Classical Poisson algebra of a vector bundle : Lie-algebraic characterization
aa r X i v : . [ m a t h . DG ] A ug Classical Poisson algebra of a vectorbundle : Lie-algebraic characterization
Lecomte P.B.A. and Zihindula Mushengezi E.September 1, 2020
Abstract
We prove that the Lie algebra S ( P ( E, M )) of symbols of linearoperators acting on smooth sections of a vector bundle E → M, char-acterizes it. To obtain this, we assume that S ( P ( E, M )) is seen as C ∞ ( M ) − module and that the vector bundle is of rank n > . We improve this result for the Lie algebra S ( P ( E, M )) of sym-bols of first-order linear operators. We obtain a Lie algebraic charac-terization of vector bundles with S ( P ( E, M )) without the hypothe-sis of being seen as a C ∞ ( M ) − module. We know that the space of all differential operators acting on the sectionsof a vector bundle is a quantum Poisson algebra as is the case for differen-tial operators acting on smooth functions. See in [13] for instance.But for differential operators acting on sections, as we will see in this arti-cle, the study of symbols gives rise to two interpretations: • The usual principal symbol linked to the order of derivation • The symbol related to the filtration of the structure of quantum Pois-son algebra.This particularity is part of the richness of the Poisson algebra structureof the symbols of these differential operators and gives rise to remarkabledevelopments. 1et E → M be a vector bundle of rank n. Denoting by Γ( E ) the space ofsmooth sections of E, let us consider the Lie algebra of linear operators ofthis vector bundle defined by D ( E, M ) = [ k ≥ D k ( E, M ) , where A ( E, M ) := D ( E, M ) = { T ∈ End (Γ( E )) : [ T, γ u ] = 0 , ∀ u ∈ C ∞ ( M ) } and for any integer k ≥ , D k ( E, M ) = { T ∈ End (Γ( E )) |∀ u ∈ C ∞ ( M ) : [ T, γ u ] ∈ D k − ( E, M ) } . Provided with the previous filtration, the Lie algebra D ( E, M ) is not aquantum Poisson algebra. But we have the result below which we willuse later. Proposition 1.1
The Lie algebra D ( E, M ) is quasi-distinguishing; that is, therelations1. ([ T, Z ( A )] = 0) ⇒ T ∈ A ( E, M ) { T ∈ D ( E, M ) : [
T, Z ( A )] ⊂ D k ( E, M ) } = D k +1 ( E, M ) , ∀ k ∈ N are both satisfied, where Z ( A ) = { γ u , u ∈ C ∞ ( M )) } is the center of the associa-tive algebra A ( E, M ) . In the rest of this article, we have assumed that the rank n of the vectorbundle E exceeds 1. In fact, Graboswski and Poncin established in [5] thatfor n = 1 , the Lie algebras D ( E, M ) and D ( M ) , the last algebra being thatof all differential operators acting on C ∞ ( M ) , are isomorphic. It is clearthat this does not allow us to consider any characterization of a vectorbundle with such a Lie algebra.In [13] we have obtained Lie-algebraic characterization with D ( E, M ) , forvector bundles of rank greater than 1. We have also provided this Lie al-gebra with appropriated filtration in order to make it a quantum Poissonalgebra.With this new filtration, D ( E, M ) becomes P ( E, M ) and we can now de-fine the space of symbols of differential operators in P ( E, M ) , as in [4, 3, 5],for instance.This is what we discuss in the following lines.2 The classical Poisson algebra S ( P ( E, M )) Let E → M be a vector bundle of rank n > . The space Γ( E ) of smoothsections being a C ∞ ( M ) − module, let us recall that γ u : Γ( E ) → Γ( E ) : s us, ∀ u ∈ C ∞ ( M ) is an endomorphism of the space Γ( E ) . We pose P ( E, M ) = [ k ≥ P k ( E, M ) , with, by definition, P ( E, M ) = { γ u : u ∈ C ∞ ( M ) } and, for any integer k, P k +1 ( E, M ) = { T ∈ End (Γ( E )) |∀ u ∈ C ∞ ( M ) : [ T, γ u ] ∈ P k ( E, M ) } . The following results are taken from [13].
