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Non-degenerate potentials on the quiver X7.

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Title: Non-degenerate potentials on the quiver X7.
Authors: Ladkani, Sefi1 (AUTHOR) ladkani.math@gmail.com
Source: Journal of Algebra. Mar2025, Vol. 666, p94-148. 55p.
Subject Terms: *C*-algebras, *Mutations (Algebra), *Algebra, *Logical prediction, *Families, *Cluster algebras
Abstract: We develop a method to compute certain mutations of quivers with potentials and use this to construct an explicit family of non-degenerate potentials on the exceptional quiver X 7. We confirm a conjecture of Geiss-Labardini-Schröer by presenting a computer-assisted proof that over a ground field of characteristic 2, the Jacobian algebra of one member W 0 of this family is infinite-dimensional, whereas that of another member W 1 is finite-dimensional, implying that these potentials are not right equivalent. As a consequence, we draw some conclusions on the associated cluster categories, and in particular obtain a representation-theoretic proof that there are no reddening mutation sequences for the quiver X 7. We also show that when the characteristic of the ground field differs from 2, the Jacobian algebras of W 0 and W 1 are both finite-dimensional. Thus W 0 seems to be the first known non-degenerate potential with the property that the finite-dimensionality of its Jacobian algebra depends upon the ground field. [ABSTRACT FROM AUTHOR]
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  – Url: https://www.doi.org/10.1016/j.jalgebra.2024.11.007?
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  Data: Non-degenerate potentials on the quiver X7.
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  Data: <searchLink fieldCode="AR" term="%22Ladkani%2C+Sefi%22">Ladkani, Sefi</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> ladkani.math@gmail.com</i>
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Algebra%22">Journal of Algebra</searchLink>. Mar2025, Vol. 666, p94-148. 55p.
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  Data: *<searchLink fieldCode="DE" term="%22C*-algebras%22">C*-algebras</searchLink><br />*<searchLink fieldCode="DE" term="%22Mutations+%28Algebra%29%22">Mutations (Algebra)</searchLink><br />*<searchLink fieldCode="DE" term="%22Algebra%22">Algebra</searchLink><br />*<searchLink fieldCode="DE" term="%22Logical+prediction%22">Logical prediction</searchLink><br />*<searchLink fieldCode="DE" term="%22Families%22">Families</searchLink><br />*<searchLink fieldCode="DE" term="%22Cluster+algebras%22">Cluster algebras</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: We develop a method to compute certain mutations of quivers with potentials and use this to construct an explicit family of non-degenerate potentials on the exceptional quiver X 7. We confirm a conjecture of Geiss-Labardini-Schröer by presenting a computer-assisted proof that over a ground field of characteristic 2, the Jacobian algebra of one member W 0 of this family is infinite-dimensional, whereas that of another member W 1 is finite-dimensional, implying that these potentials are not right equivalent. As a consequence, we draw some conclusions on the associated cluster categories, and in particular obtain a representation-theoretic proof that there are no reddening mutation sequences for the quiver X 7. We also show that when the characteristic of the ground field differs from 2, the Jacobian algebras of W 0 and W 1 are both finite-dimensional. Thus W 0 seems to be the first known non-degenerate potential with the property that the finite-dimensionality of its Jacobian algebra depends upon the ground field. [ABSTRACT FROM AUTHOR]
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      – Type: doi
        Value: 10.1016/j.jalgebra.2024.11.007
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 55
        StartPage: 94
    Subjects:
      – SubjectFull: C*-algebras
        Type: general
      – SubjectFull: Mutations (Algebra)
        Type: general
      – SubjectFull: Algebra
        Type: general
      – SubjectFull: Logical prediction
        Type: general
      – SubjectFull: Families
        Type: general
      – SubjectFull: Cluster algebras
        Type: general
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      – TitleFull: Non-degenerate potentials on the quiver X7.
        Type: main
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            NameFull: Ladkani, Sefi
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            – D: 15
              M: 03
              Text: Mar2025
              Type: published
              Y: 2025
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            – Type: issn-print
              Value: 00218693
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              Value: 666
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            – TitleFull: Journal of Algebra
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