Some q-exponential Formulas for Finite-Dimensional □q-Modules

We consider the algebra square(q) which is a mild generalization of the quantum algebra U-q(sl(2)). The algebra square(q) is defined by generators and relations. The generators are {x(i)}(i is an element of Z4), where Z(4) is the cyclic group of order 4. For i is an element of Z(4) the generators x(i), x(i+1) satisfy a q-Weyl relation, and x(i), x(i+2) satisfy a cubic q-Serre relation. For i is an element of Z(4) we show that the action of xi is invertible on every nonzero finite-dimensional square(q)-module. We view x(i)(-1) as an operator that acts on nonzero finite-dimensional square(q)-modules. For i is an element of Z(4), define n(i,i+1) = q(1 - x(i)x(i+1))/(q - q(-1)). We show that the action of n(i,i+1) is nilpotent on every nonzero finite-dimensional square(q)-module. We view the q-exponential exp(q) (n(i,i+1)) as an operator that acts on nonzero finite-dimensional square(q)-modules. In our main results, for i, j is an element of Z(4) we express each of exp(q) (n(i,i+1))x(j) exp(q) (n(i,i+1))(-1) and exp(q) (n(i,i+1))(-1)x(j)exp(q) (n(i,i+1)) as a polynomial in {x(K)(+/- 1)}(k is an element of Z4).