Lie algebras with compatible scalar products for non-homogeneous Hamiltonian operators

This paper is devoted to characterising Hamiltonian operators expressible as a sum of a non-degenerate first-order homogeneous operator and a Poisson tensor and classifying them in low dimension. It is known that in flat coordin- ates (also known as Darboux coordinates), these operators are related to Lie algebras and their compatible scalar products. In this paper, we give a novel interpretation of this finding, showing that they are uniquely determined by a triple composed of a Lie algebra, its most general non-degenerate quadratic Casimir elements, and a 2-cocycle. This allows us to use the well-known theory of Casimir elements of Lie algebras to explicitly characterise some classes of operators and give a complete description of such operators up to six compon- ents. Finally, motivated by the example of the KdV equation, we discuss the conditions for bi-Hamiltonianity of such operators.