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Higher homological algebra for one-point extensions of bipartite hereditary algebras and spectral graph theory

Karin M. Jacobsen, Mads Hustad Sandøy, Laertis Vaso

Abstract

In this article we study higher homological properties of $n$-levelled algebras and connect them to properties of the underlying graphs. Notably, to each $2$-representation-finite quadratic monomial algebra $Λ$ we associate a bipartite graph $\overline{B_Λ}$ and we classify all such algebras $Λ$ for which $\overline{B_Λ}$ is regular or edge-transitive. We also show that if $\overline{B_Λ}$ is semi-regular, then it is a reflexive graph.

Higher homological algebra for one-point extensions of bipartite hereditary algebras and spectral graph theory

Abstract

In this article we study higher homological properties of -levelled algebras and connect them to properties of the underlying graphs. Notably, to each -representation-finite quadratic monomial algebra we associate a bipartite graph and we classify all such algebras for which is regular or edge-transitive. We also show that if is semi-regular, then it is a reflexive graph.

Paper Structure

This paper contains 20 sections, 28 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Conventions and notation
  4. Spectral graph theory
  5. Bi-eigenvectors and bi-eigenvalues
  6. Koszul algebras
  7. Higher-dimensional homological algebra
  8. One-point extensions
  9. n-levelled algebras
  10. Definition and basic properties
  11. n-levelled algebras and higher homological algebra
  12. One-point extensions of bipartite hereditary algebras
  13. 2-representation-finite quadratic monomial algebras
  14. Trivial extensions of star algebras
  15. Star algebras and higher homological algebra
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\Lambda=\mathop{\mathrm{\mathbf{k}}}\nolimits S_{(r,s)}$ be a $2$-representation-finite $(r,s)$-star algebra. If the underlying graph $\overline{B_{\Lambda}}$ of $B_{\Lambda}$ is semi-regular of bidegree $(\Sigma_1,\Sigma_2)$, then

Figures (1)

  • Figure 1: On the left, a bipartite graph known as the Heawood graph. On the right, its bipartite complement.

Theorems & Definitions (63)

  • Theorem 1.1: Corollary \ref{['cor:2-RF semi-regular star algebra']}
  • Theorem 1.2: Proposition \ref{['prop: classification in the regular case']} and Proposition \ref{['prop: classification in the edge-transitive case']}
  • Theorem 1.3: Corollary \ref{['cor:the graph B_Lambda is reflexive and almost always Salem']}
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Example 2.5
  • ...and 53 more