Proposition 2.1
For any integer j, k ∈ N , we have1. P k ( E, M ) ⊂ P k +1 ( E, M ) and P j ( E, M ) · P k ( E, M ) ⊂ P j + k ( E, M ) [ P j ( E, M ) , P k ( E, M )] ⊂ P j + k − ( E, M ) The above Proposition 2.1 means the Lie algebra P ( E, M ) is a quantumPoisson one. Proposition 2.2
The quantum Poisson algebra P ( E, M ) satisfies the followingproperties.1. [ T, γ u ] = 0 , ∀ u ∈ C ∞ ( M ) ⇒ T ∈ gl ( E ) ⊂ P ( E, M ) { T ∈ P ( E, M ) | [ T, P ( E, M )] ⊂ P k ( E, M ) } = P k +1 ( E, M ) , ∀ k ∈ N The first property above is the non-singularity of P ( E, M ) . This last al-gebra is also sympletic; this meaning that its center Z ( P ( E, M )) containsonly constants, i.e, multiplication by elements of R . We also have the following surprising equality D ( E, M ) = P ( E, M ) . In what follows, we study the classical limit of the quantum Poisson alge-bra P ( E, M ) . S ( P ( E, M )) = M i ∈ Z S i ( P ( E, M )); with S i ( P ( E, M )) = P i ( E, M ) / P i − ( E, M ) . We obtain a classical Poissonalgebra whose operations are given in the following.Let us recall that for any T ∈ P i ( E, M ) , ord ( T ) = i, if T / ∈ P i − ( E, M ) . For i ≥ ord ( T ) , the i -degree symbol of T is defined by σ i ( T ) = (cid:26) if i > ord ( T ) T + P i − ( E, M ) if i = ord ( T ) The symbol related to the quantum Poisson structure of P ( E, M ) is for itspart given by σ pson : P ( E, M ) → S ( P ( E, M )) : T σ ord ( T ) . For P ∈ S i ( P ( E, M )) and Q ∈ S j ( P ( E, M )) such that P = σ i ( T ) and Q = σ j ( D ) , we set, by definition, P.Q = σ i + j ( T ◦ D ) and { P, Q } = σ i + j − ([ T, D ]) · gl ( E ) ⊂P ( E, M ) By virtue of the calculations made in the previous section, we have, bydefinition, σ pson ( γ u ) = γ u + { } , ∀ u ∈ C ∞ ( M ) and we will simply denote σ pson ( γ u ) = γ u . Likewise, for A ∈ gl ( E ) \ P ( E, M ) , we have σ pson ( A ) = A ′ + P ( E, M ) , with A ′ = A − tr ( A ) n id, tr ( A ) being the trace of A. Therefore, for any
A, B ∈ gl ( E ) , we have σ pson ([ A, B ]) = [
A, B ] + P ( E, M )= [ A ′ , B ′ ] + P ( E, M ) . A, B / ∈ P ( E, M ) , we have σ pson ( A ) · σ pson ( B ) = 0 , but if γ u ∈ P ( E, M ) , we then have σ pson ( γ u ) · σ pson ( A ) = γ u ◦ A ′ + P ( E, M ) . We therefore have the following identification of Lie algebras σ pson ( gl ( E )) ∼ = sl ( E ) ⊕ C ∞ ( M ) id, where the multiplication is commutative and defined by ( A + γ u ) · ( B + γ v ) = γ v ◦ A + γ u ◦ B + γ uv ; and the bracket being given by the following relation { A + γ u , B + γ v } = [ A, B ] . Let us state the following result which gives the local expression of theelements of P ( E, M ) in a trivialization of E. Sometimes, for convenience of writing, we simply denote γ u by u ∈ C ∞ ( M ) . Proposition 4.1
The elements of P k ( E, M ) , k ≥ , are characterized by the factthat they are written locally, in a trivialization domain U ⊂ M, in the form X | α | The proof is done by induction on k . Let T ∈ P ( E, M ) and supposethat in a domain of trivialization U of E we have T = A + m X i T i ∂ i , A, T i ∈ C ∞ ( U, gl ( n, R )) The relation [ T, γ u ] ∈ P ( E, M ) gives m X i =1 T i ◦ ∂ i ( u ) ∈ C ∞ ( U ) , ∀ u ∈ C ∞ ( M ) . i ∈ [1 , m ] ∩ N , we do have T i ∈ C ∞ ( U ) . Assume by induction that the result is true for any element of P r ( E, M ) with r < k. Let T ∈ P k ( E, M ) such that we locally have T = X | α | The application θ is well-defined. Indeed, the differential operators D , D ∈ D k − ( E, M ) ⊂ P k ( E, M ) are such that σ pson ( D ) = σ pson ( D ) , wethen do have D − D ∈ P k − ( E, M ); which means that the image of T does not depend on the choice of the operator T such that σ pson ( T ) = T. Also, θ is obviously a linear map and it is injective. Indeed, let T ∈ S k − ( M ) ⊗ sl ( E ) such that θ ( T ) = 0 . We then