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Bol operators (Bols for short) are differential operators invariant under the projective action of [Formula: see text] between spaces of weighted densities on the 1-dimensional manifold. Here, we described analogs of Bols: [Formula: see text]-invariant differential operators between spaces of tensor fields on [Formula: see text]-dimensional supermanifolds with irreducible, as [Formula: see text]-modules, fibers of arbitrary, even infinite, dimension for certain “key” values of [Formula: see text] and [Formula: see text] — the ones for which the solution is describable. We discovered many new operators for [Formula: see text] and for the case of [Formula: see text]-dimensional general superstring, which looks like a most natural superization of Bol’s result, additional to the cases of super analogs of Bols between spaces of weighted densities on the [Formula: see text]-dimensional superstrings with a contact structure we classified in arXiv:2110.10504. In the case of fibers of dimension [Formula: see text], there are [Formula: see text]-parameter families of Bols, whereas there are no non-scalar nonzero differential operators between spaces of weighted densities. These two extreme answers justify the selection of cases here.

The Bol operators are unary differential operators between spaces of weighted densities on the 1-dimensional manifold invariant under projective transformations of the manifold. On the $1|n$-dimensional supermanifold (superstring) $\mathcal{M}$, we classify analogs of Bol operators invariant under the simple maximal subalgebra $\mathfrak{h}$ of the same rank as its simple ambient superalgebra $\mathfrak{g}$ of vector fields on $\mathcal{M}$ and containing all elements of negative degree of $\mathfrak{g}$ in a $\mathbb{Z}$-grading. We also consider the Lie superalgebras of vector fields $\mathfrak{g}$ preserving a contact structure on the superstring $\mathcal{M}$. We have discovered many new operators., Comment: 23 pages

All results concern characteristic 2. Two procedures that to every simple Lie algebra assign simple Lie superalgebras, most of the latter new, are offered. We prove that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures, so we classified all simple finite-dimensional Lie superalgebras modulo non-existing at the moment classification of simple finite-dimensional Lie algebras. This result concerns Lie superalgebras considered naively, as vector spaces. To obtain classification of simple Lie superalgebras in the category of supervarieties, one should list the non-isomorphic deforms (results of deformations) with odd parameter. This problem is open bar several examples described in arXiv~0807.3054. For Lie algebras, in addition to the known ---"classical" --- restrictedness, we introduce a (2,4)-structure on the two non-alternating series: orthogonal and of Hamiltonian vector fields. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: $(2|4)$- and $(2|2)$-structures, one more analog --- a $(2,4)|4$-structure on Lie superalgebras is the analog of (2,4)-structure on Lie algebras.

For each of the exceptional (not entering infinite series) finite-dimensional modular Lie superalgebras with indecomposable Cartan matrix, we give the explicit list of its roots, and the corresponding Chevalley basis, for one of its inequivalent Cartan matrices, namely the one corresponding to the greatest number of mutually orthogonal isotropic odd simple roots (this number, called the defect of the Lie superalgebra, is important in the representation theory). Our main tools: Grozman’s Mathematica-based code SuperLie, Python, and A. Lebedev’s help.

A Lie superalgebra is called quasi-Frobenius if it admits a closed anti-symmetric non-degenerate bilinear form. We study the notion of double extensions of quasi-Frobenius Lie superalgebra when the form is either orthosymplectic or periplectic. We show that every quasi-Frobenius Lie superalgebra that satisfies certain conditions can be obtained as a double extension of a smaller quasi-Frobenius Lie superalgebra. We classify all 4-dimensional quasi-Frobenius Lie superalgebras, and show that such Lie superalgebras must be solvable. We study the notion of $T^*$-extensions (or Lagrangian extensions) of Lie superalgebras, and show that they are classified by a certain cohomology space we introduce. Several examples are provided to illustrate our construction., 36 pages; minor corrections

