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 coordinates (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 components. Finally, motivated by the example of the KdV equation, we discuss the conditions for bi-Hamiltonianity of such operators.