On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators

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Abstract

Let Vect(ℝ) be the Lie algebra of smooth vector fields on ℝ. The space of symbols Pol(T*ℝ) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(ℝ)-module that becomes trivial once the action is restricted to sl(2) ⊂ Vect(ℝ). The deformations of Pol(T*ℝ), which become trivial once the action is restricted to sl(2) and such that the Vect(ℝ)-action on them is expressed in terms of differential operators, are classified by the elements of the weight basis of H2diff(Vect(ℝ),sl[(2);D λ,μ), where Hidiff denotes the differential cohomology (i.e., we consider only cochains that are given by differential operators) and where Dλμ = Hom diff(Fλ, Fμ) is the space of differential operators acting on weighted densities. The main result of this paper is computation of this cohomology. In addition to relative cohomology, we exhibit 2-cocycles spanning H2(g; Dλ,μ) for g = Vect(ℝ) and sl(2).

Original languageEnglish (US)
Pages (from-to)112-127
Number of pages16
JournalJournal of Nonlinear Mathematical Physics
Volume14
Issue number1
DOIs
StatePublished - Feb 2007

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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