A double extension ($\mathscr{D}$ extension) of a Lie (super)algebra $\mathfrak a$ with a non-degenerate invariant symmetric bilinear form $\mathscr{B}$, briefly: a NIS-(super)algebra, is an enlargement of $\mathfrak a$ by means of a central extension and a derivation; the affine Kac-Moody algebras are the best known examples of double extensions of loops algebras. Let $\mathfrak a$ be a restricted Lie (super)algebra with a NIS $\mathscr{B}$. Suppose $\mathfrak a$ has a restricted derivation $\mathscr{D}$ such that $\mathscr{B}$ is $\mathscr{D}$-invariant. We show that the double extension of $\mathfrak a$ constructed by means of $\mathscr{B}$ and $\mathscr{D}$ is restricted. We show that, the other way round, any restricted NIS-(super)algebra with non-trivial center can be obtained as a $\mathscr{D}$-extension of another restricted NIS-(super)algebra subject to an extra condition on the central element. We give new examples of $\mathscr{D}$-extensions of restricted Lie (super)algebras, and pre-Lie superalgebras indigenous to characteristic 3., An improved version; to appear in Arnold Mathematical Journal

In 1928, at the IMC, Veblen posed the problem: classify invariant differential operators between spaces of “natural objects” (in modern terms: either tensor fields, or jets) over a real manifold of any dimension. The problem was solved by Rudakov for unary operators (no nonscalar operators except the exterior differential); by Grozman for binary operators. In dimension one, Grozman discovered an indecomposable selfdual operator of order 3 that does not exist in higher dimensions. We solve Veblen's problem in the 1-dimensional case over any field of positive characteristic. Unary invariant operators are known: these are the exterior differential and analogs of the Berezin integral. We construct new binary operators from these analogs and discovered two more (up to dualizations) types of new indecomposable operators of however high order: analogs of the Grozman operator and a completely new type of operators. Gordan's transvectants, aka Cohen–Rankin brackets, always invariant with respect to the simple 3-dimensional Lie algebra, are also invariant, in characteristic 2, with respect to the whole Lie algebra of vector fields on the line when the height of the indeterminate is equal to 2.

For the Jordan algebra of hermitian matrices of order n ≥ 2 , we let X be its submanifold consisting of rank-one semi-positive definite elements. The composition of the cotangent bundle map π X : T ∗ X → X with the canonical map X → C P n − 1 (i.e., the map that sends a given hermitian matrix to its column space), pulls back the Kahler form of the Fubini–Study metric on C P n − 1 to a real closed differential two-form ω K on T ∗ X . Let ω X be the canonical symplectic form on T ∗ X and μ a real number. A standard fact says that ω μ ≔ ω X + 2 μ ω K turns T ∗ X into a symplectic manifold, hence a Poisson manifold with Poisson bracket { , } μ . In this article we exhibit a Poisson realization of the simple real Lie algebra s u ( n , n ) on the Poisson manifold ( T ∗ X , { , } μ ) , i.e., a Lie algebra homomorphism from s u ( n , n ) to C ∞ ( T ∗ X , R ) , { , } μ . Consequently one obtains the Laplace–Runge–Lenz vector for the classical U ( 1 ) -Kepler problem of level n and magnetic charge μ . Since the McIntosh–Cisneros–Zwanziger–Kepler problems (MICZ-Kepler Problems) are the U ( 1 ) -Kepler problems of level 2 , the work presented here is a direct generalization of the work by A. Barut and G. Bornzin (1971) on the classical dynamic symmetry for the MICZ-Kepler problems.

A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form $B$ is called a nis-(super)algebra. The double extension $\mathfrak{g}$ of a nis-(super)algebra $\mathfrak{a}$ is the result of simultaneous adding to $\mathfrak{a}$ a central element and a derivation so that $\mathfrak{g}$ is a nis-algebra. Loop algebras with values in simple complex Lie algebras are most known among the Lie (super)algebras suitable to be doubly extended. In characteristic 2 the notion of double extension acquires specific features. Restricted Lie (super)algebras are among the most interesting modular Lie superalgebras. In characteristic 2, using Grozman's Mathematica-based package SuperLie, we list double extensions of restricted Lie superalgebras preserving the non-degenerate closed 2-forms with constant coefficients. The results are proved for the number of indeterminates ranging from 4 to 7 - sufficient to conjecture the pattern for larger numbers. Considering multigradings allowed us to accelerate computations up to 100 times., 18 pages

