We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space V , and with every triangulation a basis in V , such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type A, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of types E.

中文翻译：

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表面和扩展的仿射WEYL基团的簇代数

我们使用正半定二次型来刻画至少3级的有限突变簇代数。特别是，我们关联与每一个未击穿的边界表面上的半正定二次空间V，和与每一个三角测量的基础V，使得集群的任何突变（即，三角测量的翻转）变换对应的碱基到每个其他通过部分反射。此外，每个三角剖分都会产生一个扩展的A型仿射Weyl基团，该基团在翻转时不变。该构造还扩展到类型E的特殊的偏对称对称突变有限簇代数。