On the variety of Lagrangian subalgebras, II

Abstract

Motivated by Drinfeld's theorem on Poisson homogeneous spaces, we study the variety L of Lagrangian subalgebras of g ⊕ g for a complex semi-simple Lie algebra g. Let G be the adjoint group of g. We show that the (G × G)-orbit closures in L are smooth spherical varieties. We also classify the irreducible components of L and show that they are smooth. Using some methods of M. Yakimov, we give a new description and proof of Karolinsky's classification of the diagonal G-orbits in L, which, as a special case, recovers the Belavin-Drinfeld classification of quasi-triangular r-matrices on g. Furthermore, L has a canonical Poisson structure, and we compute its rank at each point and describe its symplectic leaf decomposition in terms of intersections of orbits of two subgroups of G × G. © 2006 Elsevier SAS. All rights reserved.