### 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 H^{2} _{diff}(Vect(ℝ),sl[(2);D _{λ,μ}), where H^{i} _{diff} denotes the differential cohomology (i.e., we consider only cochains that are given by differential operators) and where D_{λμ} = Hom _{diff}(F_{λ}, F_{μ}) 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 H^{2}(g; D_{λ,μ}) for g = Vect(ℝ) and sl(2).

Original language | English (US) |
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Pages (from-to) | 112-127 |

Number of pages | 16 |

Journal | Journal of Nonlinear Mathematical Physics |

Volume | 14 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2007 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators.** / Bouarroudj, Sofiane.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators

AU - Bouarroudj, Sofiane

PY - 2007/2/1

Y1 - 2007/2/1

N2 - 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 H2 diff(Vect(ℝ),sl[(2);D λ,μ), where Hi diff 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).

AB - 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 H2 diff(Vect(ℝ),sl[(2);D λ,μ), where Hi diff 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).

UR - http://www.scopus.com/inward/record.url?scp=33947154930&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33947154930&partnerID=8YFLogxK

U2 - 10.2991/jnmp.2007.14.1.9

DO - 10.2991/jnmp.2007.14.1.9

M3 - Article

AN - SCOPUS:33947154930

VL - 14

SP - 112

EP - 127

JO - Journal of Nonlinear Mathematical Physics

JF - Journal of Nonlinear Mathematical Physics

SN - 1402-9251

IS - 1

ER -