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Cluster algebras: an introduction


Author: Lauren K. Williams
Journal: Bull. Amer. Math. Soc. 51 (2014), 1-26
MSC (2010): Primary 13F60, 30F60, 82B23, 05E45
DOI: https://doi.org/10.1090/S0273-0979-2013-01417-4
Published electronically: June 10, 2013
MathSciNet review: 3119820
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Abstract: Cluster algebras are commutative rings with a set of distinguished generators having a remarkable combinatorial structure. They were introduced by Fomin and Zelevinsky in 2000 in the context of Lie theory, but have since appeared in many other contexts, from Poisson geometry to triangulations of surfaces and Teichmüller theory. In this expository paper we give an introduction to cluster algebras, and illustrate how this framework naturally arises in Teichmüller theory. We then sketch how the theory of cluster algebras led to a proof of the Zamolodchikov periodicity conjecture in mathematical physics.


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Additional Information

Lauren K. Williams
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: williams@math.berkeley.edu

DOI: https://doi.org/10.1090/S0273-0979-2013-01417-4
Received by editor(s): January 14, 2013
Published electronically: June 10, 2013
Additional Notes: The author is partially supported by a Sloan Fellowship and an NSF Career award.
Dedicated: To the memory of Andrei Zelevinsky
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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