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Functional Analysis and Its Applications

, Volume 44, Issue 1, pp 4–21 | Cite as

Filtering bases and cohomology of nilpotent subalgebras of the Witt algebra and the algebra of loops in sl 2

  • F. V. Weinstein
Article

Abstract

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.

Keywords

Laplace Operator Spectral Sequence Homology Class Stable Cycle Spectral Resolution 
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Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Institut für AnatomieUniversität BernBernGermany

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