have T ∈ P k − ( E, M ) . But byconstruction T / ∈ D k − ( E, M ) . We thus have σ ppal ( T ) ∈ S k − ( M ) id We deduce, since σ pson ( T ) = σ ppal ( T ) = T, that T = 0 . The map δ being linear and directly surjective, we show to finish that ker ( δ ) = Im ( θ ) . The inclusion ker ( δ ) ⊃ Im ( θ ) is obvious. Let’s prove the other sense ofthat inclusion. If D + P k − ( E, M ) ∈ ker ( δ ) , then, by the definition of δ, wehave D ∈ D k − ( E, M ) ∩ P k ( E, M ) . Consequently, on the one hand, σ ppal ( D ) ∈ S k − ( M ) ⊗ gl ( E ) , because D ∈ D k − ( E, M ) and on the other, like D ∈ P k ( E, M ) , we haveinstead, σ ppal ( D ) ∈ S k − ( M ) ⊗ sl ( E ) . The inclusion sought is a direct result of this. (cid:4) R − vector spaces, we have the following decomposition S k ( P ( E, M )) = P ol k − ( T ∗ M, sl ( E )) ⊕ P ol k ( T ∗ M, R ) for any integer k ∈ N . The question that arises is whether the following exact sequence of Liealgebras (but also of associative algebras), whose exactness comes fromthat given in the previous statement and from the operations previouslyperformed in (4.3) and in (4.4) is split. / / S ( M ) ⊗ sl ( E ) / / S ( P ( E, M )) / / S ( M ) / / (4.5)Note that the splitting of this sequence leads in particular to that of thefollowing exact sequence of Lie algebras / / sl ( E ) / / S ( P ( E, M )) / / V ect ( M ) / / To answer it, we make use of the following result where a split sequenceof Lie algebras is given. Proposition 4.3 Let E → M be vector bundle of rank n. With respect to a con-nection on E, the following short exact sequence of Lie algebras is split / / P ( E, M ) i −→ P ( E, M ) σ pson −→ P ( E, M ) / P ( E, M ) / / where i is the canonical injection and σ pson the map previously defined. Proof. Via a covariant derivation ∇ of E, we have the identification of R − vector spaces P ( E, M ) ∼ = V ect ( M ) ⊕ gl ( E ) . Indeed, for T ∈ P ( E, M ) ⊂ D ( E, M ) , we have σ ppal ( T ) = X ∈ V ect ( M ) , with the identification S ( M ) id ≃ V ect ( M ) . Therefore, since ∇ X ∈ P ( E, M ) , the difference T − ∇ X is an endomor-phisms field.We denote by λ : P ( E, M ) → V ect ( M ) ⊕ gl ( E ) the linear bijection therebydefined. If we have T = ∇ X + A and D = ∇ Y + B, we then get in P ( E, M )[ T, D ] = ∇ [ X,Y ] + R ∇ ( X, Y ) + ∇ X B − ∇ Y A + [ A, B ] · [( X, A ) , ( Y, B )] = ([ X, Y ] , R ∇ ( X, Y ) + ∇ X B − ∇ Y A + [ A, B ]) (4.6)in the space V ect ( M ) ⊕ gl ( E ) . Moreover, consider the following short exact sequence / / sl ( E ) / / P ( E, M ) / P ( E, M ) / / V ect ( M ) / / corresponding to the particular case k = 1 of that characterizing the spaceof symbols, in the sense "Quantum Poisson", of the differential operatorsof order k , that we gave at the beginning of this section.Thus, as in the general case at the beginning of this section, we have thelinear map δ : T + P ( E ) (cid:26) if T ∈ gl ( E ) σ ppal ( T ) if notwhich is surjective and the injection θ : A ∈ sl ( E ) A + P ( E, M ) . Seen as R − vector spaces, we therefore have the following identification P ( E, M ) / P ( E, M ) ∼ = V ect ( M ) ⊕ sl ( E ) . Consider the following commutative diagram ( X, A ) ∈ V ect ( M ) ⊕ sl ( E ) λ − (cid:15) (cid:15) µ / / P ( E, M ) / P ( E, M ) ∇ X + A ∈ P ( E, M ) σ pson ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ The linear map µ = σ pson ◦ λ − is injective because µ ( X, A ) = 0 induces ∇ X + A ∈ P ( E ); and we deduce that X = 0 and A ∈ sl ( E ) ∩ P ( E, M ) . Likewise, µ is surjective.The bracket in P ( E, M ) / P ( E, M ) is given by [[ T ] , [ D ]] = ∇ [ X,Y ] + R ∇ ( X, Y ) + ∇ X B − ∇ Y A + [ A, B ] + P ( E, M ) with [ T ] = ∇ X + A + P ( E, M ) and [ D ] = ∇ Y + B + P ( E, M ) . Therefore, the corresponding operation in V ect ( M ) ⊕ sl ( E ) , obtained bystructure transport