五维Artin-Schelter正则代数的分类问题研究

Artin-Schelter regular algebras may be thought of as homogeneous coordinate rings of quantum projective spaces, of which the classification is one of the main aim in the realm of non-commutative algebraic geometry. The classification problem of Artin-Schelter regular algebras of dimension less than five has rich achievements. The proposal concerns mainly the case of dimension five and is devoted to study the following three topics: 1) Describe and depict the minimal resolution types of the trivial module over Artin-Schelter regular algebras of dimension five; 2) Try to classify some subcases of Artin-Schelter regular algebras of dimension five by taking advantage of a combination of the A-infinity algebraic method and the Hilbert series driven computation method; 3) Describe and depict the internal relationships between the Artin-Schelter regularity of connected graded algebras and the combinatorial features of Lyndon words. The expected accomplishments of this proposal will enrich examples, help to find new features of Artin-Schelter regular algebras and enhance their connections to other mathematical fields.

Artin-Schelter正则代数被视为量子射影空间的齐次坐标环，对它们进行分类是非交换代数几何的重要研究内容之一。低维（小于5）Artin-Schelter正则代数的分类问题已取得丰富成果，本项目主要关注5维的情形，研究内容包括：1）描述并刻画5维Artin-Schelter正则代数的平凡模的极小分解型；2）联合A-无穷代数方法和Hilbert级数驱动计算方法，尝试给出5维Artin-Schelter正则代数的一些子类的完全分类；3）描述并刻画连通分次代数的Artin-Schelter正则性与Lyndon字串的组合特性之间的内在联系。本项目的预期研究成果能够丰富实例、有助于发现Artin-Schelter正则代数新的特性并增强其与其它数学领域的联系。

Artin-Schelter正则代数被视作量子射影空间的齐次坐标环，是非交换代数几何的重要研究对象。本项目旨在得到五维Artin-Schelter正则代数的一些分类结果，并刻画Artin-Schelter正则代数的一些基本性质与不变量。项目总体上按计划执行。我们得到了代数的Noetherian性、Auslander性、Cohen-Macaulay性、斜Calabi-Yau性、整体维数、内射维数、Ext-代数以及Nakayama自同构等在Ore扩张、正规扩张和扭张量积等构造下的变化；也刻画了这些性质与不变量和代数的分次结构之间的关系。这些结果为后续Koszul型双分次五维Artin-Schelter正则代数的分类奠定了基础。在Artin-Schelter正则代数的结构问题研究过程中，项目负责人与合作者利用Lyndon字串的组合特性证明了所有Gelfand-Kirillov维数有限的连通分次Hopf代数都可从底域出发通过累次Hopf Ore扩张得到。另外，在项目基金的部分支持下，项目负责人与合作者研究了带势圈图的代数与几何性质，在带势圈图与它们的Jacobi代数、Ginzburg代数之间建立了深刻的联系，这些结果已应用于表示论、代数几何等领域。