A novel chain of Lie algebras and its coalgebra symmetry

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 $n\geq4$.