Lie algebra structures for four-component Hamiltonian hydrodynamic type systems

Four-component Hamiltonian systems of hydrodynamic type induce, through the Haantjes tensor, a Lie algebra structure on tangent planes in the space of dependent variables. We show that this Lie algebra is either reductive or solvable with a nilpotent three-dimensional subalgebra. We demonstrate how the precise Lie algebraic structure is determined by the Hamiltonian structure of the system. An application to perturbations of the Benney system is presented.

► Hamiltonian systems of hydrodynamic type appear in diverse physical, biological, and geometric settings.
► For Hamiltonian systems consisting of four equations, there is associated a natural Lie algebra structure.
► In the present work, this Lie algebra structure is classified.
► An application to a physical example is presented.