Pointed Hopf代数Calabi-Yau性质的研究

Calabi-Yau代数起源于代数几何的研究。近些年来， Calabi-Yau现象在数学的各个研究分支中相继被发现, 使得Calabi-Yau代数正成为数学物理, 代数几何和非交换代数等领域中的一个热门研究课题。本课题希望将Calabi-Yau代数的研究和Hopf代数的研究相结合，将详细讨论pointed Hopf代数的Calabi-Yau性质。首先，我们将尝试给出Hecke型pointed Hopf代数是Calabi-Yau代数的充要条件。其次，以此为基础，并结合Sridharan包络代数和有限Cartan型pointed Hopf代数Calabi-Yau性质的相关结论，我们将讨论Calabi-Yau pointed Hopf代数的PBW形变以及cocycle形变，尝试刻画Calabi-Yau Hopf代数的余代数结构。

Calabi-Yau algebras arose in algebraic geometry. Now, they are experiencing a surge of popularity in different branches in mathematics, such as mathematical physics, representation theory, non-commutative algebras, etc.?We intend to study the Calabi-Yau property of pointed Hopf algebras. First, we will try to find the necessary and sufficient condition for a Hecke pointed Hopf algebra to be a Calabi-Yau algebra. Second, based on the obtained result and the facts about the Calabi-Yau property of Sridharan enveloping algebras and pointed Hopf algebras of finite Cartan type, we will study the PBW deformation and cocycle deformation of a pointed Hopf algebra. We will try to describe the comultiplication structure of a Calabi-Yau Hopf algebra.