---
abstract: |
    We compute the cohomological space $\mathrm{H}^1_\mathrm{diff}\left(\mathfrak{sl}(2),
    \mathrm{D}_{\lambda,\mu}\right)$ where $\mu\in \mathbb{R}$, $\lambda=(\lambda_1,\dots,\lambda_n)\in\mathbb{R}^n$ and $\mathrm{D}_{\lambda,\mu}$ is the space of multilinear differential operators from $\cF_{\l_1}\otimes\cdots\otimes\cF_{\l_n}$ to $\cF_\mu$. The structure of these spaces was conjectured in \[M. Ben Ammar et al. in International Journal of Geometric Methods in Modern Physics Vol. 9, No. 4 (2012) 1250033 (15 pages).\]
author:
- Mabrouk Ben Ammar
- Maha Boujelben
- 'Rabeb Sidaoui [^1]'
title: ' Cohomology of $\frak {sl}(2)$ acting on the space of $n$-ary differential operators on $\mathbb{R}$'
---

\[firstpage\]

(2000). 53D55

[**Key words** ]{} : Cohomology, Weighted Densities.

Introduction
============

The space of weighted densities of weight $\mu$ on $\mathbb{R}$ (or $\mu$-densities for short), denoted by: $${\cal F}_\mu=\left\{ fdx^{\mu}, ~f\in
C^\infty(\mathbb{R})\right\},\quad \mu\in\mathbb{R},$$is the space of sections of the line bundle $(T^*\mathbb{R})^{\otimes^\mu}.$ The Lie algebra ${\rm Vect}(\mathbb{R})$ of vector fields $X_h=h{d\over
dx}$, where $h\in C^\infty(\mathbb{R})$, acts on ${\cal F}_\mu$ by the [*Lie derivative*]{} $L^\mu$. Alternatively, this action can be written as follows: $$\label{Lie1}X_h\cdot (fdx^{\mu}) =L_{X_h}^\mu(fdx^{\mu}):=(hf'+\mu h'f)dx^\mu,$$ where $f'$, $h'$ are $\frac{df}{dx}$, $\frac{dh}{dx}$. For $\lambda=(\lambda_1,\dots,\lambda_n)\in\mathbb{R}^n$ and $\mu\in\mathbb{R}$ we denote by $\mathrm{D}_{\lambda,\mu}$ the space of multilinear differential operators $A$ from $\cF_{\l_1}\otimes\cdots\otimes\cF_{\l_n}$ to $\cF_\mu$. The Lie algebra ${\rm Vect}(\mathbb{R})$ acts on the space $\mathrm{D}_{\lambda,\mu}$ of these differential operators by: $$\label{Lieder2}X_h\cdot A=L_{X_h}^\mu\circ
A-A\circ L_{X_h}^{\lambda}$$ where $L_{X_h}^{\lambda}$ is the Lie derivative on $\cF_{\l_1}\otimes\cdots\otimes\cF_{\l_n}$ defined by the Leibnitz rule. If we restrict ourselves to the Lie algebra $\frak{ sl}(2)$ which is isomorphic to the Lie subalgebra of ${\rm Vect}(\mathbb{R})$ spanned by $$\left\{X_1,\,X_x,\,X_{x^2}\right\},$$ we have a family of infinite dimensional $\frak{ sl}(2)$-modules still denoted by $\mathrm{D}_{\lambda,\mu}$.

According to Nijenhuis-Richardson [@r1], the space $\mathrm{H}^1\left(\mathfrak{g};\mathrm{End}(V)\right)$ classifies the infinitesimal deformations of a $\mathfrak{g}$-module $V$ and the obstructions to integrability of a given infinitesimal deformation of $V$ are elements of $\mathrm{H}^2\left(\mathfrak{g};\mathrm{End}(V)\right)$. While the spaces $\mathrm{H}^1\left(\mathfrak{g};L(\otimes_s^n V,V)\right)$ appear naturally in the problem of normalization of nonlinear representations of $\mathfrak{g}$ in $V$. To be more precise, let $$T:\mathfrak{g}\rightarrow\sum_{n\geq0}L(\otimes_s^n
V,V),~X\mapsto T_X=\sum T_X^n,$$ be a nonlinear representation of $\mathfrak{g}$ in $V$, that is, $T_{[X,Y]}=[T_X,T_Y]$. In [@abs], it is proved that the representation $T$ is normalized if $T_X^n$ is in a supplementary of $\mathrm{B}^1\left(\mathfrak{g};L(\otimes_s^n V,V)\right)$ in $\mathrm{Z}^1\left(\mathfrak{g};L(\otimes_s^n V,V)\right)$.

