H. Gargoubi | Searcharxiv

arXiv2001

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

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 independant on the choice of the projective structure. We give explicit formulae of 1-cocycles generating this cohomology group.

math.CV↗

arXiv1996

Space of linear differential operators on the real line as a module over the Lie algebra of vector fields

Let ${\cal D}^k$ be the space of $k$-th order linear differential operators on ${\bf R}$: $A=a_k(x)\frac{d^k}{dx^k}+\cdots+a_0(x)$. We study a natural 1-parameter family of $\Diff(\bf R)$- (and $\Vect(\bf R)$)-modules on ${\cal D}^k$. (To define this family, one considers arguments of differential operators as tensor-densities of degree $λ$.) In this paper we solve the problem of isomorphism between $\Diff(\bf R)$-module structures on ${\cal D}^k$ corresponding to different values of $λ$. The result is as follows: for $k=3$ $\Diff(\bf R)$-module structures on ${\cal D}^3$ are isomorphic to each other for every values of $λ\not=0,\;1,\;{1\over 2},\;{1\over 2}\pm \frac{\sqrt 21}{6}$, in this case there exists a unique (up to a constant) intertwining operator $T:{\cal D}^3\to{\cal D}^3$. In the higher order case $(k\geq 4)$ $\Diff(\bf R)$-module structures on ${\cal D}^k$ corresponding to two different values of the degree: $λ$ and $λ^{\prime}$, are isomorphic if and only if $λ+λ^{\prime}=1$.

dg-ga↗