Ringel-Hall algebras beyond their quantum groups I: Restriction functor and Green's formula

TL;DR

This paper extends Lusztig's categorification of quantum groups to the entire Ringel-Hall algebra, clarifying the relation between Green's formula and restriction functors, and demonstrating a geometric proof of the Green formula.

Contribution

It generalizes Lusztig's categorification to the full Ringel-Hall algebra and links Green's formula with restriction functors through a geometric approach.

Findings

Categorification of the entire Ringel-Hall algebra achieved.

Explicit relation established between Green's formula and restriction functor.

Hopf structure of Ringel-Hall algebra can be categorified geometrically.

Abstract

In this paper, we generalize the categorifical construction of a quantum group and its canonical basis introduced by Lusztig (\cite{Lusztig,Lusztig2}) to the generic form of the whole Ringel-Hall algebra. We clarify the explicit relation between the Green formula in \cite{Green} and the restriction functor in \cite{Lusztig2}. By a geometric way to prove the Green formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig's framework.