In fact if $A$ is a differential operator on the line, $A$ can be viewed as an homomorphism from $\mathcal{F}_\tau$ to $\mathcal{F}_\mu$. If $A$ is with order $n$, we can define its symbol as an element in $\mathcal{S}_\beta^n=\bigoplus_{j=0}^n\mathcal{F}_{\beta-j}$ for $\beta=\tau-\lambda$. If $n$ goes to $+\infty$, the space $\mathcal{S}_\beta=\bigoplus_{j\geq0}\mathcal{F}_{\beta-j}$ appears as the space symbols for all differential operators. The space $\mathrm{H}^1\left(\mathfrak{sl}(2);L(\otimes_s^n
\mathcal{S}_\beta,\mathcal{S}_\beta)\right)$ can be decomposed as a sum of some spaces $\mathrm{H}^1\left(\mathfrak{sl}(2),
\mathrm{D}_{\lambda,\mu}\right)$ with $\lambda=(\lambda_1,\dots,\lambda_n)\in\mathbb{R}^n$. Thus, for $\lambda\in\mathbb{R}^n$, the knowledge of the spaces $\mathrm{H}^1\left(\mathfrak{sl}(2),
\mathrm{D}_{\lambda,\mu}\right)$ is useful to compute the terms $T^n$ of a normalized nonlinear representation $T$ of $\mathfrak{sl}(2)$ in $\mathcal{S}_\beta$.

For $\l\in\mathbb{R}$ the spaces $\mathrm{H}^1\left(\mathfrak{sl}(2),
\mathrm{D}_{\lambda,\mu}\right)$ are computed by Gargoubi [@g] and Lecomte [@lec], and for $\lambda\in \mathbb{R}^2$ the spaces $\mathrm{H}^1_\mathrm{diff}\left(\mathfrak{sl}(2),
\mathrm{D}_{\lambda,\mu}\right)$ are computed by Bouarroudj [@b], while we are interested in this paper to generalize this study to $\lambda\in \mathbb{R}^n$ solving a conjecture given by Ben Ammar et al in [@bjs].

The space $\HH^1_\mathrm{diff}(\frak {sl}(2),\rm{D}_{\lambda,\mu})$
===================================================================

Cohomology 
-----------

We will compute the first cohomology space of $\frak {sl}(2)$ with coefficients in $\rm{D}_{\lambda,\mu}$ where $\lambda\in\mathbb{R}^n$ and $\mu\in\mathbb{R}$. Let us first recall some fundamental concepts from cohomology theory (see, e.g., [@Fu]). Let $\frak{g}$ be a Lie algebra acting on a space $V$ and let $\mathfrak{h}$ be a subalgebra of $\mathfrak{g}$. (If $\frak{h}$ is omitted it assumed to be $\{0\}$.) The space of $\frak h$-relative $n$-cochains of $\frak{g}$ with values in $V$ is the $\frak{g}$-module $$C^n(\frak{g},\frak{h}; V ) := \mathrm{Hom}_{\frak
h}(\Lambda^n(\frak{g}/\frak{h});V).$$ The [*coboundary operator*]{} $ \partial^n: C^n(\frak{g},\frak{h}; V
)\longrightarrow C^{n+1}(\frak{g},\frak{h}; V )$ is a $\frak{g}$-map satisfying $\partial^n\circ\partial^{n-1}=0$. The kernel of $\partial^n$, denoted $Z^n(\mathfrak{g},\frak{h};V)$, is the space of $\frak h$-relative $n$-[*cocycles*]{}, among them, the elements in the range of $\partial^{n-1}$ are called $\frak
h$-relative $n$-[*coboundaries*]{}. We denote $B^n(\mathfrak{g},\frak{h};V)$ the space of $n$-coboundaries.

