Orthogonality graphs of real Cayley–Dickson algebras. Part II: The subgraph on pairs of basis elements

We consider zero divisors of an arbitrary real Cayley–Dickson algebra such that their components are both standard basis elements. We construct inductively the orthogonality graph on these elements. Then we show that, if we restrict our attention to at least [Formula: see text]-dimensional algebras, two algebras are isomorphic if and only if their graphs are isomorphic. We also provide an algorithm to retrieve the Cayley–Dickson parameters of an algebra from its graph.

Orthogonality graphs of real Cayley–Dickson algebras. Part I: Doubly alternative zero divisors and their hexagons

We study zero divisors whose components alternate strongly pairwise and construct oriented hexagons in the zero divisor graph of an arbitrary real Cayley–Dickson algebra. In case of the algebras of the main sequence, the zero divisor graph coincides with the orthogonality graph, and any hexagon can be extended to a double hexagon. We determine the multiplication table of the vertices of a double hexagon. Then we find a sufficient condition for three elements to generate an alternative subalgebra of an arbitrary Cayley–Dickson algebra. Finally, we consider those zero divisors whose components are both standard basis elements up to sign. We classify them and determine necessary and sufficient conditions under which two such elements are orthogonal.

Tetranacci and Tetranacci-Lucas Quaternions

The quaternions form a 4-dimensional Cayley-Dickson algebra. In this paper, we introduce the Tetranacci and Tetranacci-Lucas quaternions. Furthermore, we present some properties of these quaternions and derive relationships between them. We present the generating functions, Binet's formulas and sums formulas of these quaternions. Moreover, we give matrix formulation of Tetranacci and Tetranacci-Lucas quaternions.

Derivations with invertible values in flexible algebras

Derivations with invertible values of 0 – torsion flexible algebras satisfying x(yz) = (xz)y over an algebraically closed field are described. For this class of algebra with unit element 1 and derivation with invertible value d is either a Cayley – Dickson algebra over its center Z(A) or a factor algebra of polynomial algebra C[a]/(a2) over a Cayley – Dickson division algebra; also C is 2 – torsion, d(C) = 0 and d(a) = 1+ua for some u in center of C and d is an outer derivation. Moreover, C is a split Cayley – Dickson algebra over its center Z having a derivation with invertible value d if and only if C is obtained by means of Cayley – Dickson process from its associative division subalgebra and can be represented as a direct sum C = V ⊕ aV.

Tribonacci and Tribonacci-Lucas Sedenions

The sedenions form a 16-dimensional Cayley-Dickson algebra. In this paper, we introduce the Tribonacci and Tribonacci-Lucas sedenions. Furthermore, we present some properties of these sedenions and derive relationships between them.