Filtering bases, cohomology of infinite-dimensional Lie algebras, and Laplace operators
Let \(L_ k(1)=L_ k\) be the Lie algebra of polynomial vector fields on the line over the field of rational numbers, having trivial \(k\)-jet at the origin. The basic result of the paper is the construction of a special filtered basis in the exterior complex of \(L_ k\). This construction gives another proof of L. V. Goncharova's theorem on the cohomology of \(L_ k\) [see \textit{L. V. Goncharova}, ibid. 7, No. 2, 6--14 (1973; Zbl 0284.17006), No. 3, 33--44 (1973; Zbl 0289.57026)]. Such a special filtered basis is also constructed for the simplest Kac-Moody algebra of affine type.
scientific article; zbMATH DE number 3953020