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Sheaf realization of Bridgeland's Hall algebra of Dynkin type

Jiepeng Fang, Yixin Lan, Jie Xiao

TL;DR

This work provides a comprehensive geometric categorification of Bridgeland's Hall algebra for Dynkin type quivers by constructing induction/restriction functors on two-periodic projective complexes, and by realizing Bridgeland's algebra both through constructible and perverse sheaves. It establishes a positive, bar-invariant ${\mathcal Z}$-basis tied to IC-sheaves and shows an explicit isomorphism between the localized, reduced Bridgeland algebra and an integral Poisson form of the quantum group, linking geometric and generic Hall algebra frameworks. The results include a detailed comparison with Lusztig’s U_v^+ canonical basis, the development of a full perverse-sheaf realization, and the identification of canonical bases that respect degenerations and radical decompositions. Overall, the paper unifies categorical, geometric, and algebraic perspectives on Bridgeland-type Hall algebras and their integral, Poisson, and canonical structures, offering tools for deepening understanding of quantum groups via Dynkin quivers. The approach yields concrete bases and structural isomorphisms with potential implications for canonical bases and representation theory in related Hall-algebra contexts.

Abstract

As one of results in [6], Bridgeland realized the quantum group $\mathbf{U}_v$ via the localization of Ringel-Hall algebra for the two-periodic projective complexes of quiver representations over a finite field. In the present paper, we generalize Lusztig's categorical construction for the nilpotent part $\mathbf{U}_v^+$ to Bridgeland's Hall algebra of Dynkin type. In particular, we obtain a basis of the Ringel-Hall algebra for the two-periodic projective complexes which has the positivity, and we categorify an integral form of the generic Bridgeland's Hall algebra which is isomorphic to the Poisson integral form of $\mathbf{U}_v$, and obtain a $\mathbb{Z}[v,v^{-1}]$-basis of this integral form.

Sheaf realization of Bridgeland's Hall algebra of Dynkin type

TL;DR

This work provides a comprehensive geometric categorification of Bridgeland's Hall algebra for Dynkin type quivers by constructing induction/restriction functors on two-periodic projective complexes, and by realizing Bridgeland's algebra both through constructible and perverse sheaves. It establishes a positive, bar-invariant -basis tied to IC-sheaves and shows an explicit isomorphism between the localized, reduced Bridgeland algebra and an integral Poisson form of the quantum group, linking geometric and generic Hall algebra frameworks. The results include a detailed comparison with Lusztig’s U_v^+ canonical basis, the development of a full perverse-sheaf realization, and the identification of canonical bases that respect degenerations and radical decompositions. Overall, the paper unifies categorical, geometric, and algebraic perspectives on Bridgeland-type Hall algebras and their integral, Poisson, and canonical structures, offering tools for deepening understanding of quantum groups via Dynkin quivers. The approach yields concrete bases and structural isomorphisms with potential implications for canonical bases and representation theory in related Hall-algebra contexts.

Abstract

As one of results in [6], Bridgeland realized the quantum group via the localization of Ringel-Hall algebra for the two-periodic projective complexes of quiver representations over a finite field. In the present paper, we generalize Lusztig's categorical construction for the nilpotent part to Bridgeland's Hall algebra of Dynkin type. In particular, we obtain a basis of the Ringel-Hall algebra for the two-periodic projective complexes which has the positivity, and we categorify an integral form of the generic Bridgeland's Hall algebra which is isomorphic to the Poisson integral form of , and obtain a -basis of this integral form.
Paper Structure (31 sections, 39 theorems, 219 equations)

This paper contains 31 sections, 39 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main result
  4. Structure of the paper
  5. Preliminaries
  6. Quantized enveloping algebra
  7. Two-periodic projective complex
  8. Ringel-Hall algebra
  9. Ringel-Hall algebra of quiver representations
  10. Bridgeland's Hall algebra
  11. Mixed equivariant semisimple complex
  12. Lusztig's categorification for ${\bf U}_v^+$
  13. Induction functor and restriction functor
  14. Moduli variety of two-periodic projective complexes
  15. Induction functor
  16. ...and 16 more sections

Key Result

Theorem 1.1

The subset $\{\tilde{I}_{K_P}^*, \tilde{I}_{K_P^*}^*|P\in {\mathcal{P}}\}\subset {\mathcal{K}}^{ss,*}$ satisfies the Ore conditions, and so there is a well-defined localization with a reduced quotient such that the algebra isomorphisms (three algebra isomorphism) induce algebra isomorphisms

Theorems & Definitions (80)

  • Theorem 1.1: =Theorem \ref{["Bridgeland's algebra via perverse sheaf"]}
  • Proposition 1.2: = Proposition \ref{['positivity']}
  • Theorem 1.3: = Theorem \ref{['global basis']}
  • Lemma 2.1: Fang-Lan-Xiao-2024
  • Theorem 2.2: Ringel-1990
  • Theorem 2.3: Bridgeland-2013
  • Theorem 2.4: Lusztig-1993
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 70 more