Associative ternary algebras and ternary Lie algebras at cube roots of unity
Anti Maria Aader, Viktor Abramov, Olga Liivapuu
TL;DR
This work develops a framework for ternary ω-Lie algebras, replacing antisymmetry with $ω$-symmetry in a ternary bracket and using the $GA(1,5)$ identity as a Jacobi-like condition. It shows how a ternary $ω$-commutator, built from associative ternary algebras of the first or second kind, yields a ternary $ω$-Lie structure, and it introduces ternary $ω$-associators to encode associativity. The paper then provides concrete realizations via semiheaps and associative ternary algebras, including vector, rectangular, and cubic matrix constructions, and analyzes the 8-dimensional ternary $ω$-Lie algebra of cubic matrices of order $2$, detailing its subalgebras and direct-sum decomposition. These results offer a mathematically rich generalization of Lie theory with potential applications in mathematical physics, particularly in ternary generalizations of quantum state operations and symmetry.
Abstract
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.