# Lie bialgebroids and Poisson groupoids

@article{Mackenzie1994LieBA, title={Lie bialgebroids and Poisson groupoids}, author={Kirill C. H. Mackenzie and Ping Xu}, journal={Duke Mathematical Journal}, year={1994}, volume={73}, pages={415-452} }

Lie bialgebras arise as infinitesimal invariants of Poisson Lie groups. A Lie bialgebra is a Lie algebra g with a Lie algebra structure on the dual g∗ which is compatible with the Lie algebra g in a certain sense. For a Poisson group G, the multiplicative Poisson structure π induces a Lie algebra structure on the Lie algebra dual g∗ which makes (g, g∗) into a Lie bialgebra. In fact, there is a one-one correspondence between Poisson Lie groups and Lie bialgebras if the Lie groups are assumed to…

## 383 Citations

### Generalized Lie Bialgebroids and Strong Jacobi-Nijenhuis Structures

- Mathematics
- 2002

Roughly speaking, a Lie bialgebroid is a Lie algebroid A whose dual A is also equipped with a Lie algebroid structure which is compatible in a certain way with that on A (see [15]). An important…

### Calculus on Lie algebroids, Lie groupoids and Poisson manifolds

- Mathematics
- 2008

We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a…

### A classification theorem for Dirac homogeneous spaces of Dirac Lie groupoids

- Mathematics
- 2010

Given an integrable multiplicative Dirac structure DG with units A(DG) on a Lie groupoid GP, we show that there are Lie algebroids on P associated to the Dirac structure. We construct a vector bundle…

### LIE ALGEBROIDS AND LIE PSEUDOALGEBRAS KIRILL

- Mathematics
- 1995

Lie algebroids and Lie pseudoalgebras arise from a wide variety of constructions in differential geometry; they have been introduced repeatedly into the geometry, physics and algebra literatures…

### Universal lifting theorem and quasi-Poisson groupoids

- Mathematics
- 2005

We prove the universal lifting theorem: for an α-simply connected and α-connected Lie groupoid with Lie algebroid A, the graded Lie algebra of multi-differentials on A is isomorphic to that of…

### On the integration of LA-groupoids and duality for Poisson groupoids

- Mathematics
- 2007

In this note a functorial approach to the integration problem of an LA-groupoid to a double Lie groupoid is discussed. To do that, we study the notions of fibred products in the categories of Lie…

### On Generalized Lie Bialgebroids and Jacobi Groupoids

- Mathematics
- 2016

Generalized Lie bialgebroids are generalization of Lie bialgebroids and arises nat-
urally from Jacobi manifolds. It is known that the base of a generalized Lie bialgebroid
carries a Jacobi…

### Dirac Group(oid)s and Their Homogeneous Spaces

- Mathematics
- 2011

A theorem of Drinfel'd (Drinfel'd (1993)) classifies the Poisson homogeneous spaces of a Poisson Lie group (G,πG) via a special class of Lagrangian subalgebras of the Drinfel'd double of its Lie…

## References

SHOWING 1-10 OF 25 REFERENCES

### Poisson Lie groups, dressing transformations, and Bruhat decompositions

- Mathematics
- 1990

A Poisson Lie group is a Lie group together with a compatible Poisson structure. The notion of Poisson Lie group was first introduced by Drinfel'd [2] and studied by Semenov-Tian-Shansky [17] to…

### Poisson-Nijenhuis structures

- Mathematics
- 1990

We study the deformation, defined by a Nijenhuis operator, and the dualization, defined by a Poisson bivector, of the Lie bracket of vector fields on a manifold and, more generally, of the Lie…

### Coisotropic calculus and Poisson groupoids

- Mathematics
- 1988

Lagrangian submanif olds play a special role in the geometry of symplectic manifolds. From the point of view of quantization theory, or simply a categorical approach to symplectic geometry [Gu-S2],…

### Duality for base-changing morphisms of vector bundles, modules, Lie algebroids and Poisson structures

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1993

Abstract The main result of this paper is an extension to Poisson bundles [4] and Lie algebroids of the classical result that a linear map of Lie algebras is a morphism of Lie algebras if and only if…

### Classical solutions of the quantum Yang-Baxter equation

- Mathematics
- 1992

The classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups. Whereas previous work begins with a…

### MATCHED PAIRS OF LIE GROUPS ASSOCIATED TO SOLUTIONS OF THE YANG-BAXTER EQUATIONS

- Mathematics
- 1990

Two groups G, H are said to be a matched pair if they act on each other and these actions, (a, /?), obey a certain compatibility condition. In such a situation one may form a bicrossproduct group,…

### Tangent dirac structures

- Mathematics
- 1990

The lift of a closed 2-form Omega on a manifold Q to a closed 2-form on TQ may be achieved by pulling back the canonical symplectic structure on T*Q by the bundle map Omega : TQ to T*Q. It is also…

### ANALOGUES OF THE OBJECTS OF LIE GROUP THEORY FOR NONLINEAR POISSON BRACKETS

- Mathematics
- 1987

For general degenerate Poisson brackets, analogues are constructed of invariant vector fields, invariant forms, Haar measure and adjoint representation. A pseudogroup operation is defined that…

### LIE GROUPOIDS AND LIE ALGEBROIDS IN DIFFERENTIAL GEOMETRY

- Mathematics
- 1988

4. G. D. Mostow and P. Deligne, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Etudes Sci. Publ. Math. 46 (1983). 5. E. Picard, Sur les fonctions hyperfuchsiaes…