三类来自曲面具有丛结构的代数的研究

来自曲面的具有丛结构的代数一直是丛代数领域的热门课题，给出它们丛变量的展开公式也是Fomin-Zelevinsky引入丛代数的原始动机之一。目前已知有三类来自曲面的具有丛结构的代数：来自曲面的丛代数、来自曲面的量子丛代数以及Berenstein-Retakh代数。我们将借助组合、代数表示论以及拓扑的方法去重点研究如下问题：. 1. 针对Berenstein-Retakh代数，找出其对应于带标记的三角剖分的丛结构；. 2. 给出Berenstein-Retakh代数中带标记弧对应丛变量的展开公式；. 3. 研究来自曲面的量子丛代数与Berenstein-Retakh代数的关系；. 4. 给出来自曲面的量子丛代数的量子丛变量的展开公式。

Algebra with cluster structure arising from a surface is a hot research topic since the 2000s, giving an explicit expansion formula for the cluster variables is also one of the original motivations of Fomin and Zelevinsky. There are three classes of algebras from surfaces that having cluster structures: cluster algebras from surfaces, quantum cluster algebras from surfaces, and Berenstein-Retakh algebras. We will use combinatorial, algebraic representation theory and topology to study the following questions:. 1. Find cluster structures of Berenstein-Retakh algebras for tagged triangulations;. 2. Give expansion formulae for tagged arcs in Berenstein-Retakh algebras;. 3. The relationship between quantum cluster algebras from surfaces and Berenstein-Retakh algebras;. 4. Give expansion formulae for quantum cluster variables in quantum cluster algebras from surfaces.