Algebras with ternary law of composition and their realization by cubic matrices

TL;DR

This paper explores the structure of ternary algebras, especially those of cubic matrices, establishing conditions for associativity, constructing related Lie algebras, and identifying unique totally associative multiplications.

Contribution

It introduces a classification of totally associative ternary multiplications for cubic matrices and constructs a ternary Lie algebra analog based on j-commutators.

Findings

Identified four unique totally associative ternary multiplications of second kind for cubic matrices.

Established necessary and sufficient conditions for ternary multiplication to induce associative binary algebra structures.

Constructed a ternary analog of the Lie algebra of cubic matrices with explicit commutation relations.

Abstract

We study partially and totally associative ternary algebras of first and second kind. Assuming the vector space underlying a ternary algebra to be a topological space and a triple product to be continuous mapping we consider the trivial vector bundle over a ternary algebra and show that a triple product induces a structure of binary algebra in each fiber of this vector bundle. We find the sufficient and necessary condition for a ternary multiplication to induce a structure of associative binary algebra in each fiber of this vector bundle. Given two modules over the algebras with involutions we construct a ternary algebra which is used as a building block for a Lie algebra. We construct ternary algebras of cubic matrices and find four different totally associative ternary multiplications of second kind of cubic matrices. It is proved that these are the only totally associative ternary multiplications of second kind in the case of cubic matrices. We describe a ternary analog of Lie algebra of cubic matrices of second order which is based on a notion of j-commutator and find all commutation relations of generators of this algebra.