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- W2157134313 abstract "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 $theta_i$ (resp. $theta^*_i$) 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 $theta_i = q^{2i-d}$ and $theta^*_i = q^{d-2i}$ 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 $theta_i = q^{2i-d}$ and $theta^*_i = q^{2i-d} + c q^{d-2i}$. 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." @default.
- W2157134313 created "2016-06-24" @default.
- W2157134313 creator A5006232802 @default.
- W2157134313 date "2008-06-05" @default.
- W2157134313 modified "2023-09-27" @default.
- W2157134313 title "Tridiagonal pairs and the q-tetrahedron algebra" @default.
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