Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Schur multiplier and Schur covers of relative Rota-Baxter groups

Pragya Belwal, Nishant Rathee, Mahender Singh

TL;DR

This work extends extension theory and low-dimensional cohomology to relative Rota-Baxter groups, establishing a Hochschild–Serre type exact sequence for central extensions and defining the Schur multiplier $M_{RRB}(\mathcal{A})$ with exponent dividing $|A||B|$ in the finite case. It develops a comprehensive cohomology framework for relative RRB groups, shows a bijection between second cohomology and central extensions, and links this to the cohomology of the induced skew left braces. The authors introduce weak isoclinism and Schur covers for relative RRB groups, proving existence and a weak isoclinism classification for finite bijective cases, aligning with analogous results for skew left braces. The results unify and generalize recent findings in skew left braces and offer tools for bijective YBE solutions via relative RRB groups.

Abstract

Relative Rota-Baxter groups are generalizations of Rota-Baxter groups and share a close connection with skew left braces. These structures are well-known for offering bijective non-degenerate set-theoretical solutions to the Yang-Baxter equation. This paper builds upon the recently introduced extension theory and low-dimensional cohomology of relative Rota-Baxter groups. We prove an analogue of the Hochschild-Serre exact sequence for central extensions of relative Rota-Baxter groups. We introduce the Schur multiplier $M_{RRB}(\mathcal{A})$ of a relative Rota-Baxter group $\mathcal{A} =(A,B,β,T)$, and prove that the exponent of $M_{RRB}(\mathcal{A})$ divides $|A||B|$ when $\mathcal{A}$ is finite. We define weak isoclinism of relative Rota-Baxter groups, introduce their Schur covers, and prove that any two Schur covers of a finite bijective relative Rota-Baxter group are weakly isoclinic. The results align with recent results of Letourmy and Vendramin for skew left braces.

Schur multiplier and Schur covers of relative Rota-Baxter groups

TL;DR

This work extends extension theory and low-dimensional cohomology to relative Rota-Baxter groups, establishing a Hochschild–Serre type exact sequence for central extensions and defining the Schur multiplier with exponent dividing in the finite case. It develops a comprehensive cohomology framework for relative RRB groups, shows a bijection between second cohomology and central extensions, and links this to the cohomology of the induced skew left braces. The authors introduce weak isoclinism and Schur covers for relative RRB groups, proving existence and a weak isoclinism classification for finite bijective cases, aligning with analogous results for skew left braces. The results unify and generalize recent findings in skew left braces and offer tools for bijective YBE solutions via relative RRB groups.

Abstract

Relative Rota-Baxter groups are generalizations of Rota-Baxter groups and share a close connection with skew left braces. These structures are well-known for offering bijective non-degenerate set-theoretical solutions to the Yang-Baxter equation. This paper builds upon the recently introduced extension theory and low-dimensional cohomology of relative Rota-Baxter groups. We prove an analogue of the Hochschild-Serre exact sequence for central extensions of relative Rota-Baxter groups. We introduce the Schur multiplier of a relative Rota-Baxter group , and prove that the exponent of divides when is finite. We define weak isoclinism of relative Rota-Baxter groups, introduce their Schur covers, and prove that any two Schur covers of a finite bijective relative Rota-Baxter group are weakly isoclinic. The results align with recent results of Letourmy and Vendramin for skew left braces.
Paper Structure (11 sections, 23 theorems, 152 equations)

This paper contains 11 sections, 23 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on relative Rota-Baxter groups
  3. Central Extensions and Second cohomology
  4. Central extensions and second cohomology of groups
  5. Annhilator extensions and second cohomology of skew left braces
  6. Central extensions and second cohomology of relative Rota-Baxter groups
  7. Connection between the three cohomology groups
  8. Hochschild-Serre exact sequence for relative Rota-Baxter groups
  9. Schur multiplier of relative Rota-Baxter groups
  10. Relationship between central extensions and Schur covers
  11. 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 (59)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 49 more