## 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 2007 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics