BY 4.0 license Open Access Pre-published online by De Gruyter September 25, 2021

Fiberwise linear differential operators

Fabrizio Pugliese, Giovanni Sparano and Luca Vitagliano ORCID logo
From the journal Forum Mathematicum

Abstract

We define a new notion of fiberwise linear differential operator on the total space of a vector bundle E. Our main result is that fiberwise linear differential operators on E are equivalent to (polynomial) derivations of an appropriate line bundle over E. We believe this might represent a first step towards a definition of multiplicative (resp. infinitesimally multiplicative) differential operators on a Lie groupoid (resp. a Lie algebroid). We also discuss the linearization of a differential operator around a submanifold.

MSC 2010: 58A99; 53B99; 53C99

1 Introduction

Given a vector bundle EM, it is often interesting to look at geometric structures (functions, vector fields, differential forms etc.) on the total space E that satisfy appropriate compatibility conditions with the vector bundle structure. Such compatibility is often referred to as linearity in the literature. Accordingly, one speaks about linear functions, linear vector fields, linear differential forms etc., on the total space of a vector bundle. In the present paper, we will rather use the terminology “fiberwise linearity”, to avoid confusion with other types of linearities. Now, the vector bundle structure is completely determined by (the smooth structure on E and) the action h:×EE of the monoid (,) of multiplicative reals by fiberwise scalar multiplication: h(t,e)=te for all t and all eE (see, e.g., [4]). It follows that the fiberwise linearity of a geometric structure can usually be expressed purely in terms of h. For instance, a function f on E is fiberwise linear if and only if htf=tf for all t. Similarly, a vector field X (resp. a differential form ω) on E is fiberwise linear if and only if htX=X for all t0 (resp. htω=tω for all t). A fiberwise linear function is equivalent to a section of the dual vector bundle E, a fiberwise linear vector field is equivalent to a section of the gauge algebroid of E (see, e.g., [9]), and a fiberwise linear differential 1-form is equivalent to a section of the first jet bundle J1EM. The latter examples already show that fiberwise linear structures on E can encode interesting geometric structures on (vector bundles over) M. There are even more interesting examples. A fiberwise linear symplectic structure ω on E is equivalent to a vector bundle isomorphism ETM. To see this, notice that ω, together with the standard direct sum decomposition TE|METM of the tangent bundle of E restricted to the zero section, gives rise to a bilinear map E×MTM. As ω is non-degenerate, the induced map Iω:ETM is a vector bundle isomorphism. From closedness and fiberwise linearity, ω=IωωM, the pull-back along Iω of the canonical 2-form ωM on TM. It is clear that Iω is the unique vector bundle isomorphism with this property (see, e.g., [4], see also [11]).

Another interesting example is the following: a fiberwise linear metric is equivalent to an isomorphism ETM together with a torsion free connection in TM [12].

As a final remarkable example, we recall that a fiberwise linear Poisson structure on E is the same as a Lie algebroid structure on E [9].

In this paper, we propose the following definition of fiberwise linear scalar differential operator. An -linear differential operator Δ:C(E)C(E) of order q is fiberwise linear if htΔ=t1-qΔ for all t0. This definition might seem weird at a first glance. However, it is supported by several different facts. For instance, according to our definition, a function and a vector field are fiberwise linear if and only if they are fiberwise linear when regarded as a 0-th order and a first order scalar differential operator, respectively. Moreover, the principal symbol of a fiberwise linear differential operator is a fiberwise linear symmetric multivector field. Another supporting remark is that the Laplacian (acting on functions) of a fiberwise linear metric is a fiberwise linear differential operator. Finally, a scalar differential operator Δ can be linearized around a submanifold producing a fiberwise linear differential operator representing a first order approximation to Δ in the transverse direction with respect to the submanifold. All these facts suggest that our definition might indeed be the “correct one”. Our main result is a description of fiberwise linear differential operators in terms of somehow simpler data. More precisely, we prove the following theorem (see Theorem 6.7 for a more precise statement).

Theorem.

Let EM be a vector bundle. Then there is a degree inverting C(M)-linear bijection between fiberwise linear scalar differential operators Δ:C(E)C(E) and polynomial derivations of the line bundle E×MtopEE.

This theorem is a little surprising because it describes objects of higher order in derivatives (fiberwise linear differential operators) in terms of objects of order 1 in derivatives (derivations of an appropriate vector bundle). We hope that this result might be the starting point of a more thorough investigation of multiplicative differential operators on Lie groupoids and, at the infinitesimal level, infinitesimally multiplicative differential operators on Lie algebroids. Multiplicative (resp. infinitesimally multiplicative) structures are geometric structures on a Lie groupoid (resp. Lie algebroid) which are additionally compatible with the groupoid (resp. algebroid) structure. In the last thirty years, starting from the pioneering works of Weinstein on symplectic groupoids [13], multiplicative structures captured the interest of a large community of people working in Poisson geometry and related fields, and today we have a precise description of several different multiplicative structures and their infinitesimal counterparts: infinitesimally multiplicative structures (see [7] for a survey). However, all examples investigated so far are of order 1 in derivatives and it would be interesting to investigate the compatibility of a Lie groupoid/algebroid with structures of higher order in derivatives, e.g., higher order differential operators. This is a natural issue that might conjecturally lead to new important developments. As infinitesimally multiplicative structures are, in particular, fiberwise linear structures, this paper might be also considered as a first step in this direction.

The paper is organized as follows. In Section 2, we recall what it means for a vector field on the total space of a vector bundle EM to be fiberwise linear, i.e. compatible with the vector bundle structure. In Section 3, we discuss fiberwise linear symmetric multivector fields and we describe them in terms of simpler data. This material is well known to experts (although it is scattered in the literature and it is hard to find a universal reference) and Sections 2 and 3 are mainly intended to fix our notation. In Section 4, we recall what a derivation of a vector bundle E is and introduce what we call E-multivectors, a “derivation analogue” of plain multivector fields. We also discuss fiberwise linear E-multivectors. These objects are not exactly of our primary interest, but they play a very useful role in the proofs of our main theorems (Theorems 6.7 and 7.5). To the best of our knowledge, the material in Section 4 is mostly new. Section 5 is an extremely compact introduction to linear differential operators on vector bundles, and, in particular, scalar differential operators. Section 6 contains our main constructions and results: we define and study fiberwise linear (scalar) differential operators on the total space of a vector bundle E. Somehow surprisingly, fiberwise linear differential operators on E form a transitive Lie–Rinehart algebra over fiberwise polynomial functions on E, with abelian isotropies (Theorem 6.4). The reason is ultimately explained by our main result, the theorem above (see also Theorem 6.7 below). As already announced, E-multivectors play a prominent role in the proof. In Section 7, we discuss the linearization of a scalar differential operator Δ around a submanifold M in a larger manifold. The linearization of Δ is a fiberwise linear differential operator on the total space of the normal bundle to M, and can be seen as a first order approximation to Δ in the direction transverse to M. The existence of a linearization construction strongly supports our definition of fiberwise linear differential operators.

We will adopt systematically the Einstein convention on the sum of repeated upper-lower indexes. We will not adopt the Einstein convention for sums over (repeated) multi-indexes.

2 Core and linear vector fields on a vector bundle

As we mentioned in Section 1, the main aim of this paper is to explain what it means for a differential operator on the total space E of a vector bundle EM to be compatible with the vector bundle structure. We will reach our definition (Definition 6.1) by stages. We first need to recall what it means for a function, a vector field and, more generally, a multivector field on E, to be compatible with the vector bundle structure. We do this in the present and the next section. We adopt the general philosophy of [4, 5] where it is shown that a vector bundle structure is encoded in the fiberwise scalar multiplication, and compatibility with the vector bundle structure is expressed in terms of such multiplication.

So, let π:EM be a vector bundle. The fiberwise scalar multiplication by a real number

h:×EE

is an action of the multiplicative monoid of reals (,). The algebra Cpoly(E) of fiberwise polynomial functions on the total space E is non-negatively graded:

Cpoly(E)=k=0C(E)k,

and its k-th homogeneous piece C(E)k consists of homogeneous polynomial functions of degree k, i.e. functions fC(E) such that

ht(f)=tkf

for all t. Functions in C(E)0 are just (pull-backs via the projection π:EM of) functions on M. We call them core functions and also denote them by Ccore(E). They form a subalgebra in Cpoly(E). Functions in C(E)1 are fiberwise linear (FWL for short in what follows) functions, and identify naturally with sections of the dual vector bundle E. We denote them by Clin(E). They form a Ccore(E)-submodule in Cpoly(E). We denote by φ the linear function corresponding to the section φΓ(E). The terminology “core function” (and similarly “core vector field” etc., see below) is motivated by the theory of double vector bundles, where “core sections” are sections with an appropriate degree with respect to certain actions of (,) (see, e.g., [9]).

A vector field X𝔛(E) on E is fiberwise polynomial, or simply polynomial, if it maps (fiberwise) polynomial functions to polynomial functions. Polynomial vector fields 𝔛poly(E) form a (graded) Lie–Rinehart algebra over polynomial functions Cpoly(E):

𝔛poly(E)=k=-1𝔛(E)k.

We recall for later purposes that a Lie–Rinehart algebra over a commutative algebra A is a vector space L, which is both an A-module and a Lie algebra acting on A by derivations with the following two compatibilities:

  • The Lie algebra action map ρ:LDerA is A-linear (it is often called the anchor).

  • The Lie bracket [-,-]:L×LL is a bi-derivation, i.e. it satisfies the following Leibniz rule:

    [λ,aμ]=ρ(λ)(a)μ+a[λ,μ],λ,μL,aA.

Lie–Rinehart algebras are purely algebraic counterparts of Lie algebroids. For more on Lie–Rinehart algebras, see, e.g., [6] and the references therein.

Coming back to polynomial vector fields, the k-th homogeneous piece 𝔛(E)k of 𝔛poly(E) consists of homogeneous polynomial vector fields of degree k, i.e. vector fields X𝔛(E) such that

ht(X)=tkX

for all t0. Vector fields in 𝔛(E)-1 are vertical lifts of sections of E. We call them core vector fields and also denote them by 𝔛core(E). They form an abelian Lie–Rinehart subalgebra (note: over the subalgebra Ccore(E)=C(M)) in 𝔛poly(E). We denote by

e𝔛(E)

the vertical lift of a section eΓ(E).

Vector fields in 𝔛(E)0 are, by definition, fiberwise linear (FWL) vector fields. They can be equivalently characterized as vector fields preserving linear functions and they satisfy the following property:

[X,Y]𝔛core(E)for all Y𝔛core(E).

We denote FWL vector fields by 𝔛lin(E). They form a Lie–Rinehart subalgebra (over Ccore(E)) in 𝔛poly(E).

If (xi,uα) are vector bundle coordinates, then a function fC(E) is a core function if and only if, locally, f=f(x) and it is a linear function if and only if, locally, f=fα(x)uα. Similarly, a vector field X𝔛(E) is a core vector field if and only if, locally,

X=Xα(x)uα,

and it is a linear vector field if and only if, locally,

X=Xi(x)xi+Xβα(x)uβuα.

There is also a useful notion of FWL tensor on E. Let 𝒯Γ(TrETEs) be a tensor field of type (r,s).

Definition 2.1.

Then 𝒯 is a FWL tensor if

ht𝒯=t1-r𝒯

for all t0.

Notice that FWL tensor fields are called linear tensor fields in [1], where they are characterized in terms of the fiberwise addition in E (rather than via the fiberwise multiplication h as we do).

Example 2.2.

As an instance, consider a metric gΓ(S2TE). It is linear if htg=tg for all t, and it is easy to see that this is in turn equivalent to g being locally of the form

g=gαi(x)duαdxi+gα|ij(x)uαdxidxj.

Notice that the non-degeneracy condition then implies that the x-dependent matrix (gαi) is invertible. In particular, the dimension of M and the rank of E must agree. Even more, by denoting by TπE=kerdπ the π-vertical bundle, the composition

ETπE|MTE|MTE|MTM

is a vector bundle isomorphism. Here :TETE is the musical isomorphism, the second and the fourth arrow are those induced by the canonical direct sum decomposition, TE|M=TMTπE|M, and the first arrow is the canonical isomorphism. In other words, a non-degenerate symmetric covariant 2-tensor g can only exist on the total space of (a vector bundle isomorphic to) the cotangent bundle. Finally, in standard coordinates (xi,pi) on TM, g looks like

g=dpidxi-Γijk(x)pkdxidxj

for some appropriate local functions Γijk(x). In particular, g is necessarily of split signature. For more on FWL metrics, see [12].

3 More on FWL multivector fields

The material in this section is well known to experts, and it is partly folklore, partly scattered in the literature. For this reason, it is hard to give precise references (the reader may consult, e.g., [8, Appendix A] and the references therein, although that reference does not cover the same exact material as the present one). In any case, most of the proofs are straightforward and we omit them.

We will need to consider FWL symmetric multivector fields. According to Definition 2.1, a k-multivector field P on the total space E of a vector bundle EM is FWL if

ht(P)=t1-kP

for all t0. We denote by 𝔛sym,lin(E) FWL symmetric multivector fields.

There is a useful characterization of FWL k-multivector fields. Namely, a k-multivector field P on E is FWL if and only if

P(f1,,fk)Clin(E),
P(f1,,fk-1,h1)Ccore(E),
P(f1,,fk-2,h1,h2)=0

for all fiClin(E) and all hjCcore(E). In particular, an FWL symmetric k-multivector field P determines a pair of maps (DP,lP):

DP:Γ(E)××Γ(E)k timesΓ(E)

and

lP:Γ(E)××Γ(E)k-1 times×C(M)C(M)

via

DP(φ1,,φk)=P(φ1,,φk),
lP(φ1,,φk-1,f)=P(φ1,,φk-1,f)

for all φiΓ(E) and all fC(M). The maps DP,lP satisfy the following properties:

  1. (i)

    DP is -multilinear and symmetric.

  2. (ii)

    lP is C(M)-multilinear and symmetric in the first (k-1)-arguments.

  3. (iii)

    One has

    DP(φ1,,φk-1,fφk)=fDP(φ1,,φk)+lP(φ1,,φk-1,f)φkfor φiΓ(E) and fC(M).
  4. (iv)

    lP is a derivation in its last argument.

In particular, lP can be seen as a vector bundle map lP:Sk-1ETM, and we will often write

lP(φ1,,φk-1)(f)

instead of lP(φ1,,φk-1,f). The assignment P(DP,lP) establishes a C(M)-linear bijection between FWL symmetric k-multivector fields on E and k-multiderivations of E, i.e. pairs (D,l) consisting of a map

D:Γ(E)××Γ(E)Γ(E)

and a vector bundle map l:Sk-1ETM (equivalently, a section of Sk-1ETM) satisfying

D(φ1,,φk-1,fφk)=fD(φ1,,φk)+l(φ1,,φk-1)(f)φk.

The map l is sometimes called the symbol of D and it is completely determined by D. For this reason, we will often refer to D itself as a k-multiderivation (see, e.g., [3, 2] for a skew-symmetric version of multiderivations).

Recall that there is a natural Poisson bracket {-,-} on symmetric multivector fields given by the following (Gerstenhaber-type) formula:

{P1,P2}(f1,,fk1+k2+1)=σSk2+1,k1P1(P2(fσ(1),,fσ(k2+1)),fσ(k2+2),,fσ(k1+k2+1))
(3.1)-σSk1+1,k2P2(P1(fσ(1),,fσ(k1+1)),fσ(k1+2),,fσ(k1+k2+1))

for all (k1+1)-multivector fields P1, all (k2+1)-multivector fields P2, and all functions fi, where Sk,h denotes (k,h)-unshuffles. The Poisson bracket (3.1) preserves FWL symmetric multivector fields, and the Poisson bracket {P1,P2} of FWL Poisson multivector fields P1,P2𝔛sym,lin(E) identifies with the obvious Gerstenhaber-like bracket {D1,D2} of the associated multiderivations D1,D2.

FWL symmetric multivector fields on E do also identify with polynomial vector fields on E. To see this, it is useful to talk about core multivector fields first. A k-multivector field P on E is core if

ht(P)=t-kP

for all t0. Core k-multivector fields can be characterized as those multivector fields P such that

P(f1,,fk)Ccore(E)=C(M),
P(f1,,fk,h)=0

for all fiClin(E) and hCcore(E), and they form a subalgebra 𝔛sym,core(E) in the associative, commutative algebra 𝔛sym(E) (with the symmetric product). More precisely, 𝔛sym,core(E) is the subalgebra spanned by core functions and core vector fields. In particular, 𝔛sym,core(E) identifies with sections Γ(SE) of the symmetric algebra of E via

(e1)(ek)e1ek,

with eiΓ(E) (recall that e𝔛core(E) denotes the vertical lift of a section eΓ(E)). In its turn, Γ(SE) identifies with polynomial functions on E via the (degree preserving) algebra isomorphism

Γ(SE)Cpoly(E),e1eqe1eq,

and, in what follows, we will often understand the latter identifications. Notice that the resulting isomorphism

𝔛sym,core(E)Cpoly(E),PFP,

is given by

(3.2)FP(φx)=1k!P(φ,,φ)(x),

where P𝔛sym,corek(E), φΓ(E) and xM.

We can now go back to FWL symmetric multivector fields. The symmetric product of a core symmetric multivector field and an FWL one is an FWL multivector field, and this turns 𝔛sym,lin(E) into a Γ(SE)-module. Now, let Q𝔛sym,lin(E). It is easy to see that the Poisson bracket HQ={Q,-} preserves core multivector fields. Hence, it is a derivation of the commutative algebra 𝔛sym,core(E)Cpoly(E). In its turn, HQ extends uniquely to a polynomial vector field, also denoted by HQ, on E. The assignment

𝔛sym,lin(E)𝔛poly(E),QHQ={Q,-},

establishes a degree inverting isomorphism of Lie algebras, between the Lie algebra of linear symmetric multivector fields on E (with the Poisson bracket) and polynomial vector fields on E (with the commutator). When we equip 𝔛sym,lin(E) with the symmetric product by a core multivector field, the latter isomorphism becomes an isomorphism of Lie–Rinehart algebras. We summarize and complement the above discussion on core and FWL multivector fields with the following proposition.

Proposition 3.1.

Let EM be a vector bundle. The map

𝔛sym,core(E)Cpoly(E),PFP,

given by (3.2) is a degree inverting isomorphism of (graded) commutative algebras, and the map

𝔛sym,lin(E)𝔛poly(E),QHQ={Q,-},

is a degree inverting isomorphism of (graded) Lie–Rinehart algebras over Xsym,core(E)Cpoly(E).

Let (xi,uα) be vector bundle coordinates on E and let (xi,uα) be the corresponding dual vector bundle coordinates on E. Then every core symmetric q-multivector field PXsym,core(E) is locally of the form

P=Pα1αq(x)uα1uαq

and we have

FP=Pα1αq(x)uα1uαq.

Additionally, every FWL symmetric q-multivector field QXsym,lin(E) is locally of the form

Q=Qi|α1αq-1(x)xiuα1uαq-1+Qαβ1βq(x)uαuβ1uβq

and we have

HQ=Qi|α1αq-1(x)uα1uαq-1xi-Qαβ1βq(x)uβ1uβquα.

Proof.

The proof of those parts of the statement that have not been already discussed is straightforward, and we leave the easy details to the reader. ∎

Finally, we remark that linear symmetric multivector fields fit in the following short exact sequence:

(3.3)0Γ(SEE)𝔛sym,lin(E)𝑙Γ(S-1ETM)0,

where the second arrow identifies the section e1ekφ of SkEE with the FWL k-multivector field φe1ek.

4 More on derivations of a vector bundle

In this section, for a vector bundle VM, we introduce a notion of (symmetric) V-multivector (Definition 4.1). To the best of our knowledge, this notion is new. It will play a significant role in the description of fiberwise linear differential operators provided in Section 6. Symmetric V-multivectors are in many respects similar to plain symmetric multivector fields, so the proofs of most of the statements in this section parallel the proofs of the analogous statements for multivector fields, and we usually omit them.

We begin with a vector bundle EM and remark that the space 𝔛lin1(E)=𝔛lin(E) of linear vector fields is of particular interest. The assignment XDX establishes an isomorphism of Lie–Rinehart algebras (over C(M)) between linear vector fields on E and derivations of E, i.e. 1-multiderivations. We stress that

X(φ)=DXφ,φΓ(E).

In the following, we denote by 𝔇(V) the Lie–Rinehart algebra of derivations of a vector bundle V. It is the Lie–Rinehart algebra of sections of a Lie algebroid DVM whose Lie bracket is the commutator of derivations and whose anchor is the symbol map DlD.

Notice also that the assignment XHX (see Proposition 3.1) does also establish an isomorphism of Lie–Rinehart algebras (over C(M)) between linear vector fields on E and linear vector fields on E. Accordingly, we have a canonical Lie algebroid isomorphism

DEDE,DD,

which is explicitly given by

Dφ,e=lD(φ,e)-φ,De

for every φΓ(E) and eΓ(E), where -,-:EEM:=M× is the duality pairing. In the following, we will simply denote by D the derivation of E induced by a derivation of E (and vice-versa). It is easy to see that

[X,e]=(DXe),eΓ(E).

More generally, a derivation D of a vector bundle V induces a derivation, also denoted by D, in each component of the whole (symmetric, resp. alternating) tensor algebra of VV. The latter derivation is defined imposing the obvious Leibniz rule with respect to the tensor product and the contraction by an element in the dual.

We are now ready to define the algebra of V-multivectors, which is a “derivation analogue” of the Poisson algebra of symmetric multivector fields. So, let VM be a vector bundle, and consider the graded space 𝔇~:=𝔛sym-1(M)𝔇(V), where the tensor product is over functions on M. Notice that 𝔇~ is concentrated in positive degrees. Consider the graded subspace 𝔇~(V)sym𝔇~ consisting of elements projecting on symmetric multivector fields 𝔛sym(M)𝔛sym-1(M)𝔛(M) via

idl:𝔇~𝔛sym(M)𝔛(M).

Finally, we define the graded space 𝔇sym(V) by putting

𝔇sym0(V)=C(M)  and  𝔇symk(V)=𝔇~(V)symkfor k>0.

We denote by

L:𝔇sym(V)𝔛sym(M)

the projection (given simply by the identity id:C(M)C(M) in degree 0). Clearly, 𝔇sym(V) fits in an exact sequence:

0𝔛sym-1(M)Γ(EndV)𝔇sym(V)𝐿𝔛sym(M)0.

Definition 4.1.

Elements in 𝔇symk(V) are symmetric k-V-multivectors.

For k>0, a symmetric k-V-multivector D𝔇symk(V) will often be interpreted as an operator

D:C(M)×C(M)×Γ(V)Γ(V),(f1,,fk-1,v)D(f1,,fk-1|v).

As such it is symmetric and a derivation in the first k-1 arguments. Additionally, there exists a (necessarily unique) symmetric multivector field LD such that

D(f1,,fk-1|fv)=LD(f1,,fk-1,f)v+fD(f1,,fk-1|v)

for all f1,,fk-1,fC(M) and vΓ(V).

Lemma 4.2.

The space Dsym(V) of symmetric V-multivectors is a Poisson algebra.

  1. (i)

    The associative product is the usual C(M)-module product when one of the two factors has 0 degree. It is given by

    D1D2(f1,,fk1+k2+1|v)=σSk1+1,k2LD1(fσ(1),,fσ(k1+1))D2(fσ(k1+2),,fσ(k1+k2+1)|v)
    +σSk2+1,k1LD2(fσ(1),,fσ(k2+1))D1(fσ(k2+2),,fσ(k1+k2+1)|v)

    for all fiC(M) and vΓ(V), if D1𝔇symk1+1(V), D2𝔇symk2+1(V), in which case

    D1D2𝔇symk1+k2+2(V).
  2. (ii)

    The Lie bracket {-,-} is given by

    {D,f}=-{f,D}=D(f,-,,-|-)

    when f𝔇sym0(V)=C(M). It is given by

    (4.1){D1,D2}=D1D2-D2D1,

    where

    D1D2:C(M)××C(M)×Γ(V)Γ(V)

    is the operator given by

    D1D2(f1,,fk1+k2|v)=σSk1,k2D1(fσ(1),,fσ(k1)|D2(fσ(k1+1),,fσ(k1+k2)|v))
    +σSk1-1,k2+1D1(fσ(1),,fσ(k1-1),LD2(fσ(k1),,fσ(k1+k2))|v)

    for all fiC(M) and vΓ(V), if D1𝔇symk1+1(V), D2𝔇symk2+1(V), in which case

    {D1,D2}𝔇symk1+k2+1(V).

The map

L:𝔇sym(V)𝔛sym(M)

is a surjective Poisson algebra map.

Proof.

The proof is straightforward, but cumbersome. We have to prove associativity (and commutativity) of the product , and Jacobi identity and Leibniz rule for the bracket {-,-}. The easiest case is when one of the involved V-multivectors has degree 0 and the rest have degree 1. We skip this case. The next to the easiest case is when all involved V-multivectors have degree 1. We only treat in some (but not all) details this case. The general case is similar and it is left to the reader. We begin by noticing that 1-V-multivectors are just derivations of V. Moreover, when D𝔇sym1(V)=𝔇(V), then LD is just the symbol of D, i.e. LD=lD. Now let D1,D2𝔇sym1(V). Then

D1D2:C(M)×Γ(V)Γ(V)

satisfies

D1D2(f|gv)=lD1(f)D2(gv)+lD2(f)D1(gv)
=lD1(f)lD2(g)v+glD1(f)D2(v)+lD2(f)lD1(g)v+glD2(f)D1(v)
=lD1lD2(f,g)v+gD1D2(f|v)

for all f,gC(M) and vΓ(V). This shows that D1D2 is a 2-V-multivector with

(4.2)LD1D2=lD1lD2.

This holds similarly in higher degrees.

Next let D1,D2,D3𝔇sym1(V). Then D1(D2D3) is the 3-V-multivector given by

D1(D2D3)(f1,f2|v)=lD1(f1)D2D3(f2|v)+lD1(f2)D2D3(f1|v)+LD2D3(f1,f2)D1(v)
=lD1(f1)lD2(f2)D3(v)+lD1(f1)lD3(f2)D2(v)+lD1(f2)lD2(f1)D3(v)
+lD1(f2)lD3(f1)D2(v)+lD2lD3(f1,f2)D1(v)
=lD1lD2(f1,f2)D3(v)+lD1lD3(f1,f2)D2(v)+lD2lD3(f1,f2)D1(v)

for all f1,f2C(M) and vΓ(V), where we also used (4.2). This clearly agrees with (D1D2)D3(f1,f2|v), showing associativity (in low degree). This holds similarly in higher degrees. Commutativity is obvious. The unit is the constant function 1.

We now pass to the bracket {-,-}. When the entries have degree 1, the latter is just the commutator of derivations and the Jacobi identity is obvious in this case. For the Leibniz rule let D1,D2,D3𝔇sym1(V), fC(M) and vΓ(V), and compute

{D1,D2D3}(f|v)=D1(D2D3(f|v))-D2D3(f|D1(v))-D2D3(lD1(f)|v)
=D1(lD2(f)D3(v)+lD3(f)D2(v))-lD2(f)D3(D1(v))-lD3(f)D2(D1(v))
-lD2(lD1(f))D3(v)-lD3(lD1(f))D2(v)
=lD1(lD2(f))D3(v)+lD2(f)D1(D3(v))+lD1(lD3(f))D2(v)+lD3(f)D1(D2(v))
-lD2(f)D3(D1(v))-lD3(f)D2(D1(v))-lD2(lD1(f))D3(v)-lD3(lD1(f))D2(v)
=l{D1,D2}(f)D3(v)+lD3(f){D1,D2}(v)+l{D1,D3}(f)D2(v)+lD2(f){D1,D3}(v)
=({D1,D2}D3+{D1,D3}D2)(f|v).

Hence,

{D1,D2D3}={D1,D2}D3+{D1,D3}D2.

This concludes the proof. ∎

Remark 4.3.

When V is a line bundle, the map 𝔛sym-1(M)Γ(EndV)𝔇sym(V) embeds

𝔛sym-1(M)=𝔛sym-1(M)C(M)=𝔛sym-1(M)Γ(EndV)

into 𝔇sym(V) as an abelian Poisson subalgebra and an ideal.

There is also a notion of FWL V-multivector. In order to discuss it, it is useful to discuss derivations of pull-back vector bundles first. So, let V be a vector bundle, and consider its pull-back V𝒫:=πV along a surjective submersion π:𝒫M. Clearly, a derivation D of V𝒫 is completely determined by its symbol and its action on pull-back sections. The restriction DM:=D|Γ(V) of D to pull-back sections is a derivation along π, i.e. it is an -linear map DM:Γ(V)Γ(V𝒫) and there exists a, necessarily unique, vector field along π, denoted by lDMΓ(πTM), fitting in the Leibniz rule

DM(fv)=π(f)DM(v)+lDM(f)v,fC(M),vΓ(V).

