From $n$-Leibniz algebras and linear $n$-racks to the solutions of the (higher analogue of) Yang-Baxter equation

TL;DR

This paper explores how $n$-Leibniz algebras lead to $n$-rack structures and Yang-Baxter operators, extending classical algebraic concepts to higher arity and proposing new higher-ary Yang-Baxter operators.

Contribution

It introduces the concept of linear $n$-racks, connects $n$-Leibniz algebras to Yang-Baxter operators, and generalizes Yang-Baxter operators to higher arity.

Findings

Finite-dimensional $n$-Leibniz algebras induce $n$-rack structures.

Construction of Yang-Baxter operators from $n$-Leibniz algebras.

Proposal of higher-ary $n$-Yang-Baxter operators.

Abstract

In this paper, we first demonstrate that a finite-dimensional $n$-Leibniz algebra naturally gives rise to an $n$-rack structure on the underlying vector space. Given any $n$-Leibniz algebra, we also construct two Yang-Baxter operators on suitable vector spaces and connect them by a homomorphism. Next, we introduce linear $n$-racks as the coalgebraic version of $n$-racks and show that a cocommutative linear $n$-rack yields a linear rack structure and hence a Yang-Baxter operator. An $n$-Leibniz algebra canonically gives rise to a cocommutative linear $n$-rack and thus produces a Yang-Baxter operator. In the last part, following the well-known close connections among Leibniz algebras, (linear) racks and Yang-Baxter operators, we consider a higher-ary generalization of Yang-Baxter operators (called $n$-Yang-Baxter operators). In particular, we show that $n$-Leibniz algebras and cocommutative linear $n$-racks naturally provide $n$-Yang-Baxter operators. Finally, we consider a set-theoretical variant of $n$-Yang-Baxter operators and propose some problems.