We study the cohomology with trivial coefficients of the Lie algebras L k, k ≥ 1, of polynomial vector fields with zero k-jet on the circle and the cohomology of similar subalgebras ℒk of the algebra of polynomial loops with values in sl 2 The main result is a construction of special bases in the exterior complexes of these algebras. Using this construction, we obtain the following results. We calculate the cohomology of L k and ℒL. We obtain formulas in terms of Schur polynomials for cycles representing the homology of these algebras. We introduce "stable” filtrations of the exterior complexes of L k and ℒk, thus generalizing Goncharova's notion of stable cycles for L k, and give a polynomial description of these filtrations. We find the spectral resolutions of the Laplace operators for L 1 and ℒ
