A novel chain of Lie algebras and its coalgebra symmetry

TL;DR

This paper introduces a new class of Lie algebras generalizing known structures, explores their properties, and connects them to integrable Hamiltonian systems with varying degrees of integrability depending on the parameter n.

Contribution

It defines a novel family of Lie algebras, analyzes their structure and Casimir elements, and links them to hierarchies of Hamiltonian systems with integrability properties.

Findings

Existence of a unique non-trivial Casimir polynomial of degree n

Construction of Hamiltonian systems with integrability depending on n

Systems are integrable for n=2, quasi-integrable for n=3, and of Poincaré-Lyapunov-Nekhoroshev type for n≥4

Abstract

We study a novel $n(n+1)/2$-dimensional non-semisimple Lie algebra $\mathfrak{g}_n$, a generalisation of both $\mathfrak{sl}_2(\mathbb{K})$ and the two-photon Lie algebra $\mathfrak{h}_6$. We investigate its properties, including its structure, representations, and its Casimir elements. In particular, we prove that there exists only one non-trivial Casimir polynomial of degree $n$ given by the determinant of an $n\times n$ symmetric matrix. We then associate this Lie algebra to a hierarchy of Hamiltonian systems with integrability properties depending on $n$, and describe their first integrals as sums of squares of linear combinations of the components of the angular momentum. In particular, we obtain that these systems are integrable for $n=2$, quasi-integrable for $n=3$, and of Poincar\'e-Lyapunov-Nekhoroshev type for $n\geq4$.