The Fourth-Linear $\mathfrak{aff}\left(1\right)$-Invariant Differential Operators and the First Cohomology of the Lie Algebra of Vector Fields on ${\mathbb{RP}}^1$

Abstract

In this paper, we denote the Lie algebra of smooth vector fields on ${\mathbb{RP}}^1$ by $\mathcal{V}\left({\mathbb{RP}}^1\right)$. This paper focuses on two parts. In the beginning, we determine the cohomology space of $\mathfrak{aff}\left(1\right)$ with coefficients in ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }$. Afterward, we classify $\mathfrak{aff}\left(1\right)$-invariant fourth-linear differential operators from $\mathcal{V}\left({\mathbb{RP}}^1\right)$ to ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }$ vanishing on $\mathfrak{aff}\left(1\right)$. This result enables us to compute the $\mathfrak{aff}\left(1\right)$-relative cohomology of $\mathcal{V}\left({\mathbb{RP}}^1\right)$ with coefficients in ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }$.

1. Introduction

Throughout this paper, we denote by $\mathbb{N}$ the sets of positive integers. All vector spaces and algebras in this paper are over $\mathbb{R}$. Let $\mathfrak{C}$ be the Lie algebra, $\mathcal{I}$ be the $\mathfrak{C}$-modules, and $\mathfrak{h}$ be the subalgebra of $\mathfrak{C}$. We denote by $\mathcal{LA}$, $\mathcal{DO}$, H${}^1(\mathfrak{C};\mathcal{I})$ and H${}^1(\mathfrak{C},\mathfrak{h};\mathcal{I})$ the Lie algebra, differential operators, first cohomology space of the Lie algebra $\mathfrak{C}$ with coefficients in $\mathcal{I}$ and first $\mathfrak{h}$-relative cohomology of $\mathfrak{C}$ with coefficients in $\mathcal{I}$, respectively. Additionally, we denote by ${\mathrm{Hom}}_{\mathfrak{h}}$ the $\mathfrak{h}$-invariant differential operators.

Let $\mathcal{I}$ and $\mathcal{J}$ be two $\mathfrak{C}$-modules. The nontrivial extensions of $\mathfrak{C}$-modules

$$0⟶\mathcal{I}⟶.⟶\mathcal{J}⟶0$$

are classified by H${}^1(\mathfrak{C};\mathrm{Hom}(\mathcal{I},\mathcal{J}))$ (see [1]). Every 1-cocycle $\mathcal{L}$ generates a new action on $\mathcal{I}\oplus \mathcal{J}$ as follows: for all $c\in \mathfrak{C}$ and for all $(\ell ,k)\in \mathcal{I}\oplus \mathcal{J}$, we define $c^{*}(\ell ,k):=(c^{*}\ell +C^{st}\mathcal{L}\left(k\right),c^{*}k)$. For the space of tensor densities of weight $\alpha \in \mathbb{R}$ in ${\mathcal{F}}_{\alpha }$, viewed as a module over the $\mathcal{LA}$ of smooth vector fields $\mathcal{V}\left({\mathbb{RP}}^1\right)$, the classification of nontrivial extensions

$$0⟶{\mathcal{F}}_{\mu }⟶.⟶{\mathcal{F}}_{\lambda }⟶0,$$

leads, according to Feigin and Fuks in [2], to compute the first H${}^1(\mathcal{V}\left({\mathbb{RP}}^1\right);\mathrm{Hom}({\mathcal{F}}_{\lambda },{\mathcal{F}}_{\mu }))$. Later, Ovsienko and Bouarroudj in [3] calculated

$${\mathrm{H}}^1(\mathcal{V}({\mathbb{RP}}^1,\mathfrak{sl}(2,\mathbb{R});\mathrm{Hom}({\mathcal{F}}_{\lambda },{\mathcal{F}}_{\mu })).$$

The research into the equivariant quantization method described in [4,5] led to the study of H${}^1(\mathcal{V}({\mathbb{RP}}^1,\mathfrak{sl}(2,\mathbb{R});\mathrm{Hom}({\mathcal{F}}_{\lambda },{\mathcal{F}}_{\mu }))$. It was proved that from the space of symbols and to the space of $\mathcal{DO}$, there exists an $\mathfrak{sl}(2,\mathbb{R})$-equivariant quantization map, but this map is not $\mathcal{V}\left({\mathbb{RP}}^1\right)$-equivariant. The impediment here is given by the 1-cocycles that span H${}^1(\mathcal{V}({\mathbb{RP}}^1,\mathfrak{sl}(2,\mathbb{R});\mathrm{Hom}({\mathcal{F}}_{\lambda },{\mathcal{F}}_{\mu }))$ (see [3,6]). The calculation is based on an old result of Gordan [7] on the classification of the bilinear ${\mathrm{Hom}}_{\mathfrak{sl}(2,\mathbb{R})}$ that acts on tensor densities. Moreover, the cases of a higher-dimensional manifold and a Riemann surface have been studied in [8,9,10], respectively.

In this paper, we determine the first H${}^1(\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$, where ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }$ is the space of trilinear $\mathcal{DO}$ acting on weighted densities. We show that H${}^1(\mathfrak{aff}\left(1\right);\mathrm{Hom}({\mathcal{F}}_{\tau }\otimes {\mathcal{F}}_{\lambda }\otimes {\mathcal{F}}_{\mu },{\mathcal{F}}_{\nu }))$ is $\frac{1}{2}(k+1)(k+2)$-dimensional if $k=\nu -\tau -\lambda -\mu \in \mathbb{N}$ and zero-dimensional otherwise. Secondly, we classify fourth-linear ${\mathrm{Hom}}_{\mathfrak{aff}\left(1\right)}$ from $\mathcal{V}\left({\mathbb{RP}}^1\right)$ to ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }$ vanishing on $\mathfrak{aff}\left(1\right)$. We prove that dim${\mathrm{Hom}}_{\mathfrak{aff}\left(1\right)}=\frac{1}{6}k(k-1)(k+1)$, where $k=\nu -\tau -\lambda -\mu +1$. We use the result to compute ${\mathrm{H}}_{\mathrm{diff}}^1\left(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }\right)$. We show that nonzero ${\mathrm{H}}_{\mathrm{diff}}^1\left(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }\right)$ only appear for resonant values of weights that satisfy $\nu -\tau -\lambda -\mu <8$.

These spaces appear naturally in the problem of describing the deformations of the modules ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }$. Precisely, H${}^1(\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ classifies the infinitesimal deformations of a $\mathfrak{aff}\left(1\right)$-module ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }$. Likewise, H${}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ measures the infinitesimal deformations of a $\mathcal{V}\left({\mathbb{RP}}^1\right)$-module ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }$ that become trivial once the action is restricted to $\mathfrak{aff}\left(1\right)$($\mathfrak{aff}\left(1\right)$-trivial deformations).

Our paper is structured as follows. In Section 2, we recall some necessary definitions and preliminary results related to the cohomology space and to the space of multilinear $\mathcal{DO}$. In Section 3, we determine H${}^1(\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$. In Section 4, we classify the fourth-linear ${\mathrm{Hom}}_{\mathfrak{aff}\left(1\right)}$ from $\mathcal{V}\left({\mathbb{RP}}^1\right)$ to ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }$ vanishing on $\mathfrak{aff}\left(1\right)$. In Section 5, we determine ${\mathrm{H}}_{\mathrm{diff}}^1\left(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }\right)$. Finally, in Section 6, we offer our conclusions and describe future research.

2. Preliminaries

In this Section, we recall some necessary definitions and preliminary results related to the cohomology space and to the space of multilinear $\mathcal{DO}$.

Definition 1.

The $\mathcal{LA}$, denoted by $\mathfrak{aff}\left(1\right)$, is realized as a subalgebra of $\mathcal{V}\left({\mathbb{RP}}^1\right)$,

$$\mathfrak{aff}\left(1\right)=span\left(X_1=\frac{d}{dx},X_x=x\frac{d}{dx}\right).$$

(1)

The commutation relations are given by

$$\left[X_x,X_1\right]=-X_1\mathrm{and}\left[X_1,X_1\right]=\left[X_x,X_x\right]=0,$$

where $\left[a,b\right]=a\circ b-b\circ a$.

Definition 2.

The space of sections of the line bundle ${\left(T^{*}{\mathbb{RP}}^1\right)}^{{\otimes }^m}$, denoted by ${\mathcal{F}}_m$, is the space of tensor densities of weight m on ${\mathbb{RP}}^1$. ${\mathcal{F}}_0$ coincides with the space of functions and ${\mathcal{F}}_1$ coincides with the space of differential forms. By the Lie derivative, the $\mathcal{LA}$$\mathcal{V}\left({\mathbb{RP}}^1\right)$ acts on ${\mathcal{F}}_m$. For all $f\in {\mathcal{F}}_m$ and for all $Y\in \mathcal{V}\left({\mathbb{RP}}^1\right)$,

$$L_Y^m\left(f\right)=Y{f}^{\prime }+mf{Y}^{\prime },$$

(2)

where the superscript ${}^{\prime }$ stands for $\frac{d}{dx}$.

2.1. The Space of Trilinear $\mathcal{DO}$ as a $\mathcal{V}\left({\mathbb{RP}}^1\right)$-Module

We consider the trilinear $\mathcal{DO}$ that acts on tensor densities (see [11]):

$$\Psi :{\mathcal{F}}_m\otimes {\mathcal{F}}_n\otimes {\mathcal{F}}_p⟶{\mathcal{F}}_q.$$

(3)

The $\mathcal{LA}$$\mathcal{V}\left({\mathbb{RP}}^1\right)$ acts on the space of trilinear $\mathcal{DO}$ as follows. For all $f\in {\mathcal{F}}_m$, for all $g\in {\mathcal{F}}_n$ and for all $h\in {\mathcal{F}}_p$,

$$L_Y^{m,n,p;q}(\Psi )(f,g,h)=L_Y^q\circ \Psi (f,g,h)-\Psi (L_Y^{m}f,g,h)-\Psi (f,L_Y^{n}g,h)-\Psi (f,g,L_Y^{p}h).$$

(4)

Thus, the space of trilinear $\mathcal{DO}$ is a $\mathcal{V}\left({\mathbb{RP}}^1\right)$-module.

2.2. Cohomology of Lie Algebra

Let $C^n(\mathfrak{C},\mathfrak{h};\mathcal{I}):={\mathrm{Hom}}_{\mathfrak{h}}({\Lambda }^n(\mathfrak{C}/\mathfrak{h});\mathcal{I})$ be the space of $\mathfrak{h}$-relative n-cochains of $\mathfrak{C}$ with values in $\mathcal{I}$(see [1]). We denote by ${\partial }_n$ the coboundary $\mathcal{DO}$ of $C^n(\mathfrak{C},\mathfrak{h};\mathcal{I})$ with value in $C^{n+1}(\mathfrak{C},\mathfrak{h};\mathcal{I})$ that it satisfies ${\partial }_n\circ {\partial }_{n-1}=0$.

By definition, ${\mathrm{H}}^n(\mathfrak{C},\mathfrak{h};\mathcal{I})$ is the quotient space

$${\mathrm{H}}^n(\mathfrak{C},\mathfrak{h};\mathcal{I})=Z^n(\mathfrak{C},\mathfrak{h};\mathcal{I})/B^n(\mathfrak{C},\mathfrak{h};\mathcal{I}),$$

where $Z^n(\mathfrak{C},\mathfrak{h};\mathcal{I})$ is the kernel of ${\partial }_n$ that is called the space of $\mathfrak{h}$-relative n-cocycles, and $B^n(\mathfrak{C},\mathfrak{h};\mathcal{I})$ is the elements in the range of ${\partial }_{n-1}$ that is called the space of $\mathfrak{h}$-relative n-coboundaries. For $n=1$ and for all $\phi \in C^1(\mathfrak{C},\mathfrak{h};\mathcal{I})$,

$$\partial \left(\phi \right)(g,h):=g·\phi \left(h\right)-h·\phi \left(g\right)-\phi \left(\right[g,h\left]\right),\mathrm{for}\mathrm{any}g,h\in \mathfrak{C}.$$

3. ${\mathbf{H}}_{\mathbf{diff}}^{\mathbf{1}}\left(\mathbf{aff}\left(\mathbf{1}\right);{\mathbf{D}}_{\tau ,\lambda ,\mu ;\nu }\right)$

In this section, we will determine ${\mathrm{H}}_{\mathrm{diff}}^1(\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$. In [1,8,10,12,13,14,15], the cohomology computation steps listed below were heavily utilized.

Let $(\nu ,\tau ,\lambda ,\mu )\in {\mathbb{R}}^4$ and $\beta =({\beta }_1,{\beta }_2,{\beta }_3)\in {\mathbb{N}}^3$, we consider $\delta =\nu -\tau -\lambda -\mu$ and $\left|\beta \right|=\sum {\beta }_i.$ For $F=fd{x}^{\tau }\otimes gd{x}^{\lambda }\otimes \phi d{x}^{\mu }\in {\mathcal{F}}_{\tau }\otimes {\mathcal{F}}_{\lambda }\otimes {\mathcal{F}}_{\mu }$, we denote

$$F^{\left(\beta \right)}:=f^{\left({\beta }_1\right)}{g}^{\left({\beta }_2\right)}{\phi }^{\left({\beta }_3\right)}d{x}^{\nu }.$$

Recall that the space ${\mathcal{F}}_{\tau }\otimes {\mathcal{F}}_{\lambda }\otimes {\mathcal{F}}_{\mu }$ is a $\mathcal{V}\left({\mathbb{RP}}^1\right)$-module:

$$X_h·F:=L_{X_h}^{\tau ,\lambda ,\mu }\left(F\right)=L_{X_h}^{\tau }\left(f\right)d{x}^{\tau }\otimes gd{x}^{\lambda }\otimes \phi d{x}^{\mu }+fd{x}^{\tau }\otimes L_{X_h}^{\lambda }\left(g\right)d{x}^{\lambda }\otimes \phi d{x}^{\mu }+fd{x}^{\tau }\otimes gd{x}^{\lambda }\otimes L_{X_h}^{\mu }\left(\phi \right)d{x}^{\mu }.$$

The following theorem is the main result in this section:

Theorem 1.

(1) 

If $\delta \notin \mathbb{N}$ then ${\mathrm{H}}_{\mathrm{diff}}^1(\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })=0$.

(2) 

If $\delta =k\in \mathbb{N}$ then

$$\dim {\mathrm{H}}_{\mathrm{diff}}^1(\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })=\frac{(k+1)(k+2)}{2}.$$

Moreover, ${\mathrm{H}}_{\mathrm{diff}}^1(\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is spanned by the following 1-cocycles:

$${\omega }_j(X_h,G)=h^{\prime }{G}^{\left(j\right)},\left|j\right|=k.$$

To prove this theorem, we need the following lemma.

Lemma 1.

$${\mathrm{H}}_{\mathrm{diff}}^1(\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })={\mathrm{H}}_{\mathrm{diff}}^1(\mathfrak{aff}\left(1\right),X_1;{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }).$$

That is, up to a coboundary, any 1-cocycle is ${\mathrm{Hom}}_{X_1}$.

Proof. 

Any 1-cocycle on $\mathfrak{aff}\left(1\right)$ should retain the following general form:

$$\omega (X_h,G)=\sum _j{N}_{j}h{G}^{\left(j\right)}+\sum _j{M}_j{h}^{\prime }{G}^{\left(j\right)},$$

where $N_j$ and $M_j$ are, a priori, functions. First, we prove that the terms in h can be annihilated by adding a coboundary. Let $c:{\mathcal{F}}_{\tau }\otimes {\mathcal{F}}_{\lambda }\otimes {\mathcal{F}}_{\mu }\to {\mathcal{F}}_{\nu }$ be a 3-ary $\mathcal{DO}$ defined by

$$c\left(G\right)=\sum _i{E}_i{G}^{\left(i\right)},$$

We have

$$\begin{array}{ccc}\partial c(X_h,G)\hfill & =\hfill & h{\left(c\left(G\right)\right)}^{\prime }+\nu h^{\prime }c\left(G\right)-c(X_h·G)\hfill \\ \hfill & =\hfill & {\displaystyle \sum _j{E}_j^{\prime }h{G}^{\left(j\right)}+\sum _j(\delta -|j\left|\right)E_j{h}^{\prime }{G}^{\left(j\right)}}\hfill \end{array}$$

(5)

Thus, if $E_j^{\prime }=N_j$ then $\omega -\partial c$ does not contain terms in h. So, we can replace $\omega$ by $\omega -\partial c$. That is, on $\mathfrak{aff}\left(1\right)$, any 1-cocycle can be expressed up to a coboundary as follows:

$$\omega (X_h,G)=\sum _j{B}_j{h}^{\prime }{G}^{\left(j\right)}.$$

(6)

Now, this $\omega$ is vanishing on $X_1$, and thus, it is $X_1$-invariant. Hence, $B_j^{\prime }=0$ for all j. □

Before proving Theorem 1, we need this proposition.

Proposition 1.

If $\delta \in \mathbb{N}$ then, any 1-cocycle $\omega \in {\mathrm{Z}}_{\mathrm{diff}}^1(\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ can be expressed up to a coboundary, as follows. For all $X_h\in \mathfrak{aff}\left(1\right)$ and for all $G=f_{1}d{x}^{\tau }\otimes gd{x}^{\lambda }\otimes \phi d{x}^{\mu }\in {\mathcal{F}}_{\tau }\otimes {\mathcal{F}}_{\lambda }\otimes {\mathcal{F}}_{\mu }$:

$$\omega (X_h,G)=\sum _{\left|j\right|=\delta }{B}_j{h}^{\prime }{G}^{\left(j\right)},$$

(7)

where the $B_j$ are constants.

Proof. 

Consider the 1-cocycle $\omega$ defined by (6) and consider the operator $\partial c$ where

$$c\left(F\right)=\sum _{\left|j\right|\ne \delta }\frac{B_j}{\delta -\left|j\right|}{F}^{\left(j\right)}.$$

We can easily prove that

$$(\omega -\partial c)(X_h,G)=\sum _{\left|j\right|=\delta }{B}_j{h}^{\prime }{G}^{\left(j\right)}.$$

□

We are now able to prove Theorem 1.

Proof of Theorem 1.

(1) Using Lemma 1, we can easily verify that the 1-cocycle $\omega$ defined by (6) is nothing but the operator $\partial c$ where

$$c\left(G\right)=\sum _j\frac{B_j}{\delta -\left|j\right|}{G}^{\left(j\right)},$$

(2) By Formula (5), $\partial c(X_h,G)$ does not contain any terms in $h^{\prime }{G}^{\left(j\right)}$ for $\left|j\right|=\delta$; therefore, for $\left|j\right|=\delta$, the 1-cocycles ${\omega }_j$ are nontrivial. Thus, according to the Proposition 1, the classes of 1-cocycles ${\omega }_j$ defined by ${\omega }_j(X_h,G)=h^{\prime }{G}^{\left(j\right)}$, where $\left|j\right|=k$ provides a basis of ${\mathrm{H}}_{\mathrm{diff}}^1(\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$. Therefore, $\dim {\mathrm{H}}_{\mathrm{diff}}^1(\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is the cardinal ${\Gamma }_3^k$ of the set $\left\{j\in {\mathbb{N}}^3,\left|j\right|=k\right\}$. Using induction, we prove ${\Gamma }_3^k=\left(\begin{array}{c}3+k-1\\ k\end{array}\right)=\frac{(k+1)(k+2)}{2}$. □

4. Fourth-Linear ${\mathbf{Hom}}_{\mathbf{aff}\left(\mathbf{1}\right)}$

The following computation steps for the relative cohomology have intensively been used in [3,8,10,16,17,18]. First, we classify the fourth-linear ${\mathrm{Hom}}_{\mathfrak{aff}\left(1\right)}$, then we isolate among them those that are 1-cocycles. To do that, we need the following lemma.

Lemma 2.

Any $1-$cocycle vanishing on the subalgebra $\mathfrak{aff}\left(1\right)$ of $\mathcal{V}\left({\mathbb{RP}}^1\right)$ is ${\mathrm{Hom}}_{\mathfrak{aff}\left(1\right)}$.

Proof. 

The 1-cocycle condition of $\Lambda$ reads as follows:

$$X·\Lambda \left(Y\right)-Y·\Lambda \left(X\right)-\Lambda \left(\right[X,Y\left]\right)=0,$$

(8)

where $X,Y\in \mathcal{V}\left({\mathbb{RP}}^1\right)$. Thus, if $\Lambda \left(X\right)=0$ for all $X\in \mathfrak{aff}\left(1\right)$, Equation (8) becomes

$$\Lambda \left(\right[X,Y\left]\right)=X·\Lambda \left(Y\right)$$

expressing the $\mathfrak{aff}\left(1\right)$-invariance property of $\Lambda$. □

As our 1-cocycles vanish on $\mathfrak{aff}\left(1\right)$, we will investigate the linear ${\mathrm{Hom}}_{\mathfrak{aff}\left(1\right)}$ that vanishes on $\mathfrak{aff}\left(1\right)$.

Proposition 2.

There exist fourth-linear ${\mathrm{Hom}}_{\mathfrak{aff}\left(1\right)}$ ${{\rm Y}}_k^{\xi ,\tau ,\lambda ,\mu }:{\mathcal{F}}_{\xi }\otimes {\mathcal{F}}_{\tau }\otimes {\mathcal{F}}_{\lambda }\otimes {\mathcal{F}}_{\mu }⟶$ ${\mathcal{F}}_{\xi +\tau +\lambda +\mu +k}$, given by

$$\begin{array}{c}{\displaystyle {{\rm Y}}_k^{\xi ,\tau ,\lambda ,\mu }(f,\phi ,\varphi ,\psi )=\sum _{i+j+\ell +m=k}{\epsilon }_{i,j,\ell ,m}{f}^{\left(i\right)}{\phi }^{\left(j\right)}{\varphi }^{(\ell )}{\psi }^{\left(m\right)},}\hfill \end{array}$$

(9)

where $i+j+\ell +m=k$ and the coefficients ${\epsilon }_{i,j,\ell ,m}$ are constants.

If ξ, τ, λ and μ are generic, then the space of solutions is $\frac{1}{6}(k+1)(k+2)(k+3)$-dimensional.

Proof. 

Any differential operator ${{\rm Y}}_k^{\xi ,\tau ,\lambda ,\mu }:{\mathcal{F}}_{\xi }\otimes {\mathcal{F}}_{\tau }\otimes {\mathcal{F}}_{\lambda }\otimes {\mathcal{F}}_{\mu }⟶{\mathcal{F}}_{\xi +\tau +\lambda +\mu +k}$ is of the form

$$\begin{array}{c}{\displaystyle {{\rm Y}}_k^{\xi ,\tau ,\lambda ,\mu }(f,\phi ,\varphi ,\psi )=\sum _{i+j+\ell +m=k}{\epsilon }_{i,j,\ell ,m}{f}^{\left(i\right)}{\phi }^{\left(j\right)}{\varphi }^{(\ell )}{\psi }^{\left(m\right)},}\hfill \end{array}$$

(10)

where ${\epsilon }_{i,j,l,m}$ are functions. The $\mathfrak{aff}\left(1\right)$-invariant property of the operators ${{\rm Y}}_k^{\xi ,\tau ,\lambda ,\mu }$ reads as follows:

$$\begin{array}{cc}{L}_X^{\nu }{{\rm Y}}_k^{\xi ,\tau ,\lambda ,\mu }(f,\phi ,\varphi ,\psi )\hfill & ={{\rm Y}}_k^{\xi ,\tau ,\lambda ,\mu }(L_X^{\xi }f,\phi ,\varphi ,\psi )+{{\rm Y}}_k^{\xi ,\tau ,\lambda ,\mu }(f,L_X^{\tau }\phi ,\varphi ,\psi )+{{\rm Y}}_k^{\xi ,\tau ,\lambda ,\mu }(f,\phi ,L_X^{\lambda }\varphi ,\psi )\hfill \\ & +{{\rm Y}}_k^{\xi ,\tau ,\lambda ,\mu }(f,\phi ,\varphi ,L_X^{\mu }\psi ).\hfill \end{array}$$

The invariant property with respect to $X=\frac{d}{dx}$ (respectively $X=x\frac{d}{dx}$) implies that ${\epsilon }_{i,j,\ell }^{\prime }=0$ (respectively $\nu =\xi +\tau +\lambda +\mu +k$). Then the space of solutions is $\frac{1}{6}(k+1)(k+2)(k+3)$-dimensional for $\xi$, $\tau$, $\lambda$ and $\mu$ are generic, spanned by

$$\begin{array}{cc}\hfill & {\epsilon }_{r,s,t,v},wherer+s+t+v=k,0\le r,s,t,v\le k.\hfill \end{array}$$

(11)

□

Proposition 3.

