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Critical Cohomological Hall Algebra and Edge Contraction

Yiqiang Li, Jie Ren

Abstract

We study the behaviors of cohomological Hall algebras and relevant subjects under an edge contraction. Given a quiver with potential and fix an arrow, the edge contraction is a way to construct a new quiver with potential. We show that there is an algebra homomorphism between the cohomological Hall algebras induced by the edge contraction, which preserves the Hopf algebra structure, Drinfeld double, mutation, and the dimensional reduction, and induces relations between scattering diagram and Donaldson-Thomas series.

Critical Cohomological Hall Algebra and Edge Contraction

Abstract

We study the behaviors of cohomological Hall algebras and relevant subjects under an edge contraction. Given a quiver with potential and fix an arrow, the edge contraction is a way to construct a new quiver with potential. We show that there is an algebra homomorphism between the cohomological Hall algebras induced by the edge contraction, which preserves the Hopf algebra structure, Drinfeld double, mutation, and the dimensional reduction, and induces relations between scattering diagram and Donaldson-Thomas series.
Paper Structure (18 sections, 13 theorems, 57 equations)

This paper contains 18 sections, 13 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Cohomological Hall algebra
  3. Reminder on the equivariant critical cohomology
  4. COHA of a quiver with potential
  5. Assumption of shuffle description
  6. Drinfeld double
  7. Donaldson-Thomas invariants
  8. Scattering diagram
  9. Dimensional reduction
  10. Edge contraction
  11. Edge contraction of a quiver with potential
  12. Compatibility with multiplication and comultiplication
  13. Spherical subalgebras
  14. Compatibility of Drinfeld double
  15. Scattering diagram and DT-series
  16. ...and 3 more sections

Key Result

Theorem 1

There is an algebra homomorphism $\mathfrak c: \mathcal{H}_{Q, W}^= \to \mathcal{H}_{\widehat{Q}, \widehat{W}}$. Here $\mathcal{H}_{Q, W}^=$ is the subalgebra of $\mathcal{H}_{Q, W}$ restricted to the dimension vectors $\gamma=(\gamma^i)_{i\in I}$ such that $\gamma^{i_+}=\gamma^{i_-}$, where $I$ is

Theorems & Definitions (29)

  • Theorem
  • Theorem
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 19 more