在有限维半单Hopf代数的分类研究中，有限秩半单monoidal范畴（或fusion范畴）起着关键性的作用．进一步研究有限秩monoidal范畴显得非常重要，有望对Hopf代数的分类发挥更大的作用．本项目拟研究有限秩monoidal范畴及其相关问题．研究有限秩monoidal范畴的不变量，给出Green环以外的其它重要不变量使得这些不变量能够完全确定monoidal范畴的结构．研究有限秩monoidal范畴的重构，给出从某种给定结构出发构造有限秩monoidal范畴的一般方法．研究Taft代数的表示环的范畴化，给出有限秩monoidal范畴的分类使得它们与Taft代数有相同的表示环和Auslander代数．研究Taft代数的Drinfeld量子偶的表示环，给出张量积模的分解式和表示环的结构．研究若干有限表示型Hopf代数的Auslander代数，利用它们刻画相应monoidal范畴的结构．

The semisimple monoidal categories of finite rank (or fusion categories) play a key role in the classification of finite dimensional semisimple Hopf algebras. Therefore, it is very important to further study the monoidal categories of finite rank, which may play the more important role in the classification of finite dimensional Hopf algebras. In this project, we plan to study the monoidal categories of finite rank and the relative topics. We study the invariants of the monoidal categories of finite rank. Except Green ring, we will give the other important invariants which, together with Green ring, determine the monoidal categories of finite rank completely. Then we study the reconstruction of the monoidal categories of finite rank. We display a general precedure to construct a monoidal category of finite rank from some given structures. Next, we study the categorifications of the Green rings of Taft algebras. We classify the monoidal categories of finite rank which have the same representation rings and Auslander algebras with Taft algebras. We also study the representation rings of the Drinfeld quantum doubles of Taft algebras. We decompose the tensor product of two indecomposable modules into the direct sum of indecomposable modules, and describe the structure of the representation rings. Finally, we study the Auslander algebras of several Hopf algebras. Using Auslander algebras, we characterize the corresponding monoidal categories.