We define and compare, by model-theoretical methods, some exponentiations over the quantum algebra $U_q\left(s{l}_2\left(ℂ\right)\right)$. We discuss two cases, according to whether the parameter $q$ is a root of unity. We show that the universal enveloping algebra of $s{l}_2\left(ℂ\right)$ embeds in a non-principal ultraproduct of $U_q\left(s{l}_2\left(ℂ\right)\right)$, where $q$ varies over the primitive roots of unity.