丛代数的结构、范畴化及拓扑不变性研究

The theory of cluster algebras is an important theory developed in the new century, which is related with some important aspects of mathematics, e.g. combinatorics, graph theory, topology, Lie theory, algebraic groups and quantum groups, commutative algebras and algebraic geometry. This theory can be used to characterize the phenomena in some various fields of mathematics, whose key is the phenomenon of mutation. .Contents for research: the structure and categorification of cluster algebras and the connection wih topological invariants, including six aspects: (i) Defining higher dimensional cluster algebras corresponding to Riemann manifolds and studying it as algebraic invariants of higher Riemann manifolds; (ii) cluster algebras with relations and cluster realizations of some important structures, e.g. algebraic groups; (iii) characterizing structures of cluster algebras and studying the relationship between sub-structures of cluster algebras of geometric type in the skew-symmetric case; (iv) studying some unsolved problems on cluster algebras; (v) further research on monoidal categorification of cluster algebras and whose realization from quiver Hecke algebras and quantum groups；(vi) Under the homogeneous condition of module categories, study on the conjecture on the tame type of algebras and some related questions. .Prospective methods and goal: Building our methods for researching cluster algebras, for example: the theory of homomorphisms of cluster algebras analogous to homomorphisms of graphs; applications of Green equivalences in the theory of cluster algebras. Building the theories of higher cluster algebras and cluster algebras with relations. Solving some key open problems on cluster algebras and the conjecture on the tame type of algebras and apply them to other topics.

丛代数理论是新世纪以来产生的重要理论，涉及数学各个方面，能刻画不同领域现象。丛代数的关键是变异现象。通过丛范畴，丛代数与代数的表示型理论联系在一起。内容：丛代数的结构、范畴化以及与拓扑不变性的联系。其中包括六个方面：(i) 对应于高维黎曼面的“高维”丛代数的定义及其作为高维黎曼流形的代数不变量的研究；（ii）带关系的种子的丛代数及代数群等重要结构的丛代数实现；（iii）丛代数结构刻画和斜对称情形下几何型丛代数子结构与定向黎曼面分类的联系；(iv) 丛代数若干未解决问题研究；（v) 丛代数的张量范畴化及由箭图Hecke代数和量子群的实现的进一步研究。(vi)齐性模范畴条件下代数tame型猜想及后继问题。预期方法和目标：建立自己方法研究丛代数，比如类似图同态的丛代数同态理论；Green等价方法及应用； 建立“高维”丛代数和带关系的丛代数理论；丛代数关键公开问题和代数tame型猜想的解决和应用。

丛代数理论是新世纪以来产生的重要理论，涉及数学各个方面，能刻画不同领域现象。丛代数的关键是变异现象。通过丛范畴，丛代数与代数的表示型理论联系在一起。..本项目的研究关键是:..广义路代数、伪允许序半群代数和代数上的路代数的关系。建立广义丛代数的丛公式和D-矩阵模型。无圈符号斜对称丛代数的c-向量的符号凝聚性和极大绿色序列的存在性。斜对称丛代数和可斜对称化丛代数的分母向量的正性猜想。丛代数的结构唯一性猜想。丛代数的周期性，轨道-极大绿色序列的存在性等。丛代数自同态和自同构的关系。丛代数的C-矩阵唯一决定种子的问题。作为主系数情况下C-矩阵符号凝聚性的强化---一致符号凝聚性。表示范畴的极大绿色序列与稳定函数的相互构造。量子丛代数的二次量子化问题。..本项目主要的贡献在于：..解决了一系列丛代数中的基本问题，其中最重要的有：证明了无圈符号斜对称矩阵一定是完全的，并在此基础上证明了无圈符号斜对称丛代数的丛变量的正性；证明了丛变量关于初始变量的分母向量总是正的；证明了丛代数的结构唯一性，即组合结构可以唯一决定代数结构；建立了量子丛代数的二次量子化的结构与量子丛代数的Poisson结构的对应。