Ternary numbers and algebras. Reflexive numbers and Berger graphs

TL;DR

This paper explores the connection between Berger graphs and n-ary algebras, specifically ternary generalizations of quaternions and octonions, proposing a new algebraic framework related to Calabi-Yau spaces.

Contribution

It introduces a novel ternary algebra related to Berger graphs and extends binary division algebras to n-ary cases, providing new insights into algebraic structures linked to Calabi-Yau spaces.

Findings

Berger graph corresponds to a tetrahedron.

Ternary algebra extends SU(3) structures.

Solution for Berger graph identified as a tetrahedron.

Abstract

The Calabi-Yau spaces with SU(m) holonomy can be studied by the algebraic way through the integer lattice where one can construct the Newton reflexive polyhedra or the Berger graphs. Our conjecture is that the Berger graphs can be directly related with the $n$-ary algebras. To find such algebras we study the n-ary generalization of the well-known binary norm division algebras, ${\mathbb R}$, ${\mathbb C}$, ${\mathbb H}$, ${\mathbb O}$, which helped to discover the most important "minimal" binary simple Lie groups, U(1), SU(2) and G(2). As the most important example, we consider the case $n=3$, which gives the ternary generalization of quaternions and octonions, $3^p$, $p=2,3$, respectively. The ternary generalization of quaternions is directly related to the new ternary algebra and group which are related to the natural extensions of the binary $su(3)$ algebra and SU(3) group. Using this ternary algebra we found the solution for the Berger graph: a tetrahedron.