n-切片代数与高维代数表示理论

n-切片代数是通过箭图和关系定义的一类代数，可根据其预投射代数自然地分为有限、驯化和野型。它可看作遗传代数的一个高维推广，且具有很强的几何背景。对n-切片代数我们证明了一个Beilinson对应和BGG对应推广的代数版本，我们将在这个框架下研究n-切片代数及其表示，推广遗传代数表示理论而建立其相应的高维表示理论。通过其二次对偶和二次对偶的平凡扩张，确定与McKay箭图相关的n-切片代数的箭图和关系；研究遗传代数经典理论在n-切片代数的推广：BGP反射函子的高维推广并运用于有限型情形；研究concealed代数理论在n-切片代数的推广；研究与之相关的倾斜理论、粘合、丛倾斜和silting理论。还将研究有限复杂度理论在驯化n-切片代数的推广，对一些驯化型n-切片代数推广管范畴的理论。本研究将发展和丰富代数表示论高维表示理论，也将促进代数表示论在其他数学领域，特别是非交换代数几何的应用研究。

n-slice algebra is a class of algebras defined by quiver with relations, it is classified as finite, tame and wild type according to the properties of its preprojective algebra. It is a generalization of hereditary algebra and has strong geometric background. We recently proved an algebraic version of the Beilinson correspondence and the BGG correspondence for n-slice algebra, the proposed project is aimed at generalizing the representation theory of hereditary algebra and studying n-slice algebra in higher representation theory setting under this frame. By studying the quadratic dual and its trivial extension, the quiver and relations of an n-slice algebra related to certain McKay quiver will be determined. The generalization of classical representation theory in higher representation will be studied, higher dimensional BGP reflection functors will be studied and applied in studying n-slice algebra of finite and tame type, the theory of concealed tilted algebra will be generalized to n-sliced algebra. We will also study the recollemente of the categories related to $n$-slice algebras, study the cluster tilting theory and silting theory for these categories. The theory of complexity of representations of exterior algebras, especially that of tubes, will be generalized to n-slice algebra. These will not only enrich the higher representation theory of algebras, but also promote the application in related research fields, especially, in non-commutative geometry.