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$R$-weighted graphs and commutators

Harish Kishnani, Amit Kulshrestha

TL;DR

The paper unifies the study of commutators in groups and Lie algebras through balance equations over commutative rings and the associated $R$-weighted graphs $\,\Gamma(D)$. SOLVABILITY of $E(D)$ is characterized by a consistent vertex labeling, with a detailed graph-theoretic taxonomy (borderless graphs, nets, cycles, anchors) guiding when commutators fill the entire derived structure. It provides explicit solutions over local rings with residue field size at least $3$ and develops p-group constructions from graphs to realize or distinguish $K(G)$ and $G'$, while extending the framework to alternating bilinear maps and Lie brackets to determine surjectivity and $[L,L]=L'$. The approach yields both positive and negative results, enabling infinite families of examples and offering a dimension-free, ring-aware method to analyze commutator maps in algebraic objects of interest, including applications to the $p$-adic setting.

Abstract

In this article, we introduce balance equations over commutative rings $R$ and associate $R$-weighted graphs to them so that solving balance equations corresponds to a consistent labeling of vertices of the associated graph. Our primary focus is the case when $R$ is a commutative local ring whose residue field contains at least three elements. In this case, we provide explicit solutions of balance equations. As an application, letting $R$ to be the ring of $p$-adic integers, we examine some necessary and sufficient conditions for a $p$-group of nilpotency class $2$ to have its set of commutators coincide with its commutator subgroup. We also apply our results to study the surjectivity of the Lie bracket in Lie algebras, without any restriction on their dimension and the underlined field.

$R$-weighted graphs and commutators

TL;DR

The paper unifies the study of commutators in groups and Lie algebras through balance equations over commutative rings and the associated -weighted graphs . SOLVABILITY of is characterized by a consistent vertex labeling, with a detailed graph-theoretic taxonomy (borderless graphs, nets, cycles, anchors) guiding when commutators fill the entire derived structure. It provides explicit solutions over local rings with residue field size at least and develops p-group constructions from graphs to realize or distinguish and , while extending the framework to alternating bilinear maps and Lie brackets to determine surjectivity and . The approach yields both positive and negative results, enabling infinite families of examples and offering a dimension-free, ring-aware method to analyze commutator maps in algebraic objects of interest, including applications to the -adic setting.

Abstract

In this article, we introduce balance equations over commutative rings and associate -weighted graphs to them so that solving balance equations corresponds to a consistent labeling of vertices of the associated graph. Our primary focus is the case when is a commutative local ring whose residue field contains at least three elements. In this case, we provide explicit solutions of balance equations. As an application, letting to be the ring of -adic integers, we examine some necessary and sufficient conditions for a -group of nilpotency class to have its set of commutators coincide with its commutator subgroup. We also apply our results to study the surjectivity of the Lie bracket in Lie algebras, without any restriction on their dimension and the underlined field.
Paper Structure (12 sections, 37 theorems, 36 equations, 8 figures)

This paper contains 12 sections, 37 theorems, 36 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Commutators in groups
  3. Commutators in Lie algebras
  4. Some graphs and their properties
  5. Borderless Graphs
  6. Nets
  7. System of Balance Equations and Labeling of Graphs
  8. Balance Equations
  9. Labeling of R-weighted graphs
  10. Commutators and Commutator Subgroup of Nilpotent Groups of Class 2
  11. Constructing a $p$-group from a graph
  12. Surjectivity of bilinear maps and Lie bracket

Key Result

Theorem A

Let $p \ne 2$ and $G$ be a $p$-group of nilpotency class $2$. Let $g = \prod_{i < j}[g_i,g_j]^{d_{i,j}} \in G'$, where $g_iZ(G), g_jZ(G) \in B_G$, be such that the graph $\Gamma(D)$ does not contain bad cycles. Then $g$ is a commutator in $G$. (Theorem without bad cycle for p-groups elementwise)

Figures (8)

  • Figure 1: A borderless graph $\Gamma_1$ with $s(\Gamma_1) = 2$.
  • Figure 2: A borderless graph $\Gamma_2$ with $s(\Gamma_2) = 2$.
  • Figure 3: Example of a net $\Gamma_1$ with $\eta(\Gamma_1) =2$.
  • Figure 4: Example of a graph that is not net, but satisfies conditions (1) and (3) of Lemma \ref{['net equivalent definition']}.
  • Figure 5: Example of a graph that is not net, but satisfies conditions (1) and (2) of Lemma \ref{['net equivalent definition']}.
  • ...and 3 more figures

Theorems & Definitions (66)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 56 more