On automorphisms and universal ℛ-matrices at roots of unity

Invertible universal ?-matrices of quantum Lie algebras do not exist at roots of unity. However, quotients exist for which intertwiners of tensor products of representations always exist, i.e. ?-matrices exist in the representations. One of these quotients, which is finite-dimensional, has a universal ?-matrix. In this Letter we answer the following question: under which condition are the different quotients of U _q (sl(2)) (Hopf)-equivalent? In the case when they are equivalent, the universal ?-matrix of the one can be transformed into a universal ?-matrix of the other. We prove that this happens only whenq ⁴ = 1, and we explicitly give the expressions for the automorphisms and for the transformed universal ?-matrices in this case.