By definition, the $n^{th}$ $\frak h$-relative cohomology space is the quotient space $$\mathrm{H}^n
(\mathfrak{g},\frak{h};V)=Z^n(\mathfrak{g},\frak{h};V)/B^n(\mathfrak{g},\frak{h};V).$$ We will only need the formula of $\partial^n$ (which will be simply denoted $\partial$) in degrees 0 and 1. For $v \in
C^0(\frak{g},\,\frak{h}; V) =V^{\frak h}$, $$\partial v(g) : =
g\cdot v,$$ where $V^{\frak h}$ is the subspace of $\mathfrak{h}$-invariant elements of $V$. For $ \Upsilon\in C^1(\frak{g}, \frak{h};V )$ and $ g,\,h\in\frak{g}$, $$\partial(\Upsilon)(g,\,h):=
g\cdot \Upsilon(h)-h\cdot \Upsilon(g)-\Upsilon([g,~h]).$$ Here we consider $\mathfrak{g}=\mathfrak{sl}(2)$, $\mathfrak{h}=\mathfrak{aff}(1)$ the subalgebra of $\mathfrak{sl}(2)$ spanned by $\frac{d}{dx}$ and $x\frac{d}{dx}$ and we consider $V=D_{\lambda,\mu}$. We compute $\mathrm{H}^1
(\mathfrak{sl}(2);D_{\lambda,\mu})$ and $\mathrm{H}^1
(\mathfrak{sl}(2),\frak{aff}(1);D_{\lambda,\mu})$.

The spaces $\HH^1_\mathrm{diff}(\frak
{sl}(2),\mathrm{D}_{\lambda,\mu})$ and $\mathrm{H}^1
(\mathfrak{sl}(2),\frak{aff}(1);D_{\lambda,\mu})$
----------------------------------------------------

Let $\mu\in\mathbb{R}$, $\alpha=(\alpha_1\dots,\alpha_n)\in\mathbb{N}^n$ and $\lambda=(\lambda_1\dots,\lambda_n)\in\mathbb{R}^n$, we consider $\delta=\mu-\sum_{i=1}^n\lambda_i$ and $|\alpha|=\sum\alpha_i.$ For $F=f_1\otimes\cdots\otimes f_n\in\mathcal{F}_{\lambda_1}\otimes\cdots \otimes\mathcal{F}_{\lambda_n} $, we define $F^{(\alpha)}:=f_1^{(\alpha_1)}\dots f_n^{(\alpha_n)}.$ We also define $$\alpha^i=(\alpha_1,\dots,\alpha_i+1,\dots, \alpha_n)\quad{ and }\quad \alpha^{-i}=(\alpha_1,\dots,\alpha_i-1,\dots, \alpha_n).$$ Recall that the space $\mathcal{F}_{\lambda_1}\otimes\cdots \otimes\mathcal{F}_{\lambda_n} $ is an $\mathfrak{sl}(2)$-module: $$X_h\cdot F:=L_{X_h}^{\lambda}(F)=\sum_{i=1}^nf_1\otimes\cdots\otimes L_{X_h}^{\lambda_i}(f_i)\otimes\cdots\otimes f_n.$$

The following lemma gives the general form of any 1-cocycle.

\[lem1\] Up to a coboundary, any 1-cocycle $c\in\mathrm{Z}^1_\mathrm{diff}(\frak{sl}(2),\mathrm{D}_{\lambda,\mu})$ can be expressed as follows. For all $F=f_1\otimes\cdots\otimes f_n\in \mathcal{F}_{\lambda_1}\otimes\cdots \otimes\mathcal{F}_{\lambda_n} $ and for all $X_h\in\frak{sl}(2)$: $$\label{cocycle}c(X_h,F) =\sum_\alpha B_\alpha h' F^{(\alpha)}
+\sum_\alpha C_\alpha h''F^{(\alpha)} ,$$ where $B_\alpha$ and $C_\alpha$ are constants satisfying: $$\label{coef}
2(\delta-|\alpha|-1)C_\alpha+\sum_i(\alpha_i+1)(\alpha_i+2\lambda_i)B_{\alpha^i}=0.$$

