Hamiltonian algebras

G. K. Tolokonnikov

Teoretičeskaâ i matematičeskaâ fizika, Tome 37 (1978) no. 3, pp. 336-346

Résumé

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.