高维表示理论中的代数的周期性问题研究

近来, Iyama及其合作者们引入了高维表示理论, 被广泛的应用于表示理论和非交换几何中。 高维表示理论中的周期性问题是与遗传代数的预投射代数的周期性平行的问题. 本项目将在d-平移代数的背景下, 从斜群代数的截断代数出发, 以高维预投射代数和自入射d-表示有限代数为研究对象, 研究高维表示理论中的代数的周期性. 我们将讨论斜群代数的截断代数与d-表示有限代数之间的联系, 通过平凡扩张和Koszul对偶建立截断代数与高维预投射代数之间的联系, 继而讨论它们的几乎Koszul性和周期性, 并在此基础上研究自入射d-表示有限代数的周期性. 本项目的实现将推广Dynkin型遗传代数的预投射代数、表示有限自入射代数的周期性到高维表示理论情形, 并将得到一些新的刻画周期代数、几乎Koszul代数的方法.

Recently, Iyama and his coauthors have introduced high-dimensional representation theory, which is widely used in representational theory and non-commutative geometry. The periodic problem in high-dimensional representation theory is parallel to the periodicity of preprojective algebras of hereditary algebras. This project will study the periodicity of algebras in high-dimensional representation theory from the perspective of d-translation algebras, starting from the truncated algebras of skew-group algebras, taking higher preprojective algebras and selfinjective d-representation finite algebras as research objects. We will discuss the relationship between truncated algebras of skew-group algebras and d-representation finite algebras, establish the connection between truncated algebras and higher preprojective algebras through trivial extension and Koszul duality, and then study the almost Koszulity and periodicity of them. Basis on these, we will study the periodicity of selfinjective d-representation finite algebras. The realization of this project will generalize the periodicity of preprojective algebras of Dynkin type hereditary algebras and representation finite self-injective algebras to high-dimensional representation theory, and give some new examples and methods for characterizing of periodic algebras and almost Koszul algebras.