Such tensor categories are called "near-group categories": these are semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. In your case, the group is $\{1,a,A\}$, and is isomorphic to $\mathbb Z/3$.

In http://arxiv.org/pdf/1401.1879v2.pdf it is shown that only for finitely many values of your parameter $i$ it is the case that your fusion ring comes from a fusion category (among finite $i$'s, the maximum possible value is $i=6$). Thus, I would find it very surprising if there were a semisimple category (suitably generalised to allow for infinite direct sums of simple objects) that realises the case $i=\infty$.

Such tensor categories are called "near-group categories": these are semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. In your case, the group is $\{1,a,A\}$, and is isomorphic to $\mathbb Z/3$.

The paper http://arxiv.org/pdf/1401.1879v2.pdf shows that there's only finitely many values of your parameter $i$ such that your fusion ring comes from a fusion category (excluding $i=\infty$, the maximum possible value is $i=6$). Thus, I would find it very surprising if there were a semisimple category (suitably generalised to allow for infinite direct sums of simple objects) that realises the case $i=\infty$.

Such tensor categories are called "near-group categories": these are semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. In your case, the group is $\{1,a,A\}$, and is isomorphic to $\mathbb Z/3$.

In http://arxiv.org/pdf/1401.1879v2.pdf it is shown that only for finitely many values of your parameter $i$ it is the case that your fusion ring comes from a fusion category (among finite $i$'s, the maximum possible value is $i=6$). Thus, I would find it very surprising if there were a semisimple category (suitably generalised to allow for infinite direct sums of simple objects) that realises the case $i=\infty$, even though that's not theoretically impossible.

Such tensor categories are called "near-group categories": these are semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. In your case, the group is $\{1,a,A\}$, and is isomorphic to $\mathbb Z/3$.

In http://arxiv.org/pdf/1401.1879v2.pdf it is shown that only for finitely many values of your parameter $i$ it is the case that your fusion ring comes from a fusion category (among finite $i$'s, the maximum possible value is $i=6$). Thus, I would find it very surprising if there were a semisimple category (suitably generalised to allow for infinite direct sums of simple objects) that realises the case $i=\infty$.