The symplectic form associated to a singular Poisson algebra

Given an affine Poisson algebra, that is singular one may ask whether there is an associated symplectic form. In the smooth case the answer is obvious: for the symplectic form to exist the Poisson tensor has to be invertible. In the singular case, however, derivations do not form a projective module and the nondegeneracy condition is more subtle. For a symplectic singularity one may naively ask if there is indeed an analogue of a symplectic form. We examine an example of a symplectic singularity, namely the double cone, and show that here such a symplectic form exists. We use the naive de Rham complex of a Lie-Rinehart algebra. Our analysis of the double cone uses Gr\"obner bases calculations. We also give an alternative construction of the symplectic form that generalizes to categorical quotients of cotangent lifted representations of finite groups. We use the same formulas to construct a symplectic form on the simple cone, seen as a Poisson differential space and generalize the construction to linear symplectic orbifolds. We present useful auxiliary results that enable to explicitly determine generators for the module of derivations an affine variety. The latter may be understood as a differential space.