The correspondence D(lD,DM) establishes a C(M)-linear bijection between derivations D of V𝒫 and pairs (X,DM) consisting of a vector field X𝔛(𝒫) and a derivation along π satisfying the following additional compatibility: dπX=lDM. When 𝒫=EM is a vector bundle, it makes sense to talk about polynomial sections of VE. Namely, Γ(VE)=C(E)Γ(V), where the tensor product is over C(M), and multiplicative reals act on Γ(VE) via their action on the first factor. As for functions, we denote by h this action. A section v of Γ(VE) is homogeneous (polynomial) of degree k if ht(v)=tkv for all t; in other words, vCpoly(E)Γ(V). Denote by Γ(VE)k the space of homogeneous sections of degree k, and by

Γpoly(VE):=k=0Γ(VE)k

the space of all polynomial sections. FWL sections are degree 1 sections and they identify with sections of EV. Core sections are degree 0 sections and they identify simply with sections of V. Similarly to vector fields, a derivation D of VE is homogeneous (polynomial) of degree k if it maps homogeneous sections of degree h to homogeneous sections of degree k+h; in other words,

htD=tkD

for all t0. The space of all polynomial derivations of VE will be denoted by 𝔇poly(VE), and they correspond to pairs (X,DM) where X𝔛poly(E) and DM takes values in polynomial sections.

We can also consider polynomial symmetric VE-multivectors. We will only need core and FWL ones. A symmetric k-VE-multivector D is FWL (resp. core) if it is homogeneous of degree 1-k (resp. -k), i.e.

htD=t1-kD(resp. htD=t-kD)

for all t0. We denote by 𝔇sym,lin(VE) (resp. 𝔇sym,core(VE)) the space of FWL (resp. core) symmetric VE-multivectors. The Lie bracket {-,-} on V-multivectors preserves FWL ones. Additionally, the projection L:𝔇sym(VE)𝔛sym(E) maps FWL V-multivectors to FWL multivector fields, and we get a short exact sequence of Lie algebras:

(4.3)0𝔛sym,lin-1(E)Γ(End(V))𝔇sym,lin(VE)𝐿𝔛sym,lin(E)0.

Proposition 4.4.

A k-VE-multivector D is core if and only if

D(f1,,fk-1|v)Γcore(VE),
D(f1,,fk-1|w)=0,
D(f1,,fk-2,h|v)=0

for all fiClin(E), hCcore(E), vΓlin(VE):=Γ(VE)1, and all wΓcore(VE):=Γ(VE)0=Γ(V).

Proof.

The proof is straightforward. ∎

It easily follows from the above proposition that a core k-VE-multivector is completely determined by its symbol. More precisely, the symbol map DLD establishes a one-to-one correspondence between core k-VE-multivectors and core multivector fields. Let us discuss this in detail in the case k=1. The general case is similar. So, let D be a core 1-VE-multivector. In other words, D is a degree -1 derivation of VE. From Proposition 4.4 it maps linear sections to core sections, and core sections to 0. Now let f be an FWL function on E and let vΓ(V). Then v and fv can be seen as a core and an FWL section of VE, respectively. Hence, we have

D(fv)=lD(f)v,

with lD(f)C(M). As the action of D on linear sections determines D completely, we see that D is necessarily of the form

Xid:Γ(VE)C(E)Γ(V)Γ(VE)C(E)Γ(V)

for some core vector field X=lD on E. In particular, X=e, the vertical lift of some section eΓ(E). Conversely, every derivation of VE of the type eid, with eΓ(E), is a core derivation. We conclude that core 1-VE-multivectors identify with core vector fields on E via the symbol map: 𝔇sym,core1(VE)𝔛core(E). Similarly,

𝔇sym,core(VE)𝔛sym,core(E)Cpoly(E).

In particular, 𝔇sym,core(VE) does not really depend on V (but only on E).

Proposition 4.5.

A symmetric k-VE-multivector D is FWL if and only if

D(f1,,fk-1|v)Γlin(VE),
D(f1,,fk-1|w)Γcore(VE),
D(f1,,fk-2,h1|v)Γcore(VE),
D(f1,,fk-2,h1|w)=0,
D(f1,,fk-3,h1,h2|v)=0

for all fiClin(E), hjCcore(E), vΓlin(VE):=Γ(VE)1, and all wΓcore(VE):=Γ(VE)0=Γ(V).

Proof.

The proof is straightforward. ∎

In particular, an FWL symmetric k-VE-multivector D determines a map

ΦD:Γ(E)××Γ(E)k-1 times𝔇(V)

via

ΦD(φ1,,φk-1)(w)=D(φ1,,φk|w)

for all φiΓ(E) and all wΓ(V). The map ΦD is C(M)-multilinear and symmetric. Hence, it can be seen as a vector bundle map ΦD:Sk-1EDV or, equivalently, as a section of Sk-1EDV.

Proposition 4.6.

The assignment D(LD,ΦD) establishes a C(M)-linear bijection between FWL symmetric k-VE-multivectors DDsym,link(VE) and pairs (P,Φ) consisting of an FWL symmetric multivector field PXsym,link(E) and a vector bundle map Φ:Sk-1EDV such that lP=lΦ.

Proof.

The proof is easy and left to the reader. ∎

According to Proposition 4.6 we will sometimes call the pair (LD,ΦD) itself an FWL symmetric VE-multivector.

Now, we can combine the exact sequences (3.3) and (4.3) in one exact commutative diagram:

We only need to explain the map I. To do that, we first remark that π-vertical vector fields act naturally on sections of VE, via

X(fv)=X(f)v

for all XΓ(TπE), fC(E) and vΓ(V). Now, take e1,,ekΓ(E) and φΓ(E). Then

I(e1ekφ)(f1,,fk-1|v)=1k!φσSkeσ(1)(f1)eσ(k-1)(fk-1)eσ(k)(v).

Equivalently, we can interpret e1ek as a core VE-multivector, via the isomorphism

𝔇sym,core(VE)Γ(SE),

and then multiply by the FWL function φ to get an FWL VE-multivector.

We conclude this section showing that FWL symmetric VE-multivectors do also identify with polynomial derivations of VE. This is an easy consequence (among other things) of Proposition 4.6. Indeed, take D𝔇sym,lin(VE) and let (LD,ΦD) be the corresponding pair. Denote by π:EM the projection. We claim that ΦD can be seen as a derivation along π. Indeed, ΦD is a section of S-1EDL and, by acting on a section vΓ(V) with the DL-factor, we get a section ΦD(v) of S-1EV, i.e. a polynomial section of Γ(VE). In the following, we use this construction to interpret ΦD as a derivation along π. If we do so, the pair (HLD,ΦD) consists of a vector field on E, and a derivation ΦD along π, with the additional property that dπHLD=lΦD. Hence, it corresponds to a (polynomial) derivation D of the pull-back vector bundle VE (recall from Section 3 that HLD is the vector field on E completely determined by the property of acting as {LD,-} on polynomial functions on E, or, equivalently, core symmetric multivector fields on E, where {-,-} is the Poisson bracket of symmetric multivector fields). Finally, a tedious but straightforward computation shows that the bijection DD between linear VE-multivectors and polynomial derivations of VE obtained in this way also preserves the Lie algebra structures. When we equip 𝔇sym,lin(E) with the product by a core VE-multivector, the latter bijection becomes an isomorphism of Lie–Rinehart algebras. We have thus proved the following main result in this section.

Theorem 4.7.

Let EM and VM be vector bundles. The assignment DD establishes a degree inverting isomorphism of Lie–Rinehart algebras over Dsym,core(E)Cpoly(E) between linear VE-multivectors and polynomial derivations of VE.

5 Differential operators and their symbols

We finally come to the object of our primary interest: differential operators. This section is a super-short review of the subject.

Let V,WM be vector bundles. A (linear) differential operator (DO in the following) of order q from V to W is an -linear map

Δ:Γ(V)Γ(W)

such that

[[[Δ,f0],f1],,fq]=0

for all fiC(M). In particular, DOs of order zero are just vector bundle maps VW. We denote by DOq(V,W) the space of order q DOs from V to W. Clearly, a DO of order q is also a DO of order q+1, and we get the filtration

DO0(V,W)=Γ(Hom(V,W))DO1(V,W)DOq(V,W).

The union of all DOq(V,W) will be denoted simply by DO(V,W). A scalar DO on M is a DO acting on functions over M, i.e. a DO from the trivial line bundle M:=M× to itself. We use the symbol DOq(M) (instead of DOq(M,M)) for scalar DOs.

The composition of an order q and an order r DO is an order q+r DO. In particular, for all q, DOq(V,W) is a C(M)-module in two different ways: via composition on the left and composition on the right with a function on M (seen as an order 0 DO). We will consider the first module structure unless otherwise stated. The space DO(M) is a filtered non-commutative algebra with the composition. It is actually the universal enveloping algebra of the tangent Lie algebroid TMM. Being an associative algebra, DO(M) is also a Lie algebra with the commutator. Notice that the commutator of an order q and an order r scalar DO is an order q+r-1 scalar DO.

Given an order q DO Δ:Γ(V)Γ(W) from V to W, and functions f1,,fq, the nested commutator

[[Δ,f1],,fq]

is an order 0 DO. Additionally, it is a derivation in each of the arguments fi and it is symmetric in those arguments. In this way, we get a map

σ:DOq(V,W)Γ(SqTMHom(V,W)),Δσ(Δ),

with

σ(Δ)(f1,,fq)=[[Δ,f1],,fq].

The map σ is called the symbol and it fits in a short exact sequence of C(M)-modules

(5.1)0DOq-1(V,W)DOq(V,W)𝜎Γ(SqTMHom(V,W))0,

where the second arrow is the inclusion.

