Work

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

Year: 2025

Type: article

Abstract: 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.

Source: Journal of Physics A Mathematical and Theoretical

DOI: 10.1088/1751-8121/addb14

Authors: Giorgio Gubbiotti, Francesco Oliveri, Emanuele Sgroi, Pierandrea Vergallo

Institutions: University of Milan, Istituto Nazionale di Fisica Nucleare, Sezione di Milano, University of Messina

Language: English

Topic: Advanced Topics in Algebra

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,325

Funders: Istituto Nazionale di Alta Matematica "Francesco Severi", European Commission, Istituto Nazionale di Fisica Nucleare, Gruppo Nazionale per la Fisica Matematica