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Cluster algebras from surfaces and extended affine Weyl groups

Felikson, A.; Lawson, J.W.; Shapiro, M.; Tumarkin, P.

Cluster algebras from surfaces and extended affine Weyl groups Thumbnail


J.W. Lawson

M. Shapiro


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


Felikson, A., Lawson, J., Shapiro, M., & Tumarkin, P. (2021). Cluster algebras from surfaces and extended affine Weyl groups. Transformation Groups, 26(2), 501-535.

Journal Article Type Article
Acceptance Date Jan 17, 2021
Online Publication Date Apr 6, 2021
Publication Date 2021-06
Deposit Date Jan 17, 2021
Publicly Available Date Apr 7, 2021
Journal Transformation Groups
Print ISSN 1083-4362
Electronic ISSN 1531-586X
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 26
Issue 2
Pages 501-535
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Published Journal Article (Advance online version.) (819 Kb)

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Advance online version. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit

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