Algebras Generated by Scalar K-atic Forms and Their Linear Forms

Abstract.

A linear form with an N-elements basis set {e _i ; i = 1,...,N} generates an algebra which is that of multivectors, provided some commutation relation is defined to give a meaning to the outer product of the basis vectors. If, moreover, an inner product of sets of K basis vectors is also introduced, for a mapping ${\left\langle {e_i,e_j, \ldots } \right\rangle}_{K} \to {\user1{\mathcal{K}}}\subset {\user2{\mathbb{R}}}$ producing a 0-form, a geometric algebra is obtained. The algebra has thus two basic numbers to define its dimension: the dimension N of the basis set and the dimension K of the number of elements to be multiplied together to obtain a scalar. If the dimension K refers to the order of the power of [e _i ]^K to obtain the scalar we will say that we have a K-atic algebra, the best known example is when the scalar form is a quadratic expression; these algebras are said to have a metric which in general is either diagonal or at least symmetric. Otherwise if the dimension K refers again to the number of different basis vectors to be multiplied together in ${\left\langle {e_i,e_j, \ldots } \right\rangle }_{K} \subset {\user2{\mathbb{R}}}$ (with j ≠ i and in general all subindexes different) then we obtain a simplectic algebra where the best known case is also when K = 2 and the metric in this case is antisymmetric. In the present paper we define these sets of algebras, give the commutation relations for the algebras with a K-atic scalar form and relate the results to the best known examples of current use in the literature.