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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 $nleq 4$.
Based on the theory of Poisson vertex algebras we calculate skew-symmetry conditions and Jacobi identities for a class of third-order nonlocal operators of differential-geometric type. Hamiltonian operators within this class are defined by a Monge me
We investigate $n$-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences of lines in
Hamiltonian operators are used in the theory of integrable partial differential equations to prove the existence of infinite sequences of commuting symmetries or integrals. In this paper it is illustrated the new Reduce package cde for computations o
In this paper we study $k$-order homogeneous Rota-Baxter operators with weight $1$ on the simple $3$-Lie algebra $A_{omega}$ (over a field of characteristic zero), which is realized by an associative commutative algebra $A$ and a derivation $Delta$ a
In the paper we study homogeneous Rota-Baxter operators with weight zero on the infinite dimensional simple $3$-Lie algebra $A_{omega}$ over a field $F$ ( $ch F=0$ ) which is realized by an associative commutative algebra $A$ and a derivation $Delta$