How to Quantize the Antibracket

The uniqueness of (the class of) deformation of Poisson Lie algebra po(2n) has long been a completely accepted folklore. Actually this is wrong as stated, because its validity depends on the class of functions that generate po(2n) (e.g., it is true for polynomials but false for Laurent polynomials). We show that, unlike po(2n|m), its quotient modulo center, the Lie superalgebra h(2n|m) of Hamiltonian vector fields with polynomial coefficients, has exceptional extra deformations for (2n|m) = (2|2) and only in this superdimension. We relate this result to the complete description of deformations of the antibracket (also called the Schouten or Buttin bracket). We show that, whereas the representation of the deform (the result of deformation aka quantization) of the Poisson algebra in the Fock space coincides with the simplest space on which the Lie algebra of commutation relations acts, this coincidence is not necessary for Lie superalgebras.