## 198 Citations

### On symplectic double groupoids and the duality of Poisson groupoids

- Mathematics
- 1998

We prove that the cotangent of a double Lie groupoid S has itself a double groupoid structure with sides the duals of associated Lie algebroids, and double base the dual of the Lie algebroid of the…

### Notions of double for Lie algebroids

- Mathematics
- 2000

We define an abstract notion of double Lie algebroid, which includes as particular cases: (1) the double Lie algebroid of a double Lie groupoid in the sense of the author, such as the iterated…

### 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…

### J an 2 00 7 On the integration of LA-groupoid and duality for Poisson groupoids by

- Mathematics
- 2007

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

### Leaves of stacky Lie algebroids

- Mathematics
- 2020

We show that the leaves of an LA-groupoid which pass through the unit manifold are, modulo a connectedness issue, Lie groupoids. We illustrate this phenomenon by considering the cotangent Lie…

### From double Lie groupoids to local Lie 2-groupoids

- Mathematics
- 2011

We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger’s fundamental groupoid from the fundamental double…

### Supergroupoids, double structures, and equivariant cohomology

- Mathematics
- 2006

Q-groupoids and Q-algebroids are, respectively, supergroupoids and superalgebroids that are equipped with compatible homological vector fields. These new objects are closely related to the double…

### Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids

- Mathematics
- 2006

Abstract The word ‘double’ was used by Ehresmann to mean ‘an object X in the category of all X’. Double categories, double groupoids and double vector bundles are instances, but the notion of Lie…

## References

SHOWING 1-10 OF 35 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…

### Lie bialgebroids and Poisson groupoids

- Mathematics
- 1994

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…

### 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…

### 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…

### Affine Poisson structures

- Mathematics
- 1989

We establish a one-to-one correspondence between the set of all equivalence classes of affine Poisson structures (defined on the dual of a finite dimensional Lie algebra) and the set of all…

### 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,…

### Poisson cohomology and quantization.

- Mathematics
- 1990

Let R be a commutative ring, and let A be a Poisson algebra over R. We construct an (R,A)-Lie algebra structure, in the sense of Rinehart, on the A-module of K\"ahler differentials of A depending…

### 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],…