{"title":"统一立方根上的三元联立和三元列代数","authors":"Viktor Abramov","doi":"arxiv-2409.02557","DOIUrl":null,"url":null,"abstract":"We propose a new approach to extending the notion of commutator and Lie\nalgebra to algebras with ternary multiplication laws. Our approach is based on\nternary associativity of the first and second kind. We propose a ternary\ncommutator, which is a linear combination of six (all permutations of three\nelements) triple products. The coefficients of this linear combination are the\ncube roots of unity. We find an identity for the ternary commutator that holds\ndue to ternary associativity of the first or second kind. The form of the found\nidentity is determined by the permutations of the general affine group GA(1,5).\nWe consider the found identity as an analogue of the Jacobi identity in the\nternary case. We introduce the concept of a ternary Lie algebra at the cubic\nroot of unity and give examples of such an algebra constructed using ternary\nmultiplications of rectangular and three-dimensional matrices. We point out the\nconnection between the structure constants of a ternary Lie algebra with three\ngenerators and an irreducible representation of the rotation group.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"38 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Ternary Associativity and Ternary Lie Algebra at Cube Root of Unity\",\"authors\":\"Viktor Abramov\",\"doi\":\"arxiv-2409.02557\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We propose a new approach to extending the notion of commutator and Lie\\nalgebra to algebras with ternary multiplication laws. Our approach is based on\\nternary associativity of the first and second kind. We propose a ternary\\ncommutator, which is a linear combination of six (all permutations of three\\nelements) triple products. The coefficients of this linear combination are the\\ncube roots of unity. We find an identity for the ternary commutator that holds\\ndue to ternary associativity of the first or second kind. The form of the found\\nidentity is determined by the permutations of the general affine group GA(1,5).\\nWe consider the found identity as an analogue of the Jacobi identity in the\\nternary case. We introduce the concept of a ternary Lie algebra at the cubic\\nroot of unity and give examples of such an algebra constructed using ternary\\nmultiplications of rectangular and three-dimensional matrices. We point out the\\nconnection between the structure constants of a ternary Lie algebra with three\\ngenerators and an irreducible representation of the rotation group.\",\"PeriodicalId\":501136,\"journal\":{\"name\":\"arXiv - MATH - Rings and Algebras\",\"volume\":\"38 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Rings and Algebras\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.02557\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.02557","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Ternary Associativity and Ternary Lie Algebra at Cube Root of Unity
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.