Geodesic
Hamiltonian algebras
Teoretičeskaâ i matematičeskaâ fizika, Tome 37 (1978) no. 3, pp. 336-346 Cet article a éte moissonné depuis la source Math-Net.Ru

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Suppose we are given a Lie algebra of functions of a finite number of variables of the form $[A(x),B(x)]=\int\widetilde A(k)\widetilde B(p)\exp\{i(k+p)x\}\alpha(k\vert p)dkdp$, where $\widetilde A$ and $\widetilde B$ are the Fourier transforms of $A$ and $B$. Then the function $\alpha$ satisfies the functional equations $\alpha(k_1\vert k_2)\alpha(k_1+k_2\vert k_3)+\alpha(k_2\vert k_3)\alpha(k_2+k_3\vert k_1)+\alpha(k_3\vert k_1)\alpha(k_3+k_1\vert k_2)=0$, $\alpha(k\vert p)=-\alpha(p\vert k)$. All solutions of these equations are found under the assumption that $\frac{\partial^{n}\alpha}{\partial x^n} (x\vert 0)\not\equiv 0$ for some $n$ is $\alpha-n$ times continuously differentiable in some neighborhood of the origin. The obtained solutions give all Lie algebras of this form, in particular all algebras of polynomials. All nearly canonical Hamiltonian algebras [1] are found. 
@article{TMF_1978_37_3_a4,
     author = {G. K. Tolokonnikov},
     title = {Hamiltonian algebras},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {336--346},
     year = {1978},
     volume = {37},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1978_37_3_a4/}
}
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G. K. Tolokonnikov. Hamiltonian algebras. Teoretičeskaâ i matematičeskaâ fizika, Tome 37 (1978) no. 3, pp. 336-346. http://geodesic.mathdoc.fr/item/TMF_1978_37_3_a4/

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