Asian Journal of Mathematics

Dirac Lie Groups

David Li-Bland and Eckhard Meinrenken

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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.

Article information

Asian J. Math., Volume 18, Number 5 (2014), 779-816.

First available in Project Euclid: 2 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D17: Poisson manifolds; Poisson groupoids and algebroids
Secondary: 17B62: Lie bialgebras; Lie coalgebras 53D20: Momentum maps; symplectic reduction

Poisson Lie groups multiplicative Dirac structures multiplicative Courant algebroids Lie groupoids Lie bialgebras Manin triples multiplicative Manin pairs quasi-Poisson geometry group valued moment maps


Li-Bland, David; Meinrenken, Eckhard. Dirac Lie Groups. Asian J. Math. 18 (2014), no. 5, 779--816.

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