Coactions of cocommutative Hopf algebras on skew polynomial rings

Abstract

We classify the cocommutative Hopf algebras which coact inner-faithfully on (one-parameter) skew polynomial rings $A_q(n)=\mathbb{k}⟨x_1,\dots ,x_n⟩/(x_j{x}_i-q{x}_i{x}_j\mid i<j)$ 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 $\underline{\mathrm{aut}}(A_q(n))$, by classifying all its cocommutative quotients. We therefore also give an explicit presentation of $\underline{\mathrm{aut}}(A_q(n))$ for arbitrary $q\in {\mathbb{k}}^{\ast }$ and $n\in \mathbb{N}$.

40 pages. Comments welcome!