Example 5.1.

Vector fields are first order scalar DOs. Derivations of the vector bundle V are first order DOs D from V to itself such that σ(D) belongs to 𝔛(M)Γ(TMEndV). Additionally, we have σ(D)=lD.

For scalar DOs, the short exact sequence (5.1) becomes

0DOq-1(M)DOq(M)𝜎𝔛symq(M)0.

The symbol of scalar DOs intertwines the commutator with the Poisson bracket (of symmetric multivector fields) in the sense that

σ([Δ,Δ])={σ(Δ),σ(Δ)}

whenever ΔDOq(M) and ΔDOq(M), in which case we take [Δ,Δ]DOq+q-1(M).

We conclude this short review section by commenting briefly on the coordinate description of (scalar) DOs. To do this, we first fix our conventions on the multi-index notation for multiple partial derivatives. Let (xi), i=1,,n, be variables. A length k multi-index I is a word I=i1ik, with ij=1,,n, where words are considered modulo permutations of their letters. The length k of a multi-index I=i1ik is also denoted by |I|. Words can be composed by concatenation and we also consider the empty multi-index . If we do so, then multi-indexes are elements in the free abelian monoid generated by 1,,n. The length is then a monoid homomorphism. A length k multi-index I=i1ik, determines an order k DO

|I|xI:=kxi1xik.

Now, we go back to manifolds M (and vector bundles over them). Actually, DOq(M) (likewise DOq(V,W)) is the C(M)-module of sections of a vector bundle over M. If (xi) are coordinates on M, then DOq(M) is spanned locally (in the corresponding coordinate neighborhood) by

|I|xI,|I|=0,1,,q.

More precisely, locally, every DO ΔDOq(M) can be uniquely written in the form

(5.2)Δ=|I|qΔI(x)|I|xI,

where the ΔI(x) are local functions on M. The ΔI(x) can be recovered via formulas

Δi1ik(x)=1(i1ik)![[Δ,xi1],,xik](1),k=0,1,,q,

where, for a multi-index I, we denote by I! the product I[1]!I[n]! where I[i] is the number of times the letter i occurs in I.

Finally, if Δ is an order q scalar DO locally given by (5.2), then its symbol σ(Δ) is locally given by

σ(Δ)=1q!Δi1iqxi1xiq.

6 Core and fiberwise linear differential operators

This is the main section of the paper. We propose a notion of FWL (scalar) DO on the total space of a vector bundle. Our definition is partly motivated by the fact that the symbol of an FWL DO is an FWL multivector field. It is also motivated by the linearization construction discussed in the next section. Yet another motivating little fact is that the Laplacian of an FWL metric is an FWL DO (Example 6.2).

Let EM be a vector bundle. We have learnt from Sections 24 that, given a type 𝔗 of geometric structures on manifolds (functions, vector fields, tensors etc.) appropriate notions of core and FWL structures of the type 𝔗 on E exist, and these notions can be identified by means of the following recipe: (1) notice that the space 𝔗(E) of structures of type 𝔗 on E possesses a subspace 𝔗poly(E) which is naturally graded (via the action of multiplicative reals on E by fiberwise scalar multiplication); (2) identify the smallest degree k for which the degree k homogeneous component 𝔗(E)k of 𝔗poly(E) is non-trivial; (3) put 𝔗core(E)=𝔗(E)k and 𝔗lin(E)=𝔗(E)k+1. A quick check shows that this recipe cooks up the required definitions in all cases considered so far. Notice that we could make this recipe much more rigorous using for the rather vague “geometric structure of type 𝔗” the very precise notion of natural vector bundle 𝔗, but we will not need this level of abstraction.

We adopt the strategy described above to define core and FWL DOs on E. Consider the non-commutative algebra DO(E) of scalar DOs Δ:C(E)C(E). We begin noticing that, for each q, the space DOq(E) of DOs Δ:C(E)C(E) of order q possesses a subspace DOq,poly(E) which is naturally graded:

DOq,poly(E)=k=-qDOq(E)k,

where DOq(E)k consists of degree k DOs (of order q), i.e. DOs Δ such that

ht(Δ)=tkΔ

for all t0. The smallest degree k for which DOq(E)k is non-trivial is k=-q. So, following our recipe, we put

DOq,core(E):=DOq(E)-q,

and call them core DOs. We also put

(6.1)DOcore(E):=qDOq,core(E).

Let (xi,uα) be vector bundle coordinates on E, and let (xi,uα) be dual coordinates on E. A DO FDOq(E) is a core DO if and only if, locally,

(6.2)F=|A|=qFA(x)|A|uA,

where A=α1αq is a length q multi-index.

It follows from (6.2) that DOcore(E)DO(E) is the subalgebra generated by core functions Ccore(M) and core vector fields 𝔛core(E). Equivalently, it is the universal enveloping algebra of the abelian Lie algebroid EM. Because of the latter description, there is an algebra isomorphism

Γ(SE)DOcore(E),

mapping a monomial

e1eq,eiΓ(E),

to the DO

e1eq.

In its turn, as already mentioned, Γ(SE) identifies with polynomial functions on E. In the following, we will often identify DOcore(E) with both Γ(SE) and Cpoly(E) via the latter isomorphisms. If FDOq,core(E) is locally given by (6.1), then it identifies with

1q!Fα1αq(x)uα1uαqΓ(SqE)

and

|A|=qFA(x)uA=1q!Fα1αq(x)uα1uαqC(E)q,

where, for A=α1αq, we denote by uA the monomial uα1uαq.

We will always consider DO(E) as a DOcore(E)-module with the scalar multiplication given by the left composition.

We now pass to FWL DOs. Following our recipe again, for each q we put

DOq,lin(E):=DOq(E)-q+1.

Definition 6.1.

DOs in DOq,lin(E) are called fiberwise linear differential operators (FWL DOs) of order q.

For instance,

DO0,lin(E)=Clin(E)andDO1,lin(E)=𝔛lin(E)C(M).

It is also clear that DOq,lin(E)DOq-1,core for all q. More precisely,

DOq-1,core(E)=DOq,lin(E)DOq-1(E).

We put

DOlin(E):=qDOq(E)-q+1.

Clearly, DOcore(E)DOlin(E)DO(E).

A DO ΔDOq(E) is FWL if and only if, in vector bundle coordinates, it looks like

(6.3)Δ=|A|=q-1Δi|A(x)|A|+1xiuA+|B|=qΔαB(x)uα|B|uB+|C|=q-1ΔC(x)|C|uC.

It is easy to see from this formula that DOlin(E)DO(E) is the DOcore(E)-submodule spanned by 1, Clin(E) and 𝔛lin(E).

Example 6.2.

Let g be a metric on E, and assume it is FWL. Then the associated Laplacian operator

Δg:C(E)C(E)

is an FWL DO operator (of order 2). One can see this working in vector bundle coordinates. But there is also a (basically) coordinate free proof that we now illustrate. First of all, from g being FWL, it immediately follows that the inverse tensor g-1 is FWL as well. Now, the covariant derivative ϑ of a 1-form ϑ along the Levi-Civita connection is the covariant 2-tensor given by the formula

(6.4)ϑ=12(dϑ+(ϑ)g),

where :TETE is the musical isomorphism. Equivalently, the covariant derivative XY of a vector field Y along another vector field X is the vector field XY that acts on functions fC(M) as follows:

(6.5)XY(f)=X(Y(f))-12(gradfg)(X,Y),

where gradf=(df) is the gradient of f. By using (6.4) (or (6.5)) and the naturality of both the de Rham differential and the Lie derivative, it is easy to see that the covariant derivative of arbitrary tensor fields commutes with the pull-back along ht for all t0. As the Laplacian Δgf of a function f is obtained by contracting the covariant derivative of df with g-1, Δg decreases by one the degree of a homogeneous (fiberwise polynomial) function. So it is a second order DO of degree 1-2=-1, i.e. an FWL DO of order 2, as claimed. It might also be interesting to remark that the Levi-Civita connection of an FWL metric is an FWL connection according to a definition introduced in [10].

Lemma 6.3.

The space DOlin(E) of FWL DOs is the stabilizer of DOcore(E) in the Lie algebra DO(RE), i.e. a DO ΔDO(RE) is in DOlin(E) if and only if [Δ,F]DOcore(E) for all FDOcore(E).

Proof.

The “only if part” of the statement immediately follows from an obvious order/degree argument. For the “if part”, consider a DO Δ of order r. Locally,

Δ=|I|+|A|rΔI|A(x,u)|I|+|A|xIuA.

Assume that Δ is in the stabilizer of DOcore(E). We want to show that Δ is the sum of operators of the form (6.3) (with possibly varying qr). As xi is a core function for all i, the commutator [Δ,xi] is a core DO. But

[Δ,xi]=|J|+|A|r-1(J[i]+1)ΔJi|A(x,u)|J|+|A|-1xJuA,

so it can only be a core DO for all i if ΔI|A(x,u)=0 for |I|>1, and Δi|A(x,u)=Δi|A(x). In other words, Δ is necessarily of the form

(6.6)Δ=|A|r-1Δi|A(x)|A|+1xiuA+|B|kΔB(x,u)|B|uB.

