On the Higher Nash Blow-Up Derivation Lie Algebras of Isolated Hypersurface Singularities

Abstract

It is a natural question to ask whether there is any Lie algebra that completely characterize simple singularities? The higher Nash blow-up derivation Lie algebras ${\mathcal{L}}_k^l\left(V\right)$ associated to isolated hypersurface singularities defined to be the Lie algebra of derivations of the local Artinian algebra ${\mathcal{M}}_n^l\left(V\right):={\mathcal{O}}_l/⟨F,\mathcal{J}n⟩$, i.e., ${\mathcal{L}}_k^l\left(V\right)=Der\left({\mathcal{M}}_n^l\left(V\right)\right)$. In this paper, we construct a new conjecture for the complete characterization of simple hypersurface singularities using the Lie algebras ${\mathcal{L}}_k^l\left(V\right)$ under certain condition and prove it true for ${\mathcal{L}}_k^l\left(V\right)$ when $k,l=2$.

1. Introduction

The class of simple (ADE) singularities, comprised of $A_k:\{x_2^2+x_1^{k+1}=0\}\subset {\mathbb{C}}^2,k\ge 1,$$D_k:\{x_1^2{x}_2+x_2^{k-1}=0\}\subset {\mathbb{C}}^2,k\ge 4,$ and $E_6,E_7,E_8$, exceptional singularities defined in ${\mathbb{C}}^2$ by polynomials $x_1^3+x_2^4,x_1^3+x_1{x}_2^3,x_1^3+x_2^5,$ respectively. From the nineteenth century, characterization of ADE singularities have been studied extensively [1]. This class of singularities have many important applications in group theory, singularity theory, etc.

The root system have important role in the classification of nilpotent Lie algebras. Using the root systems, Santharoubane [2] developed a relationship between the Kac–Mody and nilpotent Lie algebras.

Yau [3,4] purposed Lie algebra $A\left(V\right):={\mathcal{O}}_n/(f,\frac{\partial f}{\partial x_1},\cdots ,\frac{\partial f}{\partial x_n})$ for isolated hypersurface singularity $(V,0)\subset ({\mathbb{C}}^n,0)$. This Lie algebra is a finite dimensional solvable Lie algebra and is called the Yau algebra of V [5,6,7]. Its study began in the 1980s [8,9,10,11,12,13,14,15,16]. Hussain, Yau, and Zuo [17] generalized the Yau algebra as the k-th Yau algebra. Hussain and collaborators further studied k-th Yau algebras [18]. In [19], authors show that the generalized Cartan matrix of Lie algebra $L^{*}\left(V\right)$ can be used to characterize the ADE singularities except the one pair ($A_6$ and $D_5$ singularities). The Lie algebra $L^{*}\left(V\right)$ also cannot completely characterize the ADE singularities [20]. So it is natural to ask whether there is any other Lie algebra that completely characterized the simple singularities?

With regards to the above question, Hussain, Yau, and Zuo [21] define the higher Nash blow-up derivation Lie algebra of isolated hypersurface singularities as ${\mathcal{L}}_k^l\left(V\right)=Der\left({\mathcal{M}}_n^l\left(V\right)\right)$ where ${\mathcal{M}}_n^l\left(V\right)$ is local Artinian algebra, $(V,0)$ defined by a polynomial $F(x_1,\cdots ,x_l)$, ${\mathcal{J}}_n\subset \mathbb{C}\left[x_1,\dots ,x_l\right]$ is an ideal generated by the $(M\times M)$-minors of ${Jac}_n\left(F\right)$ (higher-order Jacobian matrix) [22] and ${\mathcal{M}}_n^l\left(V\right):={\mathcal{O}}_l/\langle F,{\mathcal{J}}_n\rangle$ is higher Nash blow-up local algebra. The dimension of ${\mathcal{M}}_n^l\left(V\right)$ is ${\mathit{d}}_{\mathit{n}}^{\mathit{l}}\left(V\right)$ and dimension of Lie algebra ${\mathcal{L}}_k^l\left(V\right)$ denoted as ${\rho }_k^l\left(V\right)$. The Lie algebra ${\mathcal{L}}_k^l\left(V\right),k,l\ge 2$ found useful for characterization. In this work, we propose the following conjecture for Lie algebra ${\mathcal{L}}_k^l\left(V\right),k,l\ge 2$.

Conjecture 1. 

Generalized Cartan matrices $C_k^l\left(V\right)$, $k,l\ge 2$ arising from ${\mathcal{L}}_k^l\left(V\right)$ completely characterize the ADE singularities. Equivalently, for any two ADE singularities X and Y $C_k^l\left(X\right)=C_k^l\left(Y\right)$⇔X and Y are analytically isomorphic.

Theorem 1. 

The generalized Cartan matrix $C_2^2\left(V\right)$ arising from ${\mathcal{L}}_2^2\left(V\right)$ characterizes the ADE singularities. Equivalently, if X and Y are two ADE singularities, then $C_2^2\left(X\right)=C_2^2\left(Y\right)$⇔X and Y are analytically isomorphic.

2. Basic Results

Basic definitions and important results related to the higher Nash blow-up derivation Lie algebra of isolated singularities can be found in [21].

Definition 1. 

An $n\times n$ matrix with entries in $\mathbb{Z}$, $C=\left(c_{ij}\right)$ is a generalized Cartan matrix (GCM) if

(1) 

$c_{ii}=2\forall i=1,\cdots ,n,$

(2) 

$c_{ij}\le 0\forall i,j=1,\cdots ,n,i\ne j,$

(3) 

$c_{ij}=0$⇔$c_{ji}=0$$\forall i,j=1,\cdots ,n,i\ne j.$

Theorem 2 

(Mostow’s theorem, [2]). Let $T_1$ and $T_2$ be two maximal torus of $\xi \left(V\right)$, then there exist $\phi \in Aut\xi \left(V\right)$ defined as $\phi T_1{\phi }^{-1}=T_2.$

Consider the root space decomposition of $\xi \left(V\right)$ relatively to T (maximal torus) ([2]);

$$\xi \left(V\right)=\sum _{\alpha \in R\left(T\right)}\xi {\left(V\right)}^{\alpha },$$

