Leibniz $2$-algebras, linear $2$-racks and the Zamolodchikov Tetrahedron equation

TL;DR

This paper establishes a connection between Leibniz 2-algebras, linear 2-racks, and solutions to the Zamolodchikov Tetrahedron equation, expanding the algebraic structures related to higher-dimensional integrability.

Contribution

It introduces linear 2-racks, relates them to Leibniz 2-algebras, and constructs solutions to the Zamolodchikov Tetrahedron equation, advancing higher algebraic structures in mathematical physics.

Findings

Central Leibniz 2-algebras produce solutions to the Tetrahedron equation

Linear 2-racks also generate solutions to the Tetrahedron equation

A linear 2-rack can be constructed from a splittable Leibniz 2-algebra

Abstract

In this paper, first we show that a central Leibniz 2-algebra naturally gives rise to a solution of the Zamolodchikov Tetrahedron equation. Then we introduce the notion of linear 2-racks and show that a linear 2-rack also gives rise to a solution of the Zamolodchikov Tetrahedron equation. We show that a central Leibniz 2-algebra gives rise to a linear 2-rack if the underlying 2-vector space is splittable. Finally we discuss the relation between linear 2-racks and 2-racks, and show that a linear 2-rack gives rise to a 2-rack structure on the group-like category. A concrete example of strict 2-racks is constructed from an action of a strict 2-group.