3-Lie Algebras, Ternary Nambu-Lie algebras and the Yang-Baxter equation
Abstract
We construct ternary self-distributive (TSD) objects from compositions of binary Lie algebras, -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 -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.
Keywords
Cite
@article{arxiv.2103.11472,
title = {3-Lie Algebras, Ternary Nambu-Lie algebras and the Yang-Baxter equation},
author = {Viktor Abramov and Emanuele Zappala},
journal= {arXiv preprint arXiv:2103.11472},
year = {2022}
}
Comments
29 pages, 3 figures. v4: Introduction shortened for clarity, and some typos corrected in the proof of Lemma 5.2. Some figures removed because not actually used. Version accepted for publication in Journal of Geometry and Physics