Invariant symplectic and contact structures on Almost Abelian Lie groups

Abstract

An almost Abelian group is a Lie group that possesses a codimension one Abelian subgroup. A symplectic structure on a manifold is a closed, nondegenerate, differential 2-form. It is clear from the definitions that such a manifold must be even dimensional. For odd dimensions, the closely related structure is called a contact structure. In this project, we study properties of invariant symplectic and contact structures on almost Abelian Lie groups. On a connected almost Abelian group, a basis of invariant vector fields can be computed explicitly in terms of global group coordinates and any invariant tensor can be expanded in a global frame basis with constant coefficients. Here, we impose the extra conditions of symplecticity or contactness and solve the resulting equations. The class of almost Abelian Lie groups is rather wide and representative, and gives context for developing methods of non-commutative analysis on solvable Lie groups --- a subject that is still little understood. Additionally, almost Abelian Lie groups (and algebras) are related with areas such as integrable systems, linear dynamical systems, and even theoretical cosmology.