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

Giorgio Gubbiotti, Francesco Oliveri, Emanuele Sgroi, Pierandrea Vergallo

TL;DR

The work develops a Lie-algebraic framework for non-homogeneous Hamiltonian operators of hydrodynamic type, showing that in the Darboux form the operator data consist of a Lie algebra $ rak g$, a nondegenerate quadratic Casimir (defining a compatible scalar product), and a 2-cocycle that yields the linear-in-field contribution. It establishes that nondegenerate 1+0 HOs are Hamiltonian precisely when the leading homogeneous part is Dubrovin–Novikov and the lower part is a Poisson tensor compatible with the metric, with Casimir structures constrained to linear functionals. The paper then systematically constructs and classifies such operators for abelian, semi-simple, direct-sum, and nilpotent Lie algebras, providing explicit examples up to dimension six and connecting to classic integrable models like KdV via bi-Hamiltonian pairs. This framework clarifies the algebraic origin of non-homogeneous Hamiltonian structures and offers a catalog of low-dimensional operators that encode potential integrable hierarchies and their geometric underpinnings.

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.

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

TL;DR

The work develops a Lie-algebraic framework for non-homogeneous Hamiltonian operators of hydrodynamic type, showing that in the Darboux form the operator data consist of a Lie algebra , a nondegenerate quadratic Casimir (defining a compatible scalar product), and a 2-cocycle that yields the linear-in-field contribution. It establishes that nondegenerate 1+0 HOs are Hamiltonian precisely when the leading homogeneous part is Dubrovin–Novikov and the lower part is a Poisson tensor compatible with the metric, with Casimir structures constrained to linear functionals. The paper then systematically constructs and classifies such operators for abelian, semi-simple, direct-sum, and nilpotent Lie algebras, providing explicit examples up to dimension six and connecting to classic integrable models like KdV via bi-Hamiltonian pairs. This framework clarifies the algebraic origin of non-homogeneous Hamiltonian structures and offers a catalog of low-dimensional operators that encode potential integrable hierarchies and their geometric underpinnings.

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.

Paper Structure

This paper contains 31 sections, 20 theorems, 147 equations, 1 table.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Non-degenerate structures
  4. Darboux form of Hamiltonian operators
  5. Algebraic interpretation of 1+0 operators in Darboux form
  6. Final remarks on Casimir functionals of 1 + 0 HOs
  7. Some relevant classes of Lie algebras and associated operators
  8. Abelian Lie algebras
  9. Semi-simple Lie algebras
  10. The $\mathfrak{sl}(2,\mathbb{R})$ algebra.
  11. The $\mathfrak{sl}(n+1,\mathbb{R})$ algebra.
  12. The $\mathfrak{so}(3,\mathbb{R})$ algebra.
  13. The $\mathfrak{so}(n,\mathbb{R})$ algebra.
  14. Direct sums of Lie algebras
  15. The algebras $\mathfrak{sl}(2,\mathbb{R}) \oplus (k\mathfrak{n}_{1,1})$.
  16. ...and 16 more sections

Key Result

Theorem 1.1

The operator op1 is Hamiltonian if and only if $\mathcal{A}_1$ is Hamiltonian, $\mathcal{A}_0$ is Hamiltonian, and the following compatibility conditions are satisfied where $\Phi^{i j k}$ is the $(3,0)$-tensor

Theorems & Definitions (38)

  • Theorem 1.1: FerMok1mokhov98:_sympl_poiss
  • Theorem 2.1
  • Definition 2.2
  • Corollary 2.3: Mokhov, mokhov98:_sympl_poiss
  • Remark 2.4
  • Example 2.5
  • Lemma 2.6
  • Theorem 2.7
  • proof
  • Definition 2.8
  • ...and 28 more