|
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 relationsqx"ix"i"+"1-q^-^1x"i"+"1x"iq-q^-^1=1,x"i^3x"i"+"2-[3]"qx"i^2x"i"+"2x"i+[3]"qx"ix"i"+"2x"i^2-x"i"+"2x"i^3=0, where [3]"q=(q^3-q^-^3)/(q-q^-^1). There exists an automorphism @r 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 @r. 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.
|
