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Geometric realisations of the unipotent enveloping algebra of a quiver

Lucien Hennecart

Abstract

We compare and generalise the various geometric constructions (due to Ringel, Lusztig, Schofield, Bozec, Davison...) of the unipotent generalised Kac-Moody algebra associated with an arbitrary quiver. These constructions are interconnected through several geometric operations, including the stalk Euler characteristic of constructible complexes, the characteristic cycle, the Euler obstruction map, and the intersection multiplicities of Lagrangian subvarieties. We provide a proof that these geometric realisations hold for the integral form of the Lie algebra. Furthermore, by modifying the generators of the enveloping algebra, we ensure compatibility with the natural coproducts that can be defined in terms of restriction diagrams. As a result, we establish that the top cohomological Hall algebra of the strictly seminilpotent stack is isomorphic to the positive part of the enveloping algebra of the generalised Kac-Moody algebra associated with the quiver. This appears to be one of the cornerstones needed to describe the BPS algebra of very general $2$-Calabi-Yau categories, which is the subject of the author's work on BPS Lie algebras for $2$-Calabi-Yau categories with Davison and Schlegel Mejia.

Geometric realisations of the unipotent enveloping algebra of a quiver

Abstract

We compare and generalise the various geometric constructions (due to Ringel, Lusztig, Schofield, Bozec, Davison...) of the unipotent generalised Kac-Moody algebra associated with an arbitrary quiver. These constructions are interconnected through several geometric operations, including the stalk Euler characteristic of constructible complexes, the characteristic cycle, the Euler obstruction map, and the intersection multiplicities of Lagrangian subvarieties. We provide a proof that these geometric realisations hold for the integral form of the Lie algebra. Furthermore, by modifying the generators of the enveloping algebra, we ensure compatibility with the natural coproducts that can be defined in terms of restriction diagrams. As a result, we establish that the top cohomological Hall algebra of the strictly seminilpotent stack is isomorphic to the positive part of the enveloping algebra of the generalised Kac-Moody algebra associated with the quiver. This appears to be one of the cornerstones needed to describe the BPS algebra of very general -Calabi-Yau categories, which is the subject of the author's work on BPS Lie algebras for -Calabi-Yau categories with Davison and Schlegel Mejia.
Paper Structure (84 sections, 43 theorems, 189 equations)

This paper contains 84 sections, 43 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Quivers and Lie algebras
  3. The Lie algebra associated to a quiver
  4. Other categorical situations
  5. The main results
  6. The characteristic cycle map
  7. The top-CoHA of a quiver
  8. Geometric realisations of $\mathop{\mathrm{\mathbf{U}}}\nolimits^{\mathbb{Z}}(\mathfrak{n}_Q^+)$
  9. Comultiplications
  10. Cohomological Hall algebras
  11. Notations and conventions
  12. Characteristic cycle and singular support
  13. Characteristic cycle
  14. Distinguished triangles
  15. Shift
  16. ...and 69 more sections

Key Result

Theorem 1.1

Let $Q$ be a quiver. Let $\mathcal{P}$ be Lusztig's category of perverse sheaves on the stack of representations of $Q$ and $\mathfrak{N}_{\Pi_Q}^{\mathcal{SSN}}$ be the strictly seminilpotent stack. Then, the characteristic cycle map is an algebra isomorphism, where the vector space $\mathop{\mathrm{H}}\nolimits^{\mathrm{BM}}_{\mathrm{top}}(\mathfrak{N}_{\Pi_Q}^{\mathcal{SSN}},\mathbb{Z})$ has t

Theorems & Definitions (89)

  • Theorem 1.1: $\subseteq$ Theorem \ref{['theorem:maintheoremexpanded']}
  • Theorem 1.2: $\subseteq$ Theorem \ref{['theorem:maintheoremexpanded']}
  • Remark 1.3
  • Theorem 1.4: $\subseteq$ Theorem \ref{['theorem:maintheoremexpanded']}
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 2.1
  • Example 2.2: Illustration of the pullback formula
  • Lemma 2.3
  • ...and 79 more