebra D E (E) and its Lie subalgebra D 1 E (E) of first-order differential operators both characterize the vector bundle E → M .
This new Pursell-Shanks type result is thus remarkable in the sense that it is the whole fibration that is characterized here, not only the base manifold or the total space of the vector bundle. However, since the proof relies on [7, Theorem 1], our characterization inherits the same hypotheses and thus holds only for vector bundles of rank n > 1 over base manifolds M whose first space H 1 (M, Z/2) of the Cech cohomology vanishes.
In this work, we consider smooth manifolds which are assumed to be finite-dimensional, Hausdorff and second countable.
is preserved by the elements of D E (E). More precisely, if u ∈ A λ (E) and T ∈ D E (E) is an eigenvector of L E corresponding to the eigenvalue µ, then In the sequel, elements of D E (E) are usually considered as endomorphisms of the vector space A(E).
Proof . We will prove that if L E T = λT then every T α,β is homogeneous of weight |β| + λ, i.e., Observe that for all f ∈ C ∞ (E), one has Hence, for any u ∈ A(E) , It follows that Proof . If u ∈ A λ (E) for some λ = 0, then with X i ∈ Vect(M ) and v i ∈ C ∞ (M ). Since any v ∈ C ∞ (M ) can be decomposed in this way and since D(M ), the space of linear differential operators of M , is non-singular (see [4,5,6]), the conclusion follows.

The (quasi-)distinguishing property
Denote by S E (E) the set {σ(T ) : T ∈ D E (E)} where σ stands for the principal symbol operator. This set S E (E) is a subset in The Poisson bracket of A(T * E) restricts to S E (E) and endows it with a commutative algebra structure.
The algebra D E (E) would be distinguishing in the sense of [4,5,6] if for any P ∈ S E (E), we had Obviously, this is not the case. Indeed, considering the trivial bundle pr 1 : R 2 → R, it is easy to find a P ∈ S E (R 2 ) \ S 0 E (R 2 ) whose Hamiltonian vector field reads H P = ∂ u 1 and for such a P , the right-hand side of condition (1) is trivially satisfied since for any {P, −} n = H n P = ∂ n u 1 while all elements in A E (E) are finite order polynomial in u 1 .
In [4,5,6], the fact that the Lie algebra of differential operators on a manifold is non-singular and distinguishing is used to obtain that any isomorphism of such Lie algebras always have two nice properties : it is filtered and its restriction to the basic algebras is an isomorphism of associative algebras. In order to obtain similar result for our Lie algebra, we need some refinement here.
In general, see [4,5,6], for a quantum Poisson algebra D and its basic algebra A (whose unit is denoted by 1), one refers to constants for the elements of the image of the ground field K by the natural embedding k ∈ K → k1 ∈ A. The classical limit of D is the Lie algebra S(D) = ⊕ i∈Z D i /D i−1 with the bracket given by Definition 1. A quantum Poisson algebra D with basic algebra A and classical limit S is called quasi-distinguishing if for any P ∈ S, We denote the centralizer of ad A in Hom K (D, D) by Proposition 3. If D is non-singular and quasi-distinguishing, then 2) any ψ ∈ C(D) preserves the filtration of D ; 3) for any ψ ∈ C(D) and any u ∈ A, ψ(u) = ψ(1)u.
Proof . The proof is similar to the corresponding one in [4,5,6]. We give the main steps in order to show that it is enough to assume that D is quasi-distinguishing.
Using point 1) of the proposition, it follows that ψ(D) ∈ D i+1 .
The above equality reduces to for some invertible element k ∈ A 2 , central in D 2 , and some isomorphism of R-algebras Ψ : Proof . For all D ∈ D 2 and all u ∈ A 1 , w ∈ A 2 , we have
Proof . For any nowhere vanishing u ∈ A 0 (E), the function u −1 : e → 1 u(e) also belongs to A 0 (E). Since Ψ is a homomorphism, This implies that Ψ(u) and Ψ(u −1 ) are both nonvanishing fiberwise constant polynomials since their product is a nonvanishing fiberwise constant polynomial. Now, for any element u ∈ A 0 (E), u 2 + 1 E : e → u(e) 2 + 1 is a nonvanishing element in A 0 (E). It follows that which shows that Ψ(u) must be fiberwise constant.
We have proved Ψ(A 0 (E)) ⊂ A 0 (F ). The conclusion follows from applying the same arguments to the homomorphism Ψ −1 .
Lemma 2. Let {u 1 , u 2 , . . . , u r } ∪ A 0 (E) be a system of generators of the R-algebra A(E). If I ⊂ {1, 2, . . . , r} is the set of indices i for which u i is of order at most 1, then

