Cluster structures with non-integer quivers

簇代数是由Fomin和Zlevinsky在2000年引入和研究的，此后簇代数结构出现在数学和理论物理的众多不同领域。该结构是由箭图诱导的，即具有整数箭头权重的有限有向图。最近，有人提出了一种类似于箭袋的结构，这种箭袋的重量是实箭。当前工作的目的是发展对这种新结构的一些几何、组合、代数和表示理论解释，并获得其应用。

Cluster algebras were introduced and studied by Fomin and Zelevinsky in 2000, since then the structure appeared in numerous different fields of mathematics and theoretical physics. The structure is induced by a quiver, i.e. a finite oriented graph with integer weights of arrows. Recently, an analogue of the structure was proposed for quivers with real weights of arrows. The aim of the current work will be to develop a number of geometric, combinatorial, algebraic and representation theoretic interpretations of this new structure and to obtain its applications.