Contents Online

# Asian Journal of Mathematics

## Volume 18 (2014)

### Number 5

### Dirac Lie groups

Pages: 779 – 816

DOI: http://dx.doi.org/10.4310/AJM.2014.v18.n5.a2

#### Authors

#### Abstract

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.

#### Keywords

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.