$ω$-Symplectic algebra and Hamiltonian vector fields

The purpose of this paper is presenting a theoretical basis for the study of $ω$-Hamiltonian vector fields in a more general approach than the classical one. We introduce the concepts of $ω$-symplectic group and $ω$-semisymplectic group, and describe some of their properties. We show that the Lie algebra of such groups is a useful tool in the recognition of an $ω$-Hamiltonian vector field defined on a symplectic vector space $(V,ω)$ with respect to coordinates that are not necessarily symplectic.