$\xi {\left(V\right)}^{\alpha }=\{y\in \xi \left(V\right):ty=\alpha \left(t\right)y,\forall t\in T\},$ and $R\left(T\right)=\{\alpha \in T^{*}:\xi {\left(V\right)}^{\alpha }\ne \left(0\right)\}\left(\mathrm{root}\mathrm{system}\right),$

$$R^1\left(T\right)=\{\alpha \in R\left(T\right):\xi {\left(V\right)}^{\alpha }⊈[\xi \left(V\right),\xi \left(V\right)]\},$$

$$l_{\alpha }=dim\left(\frac{\xi {\left(V\right)}^{\alpha }}{[\xi \left(V\right),\xi \left(V\right)]\cap \xi {\left(V\right)}^{\alpha }}\right),\forall \alpha \in R^1\left(T\right),$$

$$d_{\alpha }=dim\left(\xi {\left(V\right)}^{\alpha }\right),\alpha \in R^1\left(T\right).$$

Then the map: $\alpha \mapsto d_{\alpha }{R}^1\left(T\right)\to {\mathbb{N}}^{*}$ gives the following partition:

$$R^1\left(T\right)=R^1{\left(T\right)}_{r_1}\cup \cdots \cup R^1{\left(T\right)}_{r_q},r_1<\cdots <r_q,R^1{\left(T\right)}_{r_j}\ne \varnothing ,$$

$$R^1{\left(T\right)}_r=\{\alpha \in R^1\left(T\right);d_{\alpha }=r\}.$$

Set $s_j=♯R^1{\left(T\right)}_{r_j}$ and $s=s_1+\cdots +s_q$. Let $d_{{\alpha }_j}=d_j$ and $n_{{\alpha }_j}=n_j.$ Let $f:\{1,\cdots ,n\}⟶\{1,\cdots ,s\}$ be defined by:

$$f_j=\left\{\begin{array}{cc}1;\hfill & 1⩽j⩽n_1,\hfill \\ 2;\hfill & n_1⩽j⩽n_1+n_2,\hfill \\ ⋮\hfill \\ s;\hfill & n_1+n_2+\cdots +n_{s-1}⩽j⩽n.\hfill \end{array}\right.$$

Theorem 3 

([2]). For $j,k\in \{1,\cdots ,n\}$, $j\ne k$, let $-c_{jk}\left(T\right)=min\{-l\in N;{\left(\mathrm{ad}v\right)}^{-l+1}w=0,\forall v\in {\xi }^{{\beta }_{f\left(j\right)}},\forall w\in {\xi }^{{\beta }_{f\left(k\right)}}\},$ with ${\left(\mathrm{ad}0\right)}^0=0$ and let $c_{jj}\left(T\right)=2$ for $j=1,\cdots ,n.$ Then, the generalized Cartan matrix define as $C\left(T\right)={\left(c_{jk}\left(T\right)\right)}_{1\le j,k\le n}$.

3. Proof of Theorem 1

The following propositions will be used to prove the main theorem.

Proposition 1. 

Let $V=\{(x_1,x_2)\in {\mathbb{C}}^2:x_1^{k+1}+x_2^2=0\}$ be the $A_k$ singularity, $k\ge 1$. Then

$$C_2^2\left(A_k\right)=\left\{\begin{array}{cc}\mathrm{is}\mathrm{not}\mathrm{defined};\hfill & k=1,2,3,\hfill \\ \left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right);\hfill & k=4,\hfill \\ \left(\begin{array}{cc}2& -(k-4)\\ -\frac{k-4}{2}& 2\end{array}\right);\hfill & when k \ge 6 is even,\hfill \\ \left(\begin{array}{cc}2& -(k-4)\\ -\frac{k-3}{2}& 2\end{array}\right);\hfill & when k \ge 5 and k is odd.\hfill \end{array}\right.$$

Proof. 

It follows that algebra ${\mathcal{M}}_2^2\left(V\right)$ has a basis of the form $x_1^{i_1},0\le i_1\le k-1$. The basis of Lie algebra ${\mathcal{L}}_2^2\left(V\right)$ is defined as:

$$\begin{array}{ccc}& {ϵ}_1^{*}=x_1{\partial }_1+2x_1^2{\partial }_2,{ϵ}_2^{*}=x_1^2{\partial }_1+2x_1^3{\partial }_2,{ϵ}_3^{*}=x_1^4{\partial }_1+2x_1^5{\partial }_2,\cdots ,{ϵ}_{k-1}^{*}=x_1^k{\partial }_1.\hfill & \hfill \end{array}$$

For $k=1$, the Lie algebra dimension are zero and for $k=2$ the dimension of nilradical $\xi \left(V\right)$ are zero. It is also noteworthy that for $k\ge 3$ the $\xi \left(V\right)$ of Lie algebra ${\mathcal{L}}_2^2\left(V\right)$ is generated by $\langle {ϵ}_2^{*},{ϵ}_3^{*},{ϵ}_4^{*},\cdots ,{ϵ}_{k-1}^{*}\rangle$.

For $A_4$ singularity, the $\xi \left(V\right)=\langle {ϵ}_2^{*},{ϵ}_3^{*}\rangle$ have multiplication table as: $[{ϵ}_2^{*},{ϵ}_3^{*}]=0$. From multiplication table the type of $A_4$ singularity are 2 and the nilpotency are 0. The torus T ([8]) of $\xi \left(V\right)$ is spanned by

$$\begin{array}{cccc}\hfill & t_1:\xi \left(V\right)⟶\xi \left(V\right)\hfill & \hfill & t_2:\xi \left(V\right)⟶\xi \left(V\right)\hfill \\ \hfill & {ϵ}_2^{*}⟶{ϵ}_2^{*}\hfill & \hfill & {ϵ}_2^{*}⟶0\hfill \\ \hfill & {ϵ}_3^{*}⟶0\hfill & \hfill & {ϵ}_3^{*}⟶{ϵ}_3^{*}.\hfill \end{array}$$

Thus, $T=\mathbb{C}{t}_1+\mathbb{C}{t}_2.$ It follows that T is maximal torus of $\xi \left(V\right)$, because dim $T=2=$ the type of $A_4$. Suppose that ${\alpha }_k:T⟶\mathbb{C}$ be a linear map with ${\alpha }_k\left(t_l\right)={\delta }_{kl}$ for $k,l=1,2.$

$$\begin{array}{cc}\hfill & \xi \left(V\right)={\xi }^{{\alpha }_1}\oplus {\xi }^{{\alpha }_2}\hfill \\ \hfill & =\mathbb{C}{ϵ}_2^{*}\oplus \mathbb{C}{ϵ}_3^{*}.\hfill \end{array}$$

Therefore, T-minimal system of generators are $({ϵ}_2^{*},{ϵ}_3^{*})$. So the generalized Cartan matrix is

$$C_2^2\left(A_4\right)=\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right).$$

