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53
Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems
 Adv. Math
, 2000
"... Abstract. We completely characterize all nonlinear partial differential equations leaving a given finitedimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinar ..."
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Cited by 8 (3 self)
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Abstract. We completely characterize all nonlinear partial differential equations leaving a given finitedimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations, and provide a nonlinear counterpart to quasiexactly solvable quantum Hamiltonians. These results rely on a useful extension of the classical Wronskian determinant condition for linear independence of functions. In addition, new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented.
Multidimensional Schwarzian derivative revisited
, 2001
"... We give intrinsic formula for the multidimensional Schwarzian derivative on a manifold M endowed with a projective connection. This Schwarzian derivative is naturally related to the space of secondorder linear differential operators acting on tensor densities on M. 1 ..."
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Cited by 6 (6 self)
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We give intrinsic formula for the multidimensional Schwarzian derivative on a manifold M endowed with a projective connection. This Schwarzian derivative is naturally related to the space of secondorder linear differential operators acting on tensor densities on M. 1
Schwarzian derivative related to modules of differential operators on a locally projective manifold
, 1998
"... ..."
Projectively Invariant Cocycles of Holomorphic Vector Fields on an Open Riemann Surface
, 2001
"... Let Σ be an open Riemann surface and Hol(Σ) be the Lie algebra of holomorphic vector fields on Σ. We fix a projective structure (i.e. a local SL2(C)−structure) on Σ. We calculate the first group of cohomology of Hol(Σ) with coefficients in the space of linear holomorphic operators acting on tensor d ..."
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Cited by 5 (4 self)
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Let Σ be an open Riemann surface and Hol(Σ) be the Lie algebra of holomorphic vector fields on Σ. We fix a projective structure (i.e. a local SL2(C)−structure) on Σ. We calculate the first group of cohomology of Hol(Σ) with coefficients in the space of linear holomorphic operators acting on tensor densities, vanishing on the Lie algebra sl2(C). The result is independent on the choice of the projective structure. We give explicit formulæ of 1cocycles generating this cohomology group. 1
Cohomology of the vector fields Lie algebras on RP 1 acting on bilinear differential operators
 Int. Jour. Geom. Methods. Mod. Phys
"... The main topic of this paper is two folds. First, we compute the first relative cohomology group of the Lie algebra of smooth vector fields on the projective line, Vect(RP 1), with coefficients in the space of bilinear differential operators that act on tensor densities, Dλ,ν;µ, vanishing on the Lie ..."
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Cited by 5 (4 self)
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The main topic of this paper is two folds. First, we compute the first relative cohomology group of the Lie algebra of smooth vector fields on the projective line, Vect(RP 1), with coefficients in the space of bilinear differential operators that act on tensor densities, Dλ,ν;µ, 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 Dλ,ν;µ. 1
Poncin N., Decomposition of symmetric tensor fields in the presence of a flat contact projective structure
 2008), 252–269, math.DG/0703922. Conformally Equivariant Quantization in Dimension 12 11
"... Abstract. Let M be an odddimensional Euclidean space endowed with a contact 1form α. We investigate the space of symmetric contravariant tensor fields over M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up of those vector fields that preserve the con ..."
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Cited by 5 (1 self)
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Abstract. Let M be an odddimensional Euclidean space endowed with a contact 1form α. We investigate the space of symmetric contravariant tensor fields over M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up of those vector fields that preserve the contact structure. If we consider symmetric tensor fields with coefficients in tensor densities (also called symbols), the vertical cotangent lift of the contact form α is a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symmetric density valued tensor fields. This generalized Hamiltonian operator on the space of symbols is invariant with respect to the action of the projective contact algebra sp(2n+2). These two operators lead to a decomposition of the space of symbols (except for some critical density weights), which generalizes a splitting proposed by V. Ovsienko in [18]. 1.
Conformal Schwarzian derivatives and conformally invariant quantization
 Internat. Math. Res. Not. 2002
"... Let (M, g) be a pseudoRiemannian manifold. We propose a new approach for defining the conformal Schwarzian derivatives. These derivatives are 1cocycles on the group of diffeomorphisms of M related to the modules of linear differential operators. As operators, these derivatives do not depend on the ..."
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Cited by 4 (4 self)
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Let (M, g) be a pseudoRiemannian manifold. We propose a new approach for defining the conformal Schwarzian derivatives. These derivatives are 1cocycles 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 O(p + 1, q + 1), where dim(M) = p + q. This work is a continuation of [2, 4] where the Schwarzian derivative was defined on a manifold endowed with a projective connection. 1
Higher dimensional uniformisation and Wgeometry
, 1994
"... We formulate the uniformisation problem underlying the geometry of W n  gravity using the differential equation approach to W algebras. We construct W n space (analogous to superspace in supersymmetry) as a (n \Gamma 1) dimensional complex manifold using isomonodromic deformations of linear diff ..."
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Cited by 3 (2 self)
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We formulate the uniformisation problem underlying the geometry of W n  gravity using the differential equation approach to W algebras. We construct W n space (analogous to superspace in supersymmetry) as a (n \Gamma 1) dimensional complex manifold using isomonodromic deformations of linear differential equations. The W n manifold is obtained by the quotient of a Fuchsian subgroup of PSL(n; R) which acts properly discontinuously on a simply connected domain in C P n\Gamma1 . The requirement that a deformation be isomonodromic furnishes relations which enable one to convert nonlinear Wdiffeomorphisms to (linear) diffeomorphisms on the W n manifold. We discuss how the Teichmuller spaces introduced by Hitchin can then be interpreted as the space of complex structures or the space of projective structures with real holonomy on the W n  manifold. The projective structures are characterised by Halphen invariants which are appropriate generalisation of the Schwarzian. This construc...
Invariant Differential Equations and the AdlerGel'fandDikii Bracket
, 1996
"... In this paper we find an explicit formula for the most general vector evolution of curves on RP n\Gamma1 invariant under the projective action of SL(n; R). When this formula is applied to the projectivization of solution curves of scalar Lax operators with periodic coefficients, one obtains a corr ..."
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Cited by 3 (0 self)
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In this paper we find an explicit formula for the most general vector evolution of curves on RP n\Gamma1 invariant under the projective action of SL(n; R). When this formula is applied to the projectivization of solution curves of scalar Lax operators with periodic coefficients, one obtains a corresponding evolution in the space of such operators. We conjecture that the formula we found gives another alternative definition of the second KdV Hamiltonian evolution under appropriate conditions; that is, both evolutions are identical. These conditions give a Hamiltonian interpretation of general vector differential invariants for the projective action of SL(n; R), namely, the SL(n; R) invariant evolution can be written so that a general vector differential invariant corresponds to the Hamiltonian pseudodifferential operator. We find common coordinates and simplify both evolutions so that one can attempt to prove the equivalence for arbitrary n.