On the variety of Lagrangian subalgebras, I

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 L of Lagrangian subalgebras carries a natural Poisson structure Π. We determine the irreducible components of L, 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 Π and show that it contains many interesting Poisson submanifolds.

A note on Poisson homogeneous spaces

“…See [8,9,16] for studies of certain varieties which can serve as moduli spaces of Poisson homogeneous spaces. In forthcoming papers, we will use results from this note to compute explicitly the Poisson cohomology and study their symplectic groupoids for certain examples of Poisson homogeneous spaces treated in [8,9,16]. Such examples included flag varieties of complex semi-simple groups [8] and semi-simple Riemannian symmetric spaces [10] (see Example 5.14).…”

Poisson structures on complex flag manifolds associated with real forms

“…m(l) = {x ∈ l : x, y = 0 ∀y ∈ n(l)} ⊂ l, and let When v is the Vogan diagram with d = 1 and no vertex painted, we have τ v = θ, so g v = k. The Poisson structure Π v in this case was first introduced in [11] and [13], and it has the property that its symplectic leaves are precisely the Bruhat cells (hence the name "Bruhat Poisson structure" in [11]). In [3] and [10] this Poisson structure was related to some earlier work of Kostant [7] and of Kostant-Kumar [8] on the Schubert calculus on X.…”

“…In this section, we will start with a Vogan diagram v for g and define a Poisson structure Π v on X such that every G v -orbit in X is a Poisson submanifold. This Poisson structure comes from an identification of X with the G-orbit through t + n inside the variety L of Lagrangian subalgebras of g, which was studied in [3]. We now recall the relevant details.…”

“…All the G v -orbits in L become G v -Poisson homogeneous spaces [3,9]. We remark that in [1], Andruskiewitsch and Jancsa classified non-triangular Lie bialgebra structures on g v using BelavinDrinfeld triples.…”

Group Orbits and Regular Partitions of Poisson Manifolds

“…Clearly, n(l) is coisotropic in d because it contains l. Thus it follows from Theorem 2.3 that Π u,u ′ is a Poisson structure on L(d), see also [7]. The following proposition now follows immediately from Theorem 2.7.…”