Case (a). For k is odd and $k\ge 5$, the $\xi \left(V\right)=\langle {ϵ}_2^{*},{ϵ}_3^{*},{ϵ}_4^{*},\cdots ,{ϵ}_{k-1}^{*}\rangle$ have following multiplication table:

$$\begin{array}{cc}\hfill & [{ϵ}_2^{*},{ϵ}_3^{*}]=-{ϵ}_4^{*},[{ϵ}_2^{*},{ϵ}_4^{*}]=-2{ϵ}_5^{*},[{ϵ}_2^{*},{ϵ}_5^{*}]=-3{ϵ}_6^{*},\cdots ,[{ϵ}_2^{*},{ϵ}_{k-2}^{*}]=-(k-4){ϵ}_{k-1}^{*},\hfill \\ \hfill & [{ϵ}_3^{*},{ϵ}_4^{*}]=-{ϵ}_6^{*},[{ϵ}_3^{*},{ϵ}_5^{*}]=-2{ϵ}_7^{*},[{ϵ}_3^{*},{ϵ}_6^{*}]=-3{ϵ}_8^{*},\cdots ,[{ϵ}_3^{*},{ϵ}_{k-3}^{*}]=-(k-6){ϵ}_{k-1}^{*},\hfill \\ \hfill & [{ϵ}_4^{*},{ϵ}_5^{*}]=-{ϵ}_8^{*},[{ϵ}_4^{*},{ϵ}_6^{*}]=-2{ϵ}_9^{*},[{ϵ}_4^{*},{ϵ}_7^{*}]=-3{ϵ}_{10}^{*},\cdots ,[{ϵ}_4^{*},{ϵ}_{k-4}^{*}]=-(k-8){ϵ}_{k-1}^{*},\hfill \\ \hfill & ⋮\hfill \\ \hfill & [{ϵ}_{\frac{k-1}{2}}^{*},{ϵ}_{\frac{k-1}{2}+1}^{*}]=-{ϵ}_{k-1}^{*}.\hfill \end{array}$$

Case (b). For k is even and $k\ge 6$, the $\xi \left(V\right)=\langle {ϵ}_2^{*},{ϵ}_3^{*},{ϵ}_4^{*},\cdots ,{ϵ}_{k-1}^{*}\rangle$ have following multiplication table:

$$\begin{array}{cc}\hfill & [{ϵ}_2^{*},{ϵ}_3^{*}]=-{ϵ}_4^{*},[{ϵ}_2^{*},{ϵ}_4^{*}]=-2{ϵ}_5^{*},[{ϵ}_2^{*},{ϵ}_5^{*}]=-3{ϵ}_6^{*},\cdots ,[{ϵ}_2^{*},{ϵ}_{k-2}^{*}]=-(k-4){ϵ}_{k-1}^{*},\hfill \\ \hfill & [{ϵ}_3^{*},{ϵ}_4^{*}]=-{ϵ}_6^{*},[{ϵ}_3^{*},{ϵ}_5^{*}]=-2{ϵ}_7^{*},[{ϵ}_3^{*},{ϵ}_6^{*}]=-3{ϵ}_8^{*},\cdots ,[{ϵ}_3^{*},{ϵ}_{k-3}^{*}]=-(k-6){ϵ}_{k-1}^{*},\hfill \\ \hfill & [{ϵ}_4^{*},{ϵ}_5^{*}]=-{ϵ}_8^{*},[{ϵ}_4^{*},{ϵ}_6^{*}]=-2{ϵ}_9^{*},[{ϵ}_4^{*},{ϵ}_7^{*}]=-3{ϵ}_{10}^{*},\cdots ,[{ϵ}_4^{*},{ϵ}_{k-4}^{*}]=-(k-8){ϵ}_{k-1}^{*},\hfill \\ \hfill & ⋮\hfill \\ \hfill & [{ϵ}_{\frac{k-2}{2}}^{*},{ϵ}_{\frac{k-2}{2}+1}^{*}]=-{ϵ}_{k-2}^{*},[{ϵ}_{\frac{k-2}{2}}^{*},{ϵ}_{\frac{k-2}{2}+2}^{*}]=-2{ϵ}_{k-1}^{*}.\hfill \end{array}$$

For the $A_5$ singularity, the nilradical $\xi \left(V\right)=\langle {ϵ}_2^{*},{ϵ}_3^{*},{ϵ}_4^{*}\rangle$ have following multiplication table $[{ϵ}_2^{*},{ϵ}_3^{*}]=-{ϵ}_4^{*}$. From the multiplication table, the type of $A_5$ singularity are 2 and the nilpotency are 1. The torus T is spanned by

$$\begin{array}{cccc}\hfill & t_1:\xi \left(V\right)⟶\xi \left(V\right)\hfill & \hfill & t_2:\xi \left(V\right)⟶\xi \left(V\right)\hfill \\ \hfill & {ϵ}_2^{*}⟶{ϵ}_2^{*}\hfill & \hfill & {ϵ}_2^{*}⟶0\hfill \\ \hfill & {ϵ}_3^{*}⟶0\hfill & \hfill & {ϵ}_3^{*}⟶{ϵ}_3^{*}\hfill \\ \hfill & {ϵ}_4^{*}⟶{ϵ}_4^{*}\hfill & \hfill & {ϵ}_4^{*}⟶{ϵ}_4^{*}.\hfill \end{array}$$

After simple calculation we obtain

$$\begin{array}{cc}\hfill & \xi \left(V\right)={\xi }^{{\alpha }_1}\oplus {\xi }^{{\alpha }_2}\oplus {\xi }^{{\alpha }_1+{\alpha }_2}\hfill \\ \hfill & =\mathbb{C}{ϵ}_2^{*}\oplus \mathbb{C}{ϵ}_3^{*}\oplus \mathbb{C}{ϵ}_4^{*}.\hfill \end{array}$$

Therefore, T-minimal system of generators are $({ϵ}_2^{*},{ϵ}_3^{*})$. So the generalized Cartan matrix is

$$C_2^2\left(A_5\right)=\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right).$$

