超立方体、Leonard三元组与反交换子自旋代数
Hypercubes, Leonard triples and the anticommutator spin algebra
摘要：
本文研究了三类对象：Leonard三元组、距离正则图和反交换子自旋代数的模。设$\K$表示特征为零的代数闭域。设$V$表示$\K$上具有有限正维数的向量空间。$V$上的伦纳德三元组是$\mathrm{End}(V)$中线性变换的有序三元组，使得对于这些变换中的每一个，存在$V$的基，关于该基，表示该变换的矩阵是对角的，并且表示其他两个变换的矩阵是不可约三对角的。我们感兴趣的伦纳德三元组据说完全是B/AB和Bannai/Ito类型。Bannai/Ito型的B/AB Leonard三元组全部与反交换子自旋代数$\mathcal{A}$、由生成元$x,y,z$和关系\[xy+yx=2z,\qquad yz+zy=2x,\qquad zx+xz=2y.\]定义的酉结合$\K$ -代数一起出现。令$D\geq0$表示整数，令$Q_{D}$表示直径为$D$的超立方体，令$\tilde{Q}_{D}$表示对跖商。令$T$（resp. $\tilde{T}$）表示$Q_{D}$的Terwilliger代数（分别为$\tilde{Q}_{D}$）。我们得到以下结果。当$D$是偶数时（odd），我们证明了在$Q_{D}$上存在唯一的$\mathcal{A}$ -模结构（分别为. $\tilde{Q}_{D}$），使得$x,y$分别充当邻接矩阵和对偶邻接矩阵。我们分类得到的不可约$\mathcal{A}$ -模同构。我们引入加权邻接矩阵的$Q_{D}$，$\tilde{Q}_{D}$。当$D$是偶数时（奇），我们证明了$Q_{D}$的邻接矩阵、对偶邻接矩阵和加权邻接矩阵的作用。$\tilde{Q}_{D}$）在任何不可约$T$ -模上（分别$\tilde{T}$ -模块）形成完全二分的（分别是几乎二分的）Bannai/Ito型的Leonard三元组，并将Leonard三元组分类到同构.
英文摘要：
This paper is about three classes of objects: Leonard triples, distance-regular graphs and the modules for the anticommutator spin algebra. Let $\K$ denote an algebraically closed field of characteristic zero. Let $V$ denote a vector space over $\K$ with finite positive dimension. A Leonard triple on $V$ is an ordered triple of linear transformations in $\mathrm{End}(V)$ such that for each of these transformations there exists a basis for $V$ with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal. The Leonard triples of interest to us are said to be totally B/AB and of Bannai/Ito type. Totally B/AB Leonard triples of Bannai/Ito type arise in conjunction with the anticommutator spin algebra $\mathcal{A}$, the unital associative $\K$-algebra defined by generators $x,y,z$ and relations\[xy+yx=2z,\qquad yz+zy=2x,\qquad zx+xz=2y.\] Let $D\geq0$ denote an integer, let $Q_{D}$ denote the hypercube of diameter $D$ and let $\tilde{Q}_{D}$ denote the antipodal quotient. Let $T$ (resp. $\tilde{T}$) denote the Terwilliger algebra for $Q_{D}$ (resp. $\tilde{Q}_{D}$). We obtain the following. When $D$ is even (resp. odd), we show that there exists a unique $\mathcal{A}$-module structure on $Q_{D}$ (resp. $\tilde{Q}_{D}$) such that $x,y$ act as the adjacency and dual adjacency matrices respectively. We classify the resulting irreducible $\mathcal{A}$-modules up to isomorphism. We introduce weighted adjacency matrices for $Q_{D}$, $\tilde{Q}_{D}$. When $D$ is even (resp. odd) we show that actions of the adjacency, dual adjacency and weighted adjacency matrices for $Q_{D}$ (resp. $\tilde{Q}_{D}$) on any irreducible $T$-module (resp. $\tilde{T}$-module) form a totally bipartite (resp. almost bipartite) Leonard triple of Bannai/Ito type and classify the Leonard triple up to isomorphism.
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