There exist fourth-linear ${\mathrm{Hom}}_{\mathfrak{aff}\left(1\right)}$ ${{\rm Y}}_k^{\tau ,\lambda ,\mu }:\mathcal{V}\left({\mathbb{RP}}^1\right)\otimes {\mathcal{F}}_{\tau }\otimes {\mathcal{F}}_{\lambda }\otimes {\mathcal{F}}_{\mu }⟶{\mathcal{F}}_{\tau +\lambda +\mu +k-1}$ that vanish on $\mathfrak{aff}\left(1\right)$ given by

$$\begin{array}{c}{\displaystyle {{\rm Y}}_k^{\tau ,\lambda ,\mu }(X,\phi ,\varphi ,\psi )=\sum _{i+j+\ell +m=k}{\epsilon }_{i,j,\ell ,m}{X}^{\left(i\right)}{\phi }^{\left(j\right)}{\varphi }^{(\ell )}{\psi }^{\left(m\right)}.}\hfill \end{array}$$

(12)

where $i+j+\ell +m=k$ and the coefficients ${\epsilon }_{i,j,\ell ,m}$ are constants, but ${\epsilon }_{0,j,\ell ,k-j-\ell }={\epsilon }_{1,j,\ell ,k-j-\ell -1}=0$. Moreover, the space of solutions is $\frac{1}{6}k(k-1)(k+1)$-dimensional, for all τ, λ and μ.

Proof. 

The proof follows the proof of Proposition 2 by putting $\xi =-1$. In this case, the space of solutions is $\frac{1}{6}k(k-1)(k+1)$-dimensional, spanned by

$$\begin{array}{cc}\hfill & {\epsilon }_{r,s,t,v},wherer+s+t+v=k,0\le s,t,v\le k-2\mathrm{and}2\le r\le k.\hfill \end{array}$$

(13)

□

5. ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$

In this section, we will compute ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$. The following theorem is our main result:

Theorem 2.

(1) If $\tau +\mu +\lambda =\nu -1$, then

$${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })\simeq \left\{\begin{array}{ccc}\hfill & \mathbb{R}\hfill & \mathit{if}(\tau ,\lambda ,\mu )=(0,0,0),\hfill \\ \hfill & 0\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

(14)

(2) If $\tau +\mu +\lambda =\nu -2$, then

$${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })\simeq \left\{\begin{array}{cc}\mathbb{R}\hfill & \mathit{if}(\tau ,\lambda ,\mu )\in \left\{\begin{array}{c}(0,0,0),(-\frac{1}{2},0,0),\hfill \\ (0,-\frac{1}{2},0),(0,0,-\frac{1}{2}),\hfill \end{array}\right\}\hfill \\ 0& \mathit{otherwise}.\end{array}\right.$$

(15)

(3) If $\tau +\mu +\lambda =\nu -3$, then

$${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })\simeq \left\{\begin{array}{ll}\mathbb{R}\hfill & \mathit{if}(\tau ,\lambda ,\mu )\in \left\{\begin{array}{c}(-\frac{1}{3},0,0),(0,-\frac{1}{3},0),(0,0,-\frac{1}{3}),\hfill \\ (-1,0,0),(0,-1,0),(0,0,-1),\hfill \\ (-\frac{1}{2},-\frac{1}{2},0),(-\frac{1}{2},0,-\frac{1}{2}),(0,-\frac{1}{2},-\frac{1}{2}),\hfill \end{array}\right\}\hfill \\ 0& \mathit{otherwise}.\end{array}\right.$$

(16)

(4) If $\tau +\mu +\lambda =\nu -4$, then

$${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })\simeq \left\{\begin{array}{ll}\mathbb{R}\hfill & \mathit{if}(\tau ,\lambda ,\mu )\in \left\{\begin{array}{c}(-\frac{2}{3},0,0),(0,-\frac{2}{3},0),(0,0,-\frac{2}{3}),\hfill \\ (-\frac{1}{3},-\frac{1}{3},0),(-\frac{1}{3},0,-\frac{1}{3}),(0,-\frac{1}{3},-\frac{1}{3}),\hfill \\ (-\frac{3}{2},0,0),(0,-\frac{3}{2},0),(0,0,-\frac{3}{2}),\hfill \\ (-1,-\frac{1}{2},0),(-1,0,-\frac{1}{2}),(-\frac{1}{2},-1,0),\hfill \\ (-\frac{1}{2},0,-1),(0,-1,-\frac{1}{2}),(0,-\frac{1}{2},-1),\hfill \\ (-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}),\hfill \end{array}\right\}\hfill \\ 0\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

(17)

(5) If $\tau +\mu +\lambda =\nu -5$, then

$${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })\simeq \left\{\begin{array}{ll}\mathbb{R}\hfill & \mathit{if}(\tau ,\lambda ,\mu )\in \left\{\begin{array}{c}(-1,0,0),(0,-1,0),(0,0,-1),\hfill \\ (-\frac{2}{3},-\frac{1}{3},0),(-\frac{2}{3},0,-\frac{1}{3}),(0,-\frac{2}{3},-\frac{1}{3}),\hfill \\ (0,-\frac{1}{3},-\frac{2}{3}),(-\frac{1}{3},0,-\frac{2}{3}),(-\frac{1}{3},-\frac{2}{3},0),\hfill \\ (-2,0,0),(0,-2,0),(0,0,-2),\hfill \\ (-\frac{3}{2},-\frac{1}{2},0),(-\frac{3}{2},0,-\frac{1}{2}),(-\frac{1}{2},-\frac{3}{2},0),\hfill \\ (-\frac{1}{2},0,-\frac{3}{2}),(0,-\frac{3}{2},-\frac{1}{2}),(-1,-1,0),\hfill \\ (-1,0,-1),(0,-1,-1),(-1,-\frac{1}{2},-\frac{1}{2}),\hfill \\ (-\frac{1}{2},-1,-\frac{1}{2}),(-\frac{1}{2},-\frac{1}{2},-1),\hfill \\ (-\frac{1}{3},-\frac{1}{3},-\frac{1}{3}),\hfill \end{array}\right\}\hfill \\ 0\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

(18)

(6) If $\tau +\mu +\lambda =\nu -6$, then

$${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })\simeq \left\{\begin{array}{ll}\mathbb{R}\hfill & \mathit{if}(\tau ,\lambda ,\mu )\in \left\{\begin{array}{c}(-2,-\frac{1}{3},0),(-\frac{2}{3},-\frac{2}{3},0),(0,-2,0),\hfill \\ (-\frac{2}{3},0,-\frac{2}{3}),(-\frac{1}{3},-2,0),(-\frac{1}{3},-\frac{2}{3},-\frac{1}{3}),\hfill \\ (-\frac{1}{3},0,-2),(-\frac{1}{3},-\frac{1}{3},-\frac{2}{3}),(0,-2,-\frac{1}{3}),\hfill \\ (0,-\frac{3}{2},-1),(0,-\frac{1}{3},-2),(-1,-1,-\frac{1}{2}),\hfill \\ (0,-\frac{2}{3},-\frac{2}{3}),(-\frac{1}{2},-\frac{3}{2},-\frac{1}{2}),(-2,-\frac{1}{2},0),\hfill \\ (-2,0,-\frac{1}{2}),(-\frac{3}{2},-1,0),(-\frac{3}{2},-\frac{1}{2},-\frac{1}{2}),\hfill \\ (-\frac{3}{2},0,-1),(-1,-\frac{3}{2},0),(0,0,-2),\hfill \\ (-1,-\frac{1}{2},-1),(-1,0,-\frac{3}{2}),(-\frac{1}{2},-2,0),\hfill \\ (-\frac{5}{2},0,0),(-\frac{1}{2},-1,-1),(0,0,-\frac{5}{2}),\hfill \\ (-\frac{1}{2},0,-2),(0,-\frac{5}{2},0),(0,-2,-\frac{1}{2}),\hfill \\ (-\frac{2}{3},-\frac{1}{3},-\frac{1}{3}),(-\frac{1}{2},-\frac{1}{2},-\frac{3}{2}),\hfill \end{array}\right\}\hfill \\ 0\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

(19)

(7) If $\tau +\mu +\lambda =\nu -7$, then

$${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })\simeq \left\{\begin{array}{ll}\mathbb{R}\hfill & \mathit{if}(\tau ,\lambda ,\mu )\in \left\{\begin{array}{c}(-\frac{1}{3},-\frac{4}{3},0),(-\frac{1}{3},0,-\frac{4}{3}),(0,-\frac{5}{3},0),\hfill \\ (0,0,-\frac{5}{3}),(0,-1,-\frac{2}{3}),(0,-\frac{2}{3},-1),\hfill \\ (0,-\frac{1}{3},-\frac{4}{3}),(-3,0,0),(-\frac{1}{2},0,-\frac{5}{2}),\hfill \\ (0,-3,0),(-\frac{5}{2},-\frac{1}{2},0),(-\frac{5}{2},0,-\frac{1}{2}),\hfill \\ (-\frac{1}{2},-\frac{5}{2},0),(-\frac{1}{2},0,-\frac{5}{2}),(-2,-1,0),\hfill \\ (-2,0,-1),(-1,-2,0),(-1,0,-2),\hfill \\ (0,-2,-1),(-\frac{1}{2},-\frac{1}{2},-2),(-\frac{1}{2},-2,-\frac{1}{2}),\hfill \\ (-2,-\frac{1}{2},-\frac{1}{2}),(-\frac{3}{2},0,-\frac{3}{2}),(-\frac{3}{2},-\frac{3}{2},0),\hfill \\ (0,-\frac{3}{2},-\frac{3}{2}),(-1,-1,-1),(-1,-\frac{3}{2},-\frac{1}{2}),\hfill \\ (-\frac{1}{2},-\frac{3}{2},-1),(-\frac{1}{2},-1,-\frac{3}{2}),(0,-1,-2),\hfill \\ (-\frac{3}{2},-1,-\frac{1}{2}),(-1,-\frac{1}{2},-\frac{3}{2}),(0,0,-3),\hfill \\ (-\frac{3}{2},-\frac{1}{2},-1),\hfill \end{array}\right\}\hfill \\ 0\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

(20)

(8) If $\nu -\tau -\mu -\lambda$ is is not like the above but τ, λ and μ are generic then,

$$dim\left({\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })\right)=0.$$

Before proving Theorem 2, we need the proposition in which we describe the trilinear ${\mathrm{Hom}}_{\mathfrak{aff}\left(1\right)}$ from ${\mathcal{F}}_{\tau }\otimes {\mathcal{F}}_{\lambda }\otimes {\mathcal{F}}_{\mu }$ to ${\mathcal{F}}_{\tau +\lambda +\mu +k}$.

Proposition 4.

