On the variety of Lagrangian subalgebras, II
| dc.creator | Evens, Sam | |
| dc.creator | Lu, Jiang-Hua | |
| dc.date | 2004-09-14 | |
| dc.date | 2006-02-28 | |
| dc.date.accessioned | 2026-07-07T06:38:47Z | |
| dc.date.available | 2026-07-07T06:38:47Z | |
| dc.description | When ${\frak g}$ is a complex semisimple Lie algebra, we study the variety ${\mathcal L}$ of subalgebras of ${\frak g}\oplus{\frak g}$ that are maximally isotropic with respect to $K_1 - K_2$, where $K_i$ 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 $G\times G$. 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 $G\times G$. 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. | |
| dc.description | revised version, 32 pages, some proofs have been made more efficient and some comments were removed. To appear in Ann. ENS | |
| dc.identifier | https://arxiv.org/abs/math/0409236 | |
| dc.identifier | http://arxiv.org/abs/math/0409236 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/100864 | |
| dc.subject | Quantum Algebra | |
| dc.subject | Symplectic Geometry | |
| dc.title | On the variety of Lagrangian subalgebras, II | |
| dc.type | text |