In 2003, Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skew-symmetric exchange matrices. Besides cluster algebras of rank 2 and cluster algebras associated with triangulations of surfaces there are exactly 11 exceptional skew-symmetric cluster algebras of finite mutation type. More precisely, 9 of them are associated with root systems $E_6$, $E_7$, $E_8$, $\widetilde E_6$, $\widetilde E_7$, $\widetilde E_8$, $E_6^{(1,1)}$, $E_7^{(1,1)}$, $E_8^{(1,1)}$; two remaining were recently found by Derksen and Owen. We also describe a criterion which determines if a skew-symmetric cluster algebra is of finite mutation type, and discuss growth rate of cluster algebras.
一个结合方案称为斜对称的，如果它没有对称的邻接关系以外的对角。本文研究了4类斜对称结合方案。在J. Ma [On the nonexistence of skew-symmetric amorphous association schemes，submitted for publication]中，我们发现它们的特征标表分为三种类型。我们现在确定它们的交集矩阵。然后，我们确定的特征表和相交数的4类反对称伪循环结合计划，其中唯一已知的例子是分圆计划。从而回答了S. Y. Song [其对称化有两个类的交换结合方案，J. Algebras Combin. 5（1）47-55，1996]。刻画并分类了4类非本原斜对称结合方案。我们还证明了任何2类Johnson格式都不允许4类斜对称裂变格式。基于上述三种类型的特征表，生成可行参数的短列表。
Nonexistence of Certain Skew-symmetric Amorphous Association Schemes
一个关联方案是无定形的，如果它有尽可能多的融合方案。对称非晶方案由A. V.伊万诺夫[A. V. Ivanov，Amorphous cellular rings II，in Investigations in algebraic theory of combinatorial objects，pages 39- 49. VNIISI，Moscow，Institute for System Studies，1985]和交换无定形格式被T. Ito，A. Munemasa和M.山田[T. Ito，A. Munemasa和M.山田，特征为4的伽罗瓦环上的非晶结合方案，欧洲J. Combin.，12（1991），513- 526]。如果对角关系是唯一的对称关系，则称一个方案是斜对称的。证明了至少有4类的斜对称无定形格式的不存在性。我们还证明了非对称无定形格式是可交换的。
