Filtering Bases and Cohomology of Nilpotent Subalgebras of the Witt Algebra and the Algebra of Loops in $sl_2$

We study the cohomology with trivial coefficients of the Lie algebras $L_k$, $k\ge 1$, of polynomial vector fields with zero $k$-jet on the circle and the cohomology of similar subalgebras $\mathcal{L}_k$ 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 $\mathcal{L}_k$. 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 $\mathcal{L}_k$, 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 $\mathcal{L}_1$.