Dirac Lie groups

A classical theorem of Drinfel'd states that the category of simply connected Poisson Lie groups H is isomorphic to the category of Manin triples (\mathfrak{d, g, h}), where \mathfrak{h} is the Lie algebra of H. In this paper, we consider Dirac Lie groups, that is, Lie groups H endowed with a m...

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Bibliographic Details
Published in:The Asian journal of mathematics 2014, Vol.18 (5), p.779-816
Main Authors: Li-Bland, David, Meinrenken, Eckhard
Format: Article
Language:English
Subjects:
17B62
53D17
53D20
group valued moment maps
Lie bialgebras
Lie groupoids
Manin triples
multiplicative Courant algebroids
multiplicative Dirac structures
multiplicative Manin pairs
Poisson Lie groups
quasi-Poisson geometry
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Summary:A classical theorem of Drinfel'd states that the category of simply connected Poisson Lie groups H is isomorphic to the category of Manin triples (\mathfrak{d, g, h}), where \mathfrak{h} is the Lie algebra of H. In this paper, we consider Dirac Lie groups, that is, Lie groups H endowed with a multiplicative Courant algebroid A and a Dirac structure E \subseteq \mathbb{A} for which the multiplication is a Dirac morphism. It turns out that the simply connected Dirac Lie groups are classified by so-called Dirac Manin triples. We give an explicit construction of the Dirac Lie group structure defined by a Dirac Manin triple, and develop its basic properties.
ISSN:1093-6106
1945-0036
DOI:10.4310/AJM.2014.v18.n5.a2