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 Z4 denote the cyclic group of order 4. Let □q denote the unital associative F-algebra defined by generators {xi}i∈Z4 and relationsqxixi+1−q−1xi+1xiq−q−1=1,xi3xi+2−[3]qxi2xi+2xi+[3]qxixi+2xi2−xi+2xi3=0, where [3]q=(q3−q−3)/(q−q−1). There exists an automorphism ρ of □q that sends xi↦xi+1 for i∈Z4. Let V denote a finite-dimensional irreducible □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: