Caldero-Chapoton algebras

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.