## Exact Lie bialgebroids and Poisson groupoids.(English)Zbl 0869.17016

The aim of this article is to define a notion of exact Lie bialgebroids. Let $$A\to P$$ be a Lie algebroid such that its dual bundle $$A^*\to P$$ also carries a Lie algebroid structure with a compatibility property. Then $$(A,A^*)$$ is called by the authors a Lie bialgebroid. Dealing with an initial Lie algebroid $$A\to P$$, they prove that if its bracket and anchor have two compatibility properties then $$A^*\to P$$ has a canonical Lie algebroid structure such that $$(A,A^*)$$ becomes a Lie bialgebroid; such a pair is called exact. As a Lie bialgebra can be integrated to a Poisson-Lie group, a Lie bialgebroid can, under certain conditions, be integrated to a Poisson groupoid; it is proved that the exactness of the Lie bialgebroid implies those conditions, so a Poisson groupoid, which can be constructed in that way will be called an exact Poisson groupoid.

### MSC:

 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16W30 Hopf algebras (associative rings and algebras) (MSC2000)

### Keywords:

exact Lie bialgebroids; exact Poisson groupoid
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### References:

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