A Novel Chain of Lie Algebras and its Coalgebra Symmetry

We study a novel n(n+1)/2-dimensional non-semisimple Lie algebra gn, a generalisation of both sl2(K) and the two-photon Lie algebra h6. 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 × 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 ≥ 4.