General Form of the $*$-Commutator on the Grassmann Algebra

The general form of the $*$-commutator on the Grassmann algebra treated as a deformation of the conventional Poisson bracket is investigated. It is shown that in addition to the Moyal $*$-commutator, there exist other deformations of the Poisson bracket on the Grassman algebra (one additional deformation for even and odd $n$, where $n$ is the number of the Grassmann algebra generators) that are not reducible to the Moyal $*$-commutator by a similarity transformation.