For $A_6$ singularity, the $\xi \left(V\right)=\langle {ϵ}_2^{*},{ϵ}_3^{*},{ϵ}_4^{*},{ϵ}_5^{*}\rangle$ have following multiplication table $[{ϵ}_2^{*},{ϵ}_3^{*}]=-{ϵ}_4^{*},[{ϵ}_2^{*},{ϵ}_4^{*}]=-2{ϵ}_5^{*}$. From multiplication table, the type for $A_6$ singularity is 2 and the nilpotency is 1. The torus T is spanned by

$$\begin{array}{cccc}\hfill & t_1:\xi \left(V\right)⟶\xi \left(V\right)\hfill & \hfill & t_2:\xi \left(V\right)⟶\xi \left(V\right)\hfill \\ \hfill & {ϵ}_2^{*}⟶{ϵ}_2^{*}\hfill & \hfill & {ϵ}_2^{*}⟶0\hfill \\ \hfill & {ϵ}_3^{*}⟶0\hfill & \hfill & {ϵ}_3^{*}⟶{ϵ}_3^{*}\hfill \\ \hfill & {ϵ}_4^{*}⟶{ϵ}_4^{*}\hfill & \hfill & {ϵ}_4^{*}⟶{ϵ}_4^{*}\hfill \\ \hfill & {ϵ}_5^{*}⟶2{ϵ}_5^{*}\hfill & \hfill & {ϵ}_5^{*}⟶{ϵ}_5^{*}.\hfill \end{array}$$

After a simple calculation, we obtain

$$\begin{array}{cc}\hfill & \xi \left(V\right)={\xi }^{{\alpha }_1}\oplus {\xi }^{{\alpha }_2}\oplus {\xi }^{{\alpha }_1+{\alpha }_2}\oplus {\xi }^{2{\alpha }_1+{\alpha }_2}\hfill \\ \hfill & =\mathbb{C}{ϵ}_2^{*}\oplus \mathbb{C}{ϵ}_3^{*}\oplus \mathbb{C}{ϵ}_4^{*}\oplus \mathbb{C}{ϵ}_5^{*}.\hfill \end{array}$$

Therefore, T-minimal system of generators are $({ϵ}_2^{*},{ϵ}_3^{*})$. So the generalized Cartan matrix is

$$C_2^2\left(A_6\right)=\left(\begin{array}{cc}2& -2\\ -1& 2\end{array}\right).$$

From above multiplication table, the type of $A_k$($k\ge 7$) singularity is 2 and the nilpotency is $k-4$. The torus T is spanned by

$$\begin{array}{cc}\hfill & t:\xi \left(V\right)⟶\xi \left(V\right)\hfill \\ \hfill & {ϵ}_2^{*}⟶{ϵ}_2^{*}\hfill \\ \hfill & {ϵ}_3^{*}⟶2{ϵ}_3^{*}\hfill \\ \hfill & {ϵ}_4^{*}⟶4{ϵ}_4^{*}\hfill \\ \hfill & {ϵ}_5^{*}⟶5{ϵ}_5^{*}\hfill \\ \hfill & ⋮\hfill \\ \hfill & {ϵ}_{k-1}^{*}⟶(k-2){ϵ}_{k-1}^{*}.\hfill \end{array}$$

After simple calculation we obtain

$$\begin{array}{cc}\hfill & \xi \left(V\right)={\xi }^{\alpha }\oplus {\xi }^{2\alpha }\oplus {\xi }^{3\alpha }\oplus {\xi }^{4\alpha }\oplus \cdots \oplus {\xi }^{(k-2)\alpha }\hfill \\ \hfill & =\mathbb{C}{ϵ}_2^{*}\oplus \mathbb{C}{ϵ}_3^{*}\oplus \mathbb{C}{ϵ}_4^{*}\oplus \cdots \oplus \mathbb{C}{ϵ}_{k-1}^{*}.\hfill \end{array}$$

Therefore, the T-minimal system of generators are $({ϵ}_2^{*},{ϵ}_3^{*})$. So the generalized Cartan matrices are:

• Case (a). For k is even and $k\ge 8$, we have $C_2^2\left(A_k\right)=\left(\begin{array}{cc}2& k-4\\ -\frac{k-4}{2}& 2\end{array}\right).$
• Case (b). For k is odd and $k\ge 7$ we have $C_2^2\left(A_k\right)=\left(\begin{array}{cc}2& k-4\\ -\frac{k-3}{2}& 2\end{array}\right).$ □

Proposition 2. 

Let $V=\{(x_1,x_2)\in {\mathbb{C}}^2:x_1^2{x}_2+x_2^{k-1}=0\}$ be the $D_k$ singularity, $k\ge 4$. Then

$$C_2^2\left(D_k\right)=\left\{\begin{array}{cc}\left(\begin{array}{ccc}2& -3& -1\\ 0& 2& 0\\ -1& 0& 2\end{array}\right);\hfill & k=\mathit{4},\hfill \\ \left(\begin{array}{ccc}2& -3& -2\\ -2& 2& -1\\ -1& -1& 2\end{array}\right);\hfill & k=\mathit{5},\hfill \\ \left(\begin{array}{cccc}2& -1& -2& -2\\ -1& 2& -1& -1\\ -1& -1& 2& 0\\ -1& -1& 0& 2\end{array}\right);\hfill & k=\mathit{6},\hfill \\ \left(\begin{array}{cccc}2& -1& -2& 0\\ -1& 2& -1& 0\\ -1& -1& 2& -2\\ 0& 0& -1& 2\end{array}\right);\hfill & k=\mathit{7},\hfill \\ \left(\begin{array}{ccccc}2& -1& -2& -2& 0\\ -1& 2& -1& -1& 0\\ -1& -1& 2& 0& -k+5\\ -1& -1& 0& 2& -k+5\\ 0& 0& \frac{-k+6}{2}& \frac{-k+6}{2}& 2\end{array}\right);\hfill & when k \ge \mathit{8} and k is even,\hfill \\ \left(\begin{array}{ccccc}2& -1& -2& -2& 0\\ -1& 2& -1& -1& 0\\ -1& -1& 2& 0& -k+5\\ -1& -1& 0& 2& -k+5\\ 0& 0& \frac{-k+5}{2}& \frac{-k+5}{2}& 2\end{array}\right);\hfill & when k \ge \mathit{9} and k is odd.\hfill \end{array}\right.$$

Proof. 

