A two-dimensional analogue of the Virasoro algebra (Q1589695)

Let \(M\) be a complex manifold with local coordinates \(z=(z_1,\ldots ,z_n)\). In the paper the cohomology of a Lie algebra \(\text{Vect}_{1,0}(M)\) (\(\text{Vect}_{0,1}(M)\)) consisting of the vector fields of the type \(\sum u_i(z,\overline{z})\frac{\partial }{\partial z_i}\) (resp. \(\sum u_i(z,\overline{z})\frac{\partial }{\partial \overline z_i}\)) where \(u_i(z,\overline z)\in C^{\infty }(M)\) is studied. In the first part of the paper the elementary calculations of \(H^q(W_1)\), \(H^q(\widetilde W_1)\), \(q=0,1,2,3\) where \(\widetilde W_1= W_1\otimes\mathbb C[[t]]\) are given. For the diagonal cohomology using the order filtration of Gelfand and Fuks with respect to \(z\), a spectral sequence with the second term \[ E_2^{p,q}\cong H_{\overline{\partial }}^{-p,0}(M)'\otimes H^q(W_n) \] is constructed. For a compact Riemann surface \(\Sigma \) of genus \(g\) it follows that \(\dim H^2_{\Delta }(\text{Vect}_{1,0}(\Sigma))=g\) with \(\Delta \) denoting the diagonal cohomology. The author shows that there are no contributions to \(H^q(\text{Vect}_{1,0}(\Sigma))\), \(q=0,1,2,3\) from other terms of the diagonal filtration of \(C^*(\text{Vect}_{1,0}(\Sigma))\). Therefore, \(\text{Vect}_{1,0}(\Sigma)\) and analogously \(\text{Vect}_{0,1}(\Sigma)\) have the universal \(g\)-dimensional extension. It follows from the results of Etingof, Frenkel and Khesin that the universal central extension of \(\text{Vect}_{0,1}(\Sigma)\) (resp. \(\text{Vect}_{0,1}(\Sigma)\oplus <\frac{\partial }{\partial z}>\) for \(g=1\)) is a two-dimensional analogue of the Virasoro algebra. A series of conjectures on the cohomology of \(W_1\), \(\widetilde W_1\), \(\text{Vect}_{1,0}(\Sigma)\) is formulated.