The space of m-ary differential operators as a module over the Lie algebra of vector fields

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Abstract

The space Dunder(λ, -) ; μ, where under(λ, -) = (λ1, ..., λm), of m-ary differential operators acting on weighted densities is a (m + 1)-parameter family of modules over the Lie algebra of vector fields. For almost all the parameters, we construct a canonical isomorphism between the space Dunder(λ, -) ; μ and the corresponding space of symbols as s l (2)-modules. This yields to the notion of the s l (2)-equivariant symbol calculus for m-ary differential operators. We show, however, that these two modules cannot be isomorphic as s l (2)-modules for some particular values of the parameters. Furthermore, we use the symbol map to show that all modules Dunder(λ, -) ; μ2 (i.e., the space of second-order operators) are isomorphic to each other, except for a few modules called singular.

Original languageEnglish (US)
Pages (from-to)1441-1456
Number of pages16
JournalJournal of Geometry and Physics
Volume57
Issue number6
DOIs
StatePublished - May 1 2007

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Keywords

  • Equivariant quantization
  • Invariant operators
  • Module of differential operators

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Geometry and Topology

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