Braid rigidity for path algebras

Path algebras are a convenient way of describing decompositions tensor powers an object in category. If the category is braided, one obtains representations braid groups $B_n$ for all $n\in \N$. We say that such rigid if they determined by path algebra and $B_2$. show besides known classical cases also 7-dimensional representation $G_2$ satisfies rigidity condition, provided $B_3$ generates $\End(V^{\otimes 3})$. obtain complete classification ribbon categories with fusion rules $\g(G_2)$ this condition satisfied.