Geodesic
Filtering Bases and Cohomology of Nilpotent Subalgebras of the Witt Algebra and the Algebra of Loops in $sl_2$
Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 1, pp. 4-26

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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 $\mathcal{L}_1$.
Keywords: Witt algebra, algebra of loops, marked partitions, filtering basis, Sylvester's identity, Laplace operator. 
F. V. Weinstein. Filtering Bases and Cohomology of Nilpotent Subalgebras of the Witt Algebra and the Algebra of Loops in $sl_2$. Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 1, pp. 4-26. http://geodesic.mathdoc.fr/item/FAA_2010_44_1_a1/
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