We review the list of non-degenerate invariant (super)symmetric bilinear forms (briefly: NIS) on the following simple (relatives of) Lie (super)algebras: (a) with symmetrizable Cartan matrix of any growth, (b) with non-symmetrizable Cartan matrix of polynomial growth, (c) Lie (super)algebras of vector fields with polynomial coefficients, (d) stringy a.k.a. superconformal superalgebras, (e) queerifications of simple restricted Lie algebras. Over algebraically closed fields of positive characteristic, we establish when the deform (i.e., the result of deformation) of the known finite-dimensional simple Lie (super)algebra has a NIS. Amazingly, in most of the cases considered, if the Lie (super)algebra has a NIS, its deform has a NIS with the same Gram matrix after an identification of bases of the initial and deformed algebras. We do not consider odd parameters of deformations. Closely related with simple Lie (super)algebras with NIS is the notion of doubly extended Lie (super)algebras of which affine Kac--Moody (super)algebras are the most known examples., 42 pages. Definitions of certain Lie (super)algebras follow previous works of some of the authors

Several improvements of the Kostrikin-Shafarevich method conjecturally producing all simple finite-dimensional Lie algebras over algebraically closed fields of any positive characteristic were rece ...

Of four types of Kaplansky algebras, type-2 and type-4 algebras have previously unobserved Z/2-gradings: nonlinear in roots. A method assigning a simple Lie superalgebra to every Z/2-graded simple Lie algebra in characteristic 2 is illustrated by seven new series. Type-2 algebras and one of the two type-4 algebras are demystified as nontrivial deforms (the results of deformations) of the alternate Hamiltonian algebras. The type-1 Kaplansky algebra is recognized as the derived of the nonalternate version of the Hamiltonian Lie algebra, the one that preserves a tensorial 2-form. Deforms corresponding to nontrivial cohomology classes can be isomorphic to the initial algebra, e.g., we confirm Grishkov's implicit claim and explicitly describe the Jurman algebra as such a semitrivial deform of the derived of the alternate Hamiltonian Lie algebra. This paper helps to sharpen the formulation of a conjecture describing all simple finite-dimensional Lie algebras over any algebraically closed field of nonzero characteristic and supports a conjecture of Dzhumadildaev and Kostrikin stating that all simple finite-dimensional modular Lie algebras are either of standard type or deforms thereof. In characteristic 2, we give sufficient conditions for the known deformations to be semitrivial.

A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form will be called a NIS-Lie (super)algebra. The double extension of a NIS-Lie (super)algebra is the result of simultaneously adding to it a central element and an outer derivation so that the larger algebra has also a NIS. Affine loop algebras, Lie (super)algebras with symmetrizable Cartan matrix over any field, Manin triples, symplectic reflection (super)algebras are among the Lie (super)algebras suitable to be doubly extended. We consider double extensions of Lie superalgebras in characteristic 2, and concentrate on peculiarities of these notions related with the possibility for the bilinear form, the center, and the derivation to be odd. Two Lie superalgebras we discovered by this method are indigenous to the characteristic 2., To appear in Journal of Algebra

We answered the old question: does there exist a mechanical system with 3 degrees of freedom, except for the Coulomb system, which has 6 first integrals generating the Lie algebra o(4) by means of the Poisson brackets? We presented a system which is not centrally symmetric, but has such 6 first integrals. We showed also that not every mechanical system with 3 degrees of freedom possesses such Lie algebra o(4)., 8 pages. A mistake in the last Subsection of version 1 is corrected together with its corollaries

In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that mathematicians study are real or (almost) complex ones, while Minkowski superspaces are completely different objects. They are what we call almost real-complex supermanifolds, i.e., real supermanifolds with a non-integrable distribution, the collection of subspaces of the tangent space, and in every subspace a complex structure is given. An almost complex structure on a real supermanifold can be given by an even or odd operator; it is complex (without "always") if the suitable superization of the Nijenhuis tensor vanishes. On almost real-complex supermanifolds, we define the circumcised analog of the Nijenhuis tensor. We compute it for the Minkowski superspaces and superstrings. The space of values of the circumcised Nijenhuis tensor splits into (indecomposable, generally) components whose irreducible constituents are similar to those of Riemann or Penrose tensors. The Nijenhuis tensor vanishes identically only on superstrings of superdimension 1|1 and, besides, the superstring is endowed with a contact structure. We also prove that all real forms of complex Grassmann algebras are isomorphic although singled out by manifestly different anti-involutions., Exposition of the same results as in v.1 is more lucid. Reference to related recent work by Witten is added