To conclude, it is enough to prove that ΔB(x,u) is of the form ΔB(x,u)=ΔαB(x)uα+ΔB(x). To do this, recall that uα is a core vector field for all α, and hence [Δ,uα] is a core DO. But, from (6.6),

[Δ,uα]=-ΔB(x,u)uα|B|uB,

which is a core DO for all α if and only if

2ΔB(x,u)uαuβ=0

for all α,β, i.e. ΔB(x,u) is a (non-necessarily homogeneous) first order polynomial in the variables u, as desired. ∎

It follows from Lemma 6.3 that DOlin(E) is a Lie subalgebra in DO(E). As already mentioned, it is also a DOcore(E)-submodule. Actually, it is a Lie–Rinehart algebra over DOcore(E), the anchor being the adjoint operator ad:Δad(Δ):=[Δ,-]. To see this, first notice that ad(Δ) is indeed a well-defined derivation of DOcore(E) for all ΔDOlin(E). Now, take Δ,ΔDOlin(E) and F,FDOcore(E), and compute

[Δ,FΔ]=ad(Δ)(F)Δ+F[Δ,Δ],
ad(FΔ)(F)=[FΔ,F]=F[Δ,F]=Fad(Δ)(F).

Finally, from DOcore(E)Cpoly(E), we see that, for every ΔDOlin(E), the derivation ad(Δ) determines a polynomial vector field (of the same degree) on E, also denoted by ad(Δ).

Theorem 6.4.

The sequence of Lie–Rinehart algebras

(6.7)0DOcore(E)DOlin(E)ad𝔛poly(E)0

is exact.

Proof.

First of all, as already remarked, DOcore(E) is in DOlin(E). Even more, as it is an abelian subalgebra in DO(E), it is actually in the kernel of ad:DOlin(E)𝔛poly(E). To see that core DOs exhaust the kernel of ad (i.e. DOcore(E) is its own centralizer), assume that [Δ,F]=0 for all FDOcore(E). Then exactly the same computation as in the proof of Lemma 6.3 shows that Δ is locally of the form (6.3) with Δi|A(x)=ΔαB(x)=0, i.e. ΔDOcore(E). For the exactness of the sequence (6.7) it remains to show that the map ad:DOlin(E)𝔛poly(E) is surjective. To do that, we work in local coordinates again. So, let (xi,uα) be vector bundle coordinates on E, and let (xi,uα) be dual coordinates on E. It is not hard to see that, if Δ is locally given by (6.3), then the vector field ad(Δ) is locally given by

(6.8)ad(Δ)=|A|=q-1Δi|A(x)uAxi-|B|=qΔαB(x)uBuα,

where, for a multi-index A=α1αs, we denote by uA the monomial uα1uαs (s=q,q-1). As (6.8) is the local expression of a generic homogeneous polynomial vector field of degree q-1, we are done. ∎

Our next aim is to prove that the Lie–Rinehart algebra DOlin(E) is canonically isomorphic to the Lie–Rinehart algebra of polynomial derivations of an appropriate line bundle on E. We begin with a simple proposition.

Proposition 6.5.

The symbol σ(Δ) of an FWL DO ΔDOq,lin(E) is an FWL symmetric q-multivector field on E. Every FWL symmetric q-multivector field is the symbol of an order q FWL DO.

Proof.

The statement of this proposition immediately follows from (6.3) and the coordinate description of symmetric q-multivector fields (see Proposition 3.1). ∎

Now let ΔDOq,lin(E). Notice that the adjoint operator ad(Δ), seen as a polynomial vector field on E, corresponds exactly to the symbol σ(Δ) via the isomorphism 𝔛sym,linq(E)𝔛(E)q-1. It is also clear that, in view of its coordinate form (6.3), Δ is completely determined by σ(Δ) or, equivalently, ad(Δ), together with the map

ΨΔ:Γ(E)××Γ(E)q-1 timesC(M),(φ1,,φq-1)[[Δ,φ1],,φq-1](1).

The map ΨΔ is clearly well-defined. Additionally, it enjoys the following properties:

  1. (i)

    ΨΔ is symmetric.

  2. (ii)

    ΨΔ is a first order DO in each entry.

More precisely, we have the following lemma.

Lemma 6.6.

The map ΨΔ satisfies

ΨΔ(φ1,,φq-2,fφq-1)=fΨΔ(φ1,,φq-2,φq-1)+lσ(Δ)(φ1,,φq-1)(f)

for all φiΓ(E) and fC(M).

Proof.

Let φi and f be as in the statement, and compute

ΨΔ(φ1,,φq-2,fφq-1)=[[[Δ,φ1],,φq-2],fφq-1](1)
=f[[[Δ,φ1],,φq-2],φq-1](1)+[[[Δ,φ1],,φq-2],f](φq-1)
=fΨΔ(φ1,,φq-2,φq-1)+[[[Δ,φ1],,φq-2],f](φq-1).

It remains to compute the last summand. So,

[[[Δ,φ1],,φq-2],f](φq-1)
=[[[[Δ,φ1],,φq-2],f],φq-1]+φq-1[[[Δ,φ1],,φq-2],f](1)
=lσ(Δ)(φ1,,φq-1)(f),

where we used that, from (6.3) again, the last summand in the second line is necessarily zero. This concludes the proof. ∎

The data (σ(Δ),ΨΔ) (determines Δ completely and) can be repackaged in a very useful way. Namely, consider the line bundle

L=topE.

Then the pair (σ(Δ),ΨΔ) determines an FWL q-LE-multivector in the following way. Recall that an FWL q-LE-multivector can be equivalently presented as a pair (P,Φ) consisting of an FWL symmetric multivector field P𝔛sym,linq(E) and a vector bundle map Φ:Sq-1EDL such that lP=lΦ. We claim that we can construct such a pair from the pair (σ(Δ),ΨΔ). Namely, we put

P=σ(Δ)

and define Φ=ΦΔ by putting

(6.9)ΦΔ(φ1,,φq-1)(U):=σ(Δ)(φ1,,φq-1,-)U+ΨΔ(φ1,,φq-1)U

for all φiΓ(E) and UΓ(L). Equation (6.9) needs some explanations. In the first summand of the right-hand side, we interpret σ(Δ) as a q-multiderivation of E, so, when contracting it with the q-1 sections φ1,,φq-1, we get a plain derivation σ(Δ)(φ1,,φq-1,-)𝔇(E)𝔇(E). As already remarked, derivations of E act on the exterior algebra of E. In our case we have

D(e1etop)=ie1Deietop

for all D𝔇(E) and all eiΓ(E).

The next theorem is the main result of the paper.

Theorem 6.7.

The assignment Δ(σ(Δ),ΦΔ) establishes a degree inverting isomorphism of Lie–Rinehart algebras A:DOlin(E)Dsym,lin(LE).

Proof.

First of all, we have to show that ΦΔ is well-defined, i.e. it is symmetric and C(M)-linear in all its arguments. The symmetry is obvious. For the linearity, let fC(M), and compute

ΦΔ(φ1,,φq-2,fφq-1)(U)=σ(Δ)(φ1,,φq-2,fφq-1,-)U+ΨΔ(φ1,,φq-2,fφq-1)U.

Let us compute the two summands separately. First of all, for every φΓ(E),

σ(Δ)(φ1,,φq-2,fφq-1,φ)=lσ(Δ)(φ1,,φq-2,φ)(f)φq-1+fσ(Δ)(φ1,,φq-1,φ)

showing that

σ(Δ)(φ1,,φq-2,fφq-1,-)=-,eφq-1+fσ(Δ)(φ1,,φq-1,-),

where eΓ(E) is the section implicitly defined by

φ,e=lσ(Δ)(φ1,,φq-2,φ)(f)

for all φΓ(E). We remark for future use that, in particular,

(6.10)φq-1,e=lσ(Δ)(φ1,,φq-1)(f).

Now

(6.11)σ(Δ)(φ1,,φq-2,fφq-1,-)U=(-,eφq-1)U+fσ(Δ)(φ1,,φq-1,-)U.

The endomorphism -,eφq-1:EE acts on E via its dual, which is minus its transpose, and hence

(6.12)(-,eφq-1)U=-trace(-,eφq-1)U=-φq-1,eU=-lσ(Δ)(φ1,,φq-1)(f)U,

where we used (6.10). Substituting (6.12) into (6.11), we find

σ(Δ)(φ1,,φq-2,fφq-1,-)U=fσ(Δ)(φ1,,φq-1,-)U-lσ(Δ)(φ1,,φq-1)(f)U.

We also have

ΨΔ(φ1,,φq-2,fφq-1)U=fΨΔ(φ1,,φq-1)U+lσ(Δ)(φ1,,φq-1)(f)U.

Putting everything together, we find

ΦΔ(φ1,,φq-2,fφq-1)U=fσ(Δ)(φ1,,φq-1,-)U+fΨΔ(φ1,,φq-1)U
=fΦΔ(φ1,,φq-1)U.

We conclude that ΦΔ is a vector bundle map Sq-1EDL as desired. Additionally, the composition lΦΔ does clearly agree with σ(Δ), so that (σ(Δ),ΦΔ) is indeed a q-LE-multivector. It is also clear that ΨΔ can be reconstructed from (σ(Δ),ΦΔ) showing that the correspondence Δ(σ(Δ),ΦΔ) is injective. Next we prove the DOcore(E)-linearity. So, take

FDOp,core(E)=Γ(SpE)=C(E)pandΔDOq,lin(E),

so that FΔDOp+q,lin(E). We want to show that (σ(FΔ),ΦFΔ)=F(σ(Δ),ΦΔ). To do this, we begin by noticing that the product F(σ(Δ),ΦΔ) is the pair (D,Φ) where D𝔛sym,linp+q(E) is the symmetric multivector field that, when interpreted as a multiderivation D:Γ(E)××Γ(E)Γ(E), is given by

D(φ1,,φp+q)=Fσ(Δ)(φ1,,φp+q)
=σSp,qF,φσ(1)φσ(p)σ(Δ)(φσ(p+1),,φσ(p+q)),

and, similarly, Φ:Sp+q-1EDL is the bundle map given by

Φ(φ1,,φp+q-1)=FΦΔ(φ1,,φp+q-1)
=σSp,q-1F,φσ(1)φσ(p)ΦΔ(φσ(p+1),,φσ(p+q-1))

for all φiΓ(E). From the properties of the symbol map, we have σ(FΔ)=Fσ(Δ), and it remains to take care of ΦFΔ. So choose φiΓ(E), and compute ΦFΔ(φ1,,φp+q-1). From symmetry, it is enough to choose φi=φ for all i and some φ. First of all, we have

ΨFΔ(φ,,φp+q-1 times)=[[FΔ,φ],,φ]p+q-1 times(1)
=l+m=p+q-11l!m![[F,φ],,φ]l times[[Δ,φ],,φ]m times(1).

