3-Lie Algebras, Ternary Nambu-Lie algebras and the Yang-Baxter equation

TL;DR

This paper constructs ternary algebraic structures from Lie and Nambu-Lie algebras, demonstrating their role in generating Yang-Baxter operators and exploring their applications in braid group representations and link invariants.

Contribution

It introduces a method to derive Yang-Baxter operators from ternary self-distributive structures built from Lie and Nambu-Lie algebras, expanding the algebraic tools for braid and link invariants.

Findings

Constructed Yang-Baxter operators from ternary Lie algebra structures.

Demonstrated these operators are not gauge equivalent to transposition.

Explored deformations to produce new solutions to the Yang-Baxter equation.

Abstract

We construct ternary self-distributive (TSD) objects from compositions of binary Lie algebras, $3$-Lie algebras and, in particular, ternary Nambu-Lie algebras. We show that the structures obtained satisfy an invertibility property resembling that of racks. We prove that these structures give rise to Yang-Baxter operators in the tensor product of the base vector space and, upon defining suitable twisting isomorphisms, we obtain representations of the infinite (framed) braid group. We consider examples for low-dimensional Lie algebras, where the ternary bracket is defined by composition of the binary ones, along with simple $3$-Lie algebras. We show that the Yang-Baxter operators obtained are not gauge equivalent to the transposition operator, and we consider the problem of deforming the operators to obtain new solutions to the Yang-Baxter equation. We discuss the applications of this deformation procedure to the construction of (framed) link invariants.