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An approach to non simply laced cluster algebras

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G. Dupont

Representation Theory · Submitted 1 Dec 2005

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that the (coefficient-free) cluster algebra $\mathcal A(\Delta/G)$ associated to the valued quotient graph $\Delta/G$ is a subalgebra of the quotient $\pi(\mathcal A(\Delta))$ of the cluster algebra associated to $\Delta$ by the action of $G$.

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arXiv IDmath/0512043 (version 5)
Pith IDGXPABE3A, Pith's permanent record of this paper and its review
CitationG. Dupont. "An approach to non simply laced cluster algebras." arXiv:math/0512043 (2005). https://pith.science/paper/GXPABE3A
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,689 input tokens.

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arXiv listing abstract v5. Results and the main claim are underlined; .

Let $\Delta$ be an oriented valued graph equipped with a group of admissible automorphisms satisfying a certain stability condition. We prove that the (coefficient-free) cluster algebra $\mathcal A(\Delta/G)$ associated to the valued quotient graph $\Delta/G$ is a subalgebra of the quotient $\pi(\mathcal A(\Delta))$ of the cluster algebra associated to $\Delta$ by the action of $G$. When $\Delta$ is a Dynkin diagram, we prove that $\mathcal A(\Delta/G)$ and $\pi(\mathcal A(\Delta))$ coincide. As an example of application, we prove that affine valued graphs are mutation-finite, giving an alternative proof to a result of Seven.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let $\Delta$ be an oriented valued graph equipped with a group of admissible automorphisms satisfying a certain stability condition. background 0.70 0.14 0.03 asserted 0.58
s1 We prove that the (coefficient-free) cluster algebra $\mathcal A(\Delta/G)$ associated to the valued quotient graph $\Delta/G$ is a subalgebra of the quotient $\pi(\mathcal A(\Delta))$ of the cluster algebra associated to $\Delta$ by the action of $G$. result 0.98 0.95 0.61 asserted 1.00
s2 When $\Delta$ is a Dynkin diagram, we prove that $\mathcal A(\Delta/G)$ and $\pi(\mathcal A(\Delta))$ coincide. result 1.00 0.96 0.26 asserted 1.00
s3 As an example of application, we prove that affine valued graphs are mutation-finite, giving an alternative proof to a result of Seven. result 0.92 0.91 0.06 asserted 1.00

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Original abstract

Let $\Delta$ be an oriented valued graph equipped with a group of admissible automorphisms satisfying a certain stability condition. We prove that the (coefficient-free) cluster algebra $\mathcal A(\Delta/G)$ associated to the valued quotient graph $\Delta/G$ is a subalgebra of the quotient $\pi(\mathcal A(\Delta))$ of the cluster algebra associated to $\Delta$ by the action of $G$. When $\Delta$ is a Dynkin diagram, we prove that $\mathcal A(\Delta/G)$ and $\pi(\mathcal A(\Delta))$ coincide. As an example of application, we prove that affine valued graphs are mutation-finite, giving an alternative proof to a result of Seven.

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5 arXiv papers cite this in 12 sentences

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  • Some of the results proven in [ref] easily carry over to the current sitation with principal coefficients (namely, Lemma [ref] and Lemma [ref]), so the proofs are omitted here.

    Quantum F-polynomials in Classical Types · Projections of $F$-polynomials and $g$-vectors · 0.96

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