\(\omega \)-Symplectic algebra and Hamiltonian vector fields

The purpose of this paper is to present an algebraic theoretical basis for the study of \(\omega \) $\omega$ -Hamiltonian vector fields defined on a symplectic vector space \((V,\omega )\) $(V,\omega )$ with respect to coordinates that are not necessarily symplectic. We introduce the concepts of \(\omega \) $\omega$ -symplectic and \(\omega \) $\omega$ -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 \(\omega \) $\omega$ -Hamiltonian vector fields.