Ringel-Hall代数是代数表示论的重要研究课题，与量子群、李代数、丛代数和几何不变量理论有着密切的联系。本项目拟研究广义Frobenius范畴的modified Ringel-Hall代数及其与几类Ringel-Hall代数的关系。首先，从正合范畴及其取定的满子范畴出发恰当定义广义Frobenius范畴，拟建立包含复形范畴、同伦范畴、导出范畴、Frobenius范畴及其稳定范畴等范畴的一个统一框架。其次，建立广义Frobenius范畴上的modified Ringel-Hall代数理论，并比较modified Ringel-Hall代数和与几类Ringel-Hall代数的关系（商代数和子代数关系）。最后，研究modified Ringel-Hall代数的导出不变性。本项目的研究成果将统一诸多范畴上的Ringel-Hall代数理论，进一步丰富Ringel-Hall代数的理论研究成果。

Ringel-Hall algebra is an important topic of representation theory of algebras and it is closely related to quantum groups, Lie algebras, cluster algebras and invariant theory in algebraic geometry. In this proposal, we will introduce modified Ringel-Hall algebras of generalized Frobenius categories and study the relations between modified Ringel-Hall algebras and several other Ringel-Hall type algebras. More precisely, we will introduce the notion of generalized Frobenius category, which unifies the category of complexes, homotopy category, derived category, Frobenius category and the stable category. We try to establish the theory of modified Ringel-Hall algebras over generalized Frobenius categories and consider the relation between modified Ringel-Hall algebras and several other Ringel-Hall type algebras. Finally, we will also study the invariance of modified Ringel-Hall algebras under derived equivalence of exact categories. The results of this proposal can unify several kinds of Ringel-Hall type algebras and enrich the theory of representation theory of algebras.