The Yang-Baxter equation, Leibniz algebras, racks and related algebraic structures

TL;DR

This paper explores the connections between Leibniz algebras, racks, and solutions to the Yang-Baxter equation, introducing new constructions and clarifying existing relationships among these algebraic structures.

Contribution

It provides new constructions of solutions to the Yang-Baxter equation from Leibniz and 3-Leibniz algebras, and clarifies their relationships with racks and trilinear racks.

Findings

A 3-Leibniz algebra induces a 3-rack, generalizing Kinyon's construction.

Trilinear racks lead to linear racks, explaining Abramov and Zappala's solutions.

Solutions of the Yang-Baxter equation are constructed via central extensions of Leibniz and 3-Leibniz algebras.

Abstract

The purpose of this paper is to clarify the relations between various constructions of solutions of the Yang-Baxter equation from Leibniz algebras, racks, 3-Leibniz algebras, 3-racks, linear racks, trilinear racks, and give new constructions of solutions of the Yang-Baxter equation. First we show that a 3-Leibniz algebra naturally gives rise to a 3-rack on the underlying vector space, which generalizes Kinyon's construction of racks from Leibniz algebras. Then we show that a trilinear rack naturally gives rise to a linear rack. Combined with Lebed's construction of solutions of the Yang-Baxter equation from linear racks, our results give an intrinsic explanation of Abramov and Zappala's construction of solutions of the Yang-Baxter equation from trilinear racks. Next we show that a 3-Leibniz algebra gives rise to a trilinear rack, which generalizes Abramov and Zappala's construction from 3-Lie algebras. Finally, we construct solutions of the Yang-Baxter equation using central extensions of 3-Leibniz algebras and Leibniz algebras. In particular, given a 3-Leibniz algebra, there are two different approaches to construct solutions of the Yang-Baxter equation, namely either consider the central extension of the Leibniz algebra on the fundamental objects, or consider the Leibniz algebra on the fundamental objects of the central extension of the 3-Leibniz algebra. We also show that there is a homomorphism between the corresponding solutions.