The quantum adjacency algebra and subconstituent algebra of a graph
The quantum adjacency algebra and subconstituent algebra of a graph
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DOI：
10.1016/j.jcta.2019.02.022
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发表时间：
2017-10
期刊：
影响因子：
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通讯作者：
Paul M. Terwilliger;A. Žitnik
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文献类型：
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作者：
Paul M. Terwilliger;A. Žitnik
设Γ表示一个有限的无向连通图，其顶点集为X.固定一个顶点x∈ X。与x相关联的是Mat X（C）的某个子代数T= T（x），称为子成分代数。代数T是半单的。Hora和Obata引入了一个特定的子代数Q T，称为量子邻接代数。代数Q是半单的。在本文中，我们研究如何Q和T的相关。在许多情况下，Q= T，但这一般不是真的。为了澄清这个问题，我们引入了拟同构不可约T-模的概念。我们证明了下列等价：（i）Q <$T;（ii）存在一对具有不同端点的拟同构不可约T-模。为了说明这个结果，我们考虑两个例子。第一个例子涉及汉明图。第二个例子是关于二部对偶极图。我们证明了对于第一个例子Q= T，对于第二个例子Q <$T。
Let Γ denote a finite, undirected, connected graph, with vertex set X. Fix a vertex x∈ X. Associated with x is a certain subalgebra T= T (x) of Mat X (C), called the subconstituent algebra. The algebra T is semisimple. Hora and Obata introduced a certain subalgebra Q⊆ 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≠ 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≠ T.