### Abstract

The space D_{under(λ, -) ; μ}, where under(λ, -) = (λ_{1}, ..., λ_{m}), 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 D_{under(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 D_{under(λ, -) ; μ}^{2} (i.e., the space of second-order operators) are isomorphic to each other, except for a few modules called singular.

Original language | English (US) |
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Pages (from-to) | 1441-1456 |

Number of pages | 16 |

Journal | Journal of Geometry and Physics |

Volume | 57 |

Issue number | 6 |

DOIs | |

State | Published - 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