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Tridiagonal pairs of $q$-Racah type and the $q$-tetrahedron algebra

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.