Version 1
: Received: 30 August 2024 / Approved: 2 September 2024 / Online: 2 September 2024 (09:47:15 CEST)
How to cite:
Abramov, V. Ternary Associativity and Ternary Lie Algebra at Cube Root of Unity. Preprints2024, 2024090063. https://doi.org/10.20944/preprints202409.0063.v1
Abramov, V. Ternary Associativity and Ternary Lie Algebra at Cube Root of Unity. Preprints 2024, 2024090063. https://doi.org/10.20944/preprints202409.0063.v1
Abramov, V. Ternary Associativity and Ternary Lie Algebra at Cube Root of Unity. Preprints2024, 2024090063. https://doi.org/10.20944/preprints202409.0063.v1
APA Style
Abramov, V. (2024). Ternary Associativity and Ternary Lie Algebra at Cube Root of Unity. Preprints. https://doi.org/10.20944/preprints202409.0063.v1
Chicago/Turabian Style
Abramov, V. 2024 "Ternary Associativity and Ternary Lie Algebra at Cube Root of Unity" Preprints. https://doi.org/10.20944/preprints202409.0063.v1
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)⊂S5. 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.
Keywords
Lie algebra; ternary associativity of the first and second kind; ternary algebra; ternary commutator; ternary Lie algebra at cube root of unity; general affine group
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
