### Abstract

Given a pair of (real or complex) Lie algebroid structures on a vector bundle A (over M) and its dual A*, and a line bundle ℒ such that ℒ ⊗ ℒ = (∧^{top} A* ⊗ ∧^{top} T*M)^{1/2} exists, there exist two canonically defined differential operators d* and ∂ on (∧A ⊗ ℒ). We prove that the pair (A, A*) constitutes a Lie bialgebroid if and only if the square of D = d* + ∂ is the multiplication by a function on M. As a consequence, we obtain that the pair (A, A*) is a Lie bialgebroid if and only if D is a Dirac generating operator as defined by Alekseev and Xu. Our approach is to establish a list of new identities relating the Lie algebroid structures on A and A*.

Original language | English (US) |
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Pages (from-to) | 399-421 |

Number of pages | 23 |

Journal | Journal of the London Mathematical Society |

Volume | 79 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2009 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

Chen, Z., & Stiénon, M. (2009). Dirac generating operators and Manin triples.

*Journal of the London Mathematical Society*,*79*(2), 399-421. https://doi.org/10.1112/jlms/jdn084