is again a system of generators of A(E).
Proof . It is obvious that the family (4) still generates A 0 (E) and A 1 (E). Since A 0 (E) and A 1 (E) generate the whole A(E), the lemma follows. Proof . Since degrees 0 and 1 generate the whole R-algebras, it is enough to show The first relation comes from Lemma 1. Let us prove the second one. The C ∞ (M )-module Γ(E * ) of global sections of the dual bundle π * is (projective and) finitely generated. Let {α 1 , α 2 , . . . , α r } be a system of generators. Passing through the isomorphism Γ(E * ) ∼ = A 1 (E) and adding fiberwise constant polynomials, we obtain a set of generators for the R-algebra A(E). Since Ψ is an isomorphism, the images generate the whole A(F ). In view of Lemma 2, this set of generators can be refined into where I ⊂ {1, 2, . . . , r} is the set of indices for which Ψ(u α i ) ∈ A 1 (F ) ⊕ A 0 (F ). Since Ψ −1 is an isomorphism, every element in A 1 (E) can now be decomposed as an R-linear combination of the generating family The conclusion is immediate.

Isomorphisms induce graded isomorphisms
For k ∈ N, denote by pr k the projection of fiberwise polynomial functions onto their homogeneous component of order k. Proposition 6. If Ψ : A(E) → A(F ) is an isomorphism of R-algebras, then for any k ∈ N, the map is a linear bijection.
Proof . First, Ψ k is surjective. Given w ∈ A k (F ), there is an u ∈ A k (E) ⊕ · · · ⊕ A 1 (E) ⊕ A 0 (E) such that w = Ψ(u). Using the fact Ψ is filtered (in view of Proposition 5), it comes Thus u = 0.
is an isomorphism of R-algebras, then the map Ψ : A(E) → A(F ) whose restriction to each A k (E) is given by Ψ k is a graded isomorphism of R-algebras.

A Lie algebraic characterization
After a technical lemma, we will be in a position to establish a new Pursell-Shanks type result for vector bundles, in terms of the Lie algebra generated by differential operators that are eigenvalues of the Lie derivative in the direction of the Euler vector field associated with each vector bundle.
where u i j ∈ A and u i j (a) = 0, Proof . Consider u ∈ A such that j l a (u) = 0 and U a domain of trivialization such that π(a) ∈ U. One has over U, From this, it may be deduced by successive primitivations in [0, t] where we put x = a + th. In the right member of this equality, factors are local, but observe that one can write with α = π * (α ) where α ∈ C ∞ (M, R) equals to 1 in a neighborhood V ⊂ U containing π(a) and with compact support in U. The term α l+1 u becomes and has the form (5). For the other term, one proceeds in the same way by using the fact that R) is of compact support in a neighborhood W ⊂ V containing π(a), and in which it is equal to 1. Proof . We use a most general result stating that the corollary is true for a linear map D : A → A which depends only on 1-jets of its arguments (in the sense that D vanishes on arguments of zero 1-jet). Indeed, this statement results from the fact that for any function f ∈ C ∞ (E, R), any point a ∈ E and any integer k ∈ N, there exists u ∈ A such that j k a (f ) = j k a (u); hence one can write Considering D ∈ Der(A) and u ∈ A such that j 1 a (u) = 0, one has D(u)(a) = 0.
Indeed, with the notations of Lemma 3, one has Therefore D(u) vanishes at a.

The main result
The proof of the main result will rely on a theorem of [7], which we shall recall before stating out main result. As this theorem requires some hypotheses on both the base manifold and the rank of the vector bundles, our result will inherit those hypotheses. Remark 3. Along the way, we have also proved that under the same hypotheses as above, two vector bundles π : E → M and η : F → N are isomorphic if and only if the R-algebras S E (E) and S E (F ) (resp. S 1 E (E) and S 1 E (F )) are isomorphic. Finally, note that without any hypothesis, Corollary 1 and Lemma 1 ensure in view of Milnor's theorem that any isomorphism between D E (E) and D E (F ) induces a diffeomorphism between the base manifolds M and N .