论文标题
$ Q $ -RACAH类型的Tridiagonal对和$ Q $ -Tetrahedron代数
Tridiagonal pairs of $q$-Racah type and the $q$-tetrahedron algebra
论文作者
论文摘要
令$ \ mathbb f $表示一个字段，让$ v $表示vector Space以$ \ Mathbb f $具有有限的正尺寸。我们考虑了一对$ \ Mathbb f $ -linear Maps $ a：v \ to v $和$ a^*：v \ to v $，因此（i）每个$ a，a^*$都是可对角线的； （ii）存在一个订购$ \ lbrace v_i \ rbrace_ {i = 0}^d $ $ a $ a $ a $的$ a^* v_i \ subseteq v_ {i-1}+ v_i+ v_i+ v_i+ v_+ v_+ v_+ v_ {i+ 1} $ for $ 0 $ v_ {d+1} = 0 $; （iii）存在$ \ lbrace v^*_ i \ rbrace_ {i = 0} $ v^*_ { - 1} = 0 $和$ v^*_ {δ+1} = 0 $; （iv）不存在$ v $的子空间$ u $，因此$ au \ subseteq u $，$ a^*u \ subseteq u $，$ u \ u \ not = 0 $，$ u \ u \ not = v $。我们称这对在$ v $上为三角对。我们假设$ a，a^*$属于一个tridiagonal对家族，据说具有$ q $ -racah类型。有一个无限维代数$ \ boxtimes_q $称为$ q $ -tetrahedron代数；它由以某种方式相关的$ u_q（\ mathfrak {sl} _2）$的四个副本生成。使用$ a，a^*$我们在$ v $上构造了两个$ \ boxtimes_q $-模块结构。在这种结构中，这两种主要成分是由于Sarah Bockting-Conrad引起的双重降低地图$ψ：V \至V $，以及某些可逆地图$ W：V \至V $由V. F. F. F. R. Jones引起的旋转模型概念动机。
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.
