$q$ -Racah型三对角对与$q$ -四面体代数
Tridiagonal pairs of $q$-Racah type and the $q$-tetrahedron algebra
摘要：
令$\mathbb F$表示域，令$V$表示$\mathbb F$上具有有限正维数的向量空间。我们考虑一对有序的$\mathbb F$ -线性映射$A: V \to V$和$A^*:V\to V$使得（i）每个$A,A^*$都是可对角化的;（ii）存在$A$的特征空间的序$\lbrace V_i\rbrace_{i=0}^d$使得$A^* V_i \subseteq V_{i-1} + V_i+ V_{i+1}$对于$0 \leq i \leq d$，其中$V_{-1} = 0$和$V_{d+1}= 0$ ;（iii）存在$A^*$的特征空间的序$\lbrace V^*_i\rbrace_{i=0}^δ$使得$A V^*_i \subseteq V^*_{i-1} + V^*_i+ V^*_{i+1} $对于$0 \leq i \leq δ$，其中$V^*_{-1} = 0$和$V^*_{δ+1}= 0$ ;（iv）不存在$V$的子空间$U$使得$AU\subseteq U$，$A^*U \subseteq U$，$U\not=0$，$U\not=V$。我们在$V$上称这样的对为三对角对。我们假设$A, A^*$属于一个家庭的三对角对说有$q$ -拉卡型。有一种无限维的代数$\boxtimes_q$称为$q$ -四面体代数;它是由$U_q(\mathfrak{sl}_2)$的四个副本以某种方式相关而生成的。使用$A, A^*$，我们构建了两个$\boxtimes_q$模块结构的$V$。在这个构造中，两个主要成分是由Sarah Bockting-Conrad提出的双降映射$ψ:V\to V$和由V.F. R.琼斯
英文摘要：
Let $\mathbb F$ denote a field, and let $V$ denote a vector space over $\mathbb F$ with finite positive dimension. We consider an ordered pair of $\mathbb F$-linear maps $A: V \to V$ and $A^*:V\to V$ such that (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i\rbrace_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_i+ V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1} = 0$ and $V_{d+1}= 0$; (iii) there exists an ordering $\lbrace V^*_i\rbrace_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_i+ V^*_{i+1} $ for $0 \leq i \leq δ$, where $V^*_{-1} = 0$ and $V^*_{δ+1}= 0$; (iv) there does not exist a subspace $U$ of $V$ such that $AU\subseteq U$, $A^*U \subseteq U$, $U\not=0$, $U\not=V$. We call such a pair a tridiagonal pair on $V$. We assume that $A, A^*$ belongs to a family of tridiagonal pairs said to have $q$-Racah type. There is an infinite-dimensional algebra $\boxtimes_q$ called the $q$-tetrahedron algebra; it is generated by four copies of $U_q(\mathfrak{sl}_2)$ that are related in a certain way. Using $A, A^*$ we construct two $\boxtimes_q$-module structures on $V$. In this construction the two main ingredients are the double lowering map $ψ:V\to V$ due to Sarah Bockting-Conrad, and a certain invertible map $W:V\to V$ motivated by the spin model concept due to V. F. R. Jones.
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