via µ, is not necessarily a Lie bracket since the term R ∇ ( X, Y ) is not always of zero trace.10o remedy this, suppose that ∇ is associated with a connection form of areduction of the frame principal bundle L ( E ) of E to the Lie subgroup O ( n ) of GL ( n, R ) . For such a derivation, R ∇ has values in sl ( E ) . We then have an isomorphism of Lie algebras µ : V ect ( M ) ⊕ sl ( E ) → P ( E, M ) / P ( E, M ) , the space V ect ( M ) ⊕ sl ( E ) being provided with the following bracket [( X, A ) , ( Y, A )] = ([ X, Y ] , R ∇ ( X, Y ) + ∇ X B − ∇ Y A + [ A, B ]) . (4.7)Given the relations (4.6) and (4.7), we conclude that the canonical injection β : V ect ( M ) ⊕ sl ( E ) → V ect ( M ) ⊕ gl ( E ) is a homomorphism of Lie algebras. Consequently, λ − ◦ β ◦ µ − : P ( E, M ) / P ( E, M ) → P ( E, M ) is a homomorphism of Lie algebras allowing to identify the Lie algebra P ( E, M ) / P ( E, M ) to a Lie subalgebra of P ( E, M ) , for the structuresspecified in the previous lines, and we can see that it is a section of σ pson . We have just shown that the short exact sequence of the statement is split. (cid:4) Note that if ∇ is a covariant derivation of E associated with the reductionin question in the previous proof, then for T = ∇ X + A, the followingdecomposition T = ( ∇ X + A − n tr ( A ) id ) + 1 n tr ( A ) id · only depends on the reduction and not on the choice of connection.Indeed, if relatively to another covariant derivation ∇ ′ , associated withthe same reduction, we consider an analogous decomposition of T, thenwe have T = ∇ ′ X + A ′ = ∇ X + ( A ′ + ( ∇ ′ X − ∇ X ))= ∇ X + S + A ′ with S = ∇ X − ∇ ′ X , and thus the trace of S is null. We conclude that A ′ = A − S .Therefore, tr ( A ) = tr ( A ′ ) and we have ∇ ′ X + A ′ − n tr ( A ′ ) id = ∇ X + S + A − S − n tr ( A ) id. (cid:15) (cid:15) (cid:15) (cid:15) gl ( E ) (cid:15) (cid:15) sl ( E ) (cid:15) (cid:15) / / P ( E, M ) / / P ( E, M ) / / (cid:15) (cid:15) P ( E, M ) / P ( E, M ) / / (cid:15) (cid:15) V ect ( M ) (cid:15) (cid:15) V ect ( M ) (cid:15) (cid:15) We have established that the horizontal sequence is split. And we deducethat the split of the vertical sequence on the right would lead to that of thevertical sequence located on the left. Now according to [11], the splittingof this sequence essentially requires the naturality of the vector bundle E. Note that a vector bundle E → M is necessarily natural if the base M issimply connected.Thus, the answer to the question of whether the exact sequence of Lie al-gebras given in (4 . is always split is negative. Let us begin by stating results of Lie-algebraic characterization of vectorbundles taken from [9]. Theorem 5.1 Let E M and F M be two vector bundles of respectiveranks n, n ′ > with H ( M, Z / 2) = 0 . The Lie algebras gl ( E ) and gl ( F ) (resp. sl ( E ) and sl ( F ) ) are isomorphic if and only if the vector bundles E and F areisomorphic. We use the above Theorem 5.1 to obtain the following result. Theorem 5.2 Let E → M, F → M be two vector bundles of respective ranks n, n ′ > with H ( M, Z / 2) = 0 . The Lie algebras S ( P ( E, M )) and S ( P ( F, M )) , seen as C ∞ ( M ) − modules, are isomorphic if, and only if, the vector bundles E and F are. roof. Let Φ : S ( P ( E, M )) → S ( P ( F, M )) be an isomorphism of Lie alge-bras. Note that Φ preserves A = C ∞ ( M ) the basis of these classical Poissonalgebras. Indeed, for any u ∈ C ∞ ( M ) , we have Φ( γ u ) = u Φ( γ ) and the conclusion comes from the fact that the quantum Poisson algebrasconsidered are symplectic.Let B ∈ gl ( E ) . We then have { B, A} = 0 . Therefore { Φ( B ) , A} = 0 and this implies Φ( B ) ∈ gl ( E ) , since D ( E, M ) isquasi-distinguishing, according to the Proposition 1.1. (cid:4) For the Lie subalgebras S ( P ( E, M )) and S ( P ( F, N )) , this result may im-prove. To prove this, we use the short exact sequence presented in theprevious Proposition 4.3. Theorem 5.3 Let E → M, F → M be two vector bundles of respective ranks n, n ′ > with H ( M, Z / 2) = 0 . The Lie algebras S ( P ( E, M )) and S ( P ( F, M )) , are isomorphic if, and only if, the vector bundles E and F are. Proof. We observe, starting from the decomposition S ( P ( E, M )) = sl ( E ) ⊕ V ect ( M ) obtained previously via a connection on E. For all T = ( ∇ X , A ) , let B ∈ sl ( E ) such that there exists r ∈ N verifying ( ad ( T )) r ( B ) = 0 . We then have ∇ X ( ∇ X · · · ( ∇ X ( B ))) = 0 , where ∇ X is r times applied. In a trivialization of E of domain U ⊂ M, considering B whose local expression is of the form ( α ij ) = ( δ u ) , u ∈ C ∞ ( U ) , i.e. having all its terms null except that of the position (1 , , wecan choose u so that we necessarily have X = 0 . We deduce that N il ( S ( P ( E )) ⊂ sl ( E ) . A ∈ sl ( E ) is such that A p = 0 , we then have A ∈ N il ( S ( P ( E, M )) . Indeed, observe that ( adA ) k ( ∇ X + B ) = ( adA ) k ( ∇ X ) + ( adA ) k ( B )= − ( adA ) k − ( X · A ) + ( adA ) k ( B ) where, by virtue of the particular case studied in the previous section 3,for all C, D ∈ sl ( E ) , we have ( ad C )( D ) = { C, D } = [ C, D ] , which allows us to conclude, since ( ad C ) k ( D ) is then a sum of the termsof the form a k C k ◦ D ◦ C k − , a k ∈ R .In the following lines, the goal is to establish that the linear envelope ofthe nilpotent endomorphism fields is the entire space sl ( E ) . Let A ∈ sl ( E ) . Over a trivialization domain U ⊂ M, we can therefore write A | U = Σ i N Ui with N Ui ∈ sl ( E | U ) , (1 ≤ i ≤ n − , which are nilpotent endomorphismfields since sl ( n, R ) admits a basis formed of nilpotent matrices. Considernow a Palais cover of M, O = O ∪ · · · ∪ O r , r ∈ N , locally finite, the elements U α,j of each O j being trivialization domainsof E ( ρ α,j ) , a partition of the unit, locally finite andsubordinate to this cover.We therefore have ρ α,j A = n − X i =1 N i,α,j with N i,α,j ∈ sl ( E | U α,j ) nilpotent and compactly supported in U α,j .Let now pose ∪O j = ∪ α U α,j = U j and consider N ji defined by N ji ( x ) = (cid:26) N i,α,j ( x ) if x ∈ U j if notWe then obtain the smoothness of N ji . Indeed, for any x / ∈ U j , consider anopen neighborhood V ∋ x of compact adherence. We know that V is onlyencountered by a finite number of supports of N α,j , whose reunion is thecompact that we agree to denote by K. V \ K is an open neighborhood of x in which N ji is identicallyzero. And we have thus established our assertion since for x ∈ U j thesmoothness of N ji is obvious. We conclude that X α ρ α,j A = n − X i =1 N ji , and as a result, A = n − X i =1 r X j =1 N ji . We have therefore just shown that i N il ( S ( P ( E, M )) h = sl ( E ) , where the usual notation i H h designates the linear envelope of the subset H of a vector space.We deduce that for any isomorphism Φ : S ( P ( E, M )) → S ( P ( F, N )) of Lie algebras is necessarily such that Φ( sl ( E )) = sl ( F ) . Hence, by virtue of the previous Theorem 5.1, we have the desired result. (cid:4) eferences [1] De Wilde M, Lecomte P, Some Characterizations of Differential operatorson Vector Bundles , In: E.B. Christoffel, Eds: Butzer P, Feher F, BrikhäuserVerlag, Basel (1981),pp. 543-549[2] Grabowski J, Isomorphisms of algebras of smooth functionsrevisited, Archiv Math. (to appear) (electronic version athttp://arXiv.org/abs/math.DG/0310295)[3] Grabowski J, Poncin N, Automorphisms of quantum and classical Poissonalgebras , Comp. Math., (2004), pp. 511-527[4] Grabowski J, Poncin N, Lie-algebraic characterizations of manifolds , Cen-tral Europ. 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