. Q A ] 2 8 Fe b 20 06 ON THE VARIETY OF LAGRANGIAN SUBALGEBRAS , II
نویسندگان
چکیده
VERSION FRANÇ AISE: Motivé par le théorème de Drinfeld sur les espaces de Poisson homogènes, nousétudions la variété L des sous-algèbres de Lie Lagrangiennes de g ⊕ g pour g, une algèbre de Lie complexe semisimple. Soit G le groupe adjointe de g. Nous montrons que les adhérences des (G × G)-orbites dans L sont les variétés sphériques et lisses. Aussi, nous classifions les composantes irréductibles de L et nous montrons qu'elles sont lisses. Nous employons quelques méthodes de M. Yakimov pour donner une nouvelle description et une nouvelle preuve de la classification de Karolinsky des orbites diagonales de G dans L, quel, comme cas spécial, donne la classification de Belavin-Drinfeld des r-matrices quasitriangulaires de g. En outre, L possède une structure de Poisson canonique, et nous calculons son rangà chaque point and nous décrivons sa décomposition en feuilles symplectiques en termes des intersections des orbites des deux sous-groupes de G × G. ENGLISH VERSION: 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.
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