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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Caldero-Chapoton algebras
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by Giovanni Cerulli Irelli, Daniel Labardini-Fragoso and Jan Schröer PDF
Trans. Amer. Math. Soc. 367 (2015), 2787-2822 Request permission

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
  • MR Author ID: 868181
  • Email: labardini@matem.unam.mx
  • Jan Schröer
  • Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
  • MR Author ID: 633566
  • Email: schroer@math.uni-bonn.de
  • Received by editor(s): December 4, 2012
  • Received by editor(s) in revised form: April 30, 2013
  • Published electronically: November 12, 2014
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 3301882