It follows that the algebra ${\mathcal{M}}_2^2\left(V\right)$ has a monomial basis of the form:

$$\{x_2^{i_2},0\le i_2\le k-2;x_1;x_1{x}_2;x_1^2\}.$$

When $k=4$, then the basis of Lie algebra ${\mathcal{L}}_2^2\left(V\right)$ define as:

$$\begin{array}{cc}& {ϵ}_1^{*}=-x_1{\partial }_1+x_2{\partial }_2,{ϵ}_2^{*}=-x_2{\partial }_1,{ϵ}_3^{*}=-x_1{\partial }_2,{ϵ}_4^{*}=-x_2{\partial }_2,{ϵ}_5^{*}=-x_1^2{\partial }_2,\hfill \\ & {ϵ}_6^{*}=-x_1{x}_2{\partial }_2,{ϵ}_7^{*}=-x_2^2{\partial }_2,{ϵ}_8^{*}=-x_1^2{\partial }_1,{ϵ}_9^{*}=-x_1{x}_2{\partial }_1,{ϵ}_{10}^{*}=-x_2^2{\partial }_1.\hfill \end{array}$$

For the $D_4$ singularity, the nilradical $\xi \left(V\right)=\langle {ϵ}_2^{*},{ϵ}_5^{*},{ϵ}_6^{*},{ϵ}_7^{*},\cdots ,{ϵ}_{10}^{*}\rangle$ have following multiplication table:

$$\begin{array}{cc}\hfill & [{ϵ}_2^{*},{ϵ}_5^{*}]=2{ϵ}_6^{*}-{ϵ}_8^{*},[{ϵ}_2^{*},{ϵ}_6^{*}]={ϵ}_7^{*}-{ϵ}_9^{*},[{ϵ}_2^{*},{ϵ}_7^{*}]=-{ϵ}_{10}^{*},[{ϵ}_2^{*},{ϵ}_8^{*}]=2{ϵ}_9^{*},[{ϵ}_2^{*},{ϵ}_9^{*}]={ϵ}_{10}^{*}.\hfill \end{array}$$

From multiplication table the type of $D_4$ singularity are 3 and the nilpotency are 3. The torus T is spanned by

$$\begin{array}{cccccc}\hfill & t_1:\xi \left(V\right)⟶\xi \left(V\right)\hfill & \hfill & t_2:\xi \left(V\right)⟶\xi \left(V\right)\hfill & \hfill & t_3:\xi \left(V\right)⟶\xi \left(V\right)\hfill \\ \hfill & {ϵ}_2^{*}⟶0\hfill & \hfill & {ϵ}_2^{*}⟶0\hfill & \hfill & {ϵ}_2^{*}⟶{ϵ}_2^{*}\hfill \\ \hfill & {ϵ}_5^{*}⟶0\hfill & \hfill & {ϵ}_5^{*}⟶{ϵ}_5^{*}\hfill & \hfill & {ϵ}_5^{*}⟶0\hfill \\ \hfill & {ϵ}_6^{*}⟶{ϵ}_6^{*}+{ϵ}_8^{*}\hfill & \hfill & {ϵ}_6^{*}⟶-{ϵ}_8^{*}\hfill & \hfill & {ϵ}_6^{*}⟶-{ϵ}_8^{*}\hfill \\ \hfill & {ϵ}_7^{*}⟶2{ϵ}_7^{*}+2{ϵ}_9^{*}\hfill & \hfill & {ϵ}_7^{*}⟶-{ϵ}_7^{*}-2{ϵ}_9^{*}\hfill & \hfill & {ϵ}_7^{*}⟶-2{ϵ}_9^{*}\hfill \\ \hfill & {ϵ}_8^{*}⟶2{ϵ}_6^{*}+2{ϵ}_8^{*}\hfill & \hfill & {ϵ}_8^{*}⟶-2{ϵ}_6^{*}-{ϵ}_8^{*}\hfill & \hfill & {ϵ}_8^{*}⟶-2{ϵ}_6^{*}-{ϵ}_8^{*}\hfill \\ \hfill & {ϵ}_9^{*}⟶{ϵ}_7^{*}+{ϵ}_9^{*}\hfill & \hfill & {ϵ}_9^{*}⟶-{ϵ}_7^{*}\hfill & \hfill & {ϵ}_9^{*}⟶-{ϵ}_7^{*}+{ϵ}_9^{*}\hfill \\ \hfill & {ϵ}_{10}^{*}⟶0\hfill & \hfill & {ϵ}_{10}^{*}⟶{ϵ}_{10}^{*}\hfill & \hfill & {ϵ}_{10}^{*}⟶3{ϵ}_{10}^{*}.\hfill \end{array}$$

After simple calculation we obtain

$$\begin{array}{cc}\hfill & \xi \left(V\right)={\xi }^{3{\alpha }_1-2{\alpha }_2-{\alpha }_3}\oplus {\xi }^{3{\alpha }_1-2{\alpha }_2-2{\alpha }_3}\oplus {\xi }^{{\alpha }_2+3{\alpha }_3}\oplus {\xi }^{{\alpha }_2+2{\alpha }_3}\oplus {\xi }^{{\alpha }_2+{\alpha }_3}\oplus {\xi }^{{\alpha }_2}\oplus {\xi }^{{\alpha }_3}\hfill \\ \hfill & =\mathbb{C}({ϵ}_7^{*}\oplus {ϵ}_9^{*})\oplus \mathbb{C}({ϵ}_6^{*}+{ϵ}_8^{*})\oplus \mathbb{C}{ϵ}_{10}^{*}\oplus \mathbb{C}(\frac{-{ϵ}_7^{*}}{2}+{ϵ}_9^{*})\oplus (-2{ϵ}_6^{*}+{ϵ}_8^{*})\oplus \mathbb{C}{ϵ}_5^{*}\oplus \mathbb{C}{ϵ}_2^{*}\hfill \end{array}$$

Therefore, T-minimal system of generators are $({ϵ}_2^{*},{ϵ}_5^{*},{ϵ}_6^{*}+{ϵ}_8^{*})$. So the generalized Cartan matrix are

$$C_2^2\left(D_4\right)=\left(\begin{array}{ccc}2& -3& -1\\ 0& 2& 0\\ -1& 0& 2\end{array}\right).$$

For $k\ge 5$, the basis of Lie algebra ${\mathcal{L}}_2^2\left(V\right)$ are:

$$\begin{array}{cc}& {ϵ}_1^{*}=x_1{\partial }_1-x_2{\partial }_2,{ϵ}_2^{*}=x_1{\partial }_2,{ϵ}_3^{*}=x_2{\partial }_2,{ϵ}_4^{*}=x_2^{k-3}{\partial }_1,{ϵ}_5^{*}=x_2^2{\partial }_2,\hfill \\ & {ϵ}_6^{*}=x_2^3{\partial }_2,{ϵ}_7^{*}=x_2^4{\partial }_2,\cdots ,{ϵ}_k^{*}=x_2^{k-3}{\partial }_2,{ϵ}_{k+1}^{*}=x_1^2{\partial }_2,{ϵ}_{k+2}^{*}=x_1{x}_2{\partial }_2,{ϵ}_{k+3}^{*}=x_2^{k-2}{\partial }_2,\hfill \\ & {ϵ}_{k+4}^{*}=x_1^2{\partial }_1,{ϵ}_{k+5}^{*}=x_1{x}_2{\partial }_1,{ϵ}_{k+6}^{*}=x_2^{k-2}{\partial }_1.\hfill \end{array}$$

