Invariantive Characterizations of Linear Algebras with the Associative Law Not Assumed

Abstract

We consider linear algebras in which neither the commutative nor the associative law of multiplication is assumed, and whose coordinates and constants of multiplication are numbers of a general field F. A rational integral invariant, or covariant, is a rational integral function of the constants of multiplication, or of the constants of multiplication and the coordinates of the general number, which has the invariantive property under the total group of linear homogeneous transformations on the units. If an invariantive function also actually involves the units, it has been called a vector covariant by Professor 0. C. Hazlett, t who shows that every rational integral vector covariant can be obtained as a covariant of the general number of the algebra and a fundamental set of ordinary covariants. In Section II of this article, it is shown how vector covariants may be formed directly from the constants of multiplication without assuming the knowledge of any ordinary covariants or invariants. To do this, the notion is introduced of a hypercomplex determinant whose elements obey neither the associative nor the commutative law of multiplication, and a few simple properties of such hypercomplex determinants are derived. From the vector covariants and the characteristic determinants of the algebra, ordinary relative invariants may easily be found. In Section III the linear algebra in three units, one of which is a principle unit, is considered. Invariants and covariants of the algebra are calculated by the method of Section II, and a set of ten of these functions is shown to form a complete system of invariants and covariants from the standpoint of Lie. In Section IV it is shown that for the example of Section III the arithmetic invariant denoting the rank can be replaced by a rational integral covariant. The generic case is defined as the case for which certain three covariants are