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 $A^{\ast }:V\to V$ such that (i) each of $A,A^{\ast }$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i\rbrace_{i=0}^d$oftheeigenspacesof$A$suchthat$A^* V_i \subseteq V_{i-1} + V_i+ V_{i+1}$for$0 \leq i \leq d$,where$V_{-1} = 0$and$V_{d+1}= 0$;(iii)thereexistsanordering$\lbrace V^i\rbrace{i=0}^{\delta}$ of the eigenspaces of $A^{\ast }$ such that $A{V}_i^{\ast }\subseteq V_{i-1}^{\ast }+V_i^{\ast }+V_{i+1}^{\ast }$ for $0\le i\le \delta$, where $V_{-1}^{\ast }=0$ and $V_{\delta +1}^{\ast }=0$; (iv) there does not exist a subspace $U$ of $V$ such that $AU\subseteq U$, $A^{\ast }U\subseteq U$, $U≠0$, $U≠V$. We call such a pair a tridiagonal pair on $V$. We assume that $A,A^{\ast }$ belongs to a family of tridiagonal pairs said to have $q$-Racah type. There is an infinite-dimensional algebra ${⊠}_q$ called the $q$-tetrahedron algebra; it is generated by four copies of $U_q({\mathfrak{s}\mathfrak{l}}_2)$ that are related in a certain way. Using $A, A^$weconstructtwo$\boxtimes_q$-modulestructureson$V$. In this construction the two main ingredients are the double lowering map $\psi :V\to V$ 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.