2006
DOI: 10.1016/j.ansens.2005.11.004
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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, recov… Show more

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Cited by 51 publications
(55 citation statements)
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“…As have been shown in [7,8], all real and complex semi-simple symmetric spaces, as well as certain of their compactifications can be embedded into suitable varieties of Lagrangian subalgebras. Out results show that all such spaces carry Poisson structures and natural partitions into regular Poisson subvarieties.…”
Section: Introductionmentioning
confidence: 97%
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“…As have been shown in [7,8], all real and complex semi-simple symmetric spaces, as well as certain of their compactifications can be embedded into suitable varieties of Lagrangian subalgebras. Out results show that all such spaces carry Poisson structures and natural partitions into regular Poisson subvarieties.…”
Section: Introductionmentioning
confidence: 97%
“…It is shown in [1] (see also [8,Corollary 3.18]) that every Belavin-Drinfeld splitting of g ⊕ g is conjugate by an element in G diag to a splitting of the form (4.24) g ⊕ g = g diag + l ′′ S 2 ,T 2 ,d 2 ,V 2 , where (S 2 , T 2 , d 2 ) is a Belavin-Drinfeld triple in the sense that S d 2 2 = {α ∈ S 2 | d n 2 α is defined and is in S 2 for n = 1, 2, · · · } = ∅, and V 2 ∈ L space (z S 2 ⊕ z T 2 ) is such that h diag ∩ (V 2 + {(x, θ d 2 (x)) | x ∈ h S 2 } = 0. In other words, (4.24) is the special case of the splitting in (4.12) with l 1 = g diag .…”
Section: The Rank Of the Poisson Structurementioning
confidence: 99%
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“…This proposition implies that R v,w and P u v,w are Poisson subvarieties of B and B × B . In fact, they are torus orbits of symplectic leaves (see [EL04,GY05]). …”
Section: The Poisson Geometry Of Stratamentioning
confidence: 99%