Associative ternary algebras and ternary Lie algebras at cube roots of unity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909871211806720 |
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| author | Aader, Anti Maria Abramov, Viktor Liivapuu, Olga |
| author_facet | Aader, Anti Maria Abramov, Viktor Liivapuu, Olga |
| contents | We propose an approach to extending the concept of a Lie algebra to ternary structures based on $ω$-symmetry, where $ω$ is a primitive cube root of unity. We give a definition of a corresponding structure, called a ternary Lie algebra at cube roots of unity, or a ternary $ω$-Lie algebra. A method for constructing ternary associative algebras has been developed. For ternary algebras, the notions of the ternary $ω$-associator and the ternary $ω$-commutator are introduced. It is shown that if a ternary algebra possesses the property of associativity of the first or second kind, then the ternary $ω$-commutator on this algebra determines the structure of a ternary $ω$-Lie algebra. Ternary algebras of cubic matrices with associative ternary multiplication of the second kind are considered. The structure of the 8-dimensional ternary $ω$-Lie algebra of cubic matrices of the second order is studied, and all its subalgebras of dimensions 2 and 3 are determined. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23433 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Associative ternary algebras and ternary Lie algebras at cube roots of unity Aader, Anti Maria Abramov, Viktor Liivapuu, Olga Rings and Algebras We propose an approach to extending the concept of a Lie algebra to ternary structures based on $ω$-symmetry, where $ω$ is a primitive cube root of unity. We give a definition of a corresponding structure, called a ternary Lie algebra at cube roots of unity, or a ternary $ω$-Lie algebra. A method for constructing ternary associative algebras has been developed. For ternary algebras, the notions of the ternary $ω$-associator and the ternary $ω$-commutator are introduced. It is shown that if a ternary algebra possesses the property of associativity of the first or second kind, then the ternary $ω$-commutator on this algebra determines the structure of a ternary $ω$-Lie algebra. Ternary algebras of cubic matrices with associative ternary multiplication of the second kind are considered. The structure of the 8-dimensional ternary $ω$-Lie algebra of cubic matrices of the second order is studied, and all its subalgebras of dimensions 2 and 3 are determined. |
| title | Associative ternary algebras and ternary Lie algebras at cube roots of unity |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2510.23433 |