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From n-systems to Lie and Courant algebroids

Liqiang Cai, Zhuo Chen, Zhixiong Chen, Yanhui Bi

TL;DR

The paper develops a systematic framework to construct Lie and Courant algebroids from data called n-systems on anchored and metric bundles. By encoding anchor, connection, and bracket data into compatible n-systems, it yields pure/dull/Lie algebroids, DG manifolds, Lie pairs, and, in the metric setting, metric algebroids and (pre-)Courant algebroids. It then extends these constructions to Courant structures, Lie bialgebroids, and Manin pairs via compatible metric n-systems, with explicit formulas for structure maps and examples. The results provide a concrete, computable pathway from differential-geometric data to rich algebroid frameworks with broad applications in generalized geometry and related fields.

Abstract

This paper introduces a method for constructing pure algebroids, dull algebroids, and Lie algebroids. The construction relies on what we deffned as n-systems on vector bundles, and we provide explicit computations for all resulting structure maps. Analogously, metric n-systems deffned on metric vector bundles allow us to construct metric algebroids, pre-Courant algebroids, and Courant algebroids.

From n-systems to Lie and Courant algebroids

TL;DR

The paper develops a systematic framework to construct Lie and Courant algebroids from data called n-systems on anchored and metric bundles. By encoding anchor, connection, and bracket data into compatible n-systems, it yields pure/dull/Lie algebroids, DG manifolds, Lie pairs, and, in the metric setting, metric algebroids and (pre-)Courant algebroids. It then extends these constructions to Courant structures, Lie bialgebroids, and Manin pairs via compatible metric n-systems, with explicit formulas for structure maps and examples. The results provide a concrete, computable pathway from differential-geometric data to rich algebroid frameworks with broad applications in generalized geometry and related fields.

Abstract

This paper introduces a method for constructing pure algebroids, dull algebroids, and Lie algebroids. The construction relies on what we deffned as n-systems on vector bundles, and we provide explicit computations for all resulting structure maps. Analogously, metric n-systems deffned on metric vector bundles allow us to construct metric algebroids, pre-Courant algebroids, and Courant algebroids.

Paper Structure

This paper contains 18 sections, 17 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. The motivation
  3. Structures and main results of the paper
  4. List of terminologies and notations
  5. From connections to Lie algebroids
  6. Connections and pure algebroids
  7. Lie algebroids from compatible $n$-systems
  8. A special type of compatible $n$-systems
  9. DG manifolds from compatible $n$-systems
  10. Construction of Lie pairs from compatible $n$-systems
  11. From metric connections to Courant algebroids
  12. Metric connections and metric algebroids
  13. Three typical examples of metric algebroids
  14. Construction of Courant algebroids from compatible metric $n$-systems
  15. Construction of Lie bialgebroids from compatible metric $n$-systems
  16. ...and 3 more sections

Key Result

Proposition 2.2

Let $(A,a_A)$ be an anchored vector bundle. Given a connection $\nabla$ on $(A,a_A)$, a pure algebroid structure $(A,[ \,\cdot\, , \,\cdot\, ]^{\nabla}, a_A)$ can be constructed, where the bracket $[ \,\cdot\, , \,\cdot\, ]^{\nabla}$ is defined by Equation pure bracket. Conversely, every pure algebr

Theorems & Definitions (49)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 39 more