On the structure of the necklace Lie algebra
| dc.creator | Alev, Jacques | |
| dc.creator | Van de Weyer, Geert | |
| dc.date | 2008-01-10 | |
| dc.date | 2008-01-22 | |
| dc.date.accessioned | 2026-07-07T08:55:27Z | |
| dc.date.available | 2026-07-07T08:55:27Z | |
| dc.description | In this note, we initiate the systematic study of the Lie algebra structure of the necklace Lie algebra n of a free algebra in 2d variables. We begin by giving a description of n as an sp(2d)-module. Specializing to d = 1, we decompose n into a direct sum of highest weight modules for sl_2, the coefficients of which are given by a closed formula. Next, we observe that n has a nontrivial center, which we link through the center C of the trace ring of couples of generic 2x2 matrices to the Poisson center of S(sl_2). The Lie algebra structure of n induces a Poisson structure on C, the symplectic leaves of which we are able to describe as coadjoint orbits for the Lie group of the semidirect product sl_2\rtimes h of sl_2 with the Heisenberg Lie algebra h. Finally, we provide a link between double Poisson algebras on one hand and Poisson orders on the other hand, showing that all trace rings of a double Poisson algebra are Poisson orders over their center. | |
| dc.description | 26 pages, 2 tables, 1 figure. Added references for Remark 3 and Theorem 6. Added a section on non-symplectic necklace Lie algebras. Corrected some minor mistakes/typos | |
| dc.identifier | https://arxiv.org/abs/0801.1621 | |
| dc.identifier | http://arxiv.org/abs/0801.1621 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/146277 | |
| dc.subject | Rings and Algebras | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 17B05; 17B63; 16G20 | |
| dc.title | On the structure of the necklace Lie algebra | |
| dc.type | text |