Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Wells-like exact sequence for abelian extensions of relative Rota--Baxter groups

Pragya Belwal, Nishant Rathee

Abstract

Relative Rota--Baxter groups, a generalization of Rota--Baxter groups, are closely connected to skew left braces, which play a fundamental role in understanding non-degenerate set-theoretical solutions to the Yang-Baxter equation. In this paper, we explore the inducibility problem for automorphisms of abelian extensions of relative Rota--Baxter groups. This problem is intricately linked to the recently introduced second cohomology of relative Rota--Baxter groups. Specifically, we prove a Wells-like exact sequence for abelian extensions of relative Rota--Baxter groups. The sequence establishes a connection among the group of derivations, certain automorphism group, and the second cohomology of relative Rota--Baxter groups, thereby giving precise structural relationships between these groups.

A Wells-like exact sequence for abelian extensions of relative Rota--Baxter groups

Abstract

Relative Rota--Baxter groups, a generalization of Rota--Baxter groups, are closely connected to skew left braces, which play a fundamental role in understanding non-degenerate set-theoretical solutions to the Yang-Baxter equation. In this paper, we explore the inducibility problem for automorphisms of abelian extensions of relative Rota--Baxter groups. This problem is intricately linked to the recently introduced second cohomology of relative Rota--Baxter groups. Specifically, we prove a Wells-like exact sequence for abelian extensions of relative Rota--Baxter groups. The sequence establishes a connection among the group of derivations, certain automorphism group, and the second cohomology of relative Rota--Baxter groups, thereby giving precise structural relationships between these groups.
Paper Structure (5 sections, 15 theorems, 91 equations)

This paper contains 5 sections, 15 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extensions and second cohomology of relative Rota--Baxter groups
  4. Wells-like exact sequence for relative Rota--Baxter groups
  5. Declaration

Key Result

Theorem 2.4

Let $(H, G, \phi, R)$ be a relative Rota--Baxter group and $(K, L, \phi|, R|)$ an ideal of $(H, G, \phi, R)$. Then there are maps $\overline{\phi}: G/L \to \operatorname{Aut} (H/K)$ and $\overline{R}: H/K \to G/L$ defined by for $\overline{g} \in G/L$ and $\overline{h} \in H/K$, such that $(H/K, G/L, \overline{\phi}, \overline{R})$ is a relative Rota--Baxter group.

Theorems & Definitions (32)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 3.1
  • Example 3.2
  • Proposition 3.3
  • ...and 22 more