Green formula in Hall algebras and cluster algebras

TL;DR

This paper surveys recent advances in applying Green's formula within Hall algebras to deepen understanding of quantum groups and cluster algebras, highlighting three key variants of Green's formula.

Contribution

It provides a comprehensive overview of three different forms of Green's formula and their applications in the context of Hall algebras, quantum groups, and cluster algebras.

Findings

Identification of three main forms of Green's formula

Connections established between Hall algebras and cluster algebras

Insights into the role of Green's formula in quantum group theory

Abstract

The objective of the present paper is to give a survey of recent progress on applications of the approaches of Ringel-Hall type algebras to quantum groups and cluster algebras via various forms of Green's formula. In this paper, three forms of Green's formula are highlighted, (1) the original form of Green's formula \cite{Green}\cite{RingelGreen}, (2) the degeneration form of Green's formula \cite{DXX} and (3) the projective form of Green's formula \cite{XX2007a} i.e. Green formula with a $\bbc^{*}$-action.