Cohomology of the vector fields lie algebras on ℝℙ1 acting on bilinear differential operators

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Abstract

The main topic of this paper is two-fold. First, we compute the first relative cohomology group of the Lie algebra of smooth vector fields on the projective line, Vect(ℝℙ1), with coefficients in the space of bilinear differential operators that act on tensor densities, Dλ,ν;μ, vanishing on the Lie algebra sl(2, ℝ). Second, we compute the first cohomology group of the Lie algebra sl(2, ℝ)with coefficients in Dλ,ν;μ.

Original languageEnglish (US)
Pages (from-to)23-40
Number of pages18
JournalInternational Journal of Geometric Methods in Modern Physics
Volume2
Issue number1
DOIs
StatePublished - Feb 1 2005

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differential operators
homology
algebra
coefficients
tensors

Keywords

  • Cohomology of Lie algebras
  • Differential operators on tensor densities
  • sl(2, ℝ)-invariant operators
  • Transvectants

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)

Cite this

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AB - The main topic of this paper is two-fold. First, we compute the first relative cohomology group of the Lie algebra of smooth vector fields on the projective line, Vect(ℝℙ1), with coefficients in the space of bilinear differential operators that act on tensor densities, Dλ,ν;μ, vanishing on the Lie algebra sl(2, ℝ). Second, we compute the first cohomology group of the Lie algebra sl(2, ℝ)with coefficients in Dλ,ν;μ.

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