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Graphs, Axial Algebras and their Automorphism Groups

Hans Cuypers

Abstract

We introduce a class of algebras over a field $\mathbb{F}$ related to directed graphs in which all edges are labeled by nonzero elements of the field $\mathbb{F}$. If all labels are different from $1$, these algebras are axial algebras. We determine their fusion laws, prove them to be simple in almost all cases, and determine their automorphism group under some conditions on the degrees and girth of the graph. A construction of a class of these graphs with prescribed automorphism group enables us to construct for each group $G$ infinitely many simple (axial) algebras (with a fixed fusion law) such that the automorphism group of the algebra is isomorphic to $G$.

Graphs, Axial Algebras and their Automorphism Groups

Abstract

We introduce a class of algebras over a field related to directed graphs in which all edges are labeled by nonzero elements of the field . If all labels are different from , these algebras are axial algebras. We determine their fusion laws, prove them to be simple in almost all cases, and determine their automorphism group under some conditions on the degrees and girth of the graph. A construction of a class of these graphs with prescribed automorphism group enables us to construct for each group infinitely many simple (axial) algebras (with a fixed fusion law) such that the automorphism group of the algebra is isomorphic to .
Paper Structure (6 sections, 28 theorems, 36 equations, 1 figure, 3 tables)

This paper contains 6 sections, 28 theorems, 36 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Algebras from graphs
  3. Idempotents and fusion rules for algebras from graphs
  4. Idempotents and elements of small rank
  5. Automorphism groups of the algebras
  6. Every group is automorphism group of a simple (axial) algebra

Key Result

Theorem 1.1

Let $\mathcal{F}$ be a subset of the field $\mathbb{F}$ containing $1$. Let $\Gamma=(X,E)$ be a weakly connected directed graph with edges labeled by elements in $\mathcal{F}$ different from $0$. Then we have the following:

Figures (1)

  • Figure 1: Replace the edge $(a,b=as)$ by a subgraph.

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Example 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 48 more