Coactions of cocommutative Hopf algebras on skew polynomial rings

We classify the cocommutative Hopf algebras which coact inner-faithfully on (one-parameter) skew polynomial rings $A_q(n) = \Bbbk \langle x_1,\dots,x_n \rangle/(x_j x_i - q x_i x_j \mid i < j)$ for $n = 2$ and $3$. As a direct corollary, we obtain a classification of group gradings on two- and three-variable skew polynomial rings, recovering a result of Crawford in the two-variable case. Our results are achieved via Manin's universal coacting Hopf algebra construction, often denoted $\underline{\operatorname{aut}}(A_q(n))$, by classifying all its cocommutative quotients. We therefore also give an explicit presentation of $\underline{\operatorname{aut}}(A_q(n))$ for arbitrary $q \in \Bbbk^*$ and $n \in \mathbb{N}$.