The finite dimensional simple modular Lie algebras with Cartan matrix cannot be deformed if the characteristic p of the ground field is equal to 0 or greater than 3. If p=3, the orthogonal Lie algebra o(5)is one of the two simple modular Lie algebras with Cartan matrix that have deformations (the Brown algebras br(2; a) are among these 10-dimensional deforms and hence are not counted separately); the 29-dimensional Brown algebra br(3) is the only other simple Lie algebra with Cartan matrix that has deformations. Kostrikin and Kuznetsov described the orbits (isomorphism classes) under the action of the group O(5) of automorphisms of o(5) on the space H^2(o(5);o(5)) and produced representatives of the isomorphism classes. Here we explicitly describe global deforms of o(5) and of the simple analog of this orthogonal Lie algebra in characteristic 2., Comment: 12 pages, no figures

Recently, Grozman and Leites returned to the original Cartan's description of Lie algebras to interpret the Melikyan algebras (for p, 10 pages; no figures

We overview the classifications of simple finite-dimensional modular Lie algebras. In characteristic 2, their list is wider than that in other characteristics; e.g., it contains desuperizations of modular analogs of complex simple vectorial Lie superalgebras. We consider odd parameters of deformations. For all 15 Weisfeiler gradings of the 5 exceptional families, and one Weisfeiler grading for each of 2 serial simple complex Lie superalgebras (with 2 exceptional subseries), we describe their characteristic-2 analogs - new simple Lie algebras. Descriptions of several of these analogs, and of their desuperizations, are far from obvious. One of the exceptional simple vectorial Lie algebras is a previously unknown deform (the result of a deformation) of the characteristic-2 version of the Lie algebra of divergence-free vector fields; this is a new simple Lie algebra with no analogs in characteristics distinct from 2. In characteristic 2, every simple Lie superalgebra can be obtained from a simple Lie algebra by one of the two methods described in arXiv:1407.1695. Most of the simple Lie superalgebras thus obtained from simple Lie algebras we describe here are new., see pdf-file with index at https://www.emis.de/journals/SIGMA/2020/089/

Let $M$ be either a projective manifold $(M,Pi)$ or a pseudo-Riemannian manifold $(M,g).$ We extend, intrinsically, the projective/conformal Schwarzian derivatives that we have introduced recently, to the space of differential operators acting on symmetric contravariant tensor fields of any degree on $M.$ As operators, we show that the projective/conformal Schwarzian derivatives depend only on the projective connection $Pi$ and the conformal class $[g]$ of the metric, respectively. Furthermore, we compute the first cohomology group of $Vect(M)$ with coefficients into the space of symmetric contravariant tensor fields valued into $delta$-densities as well as the corresponding relative cohomology group with respect to $sl(n+1,R).$, Comment: 33 pages, no figures, Latex2e. A completely rewritten version, new results have been added; we extend the projective/conformal Schwarzian derivatives to tensor fields of any degree; we compute the first-cohomology group of the Lie algebra of smooth vector fields with values into the space of differential operators acting on contravariant tensor fields valued into d-densities, generalizing a result of Lecomte-Ovsienko

The main topic of this paper is two-fold. First, we compute the first relative cohomology group of the Lie algebra of smooth vector fields on the projective line, Vect(ℝℙ1), with coefficients in the space of bilinear differential operators that act on tensor densities, [Formula: see text], vanishing on the Lie algebra sl(2, ℝ). Second, we compute the first cohomology group of the Lie algebra sl(2, ℝ) with coefficients in [Formula: see text].

Let (M, F) be a Finsler manifold. We construct a 1-cocycle on Diff(M) with values in the space of differential operators acting on sections of some bundles, by means of the Finsler fonction F. As an operator, it has several expressions: in terms of the Chern, Berwald, Cartan or Hashiguchi connection, although its cohomology class does not depend on them. This cocycle is closely related to the conformal Schwarzian derivatives introduced in our previous work. The second main result of this paper is to discuss some properties of the conformally invariant quantization map by means of a Sazaki (type) metric on the slit bundle T Minduced by F.

A Poisson realization of the simple real Lie algebra so*(4n) on the phase space of each Sp(1)-Kepler problem is exhibited. As a consequence, one obtains the Laplace-Runge-Lenz vector for each classical Sp(1)-Kepler problem. The verification of these Poisson realizations is greatly simplified via an idea of Weinstein. The totality of these Poisson realizations is shown to be equivalent to the canonical Poisson realization of so*(4n) on the Poisson manifold T*H*n/Sp(1). (Here H*n≔Hn\{0} and the Hamiltonian action of Sp(1) on T*H*n is induced from the natural right action of Sp(1) on H*n.)

