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Tridiagonal pairs and the q-tetrahedron algebra

In this paper we further develop the connection between tridiagonal pairs and the q-tetrahedron algebra $\boxtimes_q$. Let V denote a finite dimensional vector space over an algebraically closed field and let A, A^* denote a tridiagonal pair on V. For $0 \leq i \leq d$ let $θ_i$ (resp. $θ^*_i$) denote a standard ordering of the eigenvalues of A (resp. A^*). Fix a nonzero scalar q which is not a root of unity. T. Ito and P. Terwilliger have shown that when $θ_i = q^{2i-d}$ and $θ^*_i = q^{d-2i}$ there exists an irreducible $\boxtimes_q$-module structure on V such that the $\boxtimes_q$ generators x_{01}, x_{23} act as A, A^* respectively. In this paper we examine the case in which there exists a nonzero scalar c in K such that $θ_i = q^{2i-d}$ and $θ^*_i = q^{2i-d} + c q^{d-2i}$. In this case we associate to A,A^* a polynomial P and prove the following equivalence. The following are equivalent: (i) There exists a $\boxtimes_q$-module structure on V such that x_{01} acts as A and x_{30} + cx_{23} acts as A^*, where x_{01}, x_{30}, x_{23} are standard generators for $\boxtimes_q$. (ii) P(q^{2d-2} (q-q^{-1})^{-2}) \neq 0. Suppose (i),(ii) hold. Then the $\boxtimes_q$-module structure on V is unique and irreducible.