Dirac Lie groups. (English) Zbl 1320.53102

Li-Bland, David; Meinrenken, Eckhard

Asian J. Math. 18, No. 5, 779-816 (2014).

The authors extend Drinfeld’s categorical equivalence between simply connected Lie groups and Manin triples to a categorical equivalence between Dirac Lie groups and equivariant Dirac Manin triples.
After recalling the notions of a Courant algebroid \(\mathbb{A}\) over a manifold \(M\), a Dirac structure as an involutive Lagrangian subbundle \(E\) of \(\mathbb{A}\), three types of smooth relations between vector bundles, and a multiplicative Manin pair \(\left( \mathbb{A},E\right) \) over a Lie groupoid, the authors introduce the notion of a Dirac Lie group structure on a Lie group \(H\) as a multiplicative Manin pair over the “group” groupoid \(H\rightrightarrows\text{pt}\) and point out how in the case of the standard Courant algebroid \(\mathbb{T} H\rightarrow H\), a Dirac Lie group structure on \(H\) is equivalent to a Poisson-Lie group structure \(\pi\) on \(H\). A triple \(\left( \mathfrak{d} ,\mathfrak{g},\mathfrak{h}\right) _{\beta}\) of Lie algebras with \(\mathfrak{d}=\mathfrak{g\oplus h}\) as vector spaces and \(\beta\) an invariant symmetric bilinear form on \(\mathfrak{d}^{\ast}\) vanishing on \(\text{ann}\left( \mathfrak{g}\right) \subset\mathfrak{d}^{\ast}\) is called a Dirac Manin triple, and is called \(H\)-equivariant if \(\left( \mathfrak{d},\mathfrak{g},\mathfrak{h}\right) \) is an \(H\)-equivariant triple and \(\beta\) is \(H\)-invariant. A functorial construction of Dirac Lie group structures on \(H\) from \(H\)-equivariant Dirac Manin triples \(\left( \mathfrak{d},\mathfrak{g},\mathfrak{h}\right) _{\beta}\) is presented and shown to give rise to a categorical equivalence. Furthermore it is proved that the underlying Courant groupoid of a Dirac Lie group structure on \(H\) is exact if and only if \(\beta\) is nondegenerate and \(\mathfrak{g}\) is Lagrangian. The authors also classify Dirac Lie groups with their morphisms, and relate them to quasi-Poisson geometry and quasi-Lie bialgebroids.

Reviewer: Albert Sheu (Lawrence)

MSC:

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

Keywords:

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