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$\mathcal{L}$-Lie algebroids over topological ringed spaces

Mainak Poddar, Abhishek Sarkar

TL;DR

This work develops a unified, sheaf-theoretic framework for Lie algebroids over topological ringed spaces by introducing rooted constructions such as $\mathcal{L}$-Lie algebroids and $\mathcal{A}$-Gerstenhaber algebras. It extends fundamental correspondences between Lie algebroids, Gerstenhaber and BV-algebras, Chevalley–Eilenberg–de Rham complexes, and hypercohomology to the rooted setting, including a generalized PBW theorem and a universal enveloping algebroid. The authors establish dualities between homology and cohomology, define generalized connections and homology theories for $\mathcal{L}$-algebroids, and explore examples from foliations, Poisson and logarithmic geometries, and equivariant contexts. These developments enable a coherent treatment of Lie algebroids across smooth, holomorphic, and algebraic settings, with potential applications to Poisson geometry, log geometry, and representation theory. Overall, the paper broadens the algebraic and geometric toolkit for infinitesimal symmetries in singular and non-smooth contexts by systematically rooting classical structures in $\mathcal{L}$-theory.

Abstract

The notion of Lie algebroids over a topological ringed space provides a unified framework to study various geometric structures. This geometric concept is intimately connected with well-known algebraic structures, including Gerstenhaber algebras and Batalin--Vilkovisky algebras. We introduce more general concepts such as $\mathcal{L}$-Lie algebroids and $\mathcal{A}$-Gerstenhaber algebras, associated with a given Lie algebroid $\mathcal{L}$ and Gerstenhaber algebra $\mathcal{A}$ over a topological ringed space, respectively. Following this, we explore how several standard correspondences extend within this broader framework.

$\mathcal{L}$-Lie algebroids over topological ringed spaces

TL;DR

This work develops a unified, sheaf-theoretic framework for Lie algebroids over topological ringed spaces by introducing rooted constructions such as -Lie algebroids and -Gerstenhaber algebras. It extends fundamental correspondences between Lie algebroids, Gerstenhaber and BV-algebras, Chevalley–Eilenberg–de Rham complexes, and hypercohomology to the rooted setting, including a generalized PBW theorem and a universal enveloping algebroid. The authors establish dualities between homology and cohomology, define generalized connections and homology theories for -algebroids, and explore examples from foliations, Poisson and logarithmic geometries, and equivariant contexts. These developments enable a coherent treatment of Lie algebroids across smooth, holomorphic, and algebraic settings, with potential applications to Poisson geometry, log geometry, and representation theory. Overall, the paper broadens the algebraic and geometric toolkit for infinitesimal symmetries in singular and non-smooth contexts by systematically rooting classical structures in -theory.

Abstract

The notion of Lie algebroids over a topological ringed space provides a unified framework to study various geometric structures. This geometric concept is intimately connected with well-known algebraic structures, including Gerstenhaber algebras and Batalin--Vilkovisky algebras. We introduce more general concepts such as -Lie algebroids and -Gerstenhaber algebras, associated with a given Lie algebroid and Gerstenhaber algebra over a topological ringed space, respectively. Following this, we explore how several standard correspondences extend within this broader framework.

Paper Structure

This paper contains 30 sections, 16 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Lie algebroids over $(X, \mathcal{O}_X)$
  3. Lie algebroids over topological ringed spaces
  4. Topological ringed spaces
  5. Lie algebroids
  6. Gerstenhaber algebras over a topological ringed space
  7. Chevalley-Eilenberg-de Rham complex
  8. The complex of $\mathcal{L}$
  9. Hypercohomology of the complex $\Omega^{\bullet}_{\mathcal{L}}$
  10. Lie bialgebroids over a topological ringed space
  11. Strong differential Gerstenhaber algebras
  12. BV-algebras or exact Gerstenhaber algebras
  13. Atiyah algebroids and Lie algebroid connections
  14. Particular cases of $\mathcal{L}$-connections
  15. $\mathcal{L}$-Lie algebroids and $\mathcal{G}_{\mathcal{L}}$-Gerstenhaber algebras
  16. ...and 15 more sections

Key Result

Theorem 2.17

For an $\mathcal{O}_X$-module $\mathcal{E}$, there is a (sheaf of) Gerstenhaber algebra structure on the exterior algebra $\wedge_{\mathcal{O}_X}^\bullet \mathcal{E}$ if and only if $\mathcal{E}$ has a Lie algebroid structure.

Theorems & Definitions (91)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Example 2.10
  • ...and 81 more