2004/09/14 by Sam Evens, Jiang-Hua Lu, Evens, Sam +1
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.math/0409236
openalex publication_date 2004/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
When \frak g is a complex semisimple Lie algebra, we study the variety \mathcal L of subalgebras of \frak g⊕\frak g that are maximally isotropic with respect to K1 - K2, where Ki is the Killing form on the ith factor. We show the irreducible components of \mathcal L are smooth, classify them in terms of the generalized Belavin-Drinfeld triples introduced by Schiffmann, and relate them to orbits of the adjoint group G× G. Building on ideas of Yakimov, we give a new proof of Karolinsky's classification of the diagonal G-orbits in \mathcal L. Our proof enables us to compute of the normalizer in \frak g of a subalgebra in \mathcal L under the diagonal action. As a consequence, we recover the classification of Belavin-Drinfeld triples. By results of math.DG/9909005, \mathcal L is a Poisson variety and we determine the rank of the symplectic leaf at each point of \mathcal L in terms of combinatorial data and relate the symplectic leaves to intersections of orbits of subgroups of G× G. As a consequence, an intrinsically defined Poisson structure on each conjugacy class on G has an open symplectic leaf and we determine the rank at each point of the conjugacy class.