Ternary Associativity and Ternary Lie Algebra at Cube Root of Unity

TL;DR

This paper introduces a novel ternary Lie algebra framework using ternary associativity and a special commutator with cube root of unity coefficients, extending classical Lie algebra concepts.

Contribution

It develops a new ternary commutator and identity based on ternary associativity, defining a ternary Lie algebra at the cube root of unity with concrete matrix examples.

Findings

Derived a ternary commutator with cube root of unity coefficients.

Established a ternary Jacobi-like identity based on ternary associativity.

Connected ternary Lie algebra structure constants to rotation group representations.

Abstract

We propose a new approach to extending the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on ternary associativity of the first and second kind. We propose a ternary commutator, which is a linear combination of six (all permutations of three elements) triple products. The coefficients of this linear combination are the cube roots of unity. We find an identity for the ternary commutator that holds due to ternary associativity of the first or second kind. The form of the found identity is determined by the permutations of the general affine group GA(1,5). We consider the found identity as an analogue of the Jacobi identity in the ternary case. We introduce the concept of a ternary Lie algebra at the cubic root of unity and give examples of such an algebra constructed using ternary multiplications of rectangular and three-dimensional matrices. We point out the connection between the structure constants of a ternary Lie algebra with three generators and an irreducible representation of the rotation group.