On zeroth Poisson homology in positive characteristic

A Poisson algebra is a commutative algebra with a Lie bracket {, } satisfying the Leibniz rule. Such algebras appear in classical mechanics. Namely, functions on the phase space form a Poisson algebra, and Hamilton’s equation of motion is df dt = {f, H}, where H is the Hamiltonian (energy) function. Moreover, the transition from classical to quantum mechanics can be understood in terms of deformation quantization of Poisson algebras, so that Schrödinger’s equation −i~ dt = [f, H] is a deformation of Hamilton’s equation. An important invariant of a Poisson algebra A is its zeroth Poisson homology HP0(A) = A/{A, A}. It characterizes densities on the phase space invariant under all Hamiltonian flows. Also, the dimension of HP0(A) gives an upper bound for the number of irreducible representations of any quantization of A. We study HP0(A) when A is the algebra of functions on an isolated quasihomogeneous surface singularity. Over C, it’s known that HP0(A) is the Jacobi ring of the singularity whose dimension is the Milnor number. We generalize this to characteristic p. In this case, HP0(A) is a finite (although not finite dimensional) module over Ap. We give its conjectural Hilbert series for Kleinian singularities and for cones of smooth projective curves, and prove the conjecture in several cases.