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

TL;DR

This paper explores the algebraic and geometric structures of Hamiltonian operators combining Lie algebra elements and Poisson tensors, providing classifications and conditions for bi-Hamiltonianity, with applications to integrable systems like KdV.

Contribution

It introduces a novel algebraic framework for Hamiltonian operators using Lie algebras with non-degenerate quadratic Casimirs and characterizes their properties in low dimensions.

Findings

Classified Hamiltonian operators associated with specific Lie algebras.

Described conditions for bi-Hamiltonianity in these operators.

Provided explicit examples related to the KdV equation.

Abstract

We study from an algebraic and geometric viewpoint Hamiltonian operators which are sum of a non-degenerate first-order homogeneous operator and a Poisson tensor. In flat coordinates, also known as Darboux coordinates, these operators are uniquely determined by a triple composed by a Lie algebra, its most general non-degenerate quadratic Casimir and a 2-cocycle. We present some classes of operators associated to Lie algebras with non-degenerate quadratic Casimirs and we give a description of such operators in low dimensions. Finally, motivated by the example of the KdV equation we discuss the conditions of bi-Hamiltonianity of such operators.