We introduce a cluster character with coefficients for a cluster category \(\mathcal {C}\) and rather than using a Frobenius 2-Calabi-Yau realization to incorporate coefficients into the representation-theoretic model for a cluster algebra, as done by Fu and Keller, we exploit intrinsic properties of \(\mathcal {C}\). For this purpose, we define an ice quiver associated to each cluster tilting object in \(\mathcal {C}\). In Dynkin case \(\mathbb {A}_{n}\), we also prove that the mutation class of the ice quiver associated to the cluster tilting object given by the direct sum of all projective objects is in bijection with set of ice quivers of cluster tilting objects in \(\mathcal {C}\) and that the study of a class of cluster algebra with coefficients can be reduced to the case that we called biframed.

我们引入了具有簇类别\（\ mathcal {C} \）的系数的簇字符，而不是使用Frobenius 2-Calabi-Yau实现将系数纳入簇代数的表示理论模型中，如Fu所做的那样和Keller，我们利用\（\ mathcal {C} \）的固有属性。为此，我们在\（\ mathcal {C} \）中定义与每个群集倾斜对象相关的冰颤抖。在Dynkin情况\（\ mathbb {A} _ {n} \）中，我们还证明了与由所有投影对象的直接总和给出的星团倾斜对象相关的冰颤的突变类别与冰集是双射的\（\ mathcal {C} \）中簇倾斜对象的颤动 并且可以将一类具有系数的簇代数的研究简化为我们所说的二重框架。