We study a novel <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n(n+1)/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>-dimensional non-semisimple Lie algebra <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathfrak {g}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">g</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, a generalisation of both <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathfrak {sl}_2(\mathbb {K})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="fraktur">sl</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">K</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and the two-photon Lie algebra <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathfrak {h}_6$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">h</mi> <mn>6</mn> </msub> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, quasi-integrable for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, and of Poincaré–Lyapunov–Nekhoroshev type for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>.

A Novel Chain of Lie Algebras and its Coalgebra Symmetry

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2026年4月20日

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Giorgio Gubbiotti,
Danilo Latini,
Bert van Geemen

摘要

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 $n\times n$ $n \times n$ 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 $n=2$ $n = 2$ , quasi-integrable for $n=3$ $n = 3$ , and of Poincaré–Lyapunov–Nekhoroshev type for $n\ge 4$ $n \geq 4$ .