There exist trilinear ${\mathrm{Hom}}_{\mathfrak{aff}\left(1\right)}$:

$$\begin{array}{cc}{K}_k^{\tau ,\lambda ,\mu }\hfill & :{\mathcal{F}}_{\tau }\otimes {\mathcal{F}}_{\lambda }\otimes {\mathcal{F}}_{\mu }⟶{\mathcal{F}}_{\tau +\lambda +\mu +k}\hfill \\ & {\displaystyle (\phi ,\varphi ,\psi )⟼K_k^{\tau ,\lambda ,\mu }(\phi ,\varphi ,\psi )=\sum _{i+j+\ell =k}{\gamma }_{i,j,\ell }{\phi }^{\left(i\right)}{\varphi }^{\left(j\right)}{\psi }^{(\ell )},i+j+\ell =k.}\hfill \end{array}$$

(21)

where the coefficients ${\gamma }_{i,j,\ell }$ are constants.

Then the space of solutions is $\frac{1}{2}(k+1)(k+2)$-dimensional if τ, λ and μ are generic.

Proof. 

Any $\mathcal{DO}$ $K_k^{\tau ,\lambda ,\mu }:{\mathcal{F}}_{\tau }\otimes {\mathcal{F}}_{\lambda }\otimes {\mathcal{F}}_{\mu }⟶{\mathcal{F}}_{\tau +\lambda +\mu +k}$ is of the form (9), where ${\gamma }_{i,j,\ell }$ are functions. The $\mathfrak{aff}\left(1\right)$-invariant property of the operator $K_k^{\tau ,\lambda ,\mu }$ reads as follows:

$$L_X^{\nu }{K}_k^{\tau ,\lambda ,\mu }(\phi ,\varphi ,\psi ,)=K_k^{\tau ,\lambda ,\mu }(L_X^{\tau }\phi ,\varphi ,\psi )+K_k^{\tau ,\lambda ,\mu }(\phi ,L_X^{\lambda }\varphi ,\psi )+K_k^{\tau ,\lambda ,\mu }(\phi ,\varphi ,L_X^{\mu }\psi ).$$

(22)

The invariant property with respect to $X=\frac{d}{dx}$ (respectively, $X=x\frac{d}{dx}$) implies that ${\gamma }_{i,j,\ell }^{\prime }=0$ (respectively, $\nu =\tau +\lambda +\mu +k$). Hence, the space of solutions is $\frac{1}{2}(k+1)(k+2)$-dimensional if $\tau$, $\lambda$ and $\mu$ are generic, spanned by

$$\begin{array}{cc}\hfill & {\gamma }_{0,0,k},{\gamma }_{0,1,k-1},\cdots ,{\gamma }_{0,k,0},\hfill \\ \hfill & {\gamma }_{1,0,k-1},{\gamma }_{1,1,k-2},\cdots ,{\gamma }_{1,k-1,0},\hfill \\ \hfill & ⋮\hfill \\ \hfill & {\gamma }_{k-1,0,1},{\gamma }_{k-1,1,0},\hfill \\ \hfill & {\gamma }_{k,0,0}.\hfill \end{array}$$

(23)

□

Proof of Theorem 2.

We continue by performing the three steps to prove Theorem 2:

1.

The dimension of the operator space that satisfies the 1-cocycle condition will be investigated.

2.

We will look into all 1-cocycles that are trivial, specifically, operators with the form

$$L_{X}B,$$

where B is a trilinear $\mathcal{DO}$. As our 1-cocycles vanishes on $\mathfrak{aff}\left(1\right)$, it follows that B coincides with $K_k^{\tau ,\lambda ,\mu }$.

3.

Depending on $\tau$, $\lambda$ and $\mu$ and using Part 1 and Part 2, the dimension of ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);$ ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ will be equal to

$$\begin{array}{c}\dim \left(\mathcal{DO}\mathrm{that}\mathrm{are}1-\mathrm{cocycles}\right)-\dim \left(\mathcal{DO}\mathrm{of}\mathrm{the}\mathrm{form}{L}_X{K}_k^{\tau ,\lambda ,\mu }\right).\hfill \end{array}$$

□

Now, it is obvious that the coboundary $L_X{K}_k^{\tau ,\lambda ,\mu }$ has the following form:

$${\displaystyle L_X{K}_k^{\tau ,\lambda ,\mu }(X,\phi ,\varphi ,\psi )=\sum _{i+j+\ell +m=k+1}{\Xi }_{i,j,\ell ,m}{X}^{\left(i\right)}{\phi }^{\left(j\right)}{\varphi }^{(\ell )}{\psi }^{\left(m\right)},}$$

(24)

where

$${\Xi }_{0,j,\ell ,m}={\Xi }_{1,j,\ell ,m}=0.$$

The following Lemma, which is proved directly, will be useful in the proof of Theorem 2.

Lemma 3.

For $\tau ,\lambda ,\mu \in \mathbb{R}$

$$\begin{array}{cc}{\Xi }_{\alpha ,\beta ,\gamma ,k-\alpha -\beta -\gamma +1}=\hfill & -\left(\left(\genfrac{}{}{0pt}{}{\alpha +\beta -1}{\alpha }\right)+\tau \left(\genfrac{}{}{0pt}{}{\alpha +\beta -1}{\alpha -1}\right))\right){\gamma }_{\alpha +\beta -1,\gamma ,k-\alpha -\beta -\gamma +1}\hfill \\ & -\left(\left(\genfrac{}{}{0pt}{}{\alpha +\gamma -1}{\alpha }\right)+\lambda \left(\genfrac{}{}{0pt}{}{\alpha +\gamma -1}{\alpha -1}\right)\right){\gamma }_{\beta ,\alpha +\gamma -1,k-\alpha -\beta -\gamma +1}\hfill \\ & -\left(\left(\genfrac{}{}{0pt}{}{k-\beta -\gamma }{\alpha }\right)+\mu \left(\genfrac{}{}{0pt}{}{k-\beta -\gamma }{\alpha -1}\right)\right){\gamma }_{\beta ,\gamma ,k-\beta -\gamma },\hfill \end{array}$$

where $\alpha \ge 2$ and $\beta ,\gamma \ge 0$.

Proof. 

By a direct computation. □

We need also the following lemma.

Lemma 4.

Every 1-cocycle on $\mathcal{V}\left({\mathbb{RP}}^1\right)$ with values in ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }$ is differentiable.

Proof. 

See [10]. □

Now, we can prove Theorem (2). By Lemma (4), any 1-cocycle on $\mathcal{V}\left({\mathbb{RP}}^1\right)$ should retain the general form given by

$$t(X,\phi ,\varphi ,\psi )=\sum _{i+j+\ell +m=k}{t}_{i,j,\ell ,m}{X}^{\left(i\right)}{\phi }^{\left(j\right)}{\varphi }^{(\ell )}{\psi }^{\left(m\right)},$$

(25)

where $t_{i,j,\ell ,m}$ are constants. The fact that this 1-cocycle vanishes on $\mathfrak{aff}\left(1\right)$ implies that

$$t_{0,j,\ell ,m}=t_{1,j,\ell ,m}=0.$$

The 1-cocycle condition reads as follows: for all $\phi \in {\mathcal{F}}_{\tau }$, for all $\varphi \in {\mathcal{F}}_{\lambda }$, for all $\psi \in {\mathcal{F}}_{\mu }$ and for all $X\in \mathcal{V}\left({\mathbb{RP}}^1\right)$, one has

$$t([X,Y],\phi ,\varphi ,\psi )-L_X^{\tau ,\lambda ,\mu ;\nu }t(Y,\phi ,\varphi ,\psi )+L_Y^{\tau ,\lambda ,\mu ;\nu }t(X,\phi ,\varphi ,\psi )=0.$$

5.1. The Case where $\tau +\mu +\lambda =\nu -1$

By Proposition 3, the 1-cocycle (25) can be expressed as follows:

$$t(X,\phi ,\varphi ,\psi )=t_{2,0,0,0}{X}^{{}^{″}}\phi \varphi \psi .$$

Directly, it is clear that the 1-cocycle condition is always satisfied. Let us investigate the triviality of this 1-cocycle. A straightforward calculation proves that

$$L_X{K}_1^{\tau ,\lambda ,\mu }={\Xi }_{2,0,0,0}{X}^{{}^{″}}\phi \varphi \psi =-(\tau {\gamma }_{1,0,0}+\lambda {\gamma }_{0,1,0}+\mu {\gamma }_{0,0,1})X^{{}^{″}}\phi \varphi \psi .$$

We have to distinguish two subcases:

• For $(\tau ,\lambda ,\mu )=(0,0,0)$, the coefficient ${\Xi }_{2,0,0,0}$ vanishes, then the coefficient $t_{2,0,0,0}$ cannot be eliminated by adding a coboundary. Hence, ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is of dimension one.
• For $(\tau ,\lambda ,\mu )\ne (0,0,0)$, we have ${\Xi }_{2,0,0,0}\ne 0$, then we can see that the coefficient $t_{2,0,0,0}$ can be eliminated. Hence, ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is trivial.

5.1.1. The Case Where $\tau +\mu +\lambda =\nu -2$

By Proposition 3, the 1-cocycle (25) can be stated as follows:

$$t(X,\phi ,\varphi ,\psi )=t_{3,0,0,0}{X}^{{}^{‴}}\phi \varphi \psi +t_{2,1,0,0}{X}^{{}^{″}}{\phi }^{\prime }\varphi \psi +t_{2,0,1,0}{X}^{{}^{″}}\phi {\varphi }^{\prime }\psi +t_{2,0,0,1}{X}^{{}^{″}}\phi \varphi {\psi }^{\prime }.$$

Directly, it is clear that the 1-cocycle condition is always satisfied. Let us investigate the triviality of this 1-cocycle. A straightforward calculation proves that

$$L_X{K}_2^{\tau ,\lambda ,\mu }={\Xi }_{3,0,0,0}{X}^{{}^{‴}}\phi \varphi \psi +{\Xi }_{2,1,0,0}{X}^{{}^{″}}{\phi }^{\prime }\varphi \psi +{\Xi }_{2,0,1,0}{X}^{{}^{″}}\phi {\varphi }^{\prime }\psi +{\Xi }_{2,0,0,1}{X}^{{}^{″}}\phi \varphi {\psi }^{\prime },$$

where

$$\begin{array}{ccc}{\Xi }_{3,0,0,0}=-\tau {\gamma }_{2,0,0}-\lambda {\gamma }_{0,2,0}-\mu {\gamma }_{0,0,2}\hfill & ,\hfill & {\Xi }_{2,1,0,0}=-(2\tau +1){\gamma }_{2,0,0}-\lambda {\gamma }_{1,1,0}-\mu {\gamma }_{1,0,1},\hfill \\ {\Xi }_{2,0,1,0}=-\tau {\gamma }_{1,1,0}-(2\lambda +1){\gamma }_{0,2,0}-\mu {\gamma }_{0,1,1}\hfill & ,\hfill & {\Xi }_{2,0,0,1}=-\tau {\gamma }_{1,0,1}-\lambda {\gamma }_{0,1,1}-(2\mu +1){\gamma }_{0,0,2}.\hfill \end{array}$$

We have to distinguish five subcases:

