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 TMF, 1978, Volume 37, Number 3, Pages 336–346 (Mi tmf3124)

Hamiltonian algebras

G. K. Tolokonnikov

Abstract: 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.

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English version:
Theoretical and Mathematical Physics, 1978, 37:3, 1057–1065

Bibliographic databases:

Citation: G. K. Tolokonnikov, “Hamiltonian algebras”, TMF, 37:3 (1978), 336–346; Theoret. and Math. Phys., 37:3 (1978), 1057–1065

Citation in format AMSBIB
\Bibitem{Tol78} \by G.~K.~Tolokonnikov \paper Hamiltonian algebras \jour TMF \yr 1978 \vol 37 \issue 3 \pages 336--346 \mathnet{http://mi.mathnet.ru/tmf3124} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=524698} \zmath{https://zbmath.org/?q=an:0401.17005} \transl \jour Theoret. and Math. Phys. \yr 1978 \vol 37 \issue 3 \pages 1057--1065 \crossref{https://doi.org/10.1007/BF01018587} 

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This publication is cited in the following articles:
1. G. K. Tolokonnikov, “Algebras of observables of nearly canonical physical theories. I”, Theoret. and Math. Phys., 60:1 (1984), 690–693
2. G. K. Tolokonnikov, “Algebras of observables of nearly canonical physical theories. II”, Theoret. and Math. Phys., 61:2 (1984), 1072–1077
3. G. K. Tolokonnikov, “On observable algebras of a class of associative mechanical systems”, Theoret. and Math. Phys., 63:2 (1985), 433–439
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