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

Abstract

Using the unfolding method given in \cite{HL}, 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 $\mathbb Z^n$. Also, the additive categorification of an acyclic sign-skew-symmetric cluster algebra $\mathcal A(\Sigma)$ is given, which is realized as $(\mathcal C^{\widetilde Q},\Gamma)$ for a Frobenius $2$-Calabi-Yau category $\mathcal C^{\widetilde Q}$ constructed from an unfolding $(Q,\Gamma)$ of the acyclic exchange matrix $B$ of $\mathcal A(\Sigma)$.