A new formula for the multi-dimensional Schwarzian derivative on a locally projective manifold is introduced. This derivative is a 1-cocycle on the group of diffeomorphisms with values in the space of linear differential operators acting from (symmetric contravariant) twice-tensor fields to functions.

Let M be a manifold endowed with a symmetric affine connection Γ. The aim of this Letter is to describe a quantization map between the space of second-order polynomials on the cotangent bundle T* M and the space of second-order linear differential operators, both viewed as modules over the group of diffeomorphisms and the Lie algebra of vector fields on M. This map is an isomorphism, for almost all values of certain constants, and it depends only on the projective class of the affine connection Γ.

We consider a contact manifold with a pseudo-Riemannian metric and define a contact vector field intrinsically associated to this pair of structures. We call this new differential invariant the contact Riemannian curl. On a Riemannian manifold, Killing vector fields are those that annihilate the metric; a Killing $1$-form is obtained from a Killing vector field by lowering indices. We show that the contact Riemannian curl vanishes if the metric is of constant curvature and the contact structure is defined by a Killing $1$-form. We also show that the contact Riemannian curl has a strong similarity with the Schwarzian derivative since it depends only on the projective equivalence class of the metric. For the Laplace-Beltrami operator on a contact manifold, the contact Riemannian curl is proportional to the subsymbol defined in arXiv:1205.6562. We also show that the contact Riemannian curl vanishes on the (co)tangent bundle over a Riemannian manifold. This implies that the corresponding subsymbol of the Laplace-Beltrami operator is identically zero., Comment: 16 pages, Theorem 5.2.1 corrected

For modular Lie superalgebras, new notions are introduced: Divided power homology and divided power cohomology. For illustration, we give presentations (in terms of analogs of Chevalley generators) of finite dimensional Lie (super)algebras with indecomposable Cartan matrix in characteristic 2 (and in other characteristics for completeness of the picture). We correct the currently available in the literature notions of Chevalley generators and Cartan matrix in the modular and super cases, and an auxiliary notion of the Dynkin diagram. In characteristic 2, the defining relations of simple classical Lie algebras of the A, D, E types are not only Serre ones; these non-Serre relations are same for Lie superalgebras with the same Cartan matrix and any distribution of parities of the generators. Presentations of simple orthogonal Lie algebras having no Cartan matrix are also given.., Expounds math/0611391 and math/0611392, it contains new theorems, new material on relations for p=2, and - most important - on div. power (co)homology.To appear in Homol. Homot. Appl

Over n-dimensional manifolds, I classify ternary differential operators acting on the spaces of weighted densities and invariant with respect to the Lie algebra of vector fields. For n=1, some of these operators can be expressed in terms of the de Rham exterior differential, the Poisson bracket, the Grozman operator and the Feigin-Fuchs anti-symmetric operators; four of the operators are new, up to dualizations and permutations. For n>1, I list multidimensional conformal tranvectors, i.e.,operators acting on the spaces of weighted densities and invariant with respect to o(p+1,q+1), where p+q=n. Except for the scalar operator, these conformally invariant operators are not invariant with respect to the whole Lie algebra of vector fields., 13 pages, no figures, to appear in Theor. Math. Phys

Over algebraically closed fields of characteristic p>2, prolongations of the simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. Several new simple Lie superalgebras are discovered, serial and exceptional, including superBrown and superMelikyan superalgebras. Simple Lie superalgebras with Cartan matrix of rank 2 are classified., 20 pages, to appear in Funktsional. Anal. i Prilozhen

Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the corresponding Lie superal- gebra is simple otherwise the quotient of the derived Lie superalgebra modulo center is simple (if its rank is greater than 1). Eleven new exceptional simple modular Lie superalgebras are discovered. Several features of classic notions, or notions themselves, are clarified or introduced, e.g., Cartan matrix, several versions of restrictedness in characteristic 2, Dynkin diagram, Chevalley generators, and even the notion of Lie superalgebra if the characteristic is equal to 2. Interesting phenomena in characteristic 2: (1) all simple Lie superalgebras with Cartan matrix are obtained from simple Lie algebras with Cartan matrix by declaring several (any) of its Chevalley generators odd; (2) there exist simple Lie superalgebras whose even parts are solvable. The Lie superalgebras of fixed points of automorphisms corresponding to the symmetries of Dynkin diagrams are also listed and their simple subquotients described.

