Tridiagonal pairs and the q-tetrahedron algebra

Abstract

In this paper we further develop the connection between tridiagonal pairs and the q-tetrahedron algebra ${⊠}_q$. Let V denote a finite dimensional vector space over an algebraically closed field and let A, A^* denote a tridiagonal pair on V. For $0\le i\le d$ let ${\theta }_i$ (resp. ${\theta }_i^{\ast }$) 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 ${\theta }_i^{\ast }=q^{d-2i}$ there exists an irreducible ${⊠}_q$-module structure on V such that the ${⊠}_q$ 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 ${\theta }_i^{\ast }=q^{2i-d}+c{q}^{d-2i}$. In this case we associate to A,A^* a polynomial P and prove the following equivalence. The following are equivalent: (i) There exists a ${⊠}_q$-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 ${⊠}_q$. (ii) P(q^{2d-2} (q-q^{-1})^{-2}) \neq 0. Suppose (i),(ii) hold. Then the ${⊠}_q$-module structure on V is unique and irreducible.

30 pages, bibliography added (references were missing in first version), published in Linear Algebra and its Applications

References in corpus (2)

• Finite-dimensional irreducible modules for the three-point ${\mathfrak{sl}}_2$ loop algebra
• The structure of a tridiagonal pair

Cited by in corpus (2)

• How to sharpen a tridiagonal pair
• On the shape of a tridiagonal pair