一类无限维对角型Nichols代数的研究

Nichols代数是W.Nichols在研究Hopf代数的过程中提出的一类辫子Hopf代数。此类代数在Hopf代数的分类、量子群及张量范畴等领域起着越来越重要的作用。本项目旨在研究无限维对角型Nichols代数广义根系的结构和一类张量代数的商代数是Nichols代数的判定条件。我们将采用Lyndon词、PBW基等手段来刻画对角型Nichols代数广义根系。针对第二个问题我们将其转化为一类三角型Nichols代数的性质研究。我们将采用量子对称子、shuffle映射等手段给出此类代数的定义理想，并且运用它们来探索无限维对角型Nichols的结构。作为应用，我们将刻画一些Cartan矩阵为仿射型的对角型Nichols的广义根系的结构。本项目的研究结果也将有助于我们更好地理解Cartan矩阵为仿射型Nichols代数的根系与对应的仿射型李代数根系之间的联系和区别。

Nichols algebras are a class of braided Hopf algebras, which are originally introduced by W.Nichols to study the structures of finite dimensional Hopf algebras. They play more and more important roles in the theory of Hopf algebra, quantum group and tensor category..In this proposal, our main topics are to explore the structures of generalized root systems of Nichols algebras of diagonal type and the necessary and sufficient conditions for some quotient algebras of tensor algebra are Nichols algebras. We will use Lyndon words, PBW basis to determine the generalized root system of a given Nichols algebra of diagonal type. Further, we discuss the structures of infinite-dimensional Nichols algebras of diagonal type. The second question can be reduced to the study of some Nichols algebras of triangular type. We will use quantum symmetrizers and shuffle maps to describe the defining ideal of them. As an application we will use them to explore the structure of Nichols algebras of diagonal type. The results of this proposal will also help us to better understand the relations between the root system of Nichols algebra whose cartan matrix is affine and the affine Lie algebra.