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Caldero-Chapoton algebras


Authors: Giovanni Cerulli Irelli, Daniel Labardini-Fragoso and Jan Schröer
Journal: Trans. Amer. Math. Soc. 367 (2015), 2787-2822
MSC (2010): Primary 13F60; Secondary 16G10, 16G20
DOI: https://doi.org/10.1090/S0002-9947-2014-06175-8
Published electronically: November 12, 2014
MathSciNet review: 3301882
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Abstract: Motivated by the representation theory of quivers with potential introduced by Derksen, Weyman and Zelevinsky and by work of Caldero and Chapoton, who gave explicit formulae for the cluster variables of cluster algebras of Dynkin type, we associate a Caldero-Chapoton algebra $ \mathcal {A}_\Lambda $ to any (possibly infinite-dimensional) basic algebra $ \Lambda $. By definition, $ \mathcal {A}_\Lambda $ is (as a vector space) generated by the Caldero-Chapoton functions $ C_\Lambda (\mathcal {M})$ of the decorated representations $ \mathcal {M}$ of $ \Lambda $. If $ \Lambda = \mathcal {P}(Q,W)$ is the Jacobian algebra defined by a 2-acyclic quiver $ Q$ with non-degenerate potential $ W$, then we have $ \mathcal {A}_Q \subseteq \mathcal {A}_\Lambda \subseteq \mathcal {A}_Q^{\mathrm {up}}$, where $ \mathcal {A}_Q$ and $ \mathcal {A}_Q^{\mathrm {up}}$ are the cluster algebra and the upper cluster algebra associated to $ Q$. The set $ \mathcal {B}_\Lambda $ of generic Caldero-Chapoton functions is parametrized by the strongly reduced components of the varieties of representations of the Jacobian algebra $ \mathcal {P}(Q,W)$ and was introduced by Geiss, Leclerc and Schröer. Plamondon parametrized the strongly reduced components for finite-dimensional basic algebras. We generalize this to arbitrary basic algebras. Furthermore, we prove a decomposition theorem for strongly reduced components. We define $ \mathcal {B}_\Lambda $ for arbitrary $ \Lambda $, and we conjecture that $ \mathcal {B}_\Lambda $ is a basis of the Caldero-Chapoton algebra $ \mathcal {A}_\Lambda $. Thanks to the decomposition theorem, all elements of $ \mathcal {B}_\Lambda $ can be seen as generalized cluster monomials. As another application, we obtain a new proof for the sign-coherence of $ g$-vectors.


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Additional Information

Giovanni Cerulli Irelli
Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: cerulli.math@googlemail.com

Daniel Labardini-Fragoso
Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: labardini@matem.unam.mx

Jan Schröer
Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: schroer@math.uni-bonn.de

DOI: https://doi.org/10.1090/S0002-9947-2014-06175-8
Received by editor(s): December 4, 2012
Received by editor(s) in revised form: April 30, 2013
Published electronically: November 12, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.