Exact Lie bialgebroids and Poisson groupoids
type: Item
instance of: scholarly article
title: Exact Lie bialgebroids and Poisson groupoids
zbMATH Open document ID: 0869.17016
DOI: 10.1007/BF02246770
author: Zhangju Liu
author: Ping Xu
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-04-24T00:00:00Z
full work available at URL: https://eudml.org/doc/58221
review text: 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.
reviewed by: Jean-Michel Vallin
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 881248
zbMATH Keywords: exact Lie bialgebroids
zbMATH Keywords: exact Poisson groupoid
MaRDI profile type: Publication
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cites work: Generalized Yang-Baxter equations, Koszul operators and Poisson Lie groups
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description: scientific article; zbMATH DE number 881248