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

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 whereis 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).

sl(2)-Trivial Deformations of VectPol(R)-Modules of Symbols

“…We need to prove only part iii), for the other statements see, for instance, [11] or [3]. First, we recall that Vect Pol (R) is isomorphic to F −1 as Vect Pol (R)-module.…”

“…was calculated by Bouarroudj [3]. We give explicit expressions of some 2-cocycles that span the cohomology group H 2 (Vect P (R), sl(2); D λ,λ+k ).…”

“…The obstructions to extension of any infinitesimal deformation to a formal one are related to H 2 (g; End(V )). More generally, if h is a subalgebra of g, then the h-relative cohomology space H 1 (g, h; End(V )) measures the infinitesimal deformations that become trivial once the action is restricted to h (h-trivial deformations), while the obstructions to extension of any h-trivial infinitesimal deformation to a formal one are related to H 2 (g, h; End(V )) (see, e.g., [3]). …”

On the First aff(1)-Relative Cohomology of the Lie Algebra of Vector Fields and Differential Operators

“…[3,[5][6][7][8]. First, we classify aff(1)-invariant differential operators, then we isolate among them those that are 1-cocycles.…”

Cohomology of the Lie Superalgebra of Contact Vector Fields on 𝕂1|1 and Deformations of the Superspace of Symbols

“…For motivations, see Bouarroudj's paper [7] of which this work is the most natural superization, other possibilities being cohomology of polynomial versions of various infinite dimensional "stringy" Lie superalgebras (for their list, see [21]). This list contains several infinite series and several exceptional superalgebras, but to consider cohomology relative a "middle" subsuperalgebra similar, in a sense, to sl(2) is only possible when such a subsuperalgebra exists which only happens in a few cases.…”