## Projectively equivariant symbol calculus (1999)

Citations: | 26 - 2 self |

### BibTeX

@MISC{Lecomte99projectivelyequivariant,

author = {P. B. A. Lecomte and V. Yu. Ovsienko},

title = {Projectively equivariant symbol calculus},

year = {1999}

}

### OpenURL

### Abstract

The spaces of linear differential operators Dλ(R n) acting on λ-densities on R n and the space Pol(T ∗ R n) of functions on T ∗ R n which are polynomial on the fibers are not isomorphic as modules over the Lie algebra Vect(R n) of vector fields of R n. However, these modules are isomorphic as sl(n + 1, R)-modules where sl(n + 1, R) ⊂ Vect(R n) is the Lie algebra of infinitesimal projective transformations. In addition, such an sln+1-equivariant bijection is unique (up to normalization). This leads to a notion of projectively equivariant quantization and symbol calculus for a manifold endowed with a (flat) projective structure. We apply the sln+1-equivariant symbol map to study the Vect(M)-modules D k λ (M) of k-th-order linear differential operators acting on λ-densities, for an arbitrary manifold M and classify the quotient-modules D k λ (M)/Dℓ λ (M). 1

### Citations

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94 |
On a trace functional for formal pseudo–differential operators and the symplectic structure for Korteweg–de Vries type equations
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(Show Context)
Citation Context ...e involution λ ′ = 1/2 − λ, one has Bs(λ ′ ) = (−1) s Bs(λ). Remark: the duality. There exists a nondegenerate natural pairing of ΨDk /ΨDℓ and ΨD−ℓ−2 /ΨD−k−2. It is given by the so-called Adler trace =-=[1]-=-: if A ∈ ΨDk , where k ∈ Z, then ∫ tr(A) = S 1 a1(x)dx. Let now A ∈ ΨDk /ΨDℓ and B ∈ ΨD−ℓ−2 /ΨD−k−2. Put (A,B) := tr( Ã ˜ B), where Ã ∈ ΨDk , ˜ B ∈ ΨD−ℓ−2 are arbitrary lifts of A and B. Adler’s trace... |

59 |
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Citation Context ...t an sln+1-equivariant symbol is given by a differential map. 1.5 Remarks. (a) The relationship between differential operators and projective geometry has already been studied in the fundamental book =-=[19]-=-. The best known example is the Sturm-Liouville operator d 2 /dx 2 +u(x) describing a projective structure on R (or on S 1 if u(x) is periodic). (b) In the one-dimensional case (n = 1), the sl2-equiva... |

52 |
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(Show Context)
Citation Context ...λ∂iX i φ, (2.1) where ∂i = ∂/∂x i . Note, that the formula (2.1) does not depend on the choice of local coordinates. Remark. Modules Fλ are not isomorphic to each other for different values of λ (cf. =-=[8]-=-). The simplest examples of modules of tensor densities are F0 = C ∞ (R n ) and F1 = Ω n (R n ), the module F 1/2 is particularly important for geometric quantization (see [2, 11]). Definition. Consid... |

32 |
Automorphic pseudodifferential operators. In Algebraic aspects of integrable systems
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(Show Context)
Citation Context ...rators Dλ(M). The module D 1/2(M) on half-densities plays a special role (see [2, 11]). It worth noticing that modules of differential operators on tensor densities have been studied in recent papers =-=[6, 13, 9, 4]-=-. 1.2 One of the difficulties of quantization is that there is no natural quantization map. In other words, Pol(T ∗ M) and Dλ(M) are not isomorphic as modules over the group Diff(M) of diffeomorphisms... |

24 |
Une caractérisation abstraite des opérateurs différentiels
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Citation Context ...LE + (p − s + 1)Id) · · · (LE + p Id)T(Q) = (E + (p − q − s + 1)Id) · · · (E + (p − q)Id)T(Q). With s = p − q + 1, we obtain T(P)0 = 0. Hence the result. Remarks. (a) By the well-known Peetre theorem =-=[17]-=-, the map T from Theorem 5.1 is (locally) a differential operator. This theorem therefore generalizes the statement of Theorem 4.1 that every sln+1-equivariant symbol map is differential. (b) The Lie ... |

22 |
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Citation Context ...ant symbol was carried out. (c) In the (algebraic) case of global differential operators on CP n , existence and uniqueness of the sln+1-equivariant symbol is a corollary of Borho-Brylinski’s results =-=[3]-=-: D(CP n ), as a module over sl(n + 1, C), has a decomposition as a sum of irreducible submodules of multiplicity one. This implies the uniqueness result. Our explicit formulæ are valid in the holomor... |

