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

Apurba Das, Suman Majhi

TL;DR

The article develops a higher-arity generalization of the Yang–Baxter framework by connecting $n$-Leibniz algebras, $n$-racks, and linear $n$-racks to $n$-Yang–Baxter operators and their set-theoretical counterparts. It provides explicit constructions: from finite-dimensional $n$-Leibniz algebras to functorial $n$-rack structures, from central $n$-Leibniz algebras to $n$-Yang–Baxter operators, and from cocommutative linear $n$-racks to linear $n$-Yang–Baxter operators, with a unifying framework that includes the classical $n=2$ and $n=3$ cases. The paper also establishes relationships between these objects, including embeddings and functorial mappings, and extends the standard Yang–Baxter theory to a higher-arity setting, while outlining a set-theoretical variant and open problems. This work lays groundwork for potential quantum or higher-dimensional generalizations and connections to related equations like the tetrahedron equation. Overall, it provides a coherent higher-arity pipeline linking $n$-ary algebraic structures to generalized Yang–Baxter phenomena and their set-theoretical incarnations.

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.

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

TL;DR

The article develops a higher-arity generalization of the Yang–Baxter framework by connecting -Leibniz algebras, -racks, and linear -racks to -Yang–Baxter operators and their set-theoretical counterparts. It provides explicit constructions: from finite-dimensional -Leibniz algebras to functorial -rack structures, from central -Leibniz algebras to -Yang–Baxter operators, and from cocommutative linear -racks to linear -Yang–Baxter operators, with a unifying framework that includes the classical and cases. The paper also establishes relationships between these objects, including embeddings and functorial mappings, and extends the standard Yang–Baxter theory to a higher-arity setting, while outlining a set-theoretical variant and open problems. This work lays groundwork for potential quantum or higher-dimensional generalizations and connections to related equations like the tetrahedron equation. Overall, it provides a coherent higher-arity pipeline linking -ary algebraic structures to generalized Yang–Baxter phenomena and their set-theoretical incarnations.

Abstract

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

Paper Structure

This paper contains 9 sections, 26 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Yang-Baxter equation
  3. $n$-ary structures and the Yang-Baxter equation
  4. A higher analogue to the Yang-Baxter equation
  5. Organization of the paper
  6. From n-Leibniz algebras to n-racks
  7. Central n-Leibniz algebras and the Yang-Baxter equation
  8. Linear n-racks and the Yang-Baxter equation
  9. A higher analogue to the Yang-Baxter equation

Key Result

Proposition 2.4

Let $(\mathcal{L}, [-, \ldots, -])$ be an $n$-Leibniz algebra. Then (the space of fundamental elements) $\mathcal{L}^{\otimes (n-1)}$ carries a Leibniz algebra structure with the bracket operation for $x_1 \otimes \cdots \otimes x_{n-1}$, $y_1 \otimes \cdots \otimes y_{n-1} \in \mathcal{L}^{\otimes (n-1)}$.

Theorems & Definitions (73)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Proposition 2.4
  • Definition 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Remark 2.9
  • Proposition 2.10
  • ...and 63 more