Structures Symplectiques sur les Espaces de Courbes Projectives et Affines

A symplectic structure on the space of nondegenerate and nonparametrized curves in a locally affine manifold is defined. We also consider several interesting spaces of nondegenerate projective curves endowed with Poisson structures. This construction connects the Virasoro algebra and the Gel'fand-Dikii bracket with the projective differential geometry.Comment: 41 page

Poisson-Lie group of pseudodifferential symbols

We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators on the line and the circle (with scalar or matrix coefficients). This defines a Poisson--Lie structure on the dual group of pseudodifferential symbols of an arbitrary real (or complex) order. We show that the usual (second) Benney, KdV (or GL_n--Adler--Gelfand--Dickey) and KP Poisson structures are naturally realized as restrictions of this Poisson structure to submanifolds of this ``universal'' Poisson--Lie group. Moreover, the reduced (=SL_n) versions of these manifolds (W_n-algebras in physical terminology) can be viewed as subspaces of the quotient (or Poisson reduction) of this Poisson--Lie group by the dressing action of the group of functions. Finally, we define an infinite set of functions in involution on the Poisson--Lie group that give the standard families of Hamiltonians when restricted to the submanifolds mentioned above. The Poisson structure and Hamiltonians on the whole group interpolate between the Poisson structures and Hamiltonians of Benney, KP and KdV flows. We also discuss the geometrical meaning of W_\infty as a limit of Poisson algebras W_\epsilon as \epsilon goes to 0.Comment: 64 pages, no figure

Conformal geometry of the supercotangent and spinor bundles

We study the actions of local conformal vector fields X∈conf(M,g) on the spinor bundle of (M,g) and on its classical counterpart: the supercotangent bundle M of (M,g). We first deal with the classical framework and determine the Hamiltonian lift of conf(M,g) to M. We then perform the geometric quantization of the supercotangent bundle of (M,g), which constructs the spinor bundle as the quantum representation space. The Kosmann Lie derivative of spinors is btained by quantization of the comoment map. The quantum and classical actions of conf(M,g) turn, respectively, the space of differential operators acting on spinor densities and the space of their symbols into conf(M,g)-modules. They are filtered and admit a common associated graded module. In the conformally flat case, the latter helps us determine the conformal invariants of both conf(M,g)-modules, in particular the conformally odd powers of the Dirac operator.Peer reviewe