By computing certain cohomology of Vect(M) of smooth vector fields we prove that on 1-dimensional manifolds M there is no quantization map intertwining the action of non-projective embeddings of the Lie algebra sl(2) into the Lie algebra Vect(M). Contrariwise, for projective embeddings sl(2)-equivariant quantization exists., 09 pages, LaTeX2e, no figures; to appear in Journal of Nonlinear Mathematical Physics

The space of m-ary differential operators acting on weighted densities is a (m+1)-parameter family of modules over the Lie algebra of vector fields. For almost all the parameters, we construct a canonical isomorphism between this space and the corresponding space of symbols as sl(2)-modules. This yields to the notion of the sl(2)-equivariant symbol calculus for m-ary differential operators. We show, however, that these two modules cannot be isomorphic as sl(2)-modules for some particular values of the parameters. Furthermore, we use the symbol map to show that all modules of second-order m-ary differential operators are isomorphic to each other, except for few modules called singular., Comment: 20 pages; LaTeX2e; minor corrections

Let $\Sigma$ be an open Riemann surface and $Hol (\Sigma)$ be the Lie algebra of holomorphic vector fields on $\Sigma.$ We fix a projective structure (i.e. a local $SL_2(C)-$structure) on $\Sigma.$ We calculate the first group of cohomology of $Hol(\Sigma)$ with coefficients in the space of linear holomorphic operators acting on tensor densities, vanishing on the Lie algebra $SL_2 (C).$ The result is independant on the choice of the projective structure. We give explicit formulae of 1-cocycles generating this cohomology group., Comment: 8 pages, no figures, Latex2e

The first group of differentiable cohomology of $\Diff(S^1)$, vanishing on the M\"obius subgroup $PSL(2,R)\subset\Diff(S^1)$, with coefficients in modules of linear differential operators on $S^1$ is calculated. We introduce three non-trivial $PSL(2,R)$-invariant 1-cocycles on $\Diff(S^1)$ generalizing the Schwarzian derivative., Comment: 18 pages, LaTex. to appear in IMRN

Let Vect(R) be the Lie algebra of smooth vector fields on R. The space of symbols Pol(T ∗ R) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(R)-module that becomes trivial once the action is restricted to sl(2) ⊂ Vect(R). The deformations of Pol(T ∗ R), which become trivial once the action is restricted to sl(2) and such that the Vect(R)-action on them is expressed in terms of differential operators, are classified by the elements of the weight basis of H 2 (Vect(R), sl(2); D�,µ), where H i denotes the differential cohomology (i.e., we consider only cochains that are given by differential operators) and where D�,µ = Homdiff(F�, Fµ) is the space of differential operators acting on weighted densities. The main result of this paper is computation of this cohomology. In addition to relative cohomology, we exhibit 2-cocycles spanning H 2 (g; D�,µ) for g = Vect(R) and sl(2).

Let $(M,g)$ be a pseudo-Riemannian manifold. We propose a new approach for defining the conformal Schwarzian derivatives. These derivatives are 1-cocycles on the group of diffeomorphisms of $M$ related to the modules of linear differential operators. As operators, these derivatives do not depend on the rescaling of the metric $g.$ In particular, if the manifold $(M,g)$ is conformally flat, these derivatives vanish on the conformal group $\Og(p+1,q+1),$ where $\mathrm{dim} (M)=p+q.$ This work is a continuation of [1,4] where the Schwarzian derivative was defined on a manifold endowed with a projective connection.

Let $M$ be a manifold and $T^*M$ be the cotangent bundle. We introduce a 1-cocycle on the group of diffeomorphisms of $M$ with values in the space of linear differential operators acting on $C^{\infty} (T^*M).$ When $M$ is the $n$-dimensional sphere, $S^n$, we use this 1-cocycle to compute the first-cohomology group of the group of diffeomorphisms of $S^n$, with coefficients in the space of linear differential operators acting on contravariant tensor fields., Comment: arxiv version is already official