Only the terms with l=p,p-1 (and hence m=q-1,q, respectively) survive, and we get

ΨFΔ(φ,,φp+q-1 times)=1p!(q-1)![[F,φ],,φ]p times[[Δ,φ],,φ]q-1 times(1)
+1(p-1)!q![[F,φ],,φ]p-1 times[[Δ,φ],,φ]q times(1)
=1p!(q-1)!F,φφp timesΨΔ(φ,,φq-1 times)
(6.13)+1(p-1)!q!F,φφp-1 timesσ(Δ)(φ,,φq times).

Now, for all UΓ(L),

ΦFΔ(φ,,φp+q-1 times)U=σ(FΔ)(φ,,φp+q-1 times,-)U+ΨFΔ(φ,,φp+q-1 times)U.

We already computed the second summand, while the first summand is

σ(FΔ)(φ,,φp+q-1 times,-)U=1p!(q-1)!F,φφp timesσ(Δ)(φ,,φq-1 times,-)U
+1(p-1)!q!F,φφp-1 times-σ(Δ)(φ,,φq times)U
=1p!(q-1)!F,φφp timesσ(Δ)(φ,,φq-1 times,-)U
(6.14)-1(p-1)!q!F,φφp-1 timesσ(Δ)(φ,,φq times)U.

From (6.13) and (6.14) it easily follows that ΦFΔ=FΦΔ, as claimed.

The surjectivity of the map Δ(σ(Δ),ΦΔ) now follows from (local) dimension counting.

It remains to check that the isomorphism A:DOlin(E)𝔇sym,lin(LE) defined in this way is both anchor and bracket preserving. For the anchor, the anchor of (σ(Δ),ΦΔ) is the derivation of

DOcore(E)=Γ(SE)=Cpoly(E)

corresponding to the FWL multivector field σ(Δ), which is exactly ad(Δ).

For the bracket, as we already discussed Cpoly(E)-linearity and compatibility with the anchor, it is enough to discuss the brackets of generators. As already remarked, DOlin(E) is generated (over DOcore(E)) by 1, Clin(E) and 𝔛lin(E). A direct check shows that

(6.15)A(1)=(0,1),A(φ)=(-φ,0),A(X)=(X,DX)

for all φΓ(E) and all X𝔛lin(E). Here X and DX are, respectively, the linear vector field on E and the derivation of L=topE corresponding to X. It is now easy to check that the brackets are preserved on these generators, and this concludes the proof. ∎

Composing with the isomorphism 𝔇sym,lin(LE)𝔇(LE) from Theorem 4.7 (in the case V=L), we get a (degree inverting) Lie–Rinehart algebra isomorphism

DOlin(E)𝔇(LE)

that we denote by A again.

Remark 6.8.

Let (xi,uα) be vector bundle coordinates on E, and let (xi,uα) be dual coordinates on E. Denote by Volu=u1utop the local coordinate generator of Γ(L). It is easy to check using, e.g., (6.8), (6.15) and the Cpoly(E)-linearity, that, if the operator ΔDOlin(E) is locally given by (6.3), then the corresponding derivation A(Δ)𝔇(LE) maps a local section λ=f(x,u)Volu of Γ(LE) to

A(Δ)(λ)=(|A|=q-1Δi|A(x)uAfxi(x,u)-|B|=qΔαB(x)uBfuα(x,u)+|C|=q-1ΔC(x)uCf(x,u))Volu.

7 Linearization of differential operators

Let be a manifold, let M be a submanifold and let ΔDO() be a scalar DO. Denote by EM the normal bundle to M, i.e. E=T|M/TM. In this section, we show that, under appropriate linearizability conditions, the DO Δ can be linearized around M yielding an FWL differential operator ΔlinDOlin(E). The DO Δlin represents the first order approximation of Δ around M in the direction transverse to M. This linearization construction is a further motivation supporting our definition of FWL DOs.

So, let M be a submanifold and let EM be its normal bundle. We will often consider adapted coordinates on around points of M, i.e. coordinates (Xi,Uα) such that M:{Uα=0. In particular, the restrictions (xi=Xi|M) are coordinates on M. From Uα|M=0 we see that uα:=dUα|M are conormal 1-forms and (xi,uα) are vector bundle coordinates on E.

We want to explain what it means to linearize an order q DO operator ΔDOq() around M. We proceed as follows: first, we recall the linearization of a function, second we discuss the linearization of a symmetric multivector field, and finally we define the linearization of a generic DO. So, let FC(). We say that F is linearizable (around M) if F|M=0. In this case, dF|M is a conormal 1-form to M, i.e. a section of the conormal bundle ET|M. Hence, it corresponds to an FWL function on E. We put Flin=dF|M and call it the linearization of F. For instance, if (Xi,Uα) are adapted coordinates on , then the (Uα) are linearizable and the linear fiber coordinates (uα) on E are their linearizations. If F is any linearizable function on , then locally, around a point of M, F(X,U)=Fα(X)Uα+𝒪(U2) for some functions Fα(X) of the (Xi) (given by Fα(X)=FUα(X,0)), and, in this case, Flin=Fα(x)uα. Notice that every linear function φClin(E) is the linearization of a (non-unique) linearizable function FC(): φ=Flin.

We now pass to symmetric multivector fields. So, let P be a symmetric q-multivector field on . We say that P is linearizable if it belongs to the ideal M in 𝔛sym() generated by vector fields that are tangent to M. In other words, M is a coisotropic submanifold of with respect to P.

Proposition 7.1.

Let PXsymq(E) be a linearizable symmetric multivector field on E. Then there exists a unique FWL symmetric q-multivector field Plin on E such that

(7.1)Plin((F1)lin,,(Fq)lin)=P(F1,,Fq)lin

for all linearizable functions FiC(E). The linearizationPPlin preserves the Poisson bracket of symmetric multivector fields.

Proof.

We begin by remarking that, as PM, the function P(F1,,Fq) is clearly linearizable for any choice of linearizable functions Fi. Now we want to show that the right-hand side of (7.1) does only depend on (Fi)lin. We do this in coordinates. So, let (Xi,Uα) be adapted coordinates on , and let (xi,uα) be the associated vector bundle coordinates on E. Locally

(7.2)P=l+m=qPi1il,α1αm(X,U)Xi1XilUα1Uαm.

Hence,

P(F1,,Fq)=l+m=qσSl,mPi1il,α1αm(X,U)Fσ(1)Xi1Fσ(l)XilFσ(l+1)Uα1Fσ(l+m)Uαm.

It follows from PM that Pα1,,αq(X,0)=0. Now compute

P(F1,,Fq)lin=(Uα|(x,0)P(F1,,Fq))uα.

Using that Fi|M=0 and that Pα1,,αq(X,0)=0, we find

Uα|(x,0)P(F1,,Fq)
=Pα1αqUα(x,0)(F1)α1(x)(Fq)αq(x)+σS1,k-1Pi,α1αq-1(x,0)(Fσ(1))αxi(x,0)(Fσ(2))α1(Fσ(q))αq-1,

which does only depend on the (Fi)lin. As every FWL function is a linearization, this also shows that Plin is well-defined on linear functions. Finally, Plin can be uniquely extended to all functions on E, as a symmetric q-multivector field, also denoted by Plin, and locally given by

Plin=Plinα1αq(x,u)uα1uαq+Pi,α1αq-1(x,0)xiuα1uαq-1.

In particular, Plin is an FWL multivector field.

For the last part of the statement, first notice that the ideal M is preserved by the Poisson bracket. So, if P,QM, then it makes sense to linearize {P,Q}. The rest follows easily from equation (7.1). ∎

Remark 7.2.

Proposition 7.1 is a “symmetric multivector analogue” of the following well-known fact. Let (𝒫,π) be a Poisson manifold, let (T𝒫)π be its cotangent algebroid and let M𝒫 be a coisotropic submanifold. Then the conormal bundle NMT𝒫 is a subalgebroid (T𝒫)π, and hence the normal bundle NM is equipped with a linear Poisson structure πlin (see, e.g., [14, Section 3]).

By definition, the multivector field Plin in Proposition 7.1 is the linearization of P.

We finally come to a generic differential operator ΔDOq().

Definition 7.3.

An order q DO ΔDO() is order q linearizable (around M) if its symbol σ(Δ)𝔛symq() is linearizable.

Remark 7.4.

  • An order q-1 DO ΔDO() is always order q linearizable, but it might not be order q-1 linearizable.

  • A function FC()=DO0() is order 0 linearizable if and only if F|M=0, i.e. F is a linearizable function.

  • A vector field X𝔛()DO1() is order 1 linearizable if and only if it is tangent to M, i.e. it is a linearizable vector field.

Now, let ΔDOq() be an order q linearizable DO. In order to linearize it, we use Theorem 6.7. In other words, from Δ we cook up a q-LE-multivector (Plin,Φ) of LE=E×ML, with L=topE as in Section 6. The construction is inspired by the proof of Theorem 6.7 itself. We let Plin𝔛sym,linq(E) be the linearization of the symbol P=σ(Δ) of Δ. It remains to define the vector bundle map

Φ:Sq-1EDL.

For φ1,,φq-1Γ(E), we define Φ(φ1,,φq-1) as follows. Pick linearizable functions FiC() such that (Fi)lin=φi and consider the following function on M:

fF1,,Fq-1:=[[[Δ,F1],],Fq-1](1)|M.

