A novel chain of Lie algebras and its coalgebra symmetry
Giorgio Gubbiotti, Danilo Latini, Bert van Geemen
TL;DR
The paper introduces the novel chain of non-semisimple Lie algebras $\mathfrak{g}_n$ with dimension $T_n=\frac{n(n+1)}{2}$, unifying $\mathfrak{sl}_2(\mathbb{K})$ and the two-photon algebra $\mathfrak{h}_6$ and featuring a large center. It proves the existence of a single nontrivial Casimir of degree $n$, realized as $C_n=-\det(M_n)$ with $M_n$ a symmetric $n\times n$ matrix of generators, by a vector-field/coadjoint approach and SL$_n$-invariance. Using Lie–Poisson coalgebra methods, it builds a hierarchy of Hamiltonians $H_n^{(N)}$ with two families of Casimir-derived integrals, yielding integrable dynamics for $n=2$, quasi-integrable for $n=3$, and Poincaré–Lyapunov–Nekhoroshev type behavior for $n\geq 4$. The construction relies on a one-degree-of-freedom realisation and tensoring to $N$ sites, producing explicit left/right Casimirs in terms of angular-momentum-like blocks, and opens avenues for subalgebra methods and higher-rank generalisations.
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 Poincaré-Lyapunov-Nekhoroshev type for $n\geq4$.