Projectively Invariant Cocycles of Holomorphic Vector Fields on an Open Riemann Surface

Abstract

Let Σ be an open Riemann surface and Hol(Σ) be the Lie algebra of holomorphic vector fields on Σ. We fix a projective structure (i.e. a local SL 2 (C)−structure) on Σ. We calculate the first group of cohomology of Hol(Σ) with coefficients in the space of linear holomorphic operators acting on tensor densities, vanishing on the Lie algebra sl 2 (C). The result is independent on the choice of the projective structure. We give explicit formulae of 1-cocycles generating this cohomology group.

On sl(2)-equivariant quantizations

“…Recently, they have been intensively studied in a series of papers (see [1,2,3,6,8,11,17,18]). Let D k λ,µ be the space of order k differential operators A : F λ → F µ endowed with the Vect(M )-module structure by the formula…”

“…The vector field d dx spans a commutative Lie algebra isomorphic to so (2). We require that the Lie algebras we are dealing with contain so (2). This property implies that the invariant differential operators we are studying are with constant coefficients (see sec.…”

“…On R, all subalgebras that contain d dx were classified by Lie [20] (see also [9], [23]). Apart from l 0 , they are (for a fixed s ∈ R \ {0}): The Lie subalgebras l n , k 1 and k 2 correspond to various embeddings of sl (2). The vector field d dx spans a commutative Lie algebra isomorphic to so (2).…”

“…Apart from l 0 , they are (for a fixed s ∈ R \ {0}): The Lie subalgebras l n , k 1 and k 2 correspond to various embeddings of sl (2). The vector field d dx spans a commutative Lie algebra isomorphic to so (2). We require that the Lie algebras we are dealing with contain so (2).…”

COHOMOLOGY OF THE VECTOR FIELDS LIE ALGEBRAS ON ℝℙ1 ACTING ON BILINEAR DIFFERENTIAL OPERATORS

“…The computation is based on an old result of Gordan [10] on the classification of sl(2, R)-invariant bilinear differential operators that act on tensor densities. Moreover, the case of a higher-dimensional manifold has been studied in [1,11], and the case of a Riemann surface has been studied in [2]. In this paper, we will compute the first cohomology group H 1 (Vect(RP 1 ), sl(2, R); Hom diff (F λ ⊗ F ν , F µ )).…”