The Clifford algebra and the Chevalley map - a computational approach ( detailed

The theory of the Clifford algebra of a vector space with a given symmetric bilinear form is rather well-understood: One of the basic properties of the Clifford algebra gives an explicit basis for it in terms of a basis of the underlying vector space (Theorem 1 below), and another one provides a canonical vector space isomorphism between the Clifford algebra and the exterior algebra of the same vector space (the so-called Chevalley map, Theorem 2 below). While both of these properties appear in standard literature such as [1] and [2], sadly I have never seen them proven in the generality they deserve: first, the bilinear form needs not be symmetric. Besides, the properties still hold over arbitrary commutative rings rather than just fields of characteristic 0. The proofs given in literature are usually not sufficient to cover these general cases. Here we are going to present a computational proof of both of these properties, giving integral recursive formulas for the vector space isomorphism between the Clifford algebra and the exterior algebra (in both directions).