Categorification of sign-skew-symmetric cluster algebras and some conjectures on g-vectors

2017/04/25 by Peigen Cao, Min Huang, Cao, Peigen +3
Mathematics · Physics and Astronomy · #13F60 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1704.07549

openalex publication_date 2017/04/25 · openalex created_date 2017/05/05 · openalex updated_date 2026/07/28

Abstract

Using the unfolding method given in \citeHL, we prove the conjectures on sign-coherence and a recurrence formula respectively of \bf g-vectors for acyclic sign-skew-symmetric cluster algebras. As a following consequence, the conjecture is affirmed in the same case which states that the \bf g-vectors of any cluster form a basis of \mathbb Zⁿ. Also, the additive categorification of an acyclic sign-skew-symmetric cluster algebra \mathcal A(Σ) is given, which is realized as (\mathcal C^{\widetilde Q},Γ) for a Frobenius 2-Calabi-Yau category \mathcal C^{\widetilde Q} constructed from an unfolding (Q,Γ) of the acyclic exchange matrix B of \mathcal A(Σ).

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