# Quivers with relations for symmetrizable Cartan matrices V. Caldero-Chapoton formula

## TL;DR

This paper extends the Caldero-Chapoton formula to skew-symmetrizable cluster algebras by using categories of locally free modules over specific Gorenstein algebras, linking representation theory and Lie algebra structures.

## Contribution

It generalizes the Caldero-Chapoton formula to a broader class of cluster algebras using new algebraic and categorical frameworks.

## Key findings

- Generalization of Caldero-Chapoton formula to skew-symmetrizable cases
- Introduction of categories of locally free modules over Gorenstein algebras
- New result on PBW basis of convolution algebra

## Abstract

We generalize the Caldero-Chapoton formula for cluster algebras of finite type to the skew-symmetrizable case. This is done by replacing representation categories of Dynkin quivers by categories of locally free modules over certain Iwanaga-Gorenstein algebras introduced in Part I. The proof relies on the realization of the positive part of the enveloping algebra of a simple Lie algebra of the same finite type as a convolution algebra of constructible functions on representation varieties of $H$, given in Part III. Along the way, we obtain a new result on the PBW basis of this convolution algebra.