• For $(\tau ,\lambda ,\mu )=(0,0,0)$, the coefficient ${\Xi }_{3,0,0,0}$ vanishes, then the coefficient $t_{3,0,0,0}$ cannot be eliminated by adding a coboundary. Furthermore, the coefficients $t_{2,1,0,0}$, $t_{2,0,1,0}$ and $t_{2,0,0,1}$ can be eliminated because ${\Xi }_{2,1,0,0}$, ${\Xi }_{2,0,1,0}$ and ${\Xi }_{2,0,0,1}$ are nonzero. Hence, ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is of dimension one.
• For $(\tau ,\lambda ,\mu )=(-\frac{1}{2},0,0)$, the coefficient ${\Xi }_{2,1,0,0}$ vanishes, then the coefficient $t_{2,1,0,0}$ cannot be eliminated by adding a coboundary. Furthermore, we have that the coefficients ${\Xi }_{3,0,0,0}$, ${\Xi }_{2,0,1,0}$ and ${\Xi }_{2,0,0,1}$ are nonzero, then the coefficients $t_{3,0,0,0}$, $t_{2,0,1,0}$ and $t_{2,0,0,1}$ can be eliminated. Hence, ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is of dimension one.
• For $(\tau ,\lambda ,\mu )=(0,-\frac{1}{2},0)$, the coefficient ${\Xi }_{2,0,1,0}$ vanishes, then the coefficient $t_{2,0,1,0}$ cannot be eliminated by adding a coboundary. Furthermore, we have that the coefficients ${\Xi }_{3,0,0,0}$, ${\Xi }_{2,1,0,0}$ and ${\Xi }_{2,0,0,1}$ are nonzero, and then the coefficients $t_{3,0,0,0}$, $t_{2,1,0,0}$ and $t_{2,0,0,1}$ can be eliminated. Hence, ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is of dimension one.
• For $(\tau ,\lambda ,\mu )=(0,0,-\frac{1}{2})$, the coefficient ${\Xi }_{2,0,0,1}$ vanishes, then the coefficient $t_{2,0,0,1}$ cannot be eliminated by adding a coboundary. Furthermore, we have the coefficients ${\Xi }_{3,0,0,0}$, ${\Xi }_{2,1,0,0}$ and ${\Xi }_{2,0,1,0}$ are nonzero, and then the coefficients $t_{3,0,0,0}$, $t_{2,1,0,0}$ and $t_{2,0,1,0}$ can be eliminated. Hence, ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is of dimension one.
• For $(\tau ,\lambda ,\mu )\ne (0,0,0),(-\frac{1}{2},0,0),(0,-\frac{1}{2},0),(0,0,-\frac{1}{2})$, ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is trivial since the coefficients $t_{3,0,0,0}$, $t_{2,1,0,0}$, $t_{2,0,1,0}$ and $t_{2,0,0,1}$ can be eliminated because ${\Xi }_{3,0,0,0}$, ${\Xi }_{2,1,0,0}$, ${\Xi }_{2,0,1,0}$ and ${\Xi }_{2,0,0,1}$ are nonzero.

5.1.2. The Case where $\nu -\tau -\lambda -\mu \ge 3$

The 1-cocycle condition is equivalent to the system

$$\begin{array}{cc}\hfill & \left(\left(\genfrac{}{}{0pt}{}{m+q-1}{m}\right)+\tau \left(\genfrac{}{}{0pt}{}{m+q-1}{m-1}\right)\right)t_{p,m+q-1,i,j}-\left(\left(\genfrac{}{}{0pt}{}{p+q-1}{p}\right)+\tau \left(\genfrac{}{}{0pt}{}{p+q-1}{p-1}\right)\right)t_{m,p+q-1,i,j}\hfill \\ \hfill & +\left(\left(\genfrac{}{}{0pt}{}{m+i-1}{m}\right)+\lambda \left(\genfrac{}{}{0pt}{}{m+i-1}{m-1}\right)\right)t_{p,q,m+i-1,j}-\left(\left(\genfrac{}{}{0pt}{}{p+i-1}{p}\right)+\lambda \left(\genfrac{}{}{0pt}{}{p+i-1}{p-1}\right)\right)t_{m,q,p+i-1,j}\hfill \\ \hfill & +\left(\left(\genfrac{}{}{0pt}{}{m+j-1}{m}\right)+\mu \left(\genfrac{}{}{0pt}{}{m+j-1}{m-1}\right)\right)t_{p,q,i,m+j-1}-\left(\left(\genfrac{}{}{0pt}{}{p+j-1}{p}\right)+\mu \left(\genfrac{}{}{0pt}{}{p+j-1}{p-1}\right)\right)t_{m,q,i,p+j-1}\hfill \\ \hfill & +\left(\left(\genfrac{}{}{0pt}{}{m+p-1}{m}\right)-\left(\genfrac{}{}{0pt}{}{m+p-1}{m-1}\right)\right)t_{m+p-1,q,i,j}=0,\hfill \end{array}$$

(26)

where $m+p+q+i+j=k+1$ and $m>p\ge 2$, obtained from the coefficient of $X^{\left(m\right)}{Y}^{\left(p\right)}{\phi }^{\left(q\right)}{\varphi }^{\left(i\right)}{\psi }^{\left(j\right)}$.

By a simple computation, we can deduce this system. There is, of course, at least one solution to such a system, where the solutions $t_{r,s,u,v}$ are exactly the coefficients ${\Xi }_{r,s,u,v}$ of the coboundaries (24).

The case when $\tau +\mu +\lambda =\nu -3$

By Proposition 3, the space of solutions is spanned by

$$t_{4,0,0,0},t_{3,1,0,0},t_{3,0,1,0},t_{3,0,0,1},t_{2,2,0,0},t_{2,1,1,0},t_{2,1,0,1},t_{2,0,2,0},t_{2,0,1,1},t_{2,0,0,2}.$$

Furthermore, we can construct the following equation using formula (26):

$$-2t_{4,0,0,0}+\tau t_{2,2,0,0}-\tau t_{3,1,0,0}+\lambda t_{2,0,2,0}-\lambda t_{3,0,1,0}+\mu t_{2,0,0,2}-\mu t_{3,0,0,1}=0.$$

Thus, we just showed that the coefficients of each 1-cocycle are expressed in terms of

$$t_{3,1,0,0},t_{3,0,1,0},t_{3,0,0,1},t_{2,2,0,0},t_{2,1,1,0},t_{2,1,0,1},t_{2,0,2,0},t_{2,0,1,1},t_{2,0,0,2}.$$

With a direct computation, we confirm that the coefficients of $L_X{K}_3^{\tau ,\lambda ,\mu }$ are expressed in terms of

$$\begin{array}{ccc}{\Xi }_{3,1,0,0}=-(3\tau +1){\gamma }_{3,0,0}-\lambda {\gamma }_{1,2,0}-\mu {\gamma }_{1,0,2}\hfill & ,\hfill & {\Xi }_{3,0,1,0}=-\tau {\gamma }_{2,1,0}-(3\lambda +1){\gamma }_{0,3,0}-\mu {\gamma }_{0,1,2},\hfill \\ {\Xi }_{3,0,0,1}=-\tau {\gamma }_{2,0,1}-\lambda {\gamma }_{0,2,1}-(3\mu +1){\gamma }_{0,0,3}\hfill & ,\hfill & {\Xi }_{2,0,1,1}=-\tau {\gamma }_{1,1,1}-(2\lambda +1){\gamma }_{0,2,1}-(2\mu +1){\gamma }_{0,1,2},\hfill \\ {\Xi }_{2,0,2,0}=-\tau {\gamma }_{1,2,0}-3(\lambda +1){\gamma }_{0,3,0}-\mu {\gamma }_{0,2,1}\hfill & ,\hfill & {\Xi }_{2,1,0,1}=-(2\tau +1){\gamma }_{2,0,1}-\lambda {\gamma }_{1,1,1}-(2\mu +1){\gamma }_{1,0,2},\hfill \\ {\Xi }_{2,0,0,2}=-\tau {\gamma }_{1,0,2}-\lambda {\gamma }_{0,1,2}-3(\mu +1){\gamma }_{0,0,3}\hfill & ,\hfill & {\Xi }_{2,1,1,0}=-(2\tau +1){\gamma }_{2,1,0}-(2\lambda +1){\gamma }_{1,2,0}-\mu {\gamma }_{1,1,1},\hfill \\ {\Xi }_{2,2,0,0}=-3(\tau +1){\gamma }_{3,0,0}-\lambda {\gamma }_{2,1,0}-\mu {\gamma }_{2,0,1}.\hfill & \hfill \end{array}$$

Then, for $(\tau ,\lambda ,\mu )\in \left\{(-\frac{1}{3},0,0),(0,-\frac{1}{3},0),(0,0,-\frac{1}{3}),(-1,0,0),(0,-1,0),(0,0,-1),\right.$$\left.(-\frac{1}{2},-\frac{1}{2},0),(-\frac{1}{2},0,-\frac{1}{2}),(0,-\frac{1}{2},-\frac{1}{2})\right\}$, only one of the coefficients $t_{3,1,0,0},t_{3,0,1,0},t_{3,0,0,1},t_{2,2,0,0},$$t_{2,0,2,0},t_{2,0,0,2},t_{2,1,1,0},t_{2,1,0,1},$ or $t_{2,0,1,1}$ cannot be eliminated by adding a coboundary, so ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),$$\mathfrak{aff}\left(1\right);$${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is of dimension one. While $\tau$, $\lambda$ and $\mu$ are not like the above, ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);$ ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is trivial since the coefficients $t_{3,1,0,0},t_{3,0,1,0},t_{3,0,0,1},t_{2,2,0,0},t_{2,0,2,0},t_{2,0,0,2},t_{2,1,1,0},t_{2,1,0,1},$ and $t_{2,0,1,1}$ can be eliminated because ${\Xi }_{3,1,0,0},{\Xi }_{3,0,1,0},{\Xi }_{3,0,0,1},{\Xi }_{2,2,0,0},{\Xi }_{2,0,2,0},{\Xi }_{2,0,0,2},{\Xi }_{2,1,1,0},{\Xi }_{2,1,0,1},$ and ${\Xi }_{2,0,1,1}$ are nonzero.

The case when $\tau +\mu +\lambda =\nu -4$

By Proposition 3, the space of solutions is spanned by

$$t_{n,m,p,q},\mathrm{where}n+m+p+q=5,0\le m,p,q\le 3\mathrm{and}2\le n\le 5.$$

Furthermore, we can construct the following system using formula (26):

$$\left\{\begin{array}{cc}(3\tau +1)t_{2,3,0,0}-(2\tau +1)t_{3,2,0,0}+\lambda t_{2,1,2,0}-\lambda t_{3,1,1,0}+\mu t_{2,1,0,2}-\mu t_{3,1,0,1}=2t_{4,1,0,0},\hfill & \\ \tau t_{2,2,1,0}-\tau t_{3,1,1,0}+(3\lambda +1)t_{2,0,3,0}-(2\lambda +1)t_{3,0,2,0}+\mu t_{2,0,1,2}-\mu t_{3,0,1,1}=2t_{4,0,1,0},\hfill & \\ \tau t_{2,2,0,1}-\tau t_{3,1,0,1}+\lambda t_{2,0,2,1}-\lambda t_{3,0,1,1}+(3\mu +1)t_{2,0,0,3}-(2\mu +1)t_{3,0,0,2}=2t_{4,0,0,1},\hfill & \\ \tau t_{2,3,0,0}-\tau t_{4,1,0,0}+\lambda t_{2,0,3,0}-\tau t_{4,0,1,0}+\mu t_{2,0,0,3}-\mu t_{4,0,0,1}=5t_{5,0,0,0}.\hfill \end{array}\right.$$

