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

Minghui Zhao

TL;DR

The paper extends Lusztig’s geometric realization of Ringel–Hall algebras from finite quivers to continuous quivers of type A by approximating with finite quivers Q_I and taking direct limits to form K_{A_R}, together with a sheaf–function correspondence χ_{A_R}. It constructs the geometric Ringel–Hall algebra F_{A_R} and proves a surjective map from the limit algebra K_{A_R}, establishing a canonical basis for a completion bar{K}_{A_R} via IC-sheaves. Key contributions include the direct-limit framework linking continuous and finite-type realizations, and the canonical-basis structure for the completed continuum Ringel–Hall algebra. This work connects continuous quivers to continuum quantum groups and provides tools potentially useful in persistent homology and related data-analytic contexts.

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.

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

TL;DR

The paper extends Lusztig’s geometric realization of Ringel–Hall algebras from finite quivers to continuous quivers of type A by approximating with finite quivers Q_I and taking direct limits to form K_{A_R}, together with a sheaf–function correspondence χ_{A_R}. It constructs the geometric Ringel–Hall algebra F_{A_R} and proves a surjective map from the limit algebra K_{A_R}, establishing a canonical basis for a completion bar{K}_{A_R} via IC-sheaves. Key contributions include the direct-limit framework linking continuous and finite-type realizations, and the canonical-basis structure for the completed continuum Ringel–Hall algebra. This work connects continuous quivers to continuum quantum groups and provides tools potentially useful in persistent homology and related data-analytic contexts.

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 . In this paper, we shall give geometric realizations of Ringel-Hall algebras of continuous quivers of type via the geometric realizations of Lusztig by using the method of approximation given by Sala and Schiffmann.

Paper Structure

This paper contains 5 sections, 26 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Ringel-Hall algebras and canonical bases
  3. Continuous quivers of type $A$ and their representations
  4. Geometric realization of Ringel-Hall algebra of $A_{\mathbb{R}}$
  5. Canonical basis for a completion of $K_{A_{\mathbb{R}}}$

Key Result

Proposition 2.1

There is an one to one correspondence between the $G_{\mathbb V}$-orbits in $E_{{\mathbb V}}$ and the isomorphism classes of finitely dimensional representations of $Q$.

Theorems & Definitions (44)

  • Proposition 2.1: Crawley-Boevey90
  • Proposition 2.2: Lusztig_Canonical_bases_arising_from_quantized_enveloping_algebraLusztig_Quivers_perverse_sheaves_and_the_quantized_enveloping_algebrasLusztig_Introduction_to_quantum_groups
  • Proposition 2.3: Lusztig_Canonical_bases_and_Hall_algebras
  • Proposition 2.4: Schiffmann_Lectures2XXZ2022254
  • Theorem 2.5: Lusztig_Canonical_bases_arising_from_quantized_enveloping_algebraLusztig_Quivers_perverse_sheaves_and_the_quantized_enveloping_algebrasLusztig_Introduction_to_quantum_groups
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Proposition 4.1: Sala_2019Sala_2021
  • Lemma 4.2
  • ...and 34 more