  We construct a class of new Lie algebras by generalizing the one-variable Lie
algebras generated by the quadratic conformal algebras (or corresponding
Hamiltonian operators) associated to Poisson algebras and a quasi-derivation
found by Xu. These algebras can be viewed as certain twists of Xu's generalized
Hamiltonian Lie algebras. The simplicity of these algebras is completely
determined. Moreover, we construct a family of multiplicity-free
representations of these Lie algebras and prove their irreducibility.
