Symplectic Homogeneous Spaces of Six-Dimensional Nilpotent Non-Symplectic Lie Groups

УДК 514.76

Abstract

There are only 26 classes among the 34 classes of pairwise non-isomorphic six-dimensional nilpotent Lie algebras that admit invariant simpectic structures. The remaining eight classes represent non-symplectic Lie groups, i.e., groups for which every closed left-invariant 2-form is a degenerate one. In 1974, Chu Bon-Yao showed that every closed degenerate left-invariant 2-form ω on a Lie group determines a symplectic structure on a homogeneous space of this lie group when the isotropy group is Lie subgroup corresponding to the degeneration algebra of the 2-form ω. This paper considers geometric structures on symplectic homogeneous spaces of all eight non-symplectic 6-dimensional nilpotent Lie groups. It is shown that invariant complex or paracomplex structures exist in six out of eight cases. Invariant metrics on the considered homogeneous spaces are proved to exist only in four out of eight cases. Moreover, the invariant metrics are pseudo-Riemannian ones.