LAUSR

Abstract 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 □ q denote the unital associative F -algebra defined by generators { x i } i ∈ Z 4 and relations q x 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 ρ of □ q that sends x i ↦ x i + 1 for i ∈ Z 4 . 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: (i) the Drinfel'd polynomial for the □ q -module V ; (ii) the Drinfel'd polynomial for the □ q -module V twisted via ρ . Specifically, we show that the roots of (i) are the inverses of the roots of (ii). We discuss how □ q is related to the quantum loop algebra U q ( L ( sl 2 ) ) , its positive part U q + , the q -tetrahedron algebra ⊠ q , and the q -geometric tridiagonal pairs.