Artin-Schelter正则代数的量子对称性及不变子代数研究

The invariant theory of Artin-Schelter regular algebras is a noncommutative version of the classical invariant theory of polynomial algebras. It is an important subject of noncommutative algebra and noncommutative geometry. We are interested in the setting of finite dimensional Hopf algebras actions on Artin-Schelter regular algebras. The proposal concerns the following two topics: 1) Describe the properties of Hopf algebras acting on Artin-Schelter regular algebras, and characterize the quantum symmetry of high dimensional Artin-Schelter regular algebras; 2) The properties of fixed subalgebras，one of goals is to generalize noncommutative Shephard-Todd-Chevalley theorem, including the Artin-Schelter regularity and rigidity of fixed algebras; another goal is the Auslander theorem for high dimensional Artin-Schelter regular algebras by computing pertinency of Hopf actions. The smash product of two algebras, which is a construction method considered in an ongoing project (grant no. 11626215), will be integrated into the whole research. The expected accomplishments of this proposal will benefit to the understanding of the invariant theory of Artin-Schelter regular algebras, promote the development of the invariant theory of noncommutative algebras, and enhance their connections to other mathematical fields.

Artin-Schelter正则代数不变量理论是经典多项式代数不变量理论在非交换层面的延拓，它是非交换代数和非交换几何的重要研究内容之一。本项目关注Artin-Schelter正则代数在有限维Hopf代数作用下的相关问题，结合在研课题（批准号：11626215）中考虑的代数smash积构造方法，分两个部分开展研究：1）描述可作用的有限维Hopf代数性质，刻画高维Artin-Schelter正则代数的量子对称性；2）研究不变子代数的性质，一方面进一步推广非交换Shephard-Todd-Chevalley定理，讨论不变子代数的正则性和刚性，另一方面利用相关性系数，描述并刻画高维Artin-Schelter正则代数的Auslander定理。本项目的预期研究成果将有助于加深对Artin-Schelter正则代数不变量理论的理解，推进非交换不变量理论的发展，增强与其它数学领域的联系。

Artin-Schelter(简记为AS)正则代数被视作交换多项式在非交换层面的对照，它的不变量理论是经典多项式代数不变量理论在非交换层面的延拓，是非交换代数和非交换几何的重要研究内容之一。本项目结合代数扩张，关注AS正则代数在有限维Hopf代数作用下的相关问题。我们对与AS正则代数量子对称性密切相关的Nakayama自同构进行了刻画，其中得到了分次斜扩张的Ext代数是两个Ext代数的扭张量积；引入σ-散度的概念，给出了Koszul AS正则代数Ore扩张的Nakayama自同构精确描述；并讨论了正则正规扩张的斜Calabi-Yau性质，得到了Nakayama自同构的相应等式。并利用Lyndon字串良好的组合性质，证明得到了特征为0的底域上的GK维数有限的连通分次Hopf代数必然是底域的累次Hopf Ore扩张。此外，我们构造了一些新的高维AS正则代数实例以及刻画了多元Poisson-Ore扩张的Poisson包络代数。项目所得成果对AS正则代数的不变量理论及相关课题的后续研究有积极的意义。