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Manin triples on multiplicative Courant algebroids

Ana Carolina Mançur

Abstract

We extend the characterization of Lie bialgebroids via Manin triples to the context of double structures over Lie groupoids. We consider Lie bialgebroid groupoids, given by LA-groupoids in duality, and establish their correspondence with multiplicative Manin triples, i.e., CA-groupoids equipped with a pair of complementary multiplicative Dirac structures. As an application, we give a new viewpoint to the co-quadratic Lie algebroids of Lang, Qiao, and Yin (2021) and the Manin triple description of Lie bialgebroid crossed modules.

Manin triples on multiplicative Courant algebroids

Abstract

We extend the characterization of Lie bialgebroids via Manin triples to the context of double structures over Lie groupoids. We consider Lie bialgebroid groupoids, given by LA-groupoids in duality, and establish their correspondence with multiplicative Manin triples, i.e., CA-groupoids equipped with a pair of complementary multiplicative Dirac structures. As an application, we give a new viewpoint to the co-quadratic Lie algebroids of Lang, Qiao, and Yin (2021) and the Manin triple description of Lie bialgebroid crossed modules.
Paper Structure (11 sections, 8 theorems, 27 equations)

This paper contains 11 sections, 8 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Review of Manin triples for Lie bialgebroids
  3. Lie bialgebroids
  4. Manin triples for Lie bialgebroids
  5. Lie bialgebroids over Lie groupoids
  6. VB-groupoids
  7. Lie bialgebroid groupoids
  8. Manin triples for Lie bialgebroid groupoids
  9. CA-groupoids
  10. Manin triples for Lie bialgebroid groupoids
  11. Application: Co-quadratic Lie algebroids revisited

Key Result

Proposition 2.7

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 $A^* \Rightarrow M$, if and only if, $L := B \oplus ann(B) \rightarrow N$ is a Dirac structure on $A \oplus A^*$ with

Theorems & Definitions (32)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Example 2.6
  • Proposition 2.7
  • Example 2.8
  • Example 3.1
  • Proposition 3.2
  • ...and 22 more