Filtering bases and cohomology of nilpotent subalgebras of the Witt algebra and the algebra of loops in sl 2
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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 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 ℒ L . 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 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 ℒ1.
KeywordsLaplace Operator Spectral Sequence Homology Class Stable Cycle Spectral Resolution
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