Additive categorification of the monoidal $\Lambda$-invariant

TL;DR

This paper links additive and monoidal categorifications of cluster algebras via the $ extLambda$-invariant, providing an additive interpretation within Higgs categories for certain quantum affine algebra representations.

Contribution

It introduces an additive interpretation of the $ extLambda$-invariant in Higgs categories for untwisted simply-laced quantum affine algebras, connecting it to Ginzburg algebras and cluster structures.

Findings

Proves the relative Ginzburg algebra is proper for certain ice quivers with potential.

Shows cluster algebras admit a $ extLambda$-structure via negative extensions in Higgs categories.

Provides a homological formula for tropical and F-invariants.

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.