20 |
Space of second order linear differential operators as a module over the Lie algebra of vector fields
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- 1997
(Show Context)
Citation Context ...rators Dλ(M). The module D 1/2(M) on half-densities plays a special role (see [2, 11]). It worth noticing that modules of differential operators on tensor densities have been studied in recent papers =-=[6, 13, 9, 4]-=-. 1.2 One of the difficulties of quantization is that there is no natural quantization map. In other words, Pol(T ∗ M) and Dλ(M) are not isomorphic as modules over the group Diff(M) of diffeomorphisms... |

19 |
Comparison of some modules of the Lie algebra of vector fields
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(Show Context)
Citation Context ...rators Dλ(M). The module D 1/2(M) on half-densities plays a special role (see [2, 11]). It worth noticing that modules of differential operators on tensor densities have been studied in recent papers =-=[6, 13, 9, 4]-=-. 1.2 One of the difficulties of quantization is that there is no natural quantization map. In other words, Pol(T ∗ M) and Dλ(M) are not isomorphic as modules over the group Diff(M) of diffeomorphisms... |

15 |
On the cohomology of sl(m + 1, R) acting on differential operators and sl(m + 1, R)-equivariant symbol
- Lecomte
(Show Context)
Citation Context ... Vect(M) in a subsequent article. The restriction of the modules Dλ to sl(n + 1, R) leads to the sl(n + 1, R)-cohomology (with the same coefficients). The complete answer in this case was obtained in =-=[12]-=-. Acknowledgments. It is a pleasure to acknowledge numerous fruitful discussions with C. Duval and his constant interest to this work. We are grateful to J.-L. Brylinski, R. Brylinski, M. De Wilde and... |

14 | Ovsienko V, Space of linear differential operators on the real line as a module over the Lie algebra of vector
- Gargoubi
- 1996
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Citation Context |

12 |
The Schwarzian derivative for maps between manifolds with complex projective connections
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Citation Context ...nsor field: ℓk(X)(P) = 〈 ¯ ℓ(X),P 〉, where : ¯ℓ(X) h ij = ∂ijX h − 1 ( δ n + 1 h i ∂j + δ h j ∂i ) ∂lX l (6.9) This expression is a 1-cocycle on Vect(R n ) vanishing on the subalgebra sl(n+1, R) (cf. =-=[16, 15]-=- and references therein). 6.3 Computing γ λ 2 Proposition 6.4 The map γλ 2 in (6.2) is given by γ λ 2∣ Sk = 1 2(2k + n − 2)(2k + n − 3) sλk , where the operator s λ k : Vect(Rn ) → Hom(S k , S k−2 ) i... |

9 |
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(Show Context)
Citation Context ...n is given by Q 1/2(H) = ∆g + (n + 1) R . (4.17) 12(n + 2) Remark. The problem of quantization of the geodesic flow on a (pseudo)-Riemannian manifold have already been considered by many authors (see =-=[5]-=- end references therein). Various methods leads to formulæ of the type (4.17) but with different values of the multiple in front of the scalar curvature. The formulæ (4.16,4.17) is a new version of th... |

9 | Conformally equivariant quantization - Duval, Ovsienko |

8 | Projectively invariant symbol map and cohomology of vector fields Lie algebras intervening in quantization - Lecomte, Ovsienko |

2 |
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(Show Context)
Citation Context ...8 Marseille, Cedex 9, FRANCE, mailto:ovsienko@cpt.univ-mrs.frdegree λ, this leads to a Diff(M)-module of differential operators Dλ(M). The module D 1/2(M) on half-densities plays a special role (see =-=[2, 11]-=-). It worth noticing that modules of differential operators on tensor densities have been studied in recent papers [6, 13, 9, 4]. 1.2 One of the difficulties of quantization is that there is no natura... |

1 |
Géométrie projective et algèbres de Lie de dimension infinie, habilitation à difiger les recherches
- Ovsienko
- 1994
(Show Context)
Citation Context ...nsor field: ℓk(X)(P) = 〈 ¯ ℓ(X),P 〉, where : ¯ℓ(X) h ij = ∂ijX h − 1 ( δ n + 1 h i ∂j + δ h j ∂i ) ∂lX l (6.9) This expression is a 1-cocycle on Vect(R n ) vanishing on the subalgebra sl(n+1, R) (cf. =-=[16, 15]-=- and references therein). 6.3 Computing γ λ 2 Proposition 6.4 The map γλ 2 in (6.2) is given by γ λ 2∣ Sk = 1 2(2k + n − 2)(2k + n − 3) sλk , where the operator s λ k : Vect(Rn ) → Hom(S k , S k−2 ) i... |