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Towards the classification of homogeneous third-order Hamiltonian operators

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arxiv 1508.02752 v2 pith:ZWGCJQT4 submitted 2015-08-11 math-ph math.AGmath.DGmath.MPnlin.SI

classification math-phmath.AGmath.DGmath.MPnlin.SI
keywords operatorslambdaclassificationhamiltoniannaturalthird-orderactionalgebraic
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abstract

Let $V$ be a vector space of dimension $n+1$. We demonstrate that $n$-component third-order Hamiltonian operators of differential-geometric type are parametrised by the algebraic variety of elements of rank $n$ in $S^2(\Lambda^2V)$ that lie in the kernel of the natural map $S^2(\Lambda^2V)\to \Lambda^4V$. Non-equivalent operators correspond to different orbits of the natural action of $SL(n+1)$. Based on this result, we obtain a classification of such operators for $n\leq 4$.

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  1. Compatible pairs of Hamiltonian operators of the first and third orders

    nlin.SI 2026-02 conditional novelty 7.0 of 10

    Compatibility of a first-order weakly nonlocal Hamiltonian operator with a third-order Hamiltonian operator is equivalent to algebraic equations, with the first-order metric fixed by a structure formula in terms of Ha...

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