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arXiv · 2502.05137

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

Giorgio Gubbiotti·Francesco Oliveri·Emanuele Sgroi·Pierandrea Vergallo
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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.

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Giorgio Gubbiotti, Francesco Oliveri, Emanuele Sgroi, Pierandrea Vergallo. 2025-02-07. Lie algebras with compatible scalar products for non-homogeneous Hamiltonian operators. https://arxiv.org/abs/2502.05137

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