# On Lie bialgebroid crossed modules

@article{Lang2019OnLB, title={On Lie bialgebroid crossed modules}, author={Honglei Lang and Yu Qiao and Yanbin Yin}, journal={International Journal of Mathematics}, year={2019} }

We study Lie bialgebroid crossed modules which are pairs of Lie algebroid crossed modules in duality that canonically give rise to Lie bialgebroids. A one-one correspondence between such Lie bialgebroid crossed modules and co-quadratic Manin triples [Formula: see text] is established, where [Formula: see text] is a co-quadratic Lie algebroid and [Formula: see text] is a pair of transverse Dirac structures in [Formula: see text].

## One Citation

### The classical Lie-Yamaguti Yang-Baxter equation and Lie-Yamaguti bialgebras

- Mathematics
- 2022

A bstract . In this paper, we develop the bialgebra theory for Lie-Yamaguti algebras. For this purpose, we exploit two types of compatibility conditions: local cocycle condition and double…

## References

SHOWING 1-10 OF 42 REFERENCES

### Exact Gerstenhaber algebras and Lie bialgebroids

- Mathematics
- 1995

We show that to any Poisson manifold and, more generally, to any triangular Lie bialgebroid in the sense of Mackenzie and Xu, there correspond two differential Gerstenhaber algebras in duality, one…

### Matched pairs of Lie algebroids

- MathematicsGlasgow Mathematical Journal
- 1997

Abstract We extend to Lie algebroids the notion variously known as a double Lie algebra (Lu and Weinstein), matched pair of Lie algebras (Majid), or twilled extension of Lie algebras…

### Lie 2-Bialgebras

- Mathematics
- 2013

In this paper, we study Lie 2-bialgebras, paying special attention to coboundary ones, with the help of the cohomology theory of L∞-algebras with coefficients in L∞-modules. We construct examples of…

### Locally Vacant Double Lie Groupoids and the Integration of Matched Pairs of Lie Algebroids

- Mathematics
- 1999

We prove a general integrability result for matched pairs of Lie algebroids. (Matched pairs of Lie algebras are also known as double Lie algebras or twilled extensions of Lie algebras.) The method…

### Strongly Homotopy Lie Bialgebras and Lie Quasi-bialgebras

- Mathematics
- 2006

Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer–Cartan equations on corresponding governing differential graded Lie algebras…

### Higher extensions of Lie algebroids

- Mathematics
- 2017

We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie…

### Courant Algebroids and Poisson Geometry

- Mathematics
- 2009

Given a manifold M with an action of a quadratic Lie algebra , such that all stabilizer algebras are coisotropic in , we show that the product becomes a Courant algebroid over M. If the bilinear form…

### Poisson 2-groups

- Mathematics
- 2012

We prove a 2-categorical analogue of a classical result of Drinfeld: there is a one-to-one correspondence between connected, simply-connected Poisson Lie 2-groups and Lie 2-bialgebras. In fact, we…

### General theory of lie groupoids and lie algebroids

- Mathematics
- 2005

Part I. The General Theory: 1. Lie groupoids: fundamental theory 2. Lie groupoids: algebraic constructions 3. Lie algebroids: fundamental theory 4. Lie algebroids: algebraic constructions Part II.…