. Any 1-cocycle on $\frak{sl}(2)$ should retains the following general form: $$c(X_h,F) = \sum_\alpha U_\alpha h F^{(\alpha)}+\sum_\alpha V_\alpha h' F^{(\alpha)}
+\sum_\alpha W_\alpha h''F^{(\alpha)} ,$$ where $U_\alpha$, $V_\alpha$ and $W_\alpha$ are, a priori, functions. First, we prove that the terms in $h$ can be annihilated by adding a coboundary. Let $b : \mathcal{F}_{\lambda_1}\otimes\cdots \otimes\mathcal{F}_{\lambda_n}\rightarrow
\mathcal{F_\mu}$ be a multilinear differential operator defined by $$b(F) =
\sum_\alpha D_\alpha F^{(\alpha)} ,$$ We have $$\label{delta}\begin{array}{lll}
\partial b(X_h,F) &=&h(b(F))'+\mu h'b(F)-b(X_h\cdot F)\\[10pt]
&=& \displaystyle\sum_\alpha D'_\alpha h F^{(\alpha)}+\displaystyle\sum_\alpha (\delta-|\alpha|)D_\alpha h' F^{(\alpha)}\\[10pt]
&-&{1\over2}\displaystyle\sum_{\alpha}\sum_{i=1}^n\alpha_i(\alpha_i+2\lambda_i-1) D_{\alpha}h''F^{(\alpha^{-i})}. \end{array}$$ Note that the terms in $F^{(\alpha^{-i})}$ do not appear for $\alpha_i=0$.

Thus, if $D'_\alpha=U_\alpha$ then $c-\partial b$ does not contain terms in h. So, we can replace $c$ by $c-\partial b$.\
That is, up to a coboundary, any 1-cocycle on $\frak{sl}(2)$ can be expressed as follows: $$c(X_h,F) = \sum_\alpha B_\alpha h' F^{(\alpha)}
+\sum_\alpha C_\alpha h''F^{(\alpha)} ,$$ Now, consider the 1-cocycle condition: $$c([X_{h_1}, X_{h_2}],F)- X_{h_1}\cdot c(X_{h_2}, F)
+X_{h_2}\cdot c(X_{h_1}, F) = 0,$$ where $X_{h_1},\,X_{h_2} \in\frak{sl}(2)$. That is, $$\begin{array}{lll} \displaystyle\sum_\alpha B'_\alpha ( h_1 h'_2-h'_1h_2) F^{(\alpha)}+
\displaystyle\sum_\alpha C'_\alpha ( h_1 h''_2-h''_1h_2) F^{(\alpha)}\\[10pt]
+{1\over2}\displaystyle\sum_\alpha\left(2(\delta-|\alpha|-1)C_\alpha+\sum_{i=1}^n(\alpha_i+1)(\alpha_i+2\lambda_i)B_{\alpha^i}\right)( h'_1 h''_2-h''_1h'_2)F^{(\alpha)}=0.

\end{array}$$ So, for all $\alpha$, we have $
B'_\alpha=C'_\alpha=0$ and $$2(\delta-|\alpha|-1)C_\alpha+\sum_{i=1}^n(\alpha_i+1)(\alpha_i+2\lambda_i)B_{\alpha^i}=0.$$ $\Box$

\[cor\]

-   If $\delta\notin\mathbb{N}$ then ${\rm H}^1_\mathrm{diff}(\frak
    {sl}(2),\mathrm{D}_{\lambda,\mu})=0$.

-   If $\delta=k\in\mathbb{N}$ then, up to a coboundary, any 1-cocycle $c\in\mathrm{Z}^1_\mathrm{diff}(\frak{sl}(2),\mathrm{D}_{\lambda,\mu})$ can be expressed as follows. For all $F=f_1\otimes\cdots\otimes f_n\in \mathcal{F}_{\lambda_1}\otimes\cdots \otimes\mathcal{F}_{\lambda_n} $ and for all $X_h\in\frak{sl}(2)$: $$\label{cocyc}c(X_h,F) =\sum_{|\alpha|=k} B_\alpha h' F^{(\alpha)}+\sum_{|\beta|=k-1} C_\beta h''F^{(\alpha)},$$ where, for any $\alpha$ with $|\alpha|=k-1$, the $B_{\alpha^i}$ are constants satisfying: $$\label{coef1}\sum_i(\alpha_i+1)(\alpha_i+2\lambda_i)B_{\alpha^i}=0.$$

-   If the $B_\alpha$ are not all zero then the cocycles are nontrivial.

