Asian Journal of Mathematics

Volume 18 (2014)

Number 5

Dirac Lie groups

Pages: 779 – 816



David Li-Bland (Department of Mathematics, University of California at Berkeley)

Eckhard Meinrenken (Department of Mathematics, University of Toronto, Ontario, Canada)


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.


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

2010 Mathematics Subject Classification

Primary 53D17. Secondary 17B62, 53D20.

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