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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 

 

Universal geometric cluster algebras from surfaces


Author: Nathan Reading
Journal: Trans. Amer. Math. Soc. 366 (2014), 6647-6685
MSC (2010): Primary 13F60, 57Q15
Published electronically: September 4, 2014
MathSciNet review: 3267022
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Abstract: A universal geometric cluster algebra over an exchange matrix $ B$ is a universal object in the category of geometric cluster algebras over $ B$ related by coefficient specializations. (Following an earlier paper on universal geometric cluster algebras, we broaden the definition of geometric cluster algebras relative to the definition originally given by Fomin and Zelevinsky.) The universal objects are closely related to a fan $ \mathcal {F}_B$ called the mutation fan for $ B$. In this paper, we consider universal geometric cluster algebras and mutation fans for cluster algebras arising from marked surfaces. We identify two crucial properties of marked surfaces: The Curve Separation Property and the Null Tangle Property. The latter property implies the former. We prove the Curve Separation Property for all marked surfaces except once-punctured surfaces without boundary components, and as a result we obtain a construction of the rational part of $ \mathcal {F}_B$ for these surfaces. We prove the Null Tangle Property for a smaller family of surfaces and use it to construct universal geometric coefficients for these surfaces.


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Additional Information

Nathan Reading
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205

DOI: https://doi.org/10.1090/S0002-9947-2014-06156-4
Received by editor(s): October 11, 2012
Received by editor(s) in revised form: March 26, 2013, and April 17, 2013
Published electronically: September 4, 2014
Additional Notes: This material is based upon work partially supported by the Simons Foundation under Grant Number 209288 and by the National Science Foundation under Grant Number DMS-1101568.
Article copyright: © Copyright 2014 Nathan Reading