. 1) Indeed, according to Lemma \[lem1\], we can easily show the 1-cocycle $c$ defined by (\[cocycle\]) is nothing but the operator $\partial b$ where $$b(F) =
\sum_{\alpha} {B_\alpha\over \delta-|\alpha|} F^{(\alpha)} ,$$

2\) Consider the 1-cocycle $c$ defined by (\[cocycle\]) and consider the operator $\partial b$ where $$b(F) =
\sum_{|\alpha|\neq k}{1\over k-|\alpha|} B_\alpha F^{(\alpha)}.$$ We have $$\begin{array}{lll}\label{coboundary}
\partial b(X_h,F) &=&\displaystyle\sum_{|\alpha|\neq k} B_\alpha h' F^{(\alpha)}-{1\over2}\displaystyle\sum_{|\alpha|\neq k}\sum_{i=1}^n {\alpha_i(\alpha_i+2\lambda_i-1)\over k-|\alpha|} B_\alpha h''F^{(\alpha^{-i})}. \end{array}$$

Let $\beta=\alpha^{-i}$, so, we have $$\alpha=\beta^i,\quad |\beta|\neq k-1,\quad  \beta_j=\alpha_j\quad\text{for}\quad j\neq i\quad\text{and}\quad \beta_i=\alpha_i-1.$$ Therefore $$\begin{array}{lll}\label{coboundary}
\partial b(X_h,F) &=&\displaystyle\sum_{|\alpha|\neq k} B_\alpha h' F^{(\alpha)}-{1\over2}\displaystyle\sum_{|\beta|\neq k-1}\sum_{i=1}^n {(\beta_i+1)(\beta_i+2\lambda_i)\over k-|\beta|-1} B_{\beta^i} h''F^{(\beta)}. \end{array}$$ According to we have $$-{1\over2}\sum_{i=1}^n {(\beta_i+1)(\beta_i+2\lambda_i)\over k-|\beta|-1} B_{\beta^i}=C_\beta$$ Thus, $$\partial b(X_h,F)=\sum_{|\alpha|\neq k} B_\alpha h' F^{(\alpha)}+\sum_{|\alpha|\neq k-1} C_\alpha h''F^{(\alpha)}$$ and then $$(c-\partial b)(X_h,F)=\sum_{|\alpha|=k} B_\alpha h' F^{(\alpha)}+\sum_{|\alpha|=k-1} C_\alpha h''F^{(\alpha)}.$$ In this case becomes: $$\sum_i(\alpha_i+1)(\alpha_i+2\lambda_i)B_{\alpha^i}=0.$$

3\) In fact, for $\delta=\mu-|\lambda|=|\alpha|=k$ there are no terms in $h'F^{(\alpha)}$ in the expression of $\partial b(X_h, F)$ for any $b\in \mathrm{D}_{\lambda, \mu}$ (see ).

$\Box$

Now, we prove that, generically, we can annihilate the term in $h''$ in the expression of the 1-cocycle by adding a coboundary.

\[third\] If $\delta=k\in\mathbb{N}^*$ and $-2\lambda\notin\{0,\,\dots,\,k-1\}^n$ then, any 1-cocycle $c\in\mathrm{Z}^1_\mathrm{diff}(\frak{sl}(2),\mathrm{D}_{\lambda,\mu})$ can be expressed as follows. For all $F\in \mathcal{F}_{\lambda_1}\otimes\cdots \otimes\mathcal{F}_{\lambda_n} $ and for all $X_h\in\frak{sl}(2)$: $$\label{cocycle1}c(X_h,F) =\sum_{|\alpha|=k} B_\alpha h' F^{(\alpha)},$$ where, for $|\alpha|=k-1$, the $B_{\alpha^i}$ are constants satisfying: $$\label{coef2}\sum_i(\alpha_i+1)(\alpha_i+2\lambda_i)B_{\alpha^i}=0.$$

. By Theorem \[cor\], for $\delta=k\in\mathbb{N}$, any 1-cocycle $c$ can be expressed as follows: $$\label{cocycle1}c(X_h,F) =\sum_{|\alpha|=k} B_\alpha h' F^{(\alpha)}+\sum_{|\beta|=k-1} C_\beta h''F^{(\beta)}.$$ Let $b : \mathcal{F}_{\lambda_1}\otimes\cdots \otimes\mathcal{F}_{\lambda_n}\rightarrow
\mathcal{F_\mu}$ be a multilinear differential operator defined by $$b(F) =
\sum_{|\alpha|=k} D_\alpha F^{(\alpha)}.$$ We have $$\begin{array}{lll}\label{coboundary}
\partial b(X_h,F) &=&-{1\over2}\displaystyle\sum_{|\alpha|=k}\sum_{i=1}^n\alpha_i(\alpha_i+2\lambda_i-1) D_{\alpha}h''F^{(\alpha^{-i})}. \end{array}$$

