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

We study the general form of the noncommutative associative product (the $*$-product) on the Grassmann algebra; the $*$-product is treated as a deformation of the usual pointwise product. We show that up to a similarity transformation, there exists only one such product. We discuss the relation of the algebra $\mathcal F$ (the algebra of the elements of the Grassmann algebra with the $*$-product as a product) to the Clifford algebra.