With a direct computation, we confirm that the coefficients of $L_X{K}_4^{\tau ,\lambda ,\mu }$ are expressed in terms of

$$\begin{array}{ccc}{\Xi }_{3,1,1,0}=-(3\tau +1){\gamma }_{3,1,0}-(3\lambda +1){\gamma }_{1,3,0}-\mu {\gamma }_{1,1,2},\hfill & \hfill & {\Xi }_{3,2,0,0}=-2(3\tau +2){\gamma }_{4,0,0}-\lambda {\gamma }_{2,2,0}-\mu {\gamma }_{2,0,2},\hfill \\ {\Xi }_{3,1,0,1}=-(3\tau +1){\gamma }_{3,0,1}-\lambda {\gamma }_{1,2,1}-(3\mu +1){\gamma }_{1,0,3},\hfill & \hfill & {\Xi }_{3,0,2,0}=-\tau {\gamma }_{2,2,0}-2(3\lambda +2){\gamma }_{0,4,0}-\mu {\gamma }_{0,2,2},\hfill \\ {\Xi }_{3,0,1,1}=-\tau {\gamma }_{2,1,1}-(3\lambda +1){\gamma }_{0,3,1}-(3\mu +1){\gamma }_{1,0,3},\hfill & \hfill & {\Xi }_{3,0,0,2}=-\tau {\gamma }_{2,0,2}-\lambda {\gamma }_{0,2,2}-2(3\mu +2){\gamma }_{0,0,4},\hfill \\ {\Xi }_{2,2,1,0}=-3(\tau +1){\gamma }_{3,1,0}-(2\lambda +1){\gamma }_{2,2,0}-\mu {\gamma }_{2,1,1},\hfill & \hfill & {\Xi }_{2,3,0,0}=-2(2\tau +3){\gamma }_{4,0,0}-\lambda {\gamma }_{3,1,0}-\mu {\gamma }_{3,0,1},\hfill \\ {\Xi }_{2,0,2,1}=-\tau {\gamma }_{1,2,1}-3(\lambda +1){\gamma }_{0,3,1}-(2\mu +1){\gamma }_{0,2,2},\hfill & \hfill & {\Xi }_{2,0,3,0}=-\tau {\gamma }_{1,3,0}-2(2\lambda +3){\gamma }_{0,4,0}-\mu {\gamma }_{0,3,1},\hfill \\ {\Xi }_{2,2,0,1}=-3(\tau +1){\gamma }_{3,0,1}-\lambda {\gamma }_{2,1,1}-(2\mu +1){\gamma }_{2,0,2},\hfill & \hfill & {\Xi }_{2,0,0,3}=-\tau {\gamma }_{1,0,3}-\lambda {\gamma }_{0,1,3}-2(2\mu +3){\gamma }_{0,0,4},\hfill \\ {\Xi }_{2,1,0,2}=-(2\tau +1){\gamma }_{2,0,2}-\lambda {\gamma }_{1,1,2}-3(\mu +1){\gamma }_{1,0,3},\hfill & \hfill & \\ {\Xi }_{2,0,1,2}=-\tau {\gamma }_{1,1,2}-(2\lambda +1){\gamma }_{0,2,2}-3(\mu +1){\gamma }_{0,1,3},\hfill & \hfill & \\ {\Xi }_{2,1,2,0}=-(2\tau +1){\gamma }_{2,2,0}-3(\lambda +1){\gamma }_{1,3,0}-\mu {\gamma }_{1,2,1},\hfill & \hfill & \\ {\Xi }_{2,1,1,1}=-(2\tau +1){\gamma }_{2,1,1}-(2\lambda +1){\gamma }_{1,2,1}-(2\mu +1){\gamma }_{1,1,2}.\hfill & \hfill \end{array}$$

Hence, in the same way as the previous cases, we prove that ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);$${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is of dimension one for $(\tau ,\lambda ,\mu )\in \{(-\frac{2}{3},0,0),(0,-\frac{2}{3},0),(0,0,-\frac{2}{3}),(-\frac{1}{3},-\frac{1}{3},0),$ $(-\frac{1}{3},0,-\frac{1}{3}),(0,-\frac{1}{3},-\frac{1}{3}),.$ $.(-\frac{3}{2},0,0),(0,-\frac{3}{2},0),(0,0,-\frac{3}{2}),(-1,-\frac{1}{2},0),(-1,0,-\frac{1}{2}),(-\frac{1}{2},-1,0),$ $(-\frac{1}{2},0,-1),(0,-1,-\frac{1}{2}),.$ $\left.(0,-\frac{1}{2},-1),(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2})\right\}$. While $\tau$, $\lambda$ and $\mu$ are not like the above, ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is trivial.

The case when $\tau +\mu +\lambda =\nu -5$

By Proposition 3, the space of solutions is spanned by

$$t_{n,m,p,q},\mathrm{where}n+m+p+q=6,0\le m,p,q\le 4\mathrm{and}2\le n\le 6.$$

Furthermore, we can construct the following system using formula (26):

$$\left\{\begin{array}{cc}\tau t_{2,4,0,0}-\tau t_{5,1,0,0}+\lambda t_{2,0,4,0}-\lambda t_{5,0,1,0}+\mu t_{2,0,0,4}-\mu t_{5,0,0,1}=9t_{6,0,0,0},\hfill & \\ \tau t_{3,3,0,0}-\mu t_{4,2,0,0}+\lambda t_{3,0,3,0}-\mu t_{4,0,2,0}+\mu t_{3,0,0,3}-\mu t_{4,0,0,2}=5t_{6,0,0,0},\hfill & \\ (4\tau +1)t_{2,4,0,0}-(2\tau +1)t_{4,2,0,0}+\lambda t_{2,1,3,0}-\lambda t_{4,1,1,0}+\mu t_{2,1,0,3}-\mu t_{4,1,0,1}=5t_{5,1,0,0},\hfill & \\ \tau t_{2,3,1,0}-\tau t_{4,1,1,0}+(4\lambda +1)t_{2,0,4,0}-(2\lambda +1)t_{4,0,2,0}+\mu t_{2,0,1,3}-\mu t_{4,0,1,1}=5t_{5,0,1,0},\hfill & \\ \tau t_{2,3,0,1}-\tau t_{4,1,0,1}+\lambda t_{2,0,3,1}-\lambda t_{4,0,1,1}+(4\mu +1)t_{2,0,0,4}-(2\mu +1)t_{4,0,0,2}=5t_{5,0,0,1},\hfill & \\ 2(3\tau +2)t_{2,4,0,0}-3(\tau +1)t_{3,3,0,0}+\lambda t_{2,2,2,0}-\lambda t_{3,2,1,0}+\mu t_{2,2,0,2}-\mu t_{3,2,0,1}=2c_{4,2,0,0},\hfill & \\ \tau t_{2,2,2,0}-\tau t_{3,1,2,0}+2(3\lambda +2)t_{2,0,4,0}-3(\lambda +1)t_{3,0,3,0}+\mu t_{2,0,2,2}-\mu t_{3,0,2,1}=2t_{4,0,2,0},\hfill & \\ \tau t_{2,2,0,2}-\tau t_{3,1,0,2}+\lambda t_{2,0,2,2}-\lambda t_{3,0,1,2}+2(3\mu +2)t_{2,0,0,4}-3(\mu +1)t_{3,0,0,3}=2c_{4,0,0,2},\hfill & \\ (3\tau +1)t_{2,3,1,0}-(2\tau +1)t_{3,2,1,0}+(3\lambda +1)t_{2,1,3,0}-(2\lambda +1)t_{3,1,2,0}+\mu t_{2,1,1,2}-\mu t_{3,1,1,1}=2c_{4,1,1,0},\hfill & \\ (3\tau +1)t_{2,3,0,1}-(2\tau +1)t_{3,2,0,1}+\lambda t_{2,1,2,1}-\lambda t_{3,1,1,1}+(3\mu +1)t_{2,1,0,3}-(2\mu +1)t_{3,1,0,2}=2t_{4,1,0,1},\hfill & \\ \tau t_{2,2,1,1}-\tau t_{3,1,1,1}+(3\lambda +1)t_{2,0,3,1}-(2\lambda +1)t_{3,0,2,1}+(3\mu +1)t_{2,0,1,3}-(2\mu +1)t_{3,0,1,2}=2t_{4,0,1,1}.\hfill \end{array}\right.$$

With a direct computation, we confirm that the coefficients of $L_X{K}_5^{\tau ,\lambda ,\mu }$ are expressed in terms of

$$\begin{array}{ccc}{\Xi }_{2,4,0,0}=-5(\tau +2){\gamma }_{5,0,0}-\lambda {\gamma }_{4,1,0}-\mu {\gamma }_{4,0,1}\hfill & ,\hfill & {\Xi }_{2,2,2,0}=-3(\tau +1){\gamma }_{3,2,0}-3(\lambda +1){\gamma }_{2,3,0}-\mu {\gamma }_{2,2,1},\hfill \\ {\Xi }_{2,0,4,0}=-\tau {\gamma }_{1,4,0}-5(\lambda +2){\gamma }_{0,5,0}-\mu {\gamma }_{0,4,1}\hfill & ,\hfill & {\Xi }_{2,0,3,1}=-\tau {\gamma }_{1,3,1}-2(2\lambda +3){\gamma }_{0,4,1}-(2\mu +1){\gamma }_{0,3,2},\hfill \\ {\Xi }_{2,0,0,4}=-\tau {\gamma }_{1,0,4}-\lambda {\gamma }_{0,1,4}-5(\mu +2){\gamma }_{0,0,5}\hfill & ,\hfill & {\Xi }_{2,1,0,3}=-(2\tau +1){\gamma }_{2,0,3}-\lambda {\gamma }_{1,1,3}-2(2\mu +3){\gamma }_{1,0,4},\hfill \\ {\Xi }_{3,3,0,0}=-10(\tau +1){\gamma }_{5,0,0}-\lambda {\gamma }_{3,2,0}-\mu {\gamma }_{3,0,2}\hfill & ,\hfill & {\Xi }_{2,1,3,0}=-(2\tau +1){\gamma }_{2,3,0}-2(2\lambda +3){\gamma }_{1,4,0}-\mu {\gamma }_{3,1,1},\hfill \\ {\Xi }_{3,0,3,0}=-\tau {\gamma }_{2,3,0}-10(\lambda +1){\gamma }_{0,5,0}-\mu {\gamma }_{0,3,2}\hfill & ,\hfill & {\Xi }_{2,2,0,2}=-3(\tau +1){\gamma }_{3,0,2}-\lambda {\gamma }_{2,1,2}-3(\mu +1){\gamma }_{2,0,3},\hfill \\ {\Xi }_{3,0,0,3}=-\tau {\gamma }_{2,0,3}-\lambda {\gamma }_{0,2,3}-10(\mu +1){\gamma }_{0,0,5}\hfill & ,\hfill & {\Xi }_{2,0,2,2}=-\tau {\gamma }_{1,2,2}-3(\lambda +1){\gamma }_{0,3,2}-3(\mu +1){\gamma }_{0,2,3}.\hfill \end{array}$$

$$\begin{array}{cc}& {\Xi }_{3,2,1,0}=-2(3\tau +2){\gamma }_{4,1,0}-(3\lambda +1){\gamma }_{2,3,0}-\mu {\gamma }_{2,1,2},\hfill \\ & {\Xi }_{3,2,0,1}=-2(3\tau +2){\gamma }_{4,0,1}-\lambda {\gamma }_{2,2,1}-(3\mu +1){\gamma }_{2,0,3},\hfill \\ \hfill & {\Xi }_{3,1,2,0}=-(3\tau +1){\gamma }_{3,2,0}-2(3\lambda +2){\gamma }_{1,4,0}-\mu {\gamma }_{1,2,2},\hfill \\ & {\Xi }_{3,1,0,2}=-(3\tau +1){\gamma }_{3,0,2}-\lambda {\gamma }_{1,2,2}-2(3\mu +2){\gamma }_{1,0,4},\hfill \\ \hfill & {\Xi }_{3,0,2,1}=-\tau {\gamma }_{2,2,1}-2(3\lambda +2){\gamma }_{0,4,1}-(3\mu +1){\gamma }_{0,2,3},\hfill \\ & {\Xi }_{3,0,1,2}=-\tau {\gamma }_{2,1,2}-(3\lambda +1){\gamma }_{0,3,2}-2(3\mu +2){\gamma }_{0,1,4},\hfill \\ \hfill & {\Xi }_{2,3,1,0}=-2(2\tau +3){\gamma }_{4,1,0}-(2\lambda +1){\gamma }_{3,2,0}-\mu {\gamma }_{3,1,1},\hfill \\ & {\Xi }_{2,3,0,1}=-2(2\tau +3){\gamma }_{4,0,1}-\lambda {\gamma }_{3,1,1}-(2\mu +1){\gamma }_{3,0,2},\hfill \\ \hfill & {\Xi }_{3,1,1,1}=-(3\tau +1){\gamma }_{3,1,1}-(3\lambda +1){\gamma }_{1,3,1}-(3\mu +1){\gamma }_{1,1,3},\hfill \\ & {\Xi }_{2,1,1,2}=-(2\tau +1){\gamma }_{2,1,2}-(2\lambda +1){\gamma }_{1,2,2}-3(\mu +1){\gamma }_{1,1,3},\hfill \\ \hfill & {\Xi }_{2,1,2,1}=-(2\tau +1){\gamma }_{2,2,1}-3(\lambda +1){\gamma }_{1,3,1}-(2\mu +1){\gamma }_{1,2,2},\hfill \\ & {\Xi }_{2,2,1,1}=-3(\tau +1){\gamma }_{3,1,1}-(2\lambda +1){\gamma }_{2,2,1}-(2\mu +1){\gamma }_{2,1,2}.\hfill \end{array}$$

