Green's formula with $\bbc^{*}$-action and Caldero-Keller's formula for cluster algebras

TL;DR

This paper extends Green's formula to a projective setting using geometric methods and applies it to prove Caldero-Keller's multiplication formula for acyclic cluster algebras of any type.

Contribution

It introduces a geometric proof of the projective Green's formula and utilizes it to establish Caldero-Keller's formula for all acyclic cluster algebras.

Findings

Geometric derivation of the projective Green's formula

Proof of Caldero-Keller's multiplication formula for acyclic cluster algebras

Applicability to arbitrary type cluster algebras

Abstract

It is known that Green's formula over finite fields gives rise to the comultiplications of Ringel-Hall algebras and quantum groups (see\cite{Green}, also see \cite{Lusztig}). In this paper, we deduce the projective version of Green's formula in a geometric way. Then following the method of Hubery in \cite{Hubery2005}, we apply this formula to proving Caldero-Keller's multiplication formula for acyclic cluster algebras of arbitrary type.