Tridiagonal pairs and the quantum affine algebra ${\boldmath U_q({\widehat {sl}}_2)}$

Abstract

Let ${\mathbb K}$ denote an algebraically closed field and let q denote a nonzero scalar in ${\mathbb K}$ that is not a root of unity. Let V denote a vector space over ${\mathbb K}$ with finite positive dimension and let A,A* denote a tridiagonal pair on V. Let θ₀, θ₁,…, θ _d (resp. θ*₀, θ*₁,…, θ* _d ) denote a standard ordering of the eigenvalues of A (resp. A*). We assume there exist nonzero scalars a, a* in ${\mathbb K}$ such that θ _i = aq ^2i−d and θ* _i = a*q ^d−2i for 0 ≤ i ≤ d. We display two irreducible ${\boldmath U_q({\widehat {sl}}_2)}$-module structures on V and discuss how these are related to the actions of A and A*.