Thus, in the same way as the previous cases, we prove that ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);$${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is of dimension one for $(\tau ,\lambda ,\mu )\in \{(-1,0,0),(0,-1,0),(0,0,-1),(-\frac{2}{3},-\frac{1}{3},0),$ $(-\frac{2}{3},0,-\frac{1}{3}),(0,-\frac{2}{3},-\frac{1}{3}),.$ $.(0,-\frac{1}{3},-\frac{2}{3}),(-\frac{1}{3},0,-\frac{2}{3}),(-\frac{1}{3},-\frac{2}{3},0),(-2,0,0),(0,-2,0),$ $(0,0,-2),(-\frac{3}{2},-\frac{1}{2},0),(-\frac{3}{2},0,-\frac{1}{2}),.$ $.(-\frac{1}{2},-\frac{3}{2},0),(-\frac{1}{2},0,-\frac{3}{2}),(0,-\frac{3}{2},-\frac{1}{2}),(-1,-1,0),$ $(-1,0,-1),(0,-1,-1),(-1,-\frac{1}{2},-\frac{1}{2}),(-\frac{1}{2},-1,-\frac{1}{2}),.$ $\left.(-\frac{1}{2},-\frac{1}{2},-1),(-\frac{1}{3},-\frac{1}{3},-\frac{1}{3})\right\}$. While $\tau$, $\lambda$ and $\mu$ are not like the above, ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is trivial.

The case when $\tau +\mu +\lambda =\nu -6$

By Proposition 3, the space of solutions is spanned by

$$t_{n,m,p,q},\mathrm{where}n+m+p+q=7,0\le m,p,q\le 5\mathrm{and}2\le n\le 7.$$

Furthermore, we can construct the following system using formula (26):

$$\left\{\begin{array}{cc}\tau t_{2,5,0,0}-\tau t_{6,1,0,0}+\lambda t_{2,0,5,0}-\lambda t_{6,0,1,0}+\mu t_{2,0,0,5}-\mu t_{6,0,0,1}=14t_{7,0,0,0},\hfill & \\ \tau t_{3,4,0,0}-\mu t_{5,2,0,0}+\lambda t_{3,0,4,0}-\mu t_{5,0,2,0}+\mu t_{3,0,0,4}-\mu t_{5,0,0,2}=14t_{7,0,0,0},\hfill & \\ (5\tau +1)t_{2,5,0,0}-(2\tau +1)t_{5,2,0,0}+\lambda t_{2,1,4,0}-\lambda t_{5,1,1,0}+\mu t_{2,1,0,4}-\mu t_{5,1,0,1}=9t_{6,1,0,0},\hfill & \\ \tau t_{2,4,1,0}-\tau t_{5,1,1,0}+(5\lambda +1)t_{2,0,5,0}-(2\lambda +1)t_{5,0,2,0}+\mu t_{2,0,1,4}-\mu t_{5,0,1,1}=9t_{6,0,1,0},\hfill & \\ \tau t_{2,4,0,1}-\tau t_{5,1,0,1}+\lambda t_{2,0,4,1}-\lambda t_{5,0,1,1}+(5\mu +1)t_{2,0,0,5}-(2\mu +1)t_{5,0,0,2}=9t_{6,0,0,1},\hfill & \\ (4\tau +1)t_{3,4,0,0}-(3\tau +1)t_{4,3,0,0}+\lambda t_{3,1,3,0}-\lambda t_{4,1,2,0}+\mu t_{3,1,0,3}-\mu t_{4,1,0,2}=5t_{6,1,0,0},\hfill & \\ \tau t_{3,3,1,0}-\tau t_{4,2,1,0}+(4\lambda +1)t_{3,0,4,0}-(3\lambda +1)t_{4,0,3,0}+\mu t_{3,0,1,3}-\mu t_{4,0,1,2}=5t_{6,0,1,0},\hfill & \\ \tau t_{3,3,0,1}-\tau t_{4,2,0,1}+\lambda t_{3,0,3,1}-\lambda t_{4,0,2,1}+(4\mu +1)t_{3,0,0,4}-(3\mu +1)t_{4,0,0,3}=5t_{6,0,0,1},\hfill & \\ 5(\tau +1)t_{2,5,0,0}-3(\tau +1)t_{4,3,0,0}+\lambda t_{2,2,3,0}-\lambda t_{4,2,1,0}+\mu t_{2,2,0,3}-\mu t_{4,2,0,1}=t_{5,2,0,0},\hfill & \\ \tau t_{2,3,2,0}-\tau t_{4,1,2,0}+5(\lambda +1)t_{2,0,5,0}-3(\lambda +1)t_{4,0,3,0}+\mu t_{2,0,2,3}-\mu t_{4,0,2,1}=5t_{5,0,2,0},\hfill & \\ \tau t_{2,3,0,2}-\tau t_{4,1,0,2}+\lambda t_{2,0,3,2}-\lambda t_{4,0,1,2}+5(\mu +1)t_{2,0,0,5}-3(\mu +1)t_{4,0,0,3}=5t_{5,0,0,2},\hfill & \\ \tau t_{2,2,0,3}-\tau t_{3,1,0,3}+\lambda t_{2,0,2,3}-\lambda t_{3,0,1,3}+10(\mu +1)t_{2,0,0,5}-2(2\mu +3)t_{3,0,0,4}=2t_{4,0,0,3},\hfill \end{array}\right.$$

$$\left\{\begin{array}{cc}10(\tau +1)t_{2,5,0,0}-2(2\tau +3)t_{3,4,0,0}+\lambda t_{2,3,2,0}-\lambda t_{3,3,1,0}+\mu t_{2,3,0,2}-\mu t_{3,3,0,1}=2t_{4,3,0,0},\hfill & \\ \tau t_{2,2,3,0}-\tau t_{3,1,3,0}+10(\lambda +1)t_{2,0,5,0}-2(2\lambda +3)t_{3,0,4,0}+\mu t_{2,0,3,2}-\mu t_{3,0,3,1}=2t_{4,0,3,0},\hfill & \\ (4\tau +1)t_{2,4,1,0}-(2\tau +1)t_{4,2,1,0}+(4\lambda +1)t_{2,1,4,0}-(2\lambda +1)t_{4,1,2,0}+\mu t_{2,1,1,3}-\mu t_{4,1,1,1}=5t_{5,1,1,0},\hfill & \\ (4\tau +1)t_{2,4,0,1}-(2\tau +1)t_{4,2,0,1}+\lambda t_{2,1,3,1}-\lambda t_{4,1,1,1}+(4\mu +1)t_{2,1,0,4}-(2\mu +1)t_{4,1,0,2}=5t_{5,1,0,1},\hfill & \\ \tau t_{2,3,1,1}-\tau t_{4,1,1,1}+(4\lambda +1)t_{2,0,4,1}-(2\lambda +1)t_{4,0,2,1}+(4\mu +1)t_{2,0,1,4}-(2\mu +1)t_{4,0,1,2}=5t_{5,0,1,1},\hfill & \\ 2(3\tau +2)t_{2,4,1,0}-3(\tau +1)t_{3,3,1,0}+(3\lambda +1)t_{2,2,3,0}-(2\lambda +1)t_{3,2,2,0}+\mu t_{2,2,1,2}-\mu t_{3,2,1,1}=2t_{4,2,1,0},\hfill & \\ 2(3\tau +2)t_{2,4,0,1}-3(\tau +1)t_{3,3,0,1}+\lambda t_{2,2,2,1}-\lambda t_{3,2,1,1}+(3\mu +1)t_{2,2,0,3}-(2\mu +1)t_{3,2,0,2}=2t_{4,2,0,1},\hfill & \\ (3\tau +1)t_{2,3,2,0}-(2\tau +1)t_{3,2,2,0}+2(3\lambda +2)t_{2,1,4,0}-3(\lambda +1)t_{3,1,3,0}+\mu t_{2,1,2,2}-\mu t_{3,1,2,1}=2t_{4,1,2,0},\hfill & \\ (3\tau +1)t_{2,3,0,2}-(2\tau +1)t_{3,2,0,2}+\lambda t_{2,1,2,2}-\lambda t_{3,1,1,2}+2(3\mu +2)t_{2,1,0,4}-3(\mu +1)t_{3,1,0,3}=2t_{4,1,0,2},\hfill & \\ \tau t_{2,2,2,1}-\tau t_{3,1,2,1}+2(3\lambda +2)t_{2,0,4,1}-3(\lambda +1)t_{3,0,3,1}+(3\mu +1)t_{2,0,2,3}-(2\mu +1)t_{3,0,2,2}=2t_{4,0,2,1},\hfill & \\ \tau t_{2,2,1,2}-\tau t_{3,1,1,2}+(3\lambda +1)t_{2,0,3,2}-(2\lambda +1)t_{3,0,2,2}+2(3\mu +2)t_{2,0,1,4}-3(\mu +1)t_{3,0,1,3}=2t_{4,0,1,2},\hfill & \\ (3\tau +1)t_{2,3,1,1}-(2\tau +1)t_{3,2,1,1}+(3\lambda +1)t_{2,1,3,1}-(2\lambda +1)t_{3,1,2,1}+(3\mu +1)t_{2,1,1,3}-(2\mu +1)t_{3,1,1,2}\hfill & \\ =2t_{4,1,1,1}.\hfill \end{array}\right.$$

