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Tridiagonal pairs and the q-tetrahedron algebra

Abstract

In this paper we further develop the connection between tridiagonal pairs and the q-tetrahedron algebra q\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 0id0 \leq i \leq d let θi\theta_i (resp. θi\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 θi=q2id\theta_i = q^{2i-d} and θi=qd2i\theta^*_i = q^{d-2i} there exists an irreducible q\boxtimes_q-module structure on V such that the q\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 θi=q2id\theta_i = q^{2i-d} and θi=q2id+cqd2i\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 q\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 q\boxtimes_q. (ii) P(q^{2d-2} (q-q^{-1})^{-2}) \neq 0. Suppose (i),(ii) hold. Then the q\boxtimes_q-module structure on V is unique and irreducible.Comment: 30 pages, bibliography added (references were missing in first version), published in Linear Algebra and its Application

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