For $D_5$ singularity, the nilradical $\xi \left(V\right)=\langle {ϵ}_2^{*},{ϵ}_4^{*},{ϵ}_5^{*},{ϵ}_6^{*},\cdots ,{ϵ}_{11}^{*}\rangle$ have following multiplication table:

$$\begin{array}{cc}\hfill & [{ϵ}_2^{*},{ϵ}_4^{*}]=-2{ϵ}_{10}^{*}+{ϵ}_5^{*},[{ϵ}_2^{*},{ϵ}_5^{*}]=-2{ϵ}_7^{*},[{ϵ}_2^{*},{ϵ}_7^{*}]=-{ϵ}_6^{*},[{ϵ}_2^{*},{ϵ}_9^{*}]={ϵ}_6^{*},[{ϵ}_4^{*},{ϵ}_5^{*}]=2{ϵ}_{11}^{*},\hfill \\ \hfill & [{ϵ}_4^{*},{ϵ}_7^{*}]=-{ϵ}_8^{*},[{ϵ}_4^{*},{ϵ}_{10}^{*}]=-{ϵ}_{11}^{*}.\hfill \end{array}$$

From multiplication table, the type of $D_5$ singularity is 3 and the nilpotency is 3. The torus T is spanned by

$$\begin{array}{cccc}\hfill & t_1:\xi \left(V\right)⟶\xi \left(V\right)\hfill & \hfill & t_2:\xi \left(V\right)⟶\xi \left(V\right)\hfill \\ \hfill & {ϵ}_2^{*}⟶0\hfill & \hfill & {ϵ}_2^{*}⟶0\hfill \\ \hfill & {ϵ}_4^{*}⟶0\hfill & \hfill & {ϵ}_4^{*}⟶{ϵ}_4^{*}\hfill \\ \hfill & {ϵ}_5^{*}⟶{ϵ}_5^{*}\hfill & \hfill & {ϵ}_5^{*}⟶{ϵ}_5^{*}\hfill \\ \hfill & {ϵ}_6^{*}⟶3{ϵ}_6^{*}\hfill & \hfill & {ϵ}_6^{*}⟶{ϵ}_6^{*}\hfill \\ \hfill & {ϵ}_7^{*}⟶2{ϵ}_7^{*}\hfill & \hfill & {ϵ}_7^{*}⟶{ϵ}_7^{*}\hfill \\ \hfill & {ϵ}_8^{*}⟶2{ϵ}_8^{*}\hfill & \hfill & {ϵ}_8^{*}⟶2{ϵ}_8^{*}\hfill \\ \hfill & {ϵ}_9^{*}⟶2{ϵ}_9^{*}\hfill & \hfill & {ϵ}_9^{*}⟶{ϵ}_9^{*}\hfill \\ \hfill & {ϵ}_{10}^{*}⟶{ϵ}_{10}^{*}\hfill & \hfill & {ϵ}_{10}^{*}⟶{ϵ}_{10}^{*}\hfill \\ \hfill & {ϵ}_{11}^{*}⟶{ϵ}_{11}^{*}\hfill & \hfill & {ϵ}_{11}^{*}⟶2{ϵ}_{11}^{*}.\hfill \end{array}$$

After simple calculation we obtain

$$\begin{array}{cc}\hfill & \xi \left(V\right)={\xi }^{{\alpha }_1}\oplus {\xi }^{{\alpha }_2}\oplus {\xi }^{{\alpha }_1+{\alpha }_2}\oplus {\xi }^{3{\alpha }_1+{\alpha }_2}\oplus {\xi }^{2{\alpha }_1+{\alpha }_2}\oplus {\xi }^{2{\alpha }_1+2{\alpha }_2}\oplus {\xi }^{{\alpha }_1+2{\alpha }_2}\hfill \\ \hfill & =\mathbb{C}{ϵ}_2^{*}\oplus \mathbb{C}{ϵ}_4^{*}\oplus \mathbb{C}({ϵ}_5^{*}\oplus {ϵ}_{10}^{*})\oplus \mathbb{C}{ϵ}_6^{*}\oplus \mathbb{C}({ϵ}_7^{*}\oplus {ϵ}_9^{*})\oplus \mathbb{C}{ϵ}_8^{*}\oplus \mathbb{C}{ϵ}_{11}^{*}\hfill \end{array}$$

Therefore, T-minimal system of generators are $({ϵ}_2^{*},{ϵ}_4^{*},{ϵ}_5^{*})$. So the generalized Cartan matrix is

$$C_2^2\left(D_5\right)=\left(\begin{array}{ccc}2& -3& -2\\ -2& 2& -1\\ -1& -1& 2\end{array}\right).$$

For $D_6$ singularity, the nilradical $\xi \left(V\right)=\langle {ϵ}_2^{*},{ϵ}_4^{*},{ϵ}_5^{*},{ϵ}_6^{*},\cdots ,{ϵ}_{12}^{*}\rangle$ have following multiplication table:

$$\begin{array}{cc}\hfill & [{ϵ}_2^{*},{ϵ}_4^{*}]={ϵ}_6^{*},[{ϵ}_2^{*},{ϵ}_5^{*}]=-2{ϵ}_8^{*},[{ϵ}_2^{*},{ϵ}_8^{*}]=-{ϵ}_7^{*},[{ϵ}_2^{*},{ϵ}_{10}^{*}]={ϵ}_7^{*},[{ϵ}_2^{*},{ϵ}_{11}^{*}]=-{ϵ}_{10}^{*}+{ϵ}_8^{*},\hfill \\ \hfill & [{ϵ}_2^{*},{ϵ}_{12}^{*}]={ϵ}_9^{*},[{ϵ}_4^{*},{ϵ}_5^{*}]=3{ϵ}_{12}^{*},[{ϵ}_4^{*},{ϵ}_8^{*}]=-{ϵ}_9^{*},[{ϵ}_4^{*},{ϵ}_{11}^{*}]=-{ϵ}_{12}^{*},[{ϵ}_5^{*},{ϵ}_6^{*}]=-{ϵ}_9^{*}.\hfill \end{array}$$

