CLUSTER ALGEBRAS FROM SURFACES AND EXTENDED AFFINE WEYL GROUPS
CLUSTER ALGEBRAS FROM SURFACES AND EXTENDED AFFINE WEYL GROUPS
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DOI：
10.1007/s00031-021-09647-y
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发表时间：
2020-08
影响因子：
0.7
通讯作者：
A. Felikson;John W. Lawson;M. Shapiro;P. Tumarkin
中科院分区：
文献类型：
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作者：
A. Felikson;John W. Lawson;M. Shapiro;P. Tumarkin
利用半正定二次型刻画了秩至少为3的突变有限簇代数。特别地，我们将每个未穿孔的有界表面与半正定二次空间V相关联，并且将每个三角测量与基inV相关联，使得聚类的任何突变（即，三角测量的翻转）通过部分反射将相应的基底彼此变换。此外，每个三角剖分产生一个扩展的仿射Weyl群的类型A，这是不变的翻转。该构造也被推广到E型例外斜对称突变有限簇代数。
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 spaceV, and with every triangulation a basis inV, 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 typeA, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of typesE.