Filtering bases and cohomology of nilpotent subalgebras of Witt and $\widetilde{sl}_2$ Lie algebras

Abstract

We study the cohomology with trivial coefficients of Lie algebras L_k of the polynomial vector fields on the line with zero $k$-jet, (k>=1), and the cohomology of the similar subalgebras {L}_k of the polynomial loops algebra $\widetilde{sl}_2$. In both cases we construct the special bases (filtering bases) in the external complexes of these algebras. A spectral sequence based on this construction allows to completely find the cohomology of L_k and {L}_k. We also apply the filtering bases to find the spectral resolution of the Laplace operators for algebras L_1 and L_0, and obtain explicit formulas for the representing cycles of homologies for algebras L_k and {L}_k by means of the Schur polynomials.