Hecke型Nichols代数Calabi-Yau性质的研究

Calabi-Yau algebras arose in algebraic geometry. Now, they are experiencing a surge of popularity in different branches in mathematics, such as non-commutative algebras, mathematical physics, representation theory, etc. We intend to study the Calabi-Yau property of pointed Hopf algebras. Nichols algebras, the central objects in this project, appear naturally in the classification of pointed Hopf algebras by the method of Andruskiewitsch and Schneider. We will focus on finding a necessary and sufficient condition for a Nichols algebra of Hecke type to be a Calabi-Yau algebra. A Nichols algebra of Hecke type is a Koszul algebra, we will use the theory of PBW deformation of Koszul algebras to obtain more examples of Calabi-Yau algebras. Nichols algebras are endowed with the structure of braided Hopf algebras.This fact motivates us to study the special properties of the PBW deformation of the Nichols algebras of Hecke type.

Calabi-Yau 代数的研究起源于代数几何。近几年来，随着在数学不同分支都相继发现了Calabi-Yau现象，Calabi-Yau 代数的研究已经成为非交换代数、数学物理、表示论等领域的热门课题。本课题希望将Calabi-Yau代数的研究和Hopf代数的研究相结合。Pointed Hopf代数是一类重要的Hopf代数。Nichols代数在pointed Hopf代数的分类问题中发挥了重要的作用。本课题主要研究的问题是给出Hecke型Nichols代数是Calabi-Yau代数的充要条件。Hecke型Nichols代数同时也是Koszul代数。本课题一方面希望利用Koszul代数PBW形变的理论获得更多Calabi-Yau代数的例子，另一方面，将着重分析Hecke型Nichols代数作为辫子Hopf代数，它们的PBW形变有何特性。

Hecke型Nichols代数是Koszul代数，相对应的pointed Hopf代数是Nichols代数和群代数的smash积的PBW形变。换句话说，Hecke型pointed Hopf代数是Kosuzl代数和群代数的smash积的PBW形变。本项目主要研究了Kosuzl代数和Hopf代数的smash积及其PBW形变的Calabi-Yau性质。假设H是一个twisted Calabi-Yau Hopf代数，A是一个Koszul twisted Calabi-Yau代数，同时是一个分次H-模代数。我们首先得到，smash积A#H也是twisted Calabi-Yau代数，并具体计算出了Nakayama自同构，从而得到smash积是Calabi-Yau代数的充要条件。其次，当A#H的PBW形变满足线性部分为0时，我们得到A#H的PBW形变仍然是twisted Calabi-Yau代数，且它的Nakayama自同构是一个滤子自同构，对应的分次自同构恰好是A#H的Nakayama自同构。Nichols代数都是某个Hopf代数上的辫子Hopf代数。我们还给出了有限维Hopf代数上AS-Gorenstein辫子Hopf代数的刚性对偶复形的表达式。