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.


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