Let $\beta=\alpha^{-i}$, so, we have $$\alpha=\beta^i,\quad |\beta|=k-1,\quad  \beta_j=\alpha_j\quad\text{for}\quad j\neq i\quad\text{and}\quad \beta_i=\alpha_i-1.$$ Therefore, we have $$\begin{array}{lll}\label{coboundary}
\partial b(X_h,F) &=&-{1\over2}\displaystyle\sum_{|\beta|=k-1}\sum_{i=1}^n(\beta_i+1)(\beta_i+2\lambda_i) D_{\beta^i}h''F^{(\beta)}. \end{array}$$

Consider the linear system $$\label{system}{1\over2}\displaystyle\sum_{i=1}^n(\beta_i+1)(\beta_i+2\lambda_i) D_{\beta^i}=C_{\beta}, \quad |\beta|=k-1.$$ The number $N_k$ of unknowns $D_{\beta^i}$ is the cardinal of the set $\{\alpha\in\mathbb{N}^n,\,|\alpha|=k\}$ and the number of equations is the cardinal of the set $\{\alpha\in\mathbb{N}^n,\,|\alpha|=k-1\}$. We prove recurrently that $N_k=\begin{pmatrix}n+k-1\\k\end{pmatrix}$.

We consider the lexicographic order and we denote by $a_1>\,\cdots\,>a_{N_{k-1}}$ the elements of the set $\{\alpha\in\mathbb{N}^n,\,|\alpha|=k-1\}$. So, we choose an order on the unknowns $D_{\beta_i}$ such that the first are those indexed by $a_1^1,\,\dots,a_{N_{k-1}}^1$.

Thus, the matrix of the system has the following form: $$\Lambda=\label{matrix}\begin{pmatrix}&\Lambda_1^{k-1}&\Lambda_2^0&\Lambda_3^0& \dots&\Lambda_n^0&0&\dots&\dots&\dots\\&0&\Lambda_1^{k-2}&0 &\dots&0&\Lambda_2^1&\Lambda_3^0 &\dots&\dots\\&0&0&\Lambda_1^{k-2} &0&\dots &0&\Lambda_2^0&0&\dots \\&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\ &\vdots&\dots&0 &0&\Lambda_1^j&0&\dots&\dots&\dots\\&\vdots&\vdots&\vdots&\ddots&\ddots&\vdots& \vdots&\vdots&\vdots
  \end{pmatrix}$$ where $\Lambda_i^{j}=(j+1)(j+2\lambda_i)$, $j=0,\,\dots,\,k-1$ and $i=1,\,\dots,\,n$.

Without loss of generality, assume that $-2\lambda_1\notin\{0,\,\dots,\,k-1\}$ then $\Lambda_1^{j}$ does never vanish, therefore obviously the system is of rank $N_{k-1}$. Thus, we can choose the operator $b$ such that $(c+\partial b)(X_h,F)$ does not contain terms in $h''$.

For $n=2$, we have: $$\Lambda=\label{matrix2}\begin{pmatrix}&\Lambda_1^{k-1}&\Lambda_2^0&0& \dots&0\\&0&\Lambda_1^{k-2}&\Lambda_2^1 &\ddots&\vdots&\\&\vdots&\ddots&\ddots&\ddots&0\\ &0&\dots &0&\Lambda_1^0&\Lambda_2^{k-1}
  \end{pmatrix}$$ $\Box$

If $\delta=k\geq1$ and $-2\lambda\notin\{0,\,\dots,\,k-1\}^n$ then, we have $$\mathrm{dim}\mathrm{H}^1(\mathfrak{sl}(2),\mathrm{D}_{\lambda,\mu})= \begin{pmatrix}n+k-2\\k\end{pmatrix}.$$ These spaces are spanned by the cocycles: $$\label{cocycle1}c(X_h,F) =\sum_{|\alpha|=k} B_\alpha h' F^{(\alpha)},$$ where, for $|\alpha|=k-1$, the $B_{\alpha^i}$ are constants satisfying: $$\sum_i(\alpha_i+1)(\alpha_i+2\lambda_i)B_{\alpha^i}=0.$$ For $\delta=0$ the space $\mathrm{H}^1(\mathfrak{sl}(2),\mathrm{D}_{\lambda,\mu})$ is one dimensional spanned by the 1-cocycle $C_0$ defined by $$C_0(X_h,f_1\otimes\cdots\otimes f_n)=h'f_1\cdots f_n.$$

For $n=1$ and for $n=2$ we refined the results of Lecomte and Bouarroudj (see [@lec], [@b]).

