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

Abstract

The main topic of this paper is two-fold. First, we compute the first relative cohomology group of the Lie algebra of smooth vector fields on the projective line, Vect(RP 1 ), with coefficients in the space of bilinear differential operators that act on tensor densities, D λ,ν;µ , vanishing on the Lie algebra sl(2, R). Second, we compute the first cohomology group of the Lie algebra sl(2, R) with coefficients in D λ,ν;µ .

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

“…Before starting with the proof proper, we explain our strategy. This method has already been used in [3]. First, we investigate operators that belong to Z 2 (Vect(R), sl (2); D λ,µ ).…”

“…Amazingly, they appear in many contexts, especially in the computation of cohomology (cf. [3,5]). We refer to [18] for their history.…”

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…”

Cohomology of 𝔞𝔣𝔣(1|1) acting on the space of n-ary linear differential operators on S1|1

“…Forλ ∈ R the spaces H 1 diff (sl (2), Dλ ,µ ) are computed by Gargoubi [6] and Lecomte [7], the spaces H 1 diff (Vect(R), sl (2), Dλ ,µ ) are computed by Bouarroudj and Ovsienko [3] and the spaces H 1 diff (Vect(R), Dλ ,µ ) are computed by Feigen and Fuchs [4]. Forλ ∈ R 2 the spaces H 1 diff (sl (2), Dλ ,µ ) are computed by Bouarroudj [2]. Forλ ∈ R 3 the spaces H 1 diff (sl (2), Dλ ,µ ) are computed by O. Basdouri and N. Elamine [1].…”

“…Forλ ∈ R 2 the spaces H 1 diff (sl (2), Dλ ,µ ) are computed by Bouarroudj [2]. Forλ ∈ R 3 the spaces H 1 diff (sl (2), Dλ ,µ ) are computed by O. Basdouri and N. Elamine [1]. In this paper we are interested to compute the spaces H 1 diff (sl (2), Dλ ,µ ) forλ ∈ R n .…”