On the variety of Lagrangian subalgebras

Abstract

We study Lagrangian subalgebras of a semisimple Lie algebra with respect to the imaginary part of the Killing form. We show that the variety $\Lagr$ of Lagrangian subalgebras carries a natural Poisson structure $\Pi$. We determine the irreducible components of $\Lagr$, and we show that each irreducible component is a smooth fiber bundle over a generalized flag variety, and that the fiber is the product of the real points of a De Concini-Procesi compactification and a compact homogeneous space. We study some properties of the Poisson structure $\Pi$ and show that it contains many interesting Poisson submanifolds.

References in corpus (1)

• Geometry and classification of solutions of the Classical Dynamical Yang-Baxter Equation

Cited by in corpus (6)

• On the wonderful compactification
• Poisson Cohomology of Del Pezzo surfaces
• A Deodhar type stratification on the double flag variety
• Homogeneous Poisson Structures on Loop Spaces of Symmetric Spaces
• On a class of double cosets in reductive algebraic groups
• Closures of Steinberg fibers in twisted wonderful compactifications