With a direct computation, we confirm that the coefficients of $L_X{K}_6^{\tau ,\lambda ,\mu }$ are expressed in terms of

$$\begin{array}{ccc}{\Xi }_{3,0,4,0}=-\tau {\gamma }_{2,4,0}-10(\lambda +2){\gamma }_{0,6,0}-\mu {\gamma }_{0,4,2}\hfill & ,\hfill & {\Xi }_{3,2,0,2}=-2(3\tau +2){\gamma }_{4,0,2}-\lambda {\gamma }_{2,2,2}-2(3\mu +2){\gamma }_{2,0,4},\hfill \\ {\Xi }_{3,0,0,4}=-\tau {\gamma }_{2,0,4}-\lambda {\gamma }_{0,2,4}-10(\mu +2){\gamma }_{0,0,6}\hfill & ,\hfill & {\Xi }_{2,1,4,0}=-(2\tau +1){\gamma }_{2,4,0}-5(\lambda +2){\gamma }_{1,5,0}-\mu {\gamma }_{1,4,1},\hfill \\ {\Xi }_{2,5,0,0}=-3(2\tau +5){\gamma }_{6,0,0}-\lambda {\gamma }_{5,1,0}-\mu {\gamma }_{5,0,1}\hfill & ,\hfill & {\Xi }_{2,0,3,2}=-\tau {\gamma }_{1,3,2}-2(2\lambda +3){\gamma }_{0,4,2}-3(\mu +1){\gamma }_{0,3,3},\hfill \\ {\Xi }_{2,0,0,5}=-\tau {\gamma }_{1,0,5}-\lambda {\gamma }_{0,1,5}-3(2\mu +5){\gamma }_{0,0,6}\hfill & ,\hfill & {\Xi }_{2,1,0,4}=-(2\tau +1){\gamma }_{2,0,4}-\lambda {\gamma }_{1,1,4}-5(\mu +2){\gamma }_{1,0,5},\hfill \\ {\Xi }_{2,0,5,0}=-\tau {\gamma }_{1,5,0}-3(2\lambda +5){\gamma }_{0,6,0}-\mu {\gamma }_{0,5,1}\hfill & ,\hfill & {\Xi }_{2,0,4,1}=-\tau {\gamma }_{1,4,1}-5(\lambda +2){\gamma }_{0,5,1}-(2\mu +1){\gamma }_{0,4,2},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\Xi }_{3,3,1,0}=-5(\tau +2){\gamma }_{5,1,0}-(3\lambda +1){\gamma }_{3,3,0}-\mu {\gamma }_{3,1,2},\hfill \\ & {\Xi }_{3,2,2,0}=-2(3\tau +2){\gamma }_{4,2,0}-2(3\lambda +2){\gamma }_{2,4,0}-\mu {\gamma }_{2,2,2},\hfill \\ & {\Xi }_{3,1,3,0}=-(3\tau +1){\gamma }_{3,3,0}-5(\lambda +2){\gamma }_{1,5,0}-\mu {\gamma }_{1,3,2},\hfill \\ & {\Xi }_{3,1,2,1}=-(3\tau +1){\gamma }_{3,2,1}-2(3\lambda +2){\gamma }_{1,4,1}-(3\mu +1){\gamma }_{1,2,3},\hfill \\ & {\Xi }_{3,1,0,3}=-(3\tau +1){\gamma }_{3,0,3}-\lambda {\gamma }_{1,2,3}-5(\mu +2){\gamma }_{1,0,5},\hfill \\ & {\Xi }_{3,1,1,2}=-(3\tau +1){\gamma }_{3,1,2}-(3\lambda +1){\gamma }_{1,3,2}-2(3\mu +2){\gamma }_{1,1,4},\hfill \\ & {\Xi }_{3,0,3,1}=-\tau {\gamma }_{2,3,1}-5(\lambda +2){\gamma }_{0,5,1}-(3\mu +1){\gamma }_{0,3,3},\hfill \\ & {\Xi }_{3,2,1,1}=-2(3\tau +2){\gamma }_{4,1,1}-(3\lambda +1){\gamma }_{2,3,1}-(3\mu +1){\gamma }_{2,1,3},\hfill \\ & {\Xi }_{3,0,1,3}=-\tau {\gamma }_{2,1,3}-(3\lambda +1){\gamma }_{0,3,3}-5(\mu +2){\gamma }_{0,1,5},\hfill \\ & {\Xi }_{2,2,2,1}=-3(\tau +1){\gamma }_{3,2,1}-3(\lambda +1){\gamma }_{2,3,1}-(2\mu +1){\gamma }_{2,2,2},\hfill \\ & {\Xi }_{3,0,2,2}=-\tau {\gamma }_{2,2,2}-2(3\lambda +2){\gamma }_{0,4,2}-2(3\mu +2){\gamma }_{0,2,4},\hfill \\ & {\Xi }_{2,1,3,1}=-(2\tau +1){\gamma }_{2,3,1}-2(2\lambda +3){\gamma }_{1,4,1}-(2\mu +1){\gamma }_{1,3,2},\hfill \\ & {\Xi }_{2,4,1,0}=-5(\tau +2){\gamma }_{5,1,0}-(2\lambda +1){\gamma }_{4,2,0}-\mu {\gamma }_{4,1,1},\hfill \\ & {\Xi }_{2,4,0,1}=-5(\tau +2){\gamma }_{5,0,1}-\lambda {\gamma }_{4,1,1}-(2\mu +1){\gamma }_{4,0,2},\hfill \\ & {\Xi }_{2,3,2,0}=-2(2\tau +3){\gamma }_{4,2,0}-3(\lambda +1){\gamma }_{3,3,0}-\mu {\gamma }_{3,2,1},\hfill \\ & {\Xi }_{2,3,1,1}=-2(2\tau +3){\gamma }_{4,1,1}-(2\lambda +1){\gamma }_{3,2,1}-(2\mu +1){\gamma }_{3,1,2},\hfill \\ & {\Xi }_{2,3,0,2}=-2(2\tau +3){\gamma }_{4,0,2}-\lambda {\gamma }_{3,1,2}-3(\mu +1){\gamma }_{3,0,3},\hfill \\ & {\Xi }_{2,2,3,0}=-3(\tau +1){\gamma }_{3,3,0}-2(2\lambda +3){\gamma }_{2,4,0}-\mu {\gamma }_{2,3,1},\hfill \\ & {\Xi }_{2,2,0,3}=-3(\tau +1){\gamma }_{3,0,3}-\lambda {\gamma }_{2,1,3}-2(2\mu +3){\gamma }_{2,0,4},\hfill \\ & {\Xi }_{2,2,1,2}=-3(\tau +1){\gamma }_{3,1,2}-(2\lambda +1){\gamma }_{2,2,2}-3(\mu +1){\gamma }_{2,1,3},\hfill \\ & {\Xi }_{2,1,2,2}=-(2\tau +1){\gamma }_{2,2,2}-3(\lambda +1){\gamma }_{1,3,2}-3(\mu +1){\gamma }_{1,2,3},\hfill \\ & {\Xi }_{2,1,1,3}=-(2\tau +1){\gamma }_{2,1,3}-(2\lambda +1){\gamma }_{1,2,3}-2(2\mu +3){\gamma }_{1,1,4}.\hfill \end{array}$$

So, in the same way as the previous cases, we prove that ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is of dimension one for $(\tau ,\lambda ,\mu )\in \{(-\frac{1}{3},-\frac{1}{3},-\frac{2}{3}),(0,-2,-\frac{1}{3}),(0,-\frac{3}{2},-1),(0,-\frac{1}{3},-2),$ $(-1,-1,-\frac{1}{2}),.$ $.(0,-\frac{2}{3},-\frac{2}{3}),(-\frac{1}{2},-\frac{3}{2},-\frac{1}{2}),(-2,-\frac{1}{3},0),(-\frac{2}{3},-\frac{2}{3},0),(0,-2,0),(-\frac{2}{3},0,-\frac{2}{3}),$ $(-\frac{1}{3},-2,0),.$ $.(-\frac{1}{3},-\frac{2}{3},-\frac{1}{3}),(-\frac{1}{3},0,-2),(-2,-\frac{1}{2},0),(-2,0,-\frac{1}{2}),(-\frac{3}{2},-1,0),(-\frac{3}{2},-\frac{1}{2},-\frac{1}{2}),$ $(-\frac{3}{2},0,-1),.$ $.(-1,-\frac{3}{2},0),(0,0,-2),(-1,-\frac{1}{2},-1),(-1,0,-\frac{3}{2}),(-\frac{1}{2},-2,0),(-\frac{5}{2},0,0),$ $(-\frac{1}{2},-1,-1),(0,0,-\frac{5}{2}),.$$.(-\frac{1}{2},0,-2),(0,-\frac{5}{2},0),(0,-2,-\frac{1}{2}),(-\frac{2}{3},-\frac{1}{3},-\frac{1}{3}),(-\frac{1}{2},-\frac{1}{2},-\frac{3}{2}),\}$. While $\tau$, $\lambda$ and $\mu$ are not like the above, ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is trivial.

The case when $\tau +\mu +\lambda =\nu -7$

In the same way as the previous cases, we prove that ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is of dimension one for $(\tau ,\lambda ,\mu )\in \left(\right\{(-\frac{1}{3},-\frac{4}{3},0),(-\frac{1}{3},0,-\frac{4}{3}),(0,-\frac{5}{3},0),(0,0,-\frac{5}{3}),(0,-1,-\frac{2}{3}),$$(0,-\frac{2}{3},-1),(0,-\frac{1}{3},-\frac{4}{3}),).$$.(-3,0,0),(-\frac{1}{2},0,-\frac{5}{2}),(0,-3,0),(-\frac{5}{2},-\frac{1}{2},0),(-\frac{5}{2},0,-\frac{1}{2}),$ $(-\frac{1}{2},-\frac{5}{2},0),(-\frac{1}{2},0,-\frac{5}{2}),(-2,-1,0),.$ $.(-2,0,-1),(-1,-2,0),(-1,0,-2),(0,-2,-1),$ $(-\frac{1}{2},-\frac{1}{2},-2),(-\frac{1}{2},-2,-\frac{1}{2}),(-2,-\frac{1}{2},-\frac{1}{2}),.$$.(-\frac{3}{2},0,-\frac{3}{2}),(-\frac{3}{2},-\frac{3}{2},0),(0,-\frac{3}{2},-\frac{3}{2}),$ $(-1,-1,-1),(-1,-\frac{3}{2},-\frac{1}{2}),(-\frac{1}{2},-\frac{3}{2},-1),(-\frac{1}{2},-1,-\frac{3}{2}),.$$.(0,-1,-2),(-\frac{3}{2},-1,-\frac{1}{2}),$ $(-1,-\frac{1}{2},-\frac{3}{2}),(0,0,-3),(-\frac{3}{2},-\frac{1}{2},-1)\}$ and zero-dimensional otherwise.

The case when $\nu -\tau -\lambda -\mu \ge 8$

For $\nu -\tau -\lambda -\mu \ge 8$, the number of equations coming out from the condition 1-cocycle is much larger than the number of variables generating a 1-cocycle—for example, for $\nu -\tau -\lambda -\mu \ge 8$, the number of (variables)$=120$, while the number of (equations)$=125$. For generic $\tau$, $\lambda$ and $\mu$, the number of equations will generate a 1-dimensional space, which gives a unique cohomology class. This ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is indeed trivial because the expression $L_X{K}_k^{\tau ,\lambda ,\mu }$ is also a 1-cocycle.

Remark 1.

For $\nu -\tau -\lambda -\mu \ge 8$ and for particular values of τ, λ and μ, ${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);$ ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ may not be trivial. For instance, for $\nu -\tau -\mu -\lambda =8$, we have

$${\mathrm{H}}_{\mathrm{diff}}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{0,0,-\frac{7}{2};\frac{9}{2}})\simeq \mathbb{R}.$$

6. Conclusions

In the present work, we focused on the cohomology of $\mathcal{LA}$. First, we determined H${}^1(\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$, in which we proved that H${}^1(\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is zero-dimensional if $k=\nu -\tau -\lambda -\mu \notin \mathbb{N}$ and $\frac{1}{2}(k+1)(k+2)$-dimensional if $k\in \mathbb{N}$. Moreover, ${\mathrm{H}}_{\mathrm{diff}}^1(\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$ is spanned by the 1-cocycles: ${\omega }_j(X_h,G)=h^{\prime }{G}^{\left(j\right)}$ with $\left|j\right|=k\in \mathbb{N}.$ On the second hand, we classified fourth-linear ${\mathrm{Hom}}_{\mathfrak{aff}\left(1\right)}$, and we proved that the space of ${\mathrm{Hom}}_{\mathfrak{aff}\left(1\right)}$ is $\frac{1}{6}k(k-1)(k+1)$-dimensional, where $k=\nu -\tau -\lambda -\mu +1$. Finally, we calculated H${}^1(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu })$, in which we proved that ${\mathrm{H}}_{\mathrm{diff}}^1\left(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }\right)$ is one-dimensional for resonant values of weights that satisfy $\nu -\tau -\lambda -\mu <8$. If $\nu -\tau -\lambda -\mu \ge 8$ for generic $\tau$, $\lambda$ and $\mu$, we proved that each 1-cocycle is eliminated by adding a coboundary, and then ${\mathrm{H}}_{\mathrm{diff}}^1\left(\mathcal{V}\left({\mathbb{RP}}^1\right),\mathfrak{aff}\left(1\right);{\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }\right)$ is zero-dimensional.