本项目拟研究Gr-范畴中的代数及其在相关问题上的应用，特别是平方和的Hurwitz问题。Gr-范畴中的代数是张量范畴和量子代数领域近年来重要的研究对象，而平方和的Hurwitz问题则历史悠久且极富挑战，最近这两者之间被发现有密切的联系，相关的研究亟待开展。本项目将重点研究辫子Gr-范畴中的半单代数，把域上的半单代数及Brauer群理论发展到辫子Gr-范畴的情形，并利用其中的半单代数为平方和的Hurwitz问题提供系列新解，探索平方和项数上下界的渐近公式等问题。另一方面，Gr-范畴是点化有限张量范畴的基础，其中的Hopf代数是研究的关键。本项目也将系统研究有限交换群及其上非abel型3上循环确定的Gr-范畴及其量子偶中的Nichols代数和算术根系等理论，并应用到相关点化有限张量范畴的构造和分类。

In this project we shall investigate algebras in Gr-categories with applications to related subjects, especially to the Hurwitz problem on sums of squares. Algebras in Gr-categories are important research objects in the area of tensor categories and quantum algebras, while the Hurwitz problem on sums of squares is very challenging and has been standing open for more than a century. Recently a deep connection between the two subjects was found and further study is clearly in demand. We will focus on semisimple algebras in braided Gr-categories and try to develop the theory of semisimple algebras and Brauer groups over fields to this setting. The series of semisimple algebras in braided Gr-categories are applied to construct new solutions to the Hurwitz problem and to investigate asymptotic bounds for the number of composition terms. On the other hand, Gr-categories are essential in the research of pointed finite tensor categories where Hopf algebras therein play a key role. We will also consider the theory of Nichols algebras and arithmetic root systems in those Gr-categories as well as their quantum doubles determined by finite abelian groups with nonabelian 3-cocycles. This will be applied to construct and classify pointed finite tensor categories.