Two-graded absolute valued algebras

AbstractWe study two-graded absolute valued algebras. These are two-graded algebras satisfying the absolute value multiplicative property only on homogeneous elements. Thus, hexagonions (also called sedenions) and other sixteen-dimensional algebras arise as examples of these algebras. The even parts of two-graded absolute valued algebras are the absolute valued algebras, while the odd parts are exactly the absolute valued triple systems. So, in a way these two-graded algebras give a unifying viewpoint of both structures. We also study the simplicity and give several ways to construct two-graded absolute valued algebras. We also provide a description of isomorphism classes for two-graded absolute valued algebras of dimensions 1, 2 and 4