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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)
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