由于代数表示论自身研究的深入和cluster代数、非交换代数几何等相关学科发展需要，Iyama 引入了高维Auslander-Reiten理论。其中n-表示有限代数、n-表示无限代数与我们近年在研究代数表示论、McKay箭图等高维问题时引入了n-平移代数有密切联系。本项目以深入研究这些代数及其联系，在与遗传代数表示理论相关的经典理论基础上发展这些高维表示中重要的代数的结构与表示理论为目标。通过升维构造研究给出更多具有高维表示有限性质的代数的结构，并刻画相应表示范畴之间的联系；通过McKay箭图和复杂度等不变量刻画具有高维驯化特征的代数；通过n-平移代数和Koszul复形与n-1-几乎可裂序列的关系研究相关的倾斜理论。发展高维表示理论，同时将构造一批高维表示代数的例子，刻画相应的结构与表示性质。将促进高维表示及其在非交换代数几何、高维cluster代数和Calabi-Yau代数的应用研究。

Motivated by the study of cluster algebra, non-commutative geometry and the classical Auslander-Reiten, Iyama introduced and studied Higher representation theory. n-representation-finite algebra, n-representation infinite algebras introduced by Iyama are closely related to n-translation algebras introduced in studying higher McKay quivers and related algebras. The proposed project is aimed to study the relationship between these algebras and to develop a structural and representation theory for these algebras, following the classical theory of the algebras related to hereditary algebras. By studying construction which produces algebra of higher dimension in some sense for these algebras, more algebras with higher representation-finite feature will be constructed; Using McKay quiver and complexity, algebras with higher representation-tame feature will be characterized; Via the relationship between Koszul complexes and n-1-almost split sequence for n-translation algebras, tilting theory for the related algebras will be investigated. To develop higher representation theory, we’ll construct a series of new examples and describe their structure and representations. These will be of significant importance also in studying higher representation theory and its applications in non-commutative algebraic geometry, higher cluster algebras and Calabi-Yau algebras.