On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators
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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,¬µ), where Hi diff denotes the differential cohomology (i.e., we consider only cochains that are given by differential operators) and where D,¬µ = Homdiff(F, 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 H2 (g; D,¬µ) for g = Vect(R) and sl(2).
- © 2007, the Authors. Published by Atlantis Press.
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TY - JOUR AU - Sofiane Bouarroudj PY - 2007 DA - 2007/02/01 TI - On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators JO - Journal of Nonlinear Mathematical Physics SP - 112 EP - 127 VL - 14 IS - 1 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.2007.14.1.9 DO - 10.2991/jnmp.2007.14.1.9 ID - Bouarroudj2007 ER -