Additive categorification of the monoidal $\Lambda$-invariant

Abstract

In this paper, we contribute to the broad aim of relating invariants of additive and monoidal categorifications of cluster algebras. Specifically, in the setting of representations of a quantum affine algebra $U_q'(\mathfrak{g})$, Kashiwara-Kim-Oh-Park proved that the Hernandez-Leclerc categories form a monoidal categorification of their Grothendieck rings. Furthermore, these rings are $\Lambda$-cluster algebras, meaning they are equipped with a compatible Poisson structure, constructed via the $\Lambda$-invariant. Under certain natural conditions, where $U_q'(\mathfrak{g})$ is of untwisted simply-laced type, we provide an additive interpretation of the $\Lambda$-invariant within the framework of Higgs categories. More precisely, there is an ice quiver with potential associated with these cluster algebras, and a key ingredient of our work consists in proving that its relative Ginzburg algebra is proper. More generally, if the relative Ginzburg algebra associated with an arbitrary ice quiver with potential is proper, we prove that the corresponding cluster algebra admits the structure of a $\Lambda$-cluster algebra defined in terms of negative extensions in the Higgs category. Moreover, we provide a homological formula to compute the corresponding tropical and $F$-invariants introduced by Cao.