The quantum adjacency algebra and subconstituent algebra of a graph

TL;DR

This paper explores the relationship between the quantum adjacency algebra and the subconstituent algebra of a graph, revealing conditions under which they coincide or differ, with examples from Hamming and dual polar graphs.

Contribution

It introduces the notion of quasi-isomorphic irreducible modules to distinguish when the quantum adjacency algebra equals the subconstituent algebra.

Findings

For Hamming graphs, Q equals T.

For bipartite dual polar graphs, Q does not equal T.

The equivalence between Q ≠ T and the existence of quasi-isomorphic modules with different endpoints.

Abstract

Let $\Gamma$ denote a finite, undirected, connected graph, with vertex set $X$. Fix a vertex $x \in X$. Associated with $x$ is a certain subalgebra $T=T(x)$ of ${\rm Mat}_X(\mathbb C)$, called the subconstituent algebra. The algebra $T$ is semisimple. Hora and Obata introduced a certain subalgebra $Q \subseteq T$, called the quantum adjacency algebra. The algebra $Q$ is semisimple. In this paper we investigate how $Q$ and $T$ are related. In many cases $Q=T$, but this is not true in general. To clarify this issue, we introduce the notion of quasi-isomorphic irreducible $T$-modules. We show that the following are equivalent: (i) $Q \neq T$; (ii) there exists a pair of quasi-isomorphic irreducible $T$-modules that have different endpoints. To illustrate this result we consider two examples. The first example concerns the Hamming graphs. The second example concerns the bipartite dual polar graphs. We show that for the first example $Q=T$, and for the second example $Q \neq T$.