. According to Theorem \[third\] the space $\mathrm{H}^1(\mathfrak{sl}(2),\mathrm{D}_{\lambda,\mu})$ is isomorphic to the space of solutions of the system of linear equations . The linear system has $ \begin{pmatrix}n+k-2\\k-1\end{pmatrix}$ equations with $\begin{pmatrix}n+k-1\\k\end{pmatrix}$ unknowns $B_{\alpha^i}$. As in Theorem \[third\], we see that this system is with maximal rank, so, we have $$\mathrm{dim}\mathrm{H}^1(\mathfrak{sl}(2),\mathrm{D}_{\lambda,\mu})=\begin{pmatrix}n+k-1\\k\end{pmatrix}- \begin{pmatrix}n+k-2\\k-1\end{pmatrix}=\begin{pmatrix}n+k-2\\k\end{pmatrix}.$$

$\Box$

If $\delta=k\geq0$ and $-2\lambda\notin\{0,\,\dots,\,k-1\}^n$ then, we have $$\mathrm{H}^1(\mathfrak{sl}(2),\mathfrak{aff}(1);\mathrm{D}_{\lambda,\mu})= 0.$$

In fact, in this case, there are no nontrivial 1-cocycles vanishing on $\mathfrak{aff}(1)$.

\[four\] If $\delta=k$ and $-2\lambda\in\{0,\dots,k-1\}^n$ then $$\begin{pmatrix}n+k-2\\k\end{pmatrix}
\leq
\mathrm{dim}\mathrm{H}^1(\mathfrak{sl}(2),\mathrm{D}_{\lambda,\mu})\leq\begin{pmatrix}n+k-2\\k\end{pmatrix}
+2\begin{pmatrix}n+k-r-3\\k-r-1\end{pmatrix}$$ where $r=\max(-2\lambda_i)$.

. Without loss of generality, assume that $r=-2\lambda_1$, then $\Lambda_1^r=0$ and $\Lambda_1^r$ appears $\begin{pmatrix}n+k-r-3\\k-r-1\end{pmatrix}$ times in the matrix $\Lambda$ defined by . So, $$\label{rank}N_{k-1}-\begin{pmatrix}n+k-r-3\\k-r-1\end{pmatrix}\leq \mathrm{rank}(\Lambda)\leq N_{k-1}.$$ Obviously the number $\begin{pmatrix}n+k-r-3\\k-r-1\end{pmatrix}$ is minimal if $r$ is maximal.

Now, any nontrivial cocycle $$c(X_h,F) =\sum_{|\alpha|=k} B_\alpha h' F^{(\alpha)}+\sum_{|\beta|=k-1} C_\beta h''F^{(\beta)}$$ can be decomposed into two cocycles $$\label{c1}c_1(X_h,F) =\sum_{|\alpha|=k} B_\alpha h' F^{(\alpha)}$$ and $$\label{c2}c_2(X_h,F) =\sum_{|\beta|=k-1} C_\beta h''F^{(\beta)},$$ indeed, the coefficients $B_\alpha$ are independent of the coefficients $C_\beta$ (see ). The space of nontrivial cocycles is generated by the system of linear equations , so, it is with dimension $N_k-\mathrm{rank}(\Lambda)$. The space of nontrivial cocycles is managed by the system of linear equations , so, it is with dimension $N_{k-1}-\mathrm{rank}(\Lambda)$.

Thus, $$\mathrm{dim}\mathrm{H}^1(\mathfrak{sl}(2),\mathrm{D}_{\lambda,\mu})=N_k+N_{k-1}-2\mathrm{rank}(\Lambda).$$ We conclude by using . $\Box$

Obviously, the nontrivial cocycles managed by the system of linear equations are $\mathfrak{aff}(1)$-invariant vanishing on $\mathfrak{aff}(1)$, so, we have:

$$\mathrm{dim}\mathrm{H}^1(\mathfrak{sl}(2),\mathfrak{aff}(1);\mathrm{D}_{\lambda,\mu})= N_{k-1}-\mathrm{rank}(\Lambda).$$ These spaces are generated by the cocycles .

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[^1]: Université de Sfax, Faculté des Sciences, Département de Mathématiques, Laboratoire d’Algèbre, Géométrie et Théorie Spectrale (AGTS) LR11ES53, BP 802, 3038 Sfax, Tunisie. E.mail: mabrouk.benammar@fss.rnu.tn
