Supertransvectants and symplectic geometry

We consider the $osp(1\mathrm{\mid }2)$-invariant bilinear operations on weighted densities on the supercircle $S^{1\mathrm{\mid }1}$ called the supertransvectants. These operations are analogues of the famous Gordan transvectants (or Rankin-Cohen brackets). We prove that these operations coincide with the iterated Poisson and ghost Poisson brackets on ${\mathbb{R}}^{2\mathrm{\mid }1}$ and apply this result to construct star-products involving the supertransvectants.