$$\omega $$ -Symplectic algebra and Hamiltonian vector fields

The purpose of this paper is to present an algebraic theoretical basis for the study of \(\omega \)-Hamiltonian vector fields defined on a symplectic vector space \((V,\omega )\) with respect to coordinates that are not necessarily symplectic. We introduce the concepts of \(\omega \)-symplectic and \(\omega \)-semisymplectic groups, and describe some of their properties that may not coincide with the classical context. We show that the Lie algebra of such groups is a useful tool in the recognition and construction of \(\omega \)-Hamiltonian vector fields.