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Relative Rota-Baxter operators of weight 0 on groups, pre-groups, braces, the Yang-Baxter equation and $T$-structures

Yunnan Li, Yunhe Sheng, Rong Tang

TL;DR

We study weight-$0$ relative Rota-Baxter operators on groups and Lie groups, proposing two coherent definitions that align with Malcev completion for arbitrary groups and a Lie-algebra-to-group relative-operator perspective for Lie groups. These operators induce rich algebraic structures, including pre-groups, braces, and T-structures, and yield set-theoretic solutions to the Yang-Baxter equation via braid- and cocycle-based formalisms. The work clarifies how these operators connect to existing constructions (brace-derived YBE solutions, structure groups, and cocycle data) and provides explicit examples and cohomological criteria for when braces arise from weight-$0$ RBOs. The results unify multiple perspectives on RBOs, braces, and the Yang-Baxter framework, offering concrete tools to generate and classify set-theoretic YBE solutions and related algebraic structures. The findings have potential applications in integrable systems, combinatorial algebra, and the study of post-Lie and pre-Lie structures underlying braces.

Abstract

In this paper, we study relative Rota-Baxter operators of weight $0$ on groups and give various examples. In particular, we propose different approaches to study Rota-Baxter operators of weight $0$ on groups and Lie groups. We establish various explicit relations among relative Rota-Baxter operators of weight $0$ on groups, pre-groups, braces, set-theoretic solutions of the Yang-Baxter equation and $T$-structures.

Relative Rota-Baxter operators of weight 0 on groups, pre-groups, braces, the Yang-Baxter equation and $T$-structures

TL;DR

We study weight- relative Rota-Baxter operators on groups and Lie groups, proposing two coherent definitions that align with Malcev completion for arbitrary groups and a Lie-algebra-to-group relative-operator perspective for Lie groups. These operators induce rich algebraic structures, including pre-groups, braces, and T-structures, and yield set-theoretic solutions to the Yang-Baxter equation via braid- and cocycle-based formalisms. The work clarifies how these operators connect to existing constructions (brace-derived YBE solutions, structure groups, and cocycle data) and provides explicit examples and cohomological criteria for when braces arise from weight- RBOs. The results unify multiple perspectives on RBOs, braces, and the Yang-Baxter framework, offering concrete tools to generate and classify set-theoretic YBE solutions and related algebraic structures. The findings have potential applications in integrable systems, combinatorial algebra, and the study of post-Lie and pre-Lie structures underlying braces.

Abstract

In this paper, we study relative Rota-Baxter operators of weight on groups and give various examples. In particular, we propose different approaches to study Rota-Baxter operators of weight on groups and Lie groups. We establish various explicit relations among relative Rota-Baxter operators of weight on groups, pre-groups, braces, set-theoretic solutions of the Yang-Baxter equation and -structures.
Paper Structure (9 sections, 14 theorems, 113 equations)

This paper contains 9 sections, 14 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Relative Rota-Baxter operators of weight 0 on groups and examples
  3. Rota-Baxter operators of weight 0 on (Lie) groups
  4. Rota-Baxter operators of weight 0 on groups
  5. Rota-Baxter operators of weight $0$ on Lie groups
  6. Relative Rota-Baxter operators of weight $0$ and pre-groups
  7. Relative Rota-Baxter operators of weight $0$ and braces
  8. Relative Rota-Baxter operators of weight $0$ and the Yang-Baxter equation
  9. Relative Rota-Baxter operators of weight $0$ and $T$-structures

Key Result

Proposition 2.2

Given a group $G$ and a $\mathbb Z G$-module $(V,\Phi)$, a map ${\mathcal{R}}:V\to G$ is a relative Rota-Baxter operator if and only if the graph of ${\mathcal{R}}$ is a subgroup of the semidirect product $V\rtimes_\Phi G$.

Theorems & Definitions (45)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Theorem 2.6
  • proof
  • Example 2.7
  • Definition 3.1
  • ...and 35 more