# A classification of Poisson homogeneous spaces of complex reductive Poisson-Lie groups

@article{Karolinsky1999ACO, title={A classification of Poisson homogeneous spaces of complex reductive Poisson-Lie groups}, author={Eugene Karolinsky}, journal={Banach Center Publications}, year={1999}, volume={51}, pages={103-108} }

Let G be a complex reductive connected algebraic group equipped with the Sklyanin bracket. A classification of Poisson homogeneous G-spaces with connected isotropy subgroups is given. This result is based on Drinfeld’s correspondence between Poisson homogeneous G-spaces and Lagrangian subalgebras in the double D(g) (here g = LieG). A geometric interpretation of some Poisson homogeneous G-spaces is also proposed.

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## References

SHOWING 1-10 OF 11 REFERENCES

### On Poisson homogeneous spaces of Poisson-Lie groups

- Mathematics
- 1993

Poisson homogeneous spaces of a Poisson-Lie group G are described in terms of Lagrangian subalgebras of D(g), where D(g) is the double of the Lie bialgebra g corresponding to G.

### Nonlinear Poisson structures andr-matrices

- Mathematics
- 1989

We introduce quadratic Poisson structures on Lie groups associated with a class of solutions of the modified Yang-Baxter equation and apply them to the Hamiltonian description of Lax systems. The…

### Classical Dynamical r-Matrices¶and Homogeneous Poisson Structures on G/H and K/T

- Mathematics
- 2000

Abstract: Let G be a finite dimensional simple complex group equipped with the standard Poisson Lie group structure. We show that all G-homogeneous (holomorphic) Poisson structures on G/H, where H⊂G…

### Quantum Groups

- Mathematics
- 1994

Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups…

### Quantum Groups

- Mathematics
- 1993

This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditions…

### Triangle Equations and Simple Lie Algebras

- Mathematics
- 1998

PART I: Properties of Nondegenerate Solutions 1. Elliptic Solutions 2. Brief Survey of Semisimple Lie Algebras 3. The Simplest Trigonometric Solutions 4. Modified Triangle Equation for Constants 5.…