Idempotents and representations of certain algebras of Clifford

In general, Clifford algebras of quadratic forms are finite dimensional; therefore, their representations are easy to describe. However, for homogenous polynomial forms of degree d >= 2, the situation is different because their Clifford algebras are infinite dimensional. In this article, we get a finite set of pairwise orthogonal idempotents of sum I in these algebras. This permits us to obtain interesting properties for d-dimensional representations of polynomial forms of degree d, for example, we show that the image C of the Clifford algebra by such representation is an endomorphism algebra of finitely generated projective Z(C)-module of d-rank, direct sum of finitely generated projective Z(C)-modale of 1-rank. Before establishing this, we give a new proof of the Poincare-Birkhoff-Witt theorem for these algebras with the help of a general composition lemma. At the end of this work, we give a linearization of diagonal binary and ternary forms of degree d > 3.