Towards the classification of homogeneous third-order Hamiltonian operators

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 $S^2({\Lambda }^{2}V)$ that lie in the kernel of the natural map $S^2({\Lambda }^{2}V)\to {\Lambda }^{4}V$. 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 $n\le 4$.

18 pages; added references, dedication and a few comments

References in corpus (2)

• Projective-geometric aspects of homogeneous third-order Hamiltonian operators
• On the bi-Hamiltonian Geometry of WDVV Equations

Cited by in corpus (10)

• Systems of conservation laws with third-order Hamiltonian structures
• Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures
• Computing with Hamiltonian operators
• Projective geometry of homogeneous second order Hamiltonian operators
• Bi-Hamiltonian structure of the Oriented Associativity Equation
• On a class of third-order nonlocal Hamiltonian operators
• Weakly nonlocal Poisson brackets: tools, examples, computations
• Remarks on the Lagrangian representation of bi-Hamiltonian equations
• Bi-Hamiltonian structures of KdV type, cyclic Frobenius algebrae and Monge metrics
• Bi-Hamiltonian structures of WDVV-type