遗传代数的高维推广与高维丛倾斜

Hereditary algebras are a class of important ones in the representation theory of finite dimensional algebras. In particular, m-replicated algebras, as the generalization of hereditary case, give another good interpretation of (higher) cluster categories. In our project, we mainly investigate m-replicated algebras of finite dimensional hereditary ones and their applications in (higher) cluster tilting theory. Firstly, we study tilting theory of m-replicated algebras, including homological properties and torsion theory of endomorphism algebras of tilting modules. Secondly, we introduce the τ-tilting theory of m-replicated algebras by investigating τ-tilting modules, support τ-tilting modules and their mutation. At last, we generalize exceptional sequences of hereditary algebras to m-replicated case, and mainly investigate the mutation and endomorphism algebras of exceptional sequences. Also our project will study the further relationship between m-replicated algebras and (higher) cluster tilting theory. Researches on m-replicated algebras of finite dimensional hereditary ones not only generalize some classic results of hereditary and tilted algebras, but also provide a new point of view on (higher) cluster tilting theory.

遗传代数是代数表示论中重要的研究对象，特别是 m-重代数，作为遗传代数的高维推广，与当前代数表示论的热点领域- - （高维）丛范畴有着密切的联系。本项目主要研究遗传代数的 m-重代数及其与（高维）丛倾斜理论的关系。首先研究 m-重代数的倾斜理论，主要研究（广义）倾斜模的自同态代数的同调性质，挠对理论等。其次建立 m-重代数的τ-倾斜理论，主要研究τ-倾斜模，support τ-倾斜模及其mutation。最后把遗传代数的例外序列理论推广到m-重代数，主要研究例外序列的mutation以及自同态代数的同调性质。同时，本项目也将更深入的探讨 m-重代数与（高维）丛倾斜理论的关系。通过对遗传代数的 m-重代数的研究, 不仅可以推广遗传代数和倾斜代数的经典结果，同时也为当前的研究热点-(高维)丛倾斜理论提供一种新的视角和研究思路。