From multiplication table, the type of $D_6$ singularity is 4 and the nilpotency is 2. The torus T is spanned by

$$\begin{array}{cccccc}\hfill & t_1:\xi \left(V\right)⟶\xi \left(V\right)\hfill & \hfill & t_2:\xi \left(V\right)⟶\xi \left(V\right)\hfill & \hfill & t_3:\xi \left(V\right)⟶\xi \left(V\right)\hfill \\ \hfill & {ϵ}_2^{*}⟶{ϵ}_2^{*}\hfill & \hfill & {ϵ}_2^{*}⟶0\hfill & \hfill & {ϵ}_2^{*}⟶0\hfill \\ \hfill & {ϵ}_4^{*}⟶0\hfill & \hfill & {ϵ}_4^{*}⟶{ϵ}_4^{*}\hfill & \hfill & {ϵ}_4^{*}⟶0\hfill \\ \hfill & {ϵ}_5^{*}⟶0\hfill & \hfill & {ϵ}_5^{*}⟶0\hfill & \hfill & {ϵ}_5^{*}⟶{ϵ}_5^{*}\hfill \\ \hfill & {ϵ}_6^{*}⟶{ϵ}_6^{*}\hfill & \hfill & {ϵ}_6^{*}⟶{ϵ}_6^{*}\hfill & \hfill & {ϵ}_6^{*}⟶0\hfill \\ \hfill & {ϵ}_7^{*}⟶2{ϵ}_7^{*}\hfill & \hfill & {ϵ}_7^{*}⟶0\hfill & \hfill & {ϵ}_7^{*}⟶{ϵ}_7^{*}\hfill \\ \hfill & {ϵ}_8^{*}⟶{ϵ}_8^{*}\hfill & \hfill & {ϵ}_8^{*}⟶0\hfill & \hfill & {ϵ}_8^{*}⟶{ϵ}_8^{*}\hfill \\ \hfill & {ϵ}_9^{*}⟶{ϵ}_9^{*}\hfill & \hfill & {ϵ}_9^{*}⟶{ϵ}_9^{*}\hfill & \hfill & {ϵ}_9^{*}⟶{ϵ}_9^{*}\hfill \\ \hfill & {ϵ}_{10}^{*}⟶{ϵ}_{10}^{*}\hfill & \hfill & {ϵ}_{10}^{*}⟶0\hfill & \hfill & {ϵ}_{10}^{*}⟶{ϵ}_{10}^{*}\hfill \\ \hfill & {ϵ}_{11}^{*}⟶0\hfill & \hfill & {ϵ}_{11}^{*}⟶0\hfill & \hfill & {ϵ}_{11}^{*}⟶{ϵ}_{11}^{*}\hfill \\ \hfill & {ϵ}_{12}^{*}⟶0\hfill & \hfill & {ϵ}_{12}^{*}⟶{ϵ}_{12}^{*}\hfill & \hfill & {ϵ}_{12}^{*}⟶{ϵ}_{12}^{*}.\hfill \end{array}$$

After a simple calculation, we obtain

$$\begin{array}{cc}\hfill & \xi \left(V\right)={\xi }^{{\beta }_1}\oplus {\xi }^{{\beta }_2}\oplus {\xi }^{{\beta }_3}\oplus {\xi }^{{\beta }_1+{\beta }_2}\oplus {\xi }^{2{\beta }_1+{\beta }_3}\oplus {\xi }^{{\beta }_1+{\beta }_3}\oplus {\xi }^{{\beta }_1+{\beta }_2+{\beta }_3}\oplus {\xi }^{{\beta }_2+{\beta }_3}\hfill \\ \hfill & =\mathbb{C}{ϵ}_2^{*}\oplus \mathbb{C}{ϵ}_4^{*}\oplus \mathbb{C}({ϵ}_5^{*}\oplus {ϵ}_{11}^{*})\oplus \mathbb{C}{ϵ}_6^{*}\oplus \mathbb{C}{ϵ}_7^{*}\oplus \mathbb{C}({ϵ}_8^{*}\oplus {ϵ}_{10}^{*})\oplus \mathbb{C}{ϵ}_9^{*}\oplus \mathbb{C}{ϵ}_{12}^{*}\hfill \end{array}$$

Therefore, the T-minimal system of generators are $({ϵ}_2^{*},{ϵ}_4^{*},{ϵ}_5^{*},{ϵ}_{11}^{*})$. So the generalized Cartan matrix is

$$C_2^2\left(D_6\right)=\left(\begin{array}{cccc}2& -1& -2& -2\\ -1& 2& -1& -1\\ -1& -1& 2& 0\\ -1& -1& 0& 2\end{array}\right).$$

For $D_7$ singularity, the nilradical $\xi \left(V\right)=\langle {ϵ}_2^{*},{ϵ}_4^{*},{ϵ}_5^{*},{ϵ}_6^{*},\cdots ,{ϵ}_{13}^{*}\rangle$ have following multiplication table:

$$\begin{array}{cc}\hfill & [{ϵ}_2^{*},{ϵ}_4^{*}]={ϵ}_7^{*},[{ϵ}_2^{*},{ϵ}_5^{*}]=-2{ϵ}_9^{*},[{ϵ}_2^{*},{ϵ}_9^{*}]=-{ϵ}_8^{*},[{ϵ}_2^{*},{ϵ}_{11}^{*}]={ϵ}_8^{*},[{ϵ}_2^{*},{ϵ}_{12}^{*}]=-{ϵ}_{11}^{*}+{ϵ}_9^{*},\hfill \\ \hfill & [{ϵ}_2^{*},{ϵ}_{13}^{*}]={ϵ}_{10}^{*},[{ϵ}_4^{*},{ϵ}_5^{*}]=4{ϵ}_{13}^{*},[{ϵ}_4^{*},{ϵ}_9^{*}]=-{ϵ}_{10}^{*},[{ϵ}_4^{*},{ϵ}_{12}^{*}]=-{ϵ}_{13}^{*},\hfill \\ \hfill & [{ϵ}_5^{*},{ϵ}_6^{*}]=-{ϵ}_7^{*},[{ϵ}_5^{*},{ϵ}_7^{*}]=-2{ϵ}_{10}^{*}.\hfill \end{array}$$

