We consider the supercircle S 1|1 equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on S 1|1 as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra osp(1|2). We study the space of linear differential operators on weighted densities as a module over osp(1|2). We introduce the canonical isomorphism between this space and the corresponding space of symbols and find interesting resonant cases where such an isomorphism does not exist.

The algebra of hyperbolic numbers is endowed with a partial order structure. We show that this system of numbers is the only (natural) generalization of real numbers into Archimedean f -algebra of dimension two. We establish various properties of hyperbolic numbers related to the f -algebra structure. In particular, we generalize fundamental properties of real numbers and give some order interpretations for the two dimensional space-time geometry.

The 1|1-supertransvectants are the osp(1|2)-invariant bilinear operations on weighted densities on the supercircle S 1|1 , the projective version of R 2|1 . These operations are analogues of the famous Gordan transvectants (or Rankin-Cohen brackets). We prove that supertransvectants coincide with the iterated Poisson and ghost Poisson brackets on R 2|1 and apply this result to construct star-products. ‡ I.P.E.I.T., 2 Rue Jawaher Lel Nehru, Monfleury 1008 Tunis, TUNISIE; hichem.gargoubi@ipeit.rnu.tn,

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 SL 2 (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 sl 2 (C). The result is independent on the choice of the projective structure. We give explicit formulae of 1-cocycles generating this cohomology group.

Let F λ (S 1 ) be the space of tensor densities of degree (or weight) λ on the circle) is a natural module over Diff(S 1 ), the diffeomorphism group of S 1 . We determine the algebra of symmetries of the modules D