Advanced Search

Let $H_8$ be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra $\widetilde{H_8}$ based on $H_8$, then we investigate the structure of the representation ring of $\widetilde{H_8}$. Finally, we prove that the automorphism group of $r\left(\widetilde{H_8}\right)$ is just isomorphic to $D_6$, where $D_6$ is the dihedral group with order 12.

Let $r\left({𝔴}_2^0\right)$ be the Green ring of the weak Hopf algebra ${𝔴}_2^0$ corresponding to Sweedler’s 4-dimensional Hopf algebra $H_2$, and let $\mathrm{Aut}\left(R\left({𝔴}_2^0\right)\right)$ be the automorphism group of the Green algebra $R\left({𝔴}_2^0\right)=r\left({𝔴}_2^0\right){\otimes }_{\mathbb{Z}}ℂ$. We show that the quotient group $\mathrm{Aut}\left(R\left({𝔴}_2^0\right)\right)/C_2\cong S_3$, where $C_2$ contains the identity map and is isomorphic to the infinite group $({ℂ}^{*},\times )$ and $S_3$ is the symmetric group of order 6.