ABSTRACT
Abstract We will discuss the mutual relationships between triple and bilinear products for many triple systems. Especially, the octonionic triple product can be shown to be realizable in terms of the bilinear Hurwitz product. We also study an analog of the Hurwitz theorem for composition triple systems. As an application, the existence of quintuple octonionic identities is demonstrated. Keywords: composition triple systems, nonassociative algebras, Yang-Baxter equations 2000 MSC: 17B25; 17A75
In the authors’ judgment, it seems that composition algebra was first introduced by Hurwitz at the end of the 19th century. The composition algebra is characterized by quaternion and octonion algebras permitting composition
N(xy) = N(x)N(y)
called quadratic norm N (the square of the absolute value), that is, describing for the inner product < a|b > of a vector space, this norm is denoted by
< xy|xy >=< x|x >< y|y > . Then Zorn studied them in the framework of alternative algebras, subsequently many mathematicians and physicists are investigating these subjects called nonassociative algebras (e.g., quadratic, alternative, flexible, involutive and Jordan algebras). For these nonassociative algebras, we will mainly refer to the books [J], [O.5], [Sch], and [Z-S-S-S].
