Towards reconstruction of finite tensor categories

We take a first step towards a reconstruction of finite tensor categories using finitely many $F$-matrices. The goal is to reconstruct a finite tensor category from its projective ideal. Here we set up the framework for an important concrete example--the $8$-dimensional Nicholas Hopf algebra $K_2$. Of particular importance is to determine its Green ring and tensor ideals. The Hopf algebra $K_2$ allows the recovery of $(2+1)$-dimensional Seiberg-Witten TQFT from Hennings TQFT based on $K_2$. This powerful result convinced us that it is interesting to study the Green ring of $K_2$ and its tensor ideals in more detail. Our results clearly illustrate the difficulties arisen from the proliferation of non-projective reducible indecomposable objects in finite tensor categories.