### 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) |
---|---|

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 |

### Fingerprint

### Keywords

- Equivariant quantization
- Invariant operators
- Module of differential operators

### ASJC Scopus subject areas

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

### Cite this

**The space of m-ary differential operators as a module over the Lie algebra of vector fields.** / Bouarroudj, Sofiane.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The space of m-ary differential operators as a module over the Lie algebra of vector fields

AU - Bouarroudj, Sofiane

PY - 2007/5/1

Y1 - 2007/5/1

N2 - 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.

AB - 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.

KW - Equivariant quantization

KW - Invariant operators

KW - Module of differential operators

UR - http://www.scopus.com/inward/record.url?scp=33846665882&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846665882&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2006.12.002

DO - 10.1016/j.geomphys.2006.12.002

M3 - Article

AN - SCOPUS:33846665882

VL - 57

SP - 1441

EP - 1456

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

IS - 6

ER -