Dirac Lie Groups

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.