Supertrace and superquadratic Lie structure on the Weyl algebra, with   applications to formal inverse Weyl transform
  Using the Moyal *-product and orthosymplectic supersymmetry, we construct a natural non trivial supertrace and an associated non degenerate invariant supersymmetric bilinear form for the Lie superalgebra structure of the Weyl algebra. We decompose adjoint and twisted adjoint actions. We define a renormalized supertrace and a formal inverse Weyl transform in a deformation quantization framework and develop some examples. 
Georges Pinczon, Rosane Ushirobira