Finite-dimensional irreducible □q-modules and their Drinfel'd polynomials

Let F denote an algebraically closed field with characteristic 0, and let q denote a nonzero scalar in F that is not a root of unity. Let Z(4) denote the cyclic group of order 4. Let square(q) denote the unital associative F-algebra defined by generators {x(i)}(i is an element of z4), and relations qx(i)x(i+1) - q(-1)x(i+1)x(i)/q - q(-1) = 1, x(i)(3)x(i+2) - [3](q)x(i)(2)x(i+2)x(i) + [3](q)x(i)x(i+2)x(i)(2) - x(i+2)x(i)(3) = 0 where [3](q) = (q(3) - q(-3))/(q - q(-1)). There exists an automorphism rho of square(q) that sends xi bar rigt arrow x(i+1) for i is an element of Z(4). Let V denote a finite-dimensional irreducible square(q)-module of type 1. To V we attach a polynomial called the Drinfel'd polynomial. In our main result, we explain how the following are related: (i) the Drinfel'd polynomial for the square(q)-module V; (ii) the Drinfel'd polynomial for the square(q)-module V twisted via rho. Specifically, we show that the roots of (i) are the inverses of the roots of (ii). We discuss how square(q) is related to the quantum loop algebra U-q(L(sl(2))), its positive part U-q(+) the q-tetrahedron algebra boxed plus(4), and the q-geometric tridiagonal pairs. (C) 2017 Elsevier Inc. All rights reserved.