Symplectic Lie groups. Symplectic reduction, Lagrangian extensions, and existence of Lagrangian normal subgroups (Q2811905)

This book in fact is a big informative article about the structure of symplectic Lie groups and symplectic Lie algebras. Some geometric applications also are considered.NEWLINENEWLINEIf a symplectic Lie group \(G\) has a lattice \(D\), then the compact manifold \(G/D\) has the symplectic structure; in this case the Lie group \(G\) is unimodular. It is known that all unimodular symplectic Lie groups are solvable, so the main theme of this article is the structure of symplectic solvable Lie groups (and the corresponding symplectic Lie algebras), especially nilpotent ones. As important examples of symplectic Lie groups the cotangent Lie groups \(T^\ast(G)\) are investigated.NEWLINENEWLINEThere are close interrelations between symplectic Lie groups and flat Lie groups. Flat Lie groups play an important role in this paper. These Lie groups arise as quotients with respect to Lagrangian normal subgroups of symplectic Lie groups. In connection with cotangent symplectic Lie groups the theory of Lagrangian extensions of flat Lie groups is developed.NEWLINENEWLINEThe main method of investigation in the article is a symplectic reduction with respect to isotropic normal subgroups. If a symplectic Lie group does not admit any proper isotropic normal subgroup, it is called irreducible symplectic Lie group. It is proved in this article that every irreducible symplectic Lie group is metabelian, solvable of imaginary type, and has the additional structure of a flat Kähler Lie group. The complete local classification of irreducible symplectic Lie groups is described.NEWLINENEWLINEThe existence problem for Lagrangian normal subgroups in solvable and nilpotent symplectic Lie groups is investigated in details. In particular, it is proved that every real symplectic Lie algebra of dimension \(\leq 6\) admits a Lagrangian subalgebra. But there is a 6-dimensional completely solvable symplectic Lie algebra without Lagrangian ideal. Also there exist irreducible real symplectic Lie algebras of dimension 8, which do not have any Lagrangian subalgebra, and a 10-dimensional 3-step nilpotent Lie algebra without Lagrangian ideals. It is proved that every completely solvable symplectic Lie algebra admits a Lagrangian subalgebra.NEWLINENEWLINEIn this article, several claims and conjectures are disproved. In particular, it was asked by Guan, if it is true that for any compact symplectic solvmanifold \(G/D\) the solvability degree of a Lie group \(G\) is always bounded by 3. This conjecture has been verified in dimensions \(\leq 6\). But in this article it is shown that, contrary to the conjecture, the solvability degree of symplectic solvmanifolds is unbounded with increasing dimension. A series of symplectic nilmanifolds, which has unbounded solvability degree, is presented. In particular, there is a 72-dimensional example of solvability degree 4. Also many other examples of various kinds are presented.