Modified Ringel–Hall algebras, Green's formula and derived Hall algebras

Green Formula in Hall Algebras and Cluster Algebras

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 [Gre][Rin2], (2) the degeneration form of Green’s formula [DXX] and (3) the pro...

The product formula in cohomological Hall algebras

This is an expository article. It gives a detailed explanation of the product formula of Cohomological Hall algebra.

Double Hall Algebras and Derived Equivalences

The resulting algebra is known as the Hall algebra HA of A. Hall algebras first appeared in the work of Steinitz [S] and Hall [H] in the case where A is the category of Abelian p-groups. They reemerged in the work of Ringel [R1]-[R3], who showed in [R1] that when A is the category of quiver representations of an A-D-E quiver ~ Q over a finite field Fq, the Hall algebra of A provides a realizati...

Lectures on Hall Algebras

These notes represent the written, expanded and improved version of a series of lectures given at the winter school “Representation theory and related topics” held at the ICTP in Trieste in January 2006. The topic for the lectures was “Hall algebras” and I have tried to give a survey of what I believe are the most fundamental results and examples in this area. The material was divided into five...

On Ringel-hall Algebras of Tame Hereditary Algebras