We put

Φ(φ1,,φk-1)(U)=Plin(φ1,,φq-1,-)U+fF1,,Fk-1U

for all UΓ(L). On the right-hand side, we interpret Plin as a multiderivation of E. We have to show that Φ is well defined, i.e. fF1,,Fq-1 does only depend on the φi, and Φ is indeed a vector bundle map Φ:Sq-1EDL as desired, i.e. it is (symmetric and) C(M)-linear in its arguments. To see that fF1,,Fq-1 does not depend on the choice of the Fi, assume that (Fq-1)lin=0. Hence, we have Fq-1|M=0 and dFq-1|M=0. Additionally,

fF1,,Fq-1=[[[Δ,F1],],Fq-1](1)|M
=(Fq-1[[[Δ,F1],],Fq-2](1)-[[[Δ,F1],],Fq-2](Fq-1))|M
=-[[[Δ,F1],],Fq-2](Fq-1)|M.

Now, [[[Δ,F1],],Fq-2] is a second order DO. As the first jet of Fq-1 vanishes on M, the expression [[[Δ,F1],],Fq-2](Fq-1)|M does only depend on the symbol of [[[Δ,F1],],Fq-2] (and the Hessian of Fq-1). But

σ([[[Δ,F1],],Fq-2])=σ(Δ)(F1,,Fq-2,-,-).

So, if P=σ(Δ) is locally given by (7.2), using Pα1αq(X,0)=0, we find that, locally,

fF1,,Fq-1Pi,α1αq-2αq-1(x,0)(F1)α1(x)(Fq-2)αq-2(x)xi(Fq-1)αq-1(x)=0.

We conclude that Φ(φ1,,φq-1) is well-defined. It remains to see that Φ is C(M)-linear in one, and hence in all, of its arguments. So, take fC(M). The simple formula

(7.3)(FG)lin=F|MGlinfor all FC() and all linearizable GC()

shows that, when we replace φq-1 with fφq-1 in Φ(φ1,,φq-1), we can replace Fq-1 with FFq-1, where FC() is any function such that F|M=f. So, we have

Φ(φ1,,φq-2,fφq-1)U=Plin(φ1,,φq-2,fφq-1,-)U+fF1,,Fq-2,FFq-1U.

For the first summand, recall from the proof of Theorem 6.7 (equations (6.11) and (6.12)) that

Plin(φ1,,φq-2,fφq-1,-)U=fPlin(φ1,,φq-2,φq-1,-)U-lPlin(φ1,,φq-1)(f)U

For the second summand, compute

fF1,,Fq-2,FFq-1=[[[Δ,F1],,Fq-2],FFq-1](1)|M
=F[[[[Δ,F1],],Fq-2],Fq-1](1)|M+[[[[Δ,F1],],Fq-2],F](Fq-1)|M
=ffF1,,Fq-1+[[[[[Δ,F1],],Fq-2],F],Fq-1](1)|M
=FfF1,,Fq-1+σ(Δ)(F1,,Fq-1,F)|M.

Now, to conclude the proof of the C(M)-multilinearity, it is enough to show that

(7.4)σ(Δ)(F1,,Fq-1,F)|M=lPlin(φ1,,φq-1)(f).

But this follows easily from the fact that Plin is the linearization of σ(Δ) and formula (7.3) again.

Finally, by construction, lΦ=lPlin, so the pair (Plin,Φ) corresponds to a q-LE-multivector. We call the linear DO A-1(Plin,Φ) the linearization of Δ and denote it Δlin.

The above discussion leads to the following theorem.

Theorem 7.5.

The assignment ΔΔlin is a well-defined (linearization) map from order q linearizable DOs on E and order q FWL DOs on E. The linearization preserves the commutator of DOs in the following sense: let ΔDOq+1(E) and ΔDOq+1(E) be an order q+1 linearizable and an order q+1 linearizable DO, respectively. Then [Δ,Δ] is order q+q+1 linearizable, and its linearization is [Δlin,Δ]lin.

Proof.

We only need to prove the last part of the statement. To do that, recall that the commutator [Δ,Δ] is a DO of order q+q+1 and its symbol is the Poisson bracket {σ(Δ),σ(Δ)}. Let Δlin=A-1(Plin,Φ) and Δ=linA-1(P,linΦ) be their linearizations. As the ideal M in 𝔛sym() is preserved by the Poisson bracket, [Δ,Δ] is order q+q+1 linearizable as desired. Let [Δ,Δ]lin=A-1(P′′,linΦ′′) be its linearization. As the linearization of multivector fields preserves the Poisson bracket, P′′=lin{Plin,P}lin. Finally, we have to take care of Φ′′. It is a straightforward computation, which we sketch to point out the possible subtleties related to the properties of the various objects involved. Recall from the discussion preceding the statement of the theorem that

Φ(φ1,,φq)U=Plin(φ1,,φq)U+fF1,,FqU,

where

fF1,,Fq=[[Δ,F1],,Fq](1)|M,

and likewise for Φ,Φ′′. Here Fi is any linearizable function on such that (Fi)lin=φi. In the following, to stress that, actually, the function fF1,,Fq does only depend on the φi, we write fφ1,,φq (instead of fF1,,Fq). Additionally, we call f (resp. f,f′′) the f-component of the FWL DO Δlin (resp. Δ,lin[Δlin,Δ]lin). We have to show that the f-component f[Δ,Δ] of [Δ,Δ]lin agrees with f′′. To do this, we compute f[Δ,Δ] explicitly. By symmetry and -multilinearity, it is enough to evaluate it on q+q equal sections φ1==φq+q=φΓ(E). So, let FC() be a linearizable function such that φ=Flin. We have

fφ,,φq+q times[Δ,Δ]=[[[Δ,Δ],F],,F]q+q times(1)|M
=i+j=q+q1i!j![[[Δ,F],,F]i times,[[Δ,F],,F]j times](1)|M.

The only terms that survive are those with i=q-1,q,q+1 (and hence j=q+1,q,q-1, respectively). We call them Tq-1,Tq,Tq+1, respectively. The first one is given by

Tq-1=1(q-1)!(q+1)![[[Δ,F],,F]q-1 times,[[Δ,F],,F]q+1 times](1)|M
=1(q-1)!(q+1)!fφ,,φq-1 times,P(φ,,φq+1 times)lin.

Similarly, the third one is

Tq+1=-1(q+1)!(q-1)!fφ,,φq-1 times,Plin(φ,,φq+1 times).

To compute Tq, we use a simple trick: for every two scalar DOs , we have

[,](1)=[,(1)](1)-[,(1)](1).

After using this formula, we get

Tq=1q!q![[[Δ,F],,F]q times,[[Δ,F],,F]q times](1)|M
=1q!q!σ(Δ)(F,,Fq times,[[Δ,F],,F]q times(1))|M-1q!q!σ(Δ)(F,,Fq times,[[Δ,F],,F]q times(1))|M.

Now we use that both Δ and F are linearizable to replace [[Δ,F],,F](1) with its restriction to M in the first summand (likewise for Δ in the second summand). We get

Tq=1q!q!σ(Δ)(F,,Fq times,fφ,,φq times)|M-1q!q!σ(Δ)(F,,Fq times,fφ,,φq times)|M
=1q!q!lPlin(φ,,φq times)(fφ,,φq times)-1q!q!lPlin(φ,,φq times)(fφ,,φq times),

where we also used (7.4). At this point, it can be checked directly, exploiting the explicit formula (4.1) for the Poisson bracket of LE-multivectors, that Tq-1+Tq+Tq+1 agrees with the f-component f′′ of [Δlin,Δ]lin. We conclude that [Δlin,Δ]lin=[Δ,Δ]lin, as desired. ∎

Remark 7.6.

Let be the total space of a vector bundle EM and interpret M as a submanifold in =E via the zero section. In this situation, the normal bundle NM to M identifies canonically with E itself. Clearly, an order q FWL DO ΔDOq,lin(E) is automatically order q linearizable, and it easily follows from the proof of Theorem 7.5 that the vector bundle isomorphism ENM identifies Δ with its own linearization Δlin.

Remark 7.7.

As mentioned in Section 1, the results in the present paper should be considered as a first step towards a theory of multiplicative/infinitesimally multiplicative DOs on Lie groupoids/Lie algebroids and their Lie theory. The reader may consult, e.g., [7] for a recent review on multiplicative structures on Lie groupoids and their infinitesimal counterparts: infinitesimally multiplicative structures on Lie algebroids. We speculate that the ultimate definition of a multiplicative DO on a Lie groupoid (which we do not know yet) will fit into a theorem of the following form: Let 𝒢M be a Lie groupoid with Lie algebroid AM and let ΔDO(𝒢) be a multiplicative DO on 𝒢. Then Δ is linearizable around M and its linearization ΔlinDOlin(A) is an infinitesimally multiplicative DO on A; if 𝒢 is source simply connected, then this correspondence between multiplicative and infinitesimally multiplicative DO can be inverted. Finally, we expect that non-trivial higher order examples will come from the theory of multiplicative metrics on Lie groupoids. Namely, we speculate that the Laplacian operator (on functions) determined by a multiplicative metric on a Lie groupoid 𝒢 is a multiplicative DO, and hence its linearization is an infinitesimally multiplicative DO on the Lie algebroid of A. Exploring this class of examples requires a lot of work, including finding the appropriate definition of multiplicative/infinitesimally multiplicative DO and this goes far beyond the scopes of the present paper.


Communicated by Jan Frahm


Acknowledgements

L. Vitagliano is a member of the GNSAGA of INdAM. We thank the referee for numerous suggestions that improved the presentation.

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Received: 2021-04-27
Revised: 2021-08-04
Published Online: 2021-09-25

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