How to Quantize the Antibracket

We show that in contrast to $\mathfrak{po}(2n|m)$, its quotient modulo center, the Lie superalgebra $\mathfrak{h}(2n|m)$ of Hamiltonian vector fields with polynomial coefficients, has exceptional additional deformations for $(2n|m)=(2|2)$ and only for this superdimension. We relate this result to the complete description of deformations of the antibracket (also called the Schouten or Buttin bracket). It turns out that the space in which the deformed Lie algebra (result of quantizing the Poisson algebra) acts coincides with the simplest space in which the Lie algebra of commutation relations acts. This coincidence is not necessary for Lie superalgebras.