Geometric realizations of Ringel-Hall algebras of continuous quivers of type $A$

TL;DR

This paper extends the geometric realization of Ringel-Hall algebras to continuous quivers of type A, building on Lusztig's and Sala-Schiffmann's work with finite and line quivers.

Contribution

It provides a new geometric realization for Ringel-Hall algebras of continuous quivers of type A using approximation methods.

Findings

Established geometric realizations for continuous quivers of type A.

Connected the new realizations with Lusztig's framework.

Extended Sala-Schiffmann's approximation approach.

Abstract

Lusztig introduced the geometric realizations of quantum groups associated to finite quivers and defined their canonical bases. Sala and Schiffmann introduced the Ringel-Hall algebra of line and realized it as the direct limit of Ringel-Hall algebras of finite quivers of type $A$. In this paper, we shall give geometric realizations of Ringel-Hall algebras of continuous quivers of type $A$ via the geometric realizations of Lusztig by using the method of approximation given by Sala and Schiffmann.