Double Extension Symplectique d'un Groupe de Lie Symplectique

Abstract

Lie groups carrying a left invariant symplectic form (symplectic groups) are described in terms of semi-direct product of Lie groups or symplectic reduction and principal fiber bundles with affine fiber. We give a generalization of Medina and Revoy's symplectic double extension, which realizes a symplectic group as the reduction of another symplectic group. We show that every group obtained by this process carries an invariant Lagrangian foliation such that the affine structure defined by the simplectic form over each leaf is complete.

Generalized Lie bialgebras and Jacobi structures on Lie groups

“…ii) A complete description of symplectic Lie algebras of dimension 4 was obtained in [27] (for a detailed study of symplectic Lie algebras, see also [5,21]). Thus, one can determine all contact Lie algebras of dimension 5 with center of dimension 1 and from there, using Theorem 4.1, to obtain different examples of generalized Lie bialgebras.…”

Rota–Baxter Operators on Pre-Lie Superalgebras

“…They were first mentioned by Cayley in 1890 [20] as a kind of rooted tree algebra and later arose again from the study of convex homogeneous cones [53], affine manifold and affine structures on Lie groups [40], and deformation of associative algebras [34]. They play an important role in the study of symplectic and complex structures on Lie groups and Lie algebras [5,22,24,25,44], phases spaces of Lie algebras [8,42], certain integrable systems [16], classical and quantum Yang-Baxter equations [26], combinatorics [27], quantum field theory [23] and operads [19]. See [17] for a survey.…”

Symplectic structures on quadratic Lie algebras

“…Aubert [2] proved that every quadratic symplectic Lie algebra may be obtained from an abelian quadratic symplectic Lie algebra by a sequence of (generalised) symplectic double extensions by a line or a plane (see [5,13] for the formal definition of symplectic double extensions). For the sake of completeness and since they will be used in Section 5, we give, without proof, a particular case of such result (Theorem 4.2) but, as for our purposes concerning Manin triples of Section 5 it is more appropriate to describe quadratic symplectic Lie algebras in terms of quadratic double extension, we have restated this particular case in such terms.…”