A Novel Chain of Lie Algebras and its Coalgebra Symmetry

Abstract We study a novel $$n(n+1)/2$$ n ( n + 1 ) / 2 -dimensional non-semisimple Lie algebra $$\mathfrak {g}_n$$ g n , a generalisation of both $$\mathfrak {sl}_2(\mathbb {K})$$ sl 2 ( K ) and the two-photon Lie algebra $$\mathfrak {h}_6$$ 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 $$n\times n$$ 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 $$n=2$$ n = 2 , quasi-integrable for $$n=3$$ n = 3 , and of Poincaré–Lyapunov–Nekhoroshev type for $$n\ge 4$$ n ≥ 4 .