Leibniz $2$-algebras, linear $2$-racks and the Zamolodchikov Tetrahedron equation
Nanyan Xu, Yunhe Sheng
TL;DR
The work constructs and compares three categorified sources of ZTE solutions: central Leibniz 2-algebras and linear 2-racks. It provides explicit formulas for the ZTE data, shows invertibility and naturality conditions, and reveals how splittability lets one pass from central Leibniz 2-algebras to linear 2-racks while preserving the ZTE solutions. By studying the group-like category of a linear 2-rack, the authors connect linear 2-racks to 2-racks and give a concrete strict 2-rack example from a strict 2-group. Collectively, the results illuminate the interplay between categorified algebra, decategorification, and higher-dimensional integrable structures, yielding new algebraic pathways to 3D integrable systems.
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.