Projective and Conformal Schwarzian Derivatives and Cohomology of Lie Algebras Vector Fields Related to Differential Operators

Abstract

Let M be either a projective manifold (M, Π) or a pseudo-Riemannian manifold (M, g). We extend, intrinsically, the projective/conformal Schwarzian derivatives that we have introduced recently, to the space of differential operators acting on symmetric contravariant tensor fields of any degree on M. As operators, we show that the projective/conformal Schwarzian derivatives depend only on the projective connection Π and the conformal class [g] of the metric, respectively. Furthermore, we compute the first cohomology group of Vect(M ) with coefficients into the space of symmetric contravariant tensor fields valued into δ-densities as well as the corresponding relative cohomology group with respect to sl(n + 1, R).

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

“…We have just proved that the coefficients of every 2-cocycle is expressed in terms of the two constants c 3,4 and c 4,5 . But this general formula may contain coboundaries.…”

“…According to Proposition 1, the space of solutions is generated by c 3,4 and c 4,5 . Note that the coefficients c 4,i , where i ≥ 6, are zero.…”

“…On the other hand, the constant c 4,5 cannot be eliminated because β 4,5 = 0. It follows that the coefficients of the 2-cocycle are generated by c 4,5 . Therefore the cohomology is onedimensional.…”

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

“…Bouarroudj and Ovsienko [9] computed H 1 diff (vect(1), sl(2); D λ,λ ), and Bouarroudj [8] solved a multi-dimensional version of the same problem on manifolds.…”