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Manin triples for double Lie bialgebroids

Ana Carolina Mançur

TL;DR

This work generalizes the Manin triple paradigm to double Lie bialgebroids by introducing LA-Courant algebroids and LA-Manin triples. It proves that double Lie bialgebroids are in one-to-one correspondence with LA-Courant algebroids equipped with two complementary LA-Dirac structures (via the Drinfeld double), and conversely that an LA-Manin triple yields a double Lie bialgebroid. A key technical device is the relation $oldsymbol{ ext Pi_A}$ derived from linear Poisson geometry, which encodes the LA-Courant compatibility conditions. The findings also establish a bridge between infinitesimal and global double structures by linking Drinfeld doubles to CA-groupoids through differentiation and integration, offering a unified framework for double Courant-type objects and their representations. Collectively, the results unify double Lie bialgebroids, LA-Courant algebroids, and Drinfeld doubles within a coherent Manin-triple perspective with concrete integration/differentiation consequences.

Abstract

We verify that LA-Courant algebroids provide the Manin triple framework for double Lie bialgebroids. Specifically, we establish a correspondence between double Lie bialgebroids and LA-Manin triples, i.e., LA-Courant algebroids equipped with a pair of complementary LA-Dirac structures. As an application, LA-Courant algebroids and CA-groupoids given by Drinfeld doubles are shown to correspond via integration and differentiation.

Manin triples for double Lie bialgebroids

TL;DR

This work generalizes the Manin triple paradigm to double Lie bialgebroids by introducing LA-Courant algebroids and LA-Manin triples. It proves that double Lie bialgebroids are in one-to-one correspondence with LA-Courant algebroids equipped with two complementary LA-Dirac structures (via the Drinfeld double), and conversely that an LA-Manin triple yields a double Lie bialgebroid. A key technical device is the relation derived from linear Poisson geometry, which encodes the LA-Courant compatibility conditions. The findings also establish a bridge between infinitesimal and global double structures by linking Drinfeld doubles to CA-groupoids through differentiation and integration, offering a unified framework for double Courant-type objects and their representations. Collectively, the results unify double Lie bialgebroids, LA-Courant algebroids, and Drinfeld doubles within a coherent Manin-triple perspective with concrete integration/differentiation consequences.

Abstract

We verify that LA-Courant algebroids provide the Manin triple framework for double Lie bialgebroids. Specifically, we establish a correspondence between double Lie bialgebroids and LA-Manin triples, i.e., LA-Courant algebroids equipped with a pair of complementary LA-Dirac structures. As an application, LA-Courant algebroids and CA-groupoids given by Drinfeld doubles are shown to correspond via integration and differentiation.

Paper Structure

This paper contains 11 sections, 11 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Review of Manin triples for Lie bialgebroids
  3. Lie bialgebroids over Lie algebroids
  4. Manin triples for double Lie bialgebroids
  5. LA-Courant algebroids
  6. The relation in the definition of LA-Courant algebroids
  7. Manin triples for double Lie bialgebroids
  8. Lie theory for Courant double structures given by Drinfeld doubles
  9. Double and triple vector bundles
  10. Double vector bundles
  11. Triple vector bundles

Key Result

Proposition 2.10

Let $(A \Rightarrow M, A^* \Rightarrow M)$ be a Lie bialgebroid. Then, the vector bundles $B \rightarrow N$ and $ann(B) \rightarrow N$ are Lie subalgebroids of $A \Rightarrow M$ and $A^* \Rightarrow M$, respectively, if and only if, $L := B \oplus ann(B) \rightarrow N$ is a Dirac structure on $A \op

Theorems & Definitions (42)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • Example 2.7
  • Definition 2.8
  • Example 2.9
  • Proposition 2.10
  • ...and 32 more