Equivalence of conformal superalgebras to Hamiltonian superoperators (Q2747694)

A notion of a Hamiltonian superoperator was introduced by \textit{Yu. I. Daletsky} [see ``Lie superalgebras in a Hamiltonian operator theory'' in: Nonlinear and turbulent processes in physics, V. E. Zakharov (ed.), Vol. 3, Kiev 1983, 1289-1295 (1984)]. This notion in the paper was not related to some other fields such as mathematical physics. In the present paper the author establishes an equivalence between Hamiltonian superoperators and conformal superalgebras.NEWLINENEWLINENEWLINELet \(\mathcal G \) be a complex Lie superalgebra and \(M\) a \(\mathcal G\)-module. A complex of \(q\)-multilinear forms \(c^q(\mathcal G, M), q\geq 1,\) with a differential \(d:c^q(\mathcal G, M)\to c^{q+1}(\mathcal G, M)\) is constructed. Let \(\Omega\) be a subspace in \(c^1(\mathcal G, M)\) such that \(d(M)\subseteq \Omega\). A linear map \(H:\Omega\to \mathcal G\) is graded if \(H(\Omega)=H(\Omega)_0\oplus H(\Omega)_1\) where \(H(\Omega)_i=H(\Omega)\cap \mathcal G_i\). Suppose that \(H\) is super skew-symmetric that is NEWLINE\[NEWLINE \varphi_1(H\varphi_2)=(-1)^{i_1i_2+1}\varphi_2(H\varphi_1), NEWLINE\]NEWLINE provided \(\varphi_j\in H(\Omega)_{i_j}.\) Then the form \(\omega_h(\varphi_1,\varphi_2)=\varphi_1(H\varphi_2)\) is super skew-symmetric. \(H\) is a Hamiltonian superoperator if \(H(\Omega)\) is a subalgebra in \(\mathcal G\), the left and the right kernels of \(\omega_H\) are graded, and \(d\omega_H\) vanishes on \(H(\Omega)\).NEWLINENEWLINENEWLINESuppose now that \(R\) is a super \(\mathbb C[\partial]\)-module over a superspace \(V\). Let NEWLINE\[NEWLINE Y^+(-,z):V\to \hom(V,R[z^{-1}]) NEWLINE\]NEWLINE be a linear map such that NEWLINE\[NEWLINE Y^+(V_{i_1},z)V_{i_2}\subseteq R_{i_1+i_2}[z^{-1}]z^{-1},\quad i_1,i_2\in \mathbb Z_2. NEWLINE\]NEWLINE Here \(\hom(-,-,)\) means the space of all linear maps. Then \(Y^+(-,z)\) can be extended to a linear map \(Y^+(-,z):R\to \hom(R,R[z^{-1}]z^{-1})\) and a matrix differential operator \(H:\Omega_0\to \mathcal G\) can be introduced. The main result of the paper shows that the triple \((R,\partial, Y^+(-,z))\) forms a conformal superalgebra if and only if \(H\) is a Hamiltonian superoperator.