量子群、Hopf代数及张量范畴理论与数学物理的众多分支联系日益紧密因而受到人们的广泛关注。在这些理论中，量子群和张量范畴的形变理论处于重要地位，而张量范畴的Green环作为张量等价意义下的不变量在该理论的研究中发挥关键作用。本项目主要研究量子包络代数的中心子代数的结构、自同构以及Hopf代数与张量范畴的形变等理论。首先通过揭示量子包络代数的Green环与中心子代数的关系，研究中心子代数的结构与量子包络代数的自同构；其次研究张量范畴，特别是某些Hopf代数的表示范畴的形变、分类以及相应Green环的表示理论；最后研究有限张量范畴的Casimir数与张量范畴的Green环、Auslander代数等不变量之间的联系。这些研究内容涉及量子群、Hopf代数、Lie理论、代数表示理论及数学物理等研究领域。项目预期成果不仅能促进这些研究领域的进一步融合与发展，而且有望应用于有限张量范畴的分类。

The theory of quantum groups, Hopf algebras and tensor categories has been found with close connections to many branches of mathematical physics and thus has been paid close attention. Among these research areas, the theory of quantum groups and the deformation theory of tensor categories plays an important role, especially the Green ring of a tensor category as an invariant up to tensor equivalence plays a key role. The present research proposal mainly focusses on structure, automorphisms of quantized enveloping algebras and deformations of Hopf algebras and tensor categories. Firstly, the relationship between the central subalgebras and the Green rings of quantized enveloping algebras is revealed so as to study the structure central subalgebras and automorphisms of the quantized enveloping algebras. Secondly, for tensor categories, especially for the representation categories of some Hopf algebras, we study their deformations, classifications and the representations of their Green rings. Finally, the relationship among the Casimir number, the Green ring and the Auslander algebra of a tensor category is studied. This research proposal interacts several areas of quantum groups, Hopf algebras, Lie theory, the represention theory of algebras and the mathematical physics. So we expect that the results from this project find applications not only back in those areas, but also in the classification of finite tensor categories.