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Let $R$ be a fusion ring and $R_{ℂ}:=R{\otimes }_{\mathbb{Z}}ℂ$ be the corresponding fusion algebra. We first show that the algebra $R_{ℂ}$ has only one left (right, two-sided) cell and the corresponding left (right, two-sided) cell module. Then we prove that, up to isomorphism, $R_{ℂ}$ admits a unique special module, which is 1-dimensional and given by the Frobenius-Perron homomorphism FPdim. Moreover, as an example, we explicitly determine the special module of the interpolated fusion algebra $R\left(\mathrm{PSL}(2,q)\right):=r\left(\mathrm{PSL}(2,q)\right){\otimes }_{\mathbb{Z}}ℂ$ up to isomorphism, where $r\left(\mathrm{PSL}\right(2,q\left)\right)$ is the interpolated...

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.