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Ternary associator, ternary commutator and ternary Lie algebra at cube roots of unity

Viktor Abramov

TL;DR

This work develops a ternary analogue of Lie theory by introducing $\omega$-structures at cube roots of unity. It defines the $\omega$-commutator and a corresponding $GA(1,5)$-based identity, establishing a ternary Jacobi-like framework that holds in associative ternary algebras. The authors construct broad families of ternary $\omega$-Lie algebras from associative ternary multiplications on rectangular and cubic matrices and provide a complete 2‑dimensional classification, identifying four isomorphism classes, including Abelian, simple, and mixed-type algebras. The results advance higher-arity generalizations of Lie theory with concrete matrix-based realizations relevant to noncommutative geometry and $n$-ary algebra research.

Abstract

We extend the concepts of the associator and commutator from algebras with a binary multiplication law to algebras with a ternary multiplication law using cube roots of unity. By analogy with the Jacobi identity for the binary commutator, we derive an identity for the proposed ternary commutator. While the Jacobi identity is based on the cyclic permutation group of three elements $\mathbb Z_3$, the identity we establish for the ternary commutator is based on the general affine group $GA(1,5)$. We introduce the notion of a ternary Lie algebra at cube roots of unity. A broad class of such algebras is constructed using associative ternary multiplications of rectangular and cubic matrices. Furthermore, a complete classification of non-isomorphic two-dimensional ternary Lie algebras at cube roots of unity is obtained.

Ternary associator, ternary commutator and ternary Lie algebra at cube roots of unity

TL;DR

This work develops a ternary analogue of Lie theory by introducing -structures at cube roots of unity. It defines the -commutator and a corresponding -based identity, establishing a ternary Jacobi-like framework that holds in associative ternary algebras. The authors construct broad families of ternary -Lie algebras from associative ternary multiplications on rectangular and cubic matrices and provide a complete 2‑dimensional classification, identifying four isomorphism classes, including Abelian, simple, and mixed-type algebras. The results advance higher-arity generalizations of Lie theory with concrete matrix-based realizations relevant to noncommutative geometry and -ary algebra research.

Abstract

We extend the concepts of the associator and commutator from algebras with a binary multiplication law to algebras with a ternary multiplication law using cube roots of unity. By analogy with the Jacobi identity for the binary commutator, we derive an identity for the proposed ternary commutator. While the Jacobi identity is based on the cyclic permutation group of three elements , the identity we establish for the ternary commutator is based on the general affine group . We introduce the notion of a ternary Lie algebra at cube roots of unity. A broad class of such algebras is constructed using associative ternary multiplications of rectangular and cubic matrices. Furthermore, a complete classification of non-isomorphic two-dimensional ternary Lie algebras at cube roots of unity is obtained.

Paper Structure

This paper contains 6 sections, 3 theorems, 76 equations, 1 table.

Table of Contents

  1. Introduction
  2. Ternary $\omega$-commutator
  3. General affine group of degree 1 and identity
  4. Ternary Lie algebra at cube roots of unity
  5. Associative ternary algebras of rectangular and cubic matrices
  6. Classification of 2-dimensional ternary Lie algebras at cube roots of unity

Key Result

theorem 1

Let $\cal A$ be an associative ternary algebra. Then for any five elements $a_1,a_2,a_3,a_4,a_5\in\cal A$ we have $\Omega(a_1,a_2,a_3,a_4,a_5)=0$.

Theorems & Definitions (10)

  • definition 1
  • definition 2
  • theorem 1
  • proof
  • definition 3
  • definition 4
  • proposition 1
  • proof
  • theorem 2
  • proof