Automorphism groups of representation rings of the weak Sweedler Hopf algebras

Let ${\mathfrak{w}}_{2,2}^s(s=0,1)$ be two classes of weak Hopf algebras corresponding to the Sweedler Hopf algebra, and $r({\mathfrak{w}}_{2,2}^s)$ be the representation rings of ${\mathfrak{w}}_{2,2}^s$. In this paper, we investigate the automorphism groups $\mathrm{A}\mathrm{u}\mathrm{t}(r({\mathfrak{w}}_{2,2}^s))$ of $r({\mathfrak{w}}_{2,2}^s)$, and discuss some properties of $\mathrm{A}\mathrm{u}\mathrm{t}(r({\mathfrak{w}}_{2,2}^s))$. We obtain that $\mathrm{A}\mathrm{u}\mathrm{t}(r({\mathfrak{w}}_{2,2}^0))$ is isomorphic to $K_4$, where $K_4$ is the Klein four-group. It is shown that $\mathrm{A}\mathrm{u}\mathrm{t}(r({\mathfrak{w}}_{2,2}^1))$ is a non-commutative infinite solvable group, but it is not nilpotent. In addition, $\mathrm{A}\mathrm{u}\mathrm{t}(r({\mathfrak{w}}_{2,2}^1))$ is isomorphic to $(\mathbb{Z}\times {\mathbb{Z}}_2)⋊{\mathbb{Z}}_2$, and its centre is isomorphic to ${\mathbb{Z}}_2$.