From multiplication table the type of $D_7$ singularity is 5 and the nilpotency is 2. The torus T is spanned by

$$\begin{array}{cccccc}\hfill & t_1:\xi \left(V\right)⟶\xi \left(V\right)\hfill & \hfill & t_2:\xi \left(V\right)⟶\xi \left(V\right)\hfill & \hfill & t_3:\xi \left(V\right)⟶\xi \left(V\right)\hfill \\ \hfill & {ϵ}_2^{*}⟶{ϵ}_2^{*}\hfill & \hfill & {ϵ}_2^{*}⟶0\hfill & \hfill & {ϵ}_2^{*}⟶0\hfill \\ \hfill & {ϵ}_4^{*}⟶0\hfill & \hfill & {ϵ}_4^{*}⟶{ϵ}_4^{*}\hfill & \hfill & {ϵ}_4^{*}⟶0\hfill \\ \hfill & {ϵ}_5^{*}⟶{ϵ}_5^{*}\hfill & \hfill & {ϵ}_5^{*}⟶0\hfill & \hfill & {ϵ}_5^{*}⟶{ϵ}_5^{*}\hfill \\ \hfill & {ϵ}_6^{*}⟶{ϵ}_6^{*}\hfill & \hfill & {ϵ}_6^{*}⟶{ϵ}_6^{*}\hfill & \hfill & {ϵ}_6^{*}⟶-{ϵ}_6^{*}\hfill \\ \hfill & {ϵ}_7^{*}⟶{ϵ}_7^{*}\hfill & \hfill & {ϵ}_7^{*}⟶{ϵ}_7^{*}\hfill & \hfill & {ϵ}_7^{*}⟶0\hfill \\ \hfill & {ϵ}_8^{*}⟶2{ϵ}_8^{*}\hfill & \hfill & {ϵ}_8^{*}⟶0\hfill & \hfill & {ϵ}_8^{*}⟶{ϵ}_8^{*}\hfill \\ \hfill & {ϵ}_9^{*}⟶{ϵ}_9^{*}\hfill & \hfill & {ϵ}_9^{*}⟶0\hfill & \hfill & {ϵ}_9^{*}⟶{ϵ}_9^{*}\hfill \\ \hfill & {ϵ}_{10}^{*}⟶{ϵ}_{10}^{*}\hfill & \hfill & {ϵ}_{10}^{*}⟶{ϵ}_{10}^{*}\hfill & \hfill & {ϵ}_{10}^{*}⟶{ϵ}_{10}^{*}\hfill \\ \hfill & {ϵ}_{11}^{*}⟶{ϵ}_{11}^{*}\hfill & \hfill & {ϵ}_{11}^{*}⟶0\hfill & \hfill & {ϵ}_{11}^{*}⟶{ϵ}_{11}^{*}\hfill \\ \hfill & {ϵ}_{12}^{*}⟶0\hfill & \hfill & {ϵ}_{12}^{*}⟶0\hfill & \hfill & {ϵ}_{12}^{*}⟶{ϵ}_{12}^{*}\hfill \\ \hfill & {ϵ}_{13}^{*}⟶0\hfill & \hfill & {ϵ}_{12}^{*}⟶{ϵ}_{13}^{*}\hfill & \hfill & {ϵ}_{13}^{*}⟶{ϵ}_{13}^{*}.\hfill \end{array}$$

After a simple calculation, we obtain

$$\begin{array}{cc}\hfill & \xi \left(V\right)={\xi }^{{\alpha }_1}\oplus {\xi }^{{\alpha }_2}\oplus {\xi }^{{\alpha }_1+{\alpha }_3}\oplus {\xi }^{{\alpha }_1+{\alpha }_2-{\alpha }_3}\oplus {\xi }^{{\alpha }_1+{\alpha }_2}\oplus {\xi }^{2{\alpha }_1+{\alpha }_3}\oplus {\xi }^{{\alpha }_1+{\alpha }_2+{\alpha }_3}\oplus {\xi }^{{\alpha }_3}\oplus {\xi }^{{\alpha }_2+{\alpha }_3}\hfill \\ \hfill & =\mathbb{C}{ϵ}_2^{*}\oplus \mathbb{C}{ϵ}_4^{*}\oplus \mathbb{C}({ϵ}_5^{*}\oplus {ϵ}_9^{*}\oplus {ϵ}_{11}^{*})\oplus \mathbb{C}{ϵ}_6^{*}\oplus \mathbb{C}{ϵ}_7^{*}\oplus \mathbb{C}{ϵ}_8^{*}\oplus \mathbb{C}{ϵ}_{10}^{*}\oplus \mathbb{C}{ϵ}_{12}^{*}\oplus \mathbb{C}{ϵ}_{13}^{*}\hfill \end{array}$$

Therefore, the T-minimal system of generators are $({ϵ}_2^{*},{ϵ}_4^{*},{ϵ}_5^{*},{ϵ}_6^{*},{ϵ}_{12}^{*})$. So the generalized Cartan matrix is

$$C_2^2\left(D_7\right)=\left(\begin{array}{cccc}2& -1& -2& 0\\ -1& 2& -1& 0\\ -1& -1& 2& -2\\ 0& 0& -1& 2\end{array}\right).$$

Case (a). For $k\ge 8$ and k is even the nilradical $\xi \left(V\right)=\langle {ϵ}_2^{*},{ϵ}_4^{*},{ϵ}_5^{*},{ϵ}_6^{*},\cdots ,{ϵ}_{k+6}^{*}\rangle$ have following multiplication table:

$$\begin{array}{cc}\hfill & [{ϵ}_2^{*},{ϵ}_4^{*}]={ϵ}_k^{*},[{ϵ}_2^{*},{ϵ}_5^{*}]=-2{ϵ}_{k+2}^{*},[{ϵ}_2^{*},{ϵ}_{k+2}^{*}]=-{ϵ}_{k+1}^{*},[{ϵ}_2^{*},{ϵ}_{k+4}^{*}]={ϵ}_{k+1}^{*},[{ϵ}_2^{*},{ϵ}_{k+5}^{*}]=\hfill \\ & {ϵ}_{k+2}^{*}-{ϵ}_{k+4}^{*},[{ϵ}_2^{*},{ϵ}_{k+6}^{*}]={ϵ}_{k+3}^{*},[{ϵ}_4^{*},{ϵ}_5^{*}]=(k-3){ϵ}_{k+6}^{*},[{ϵ}_4^{*},{ϵ}_{k+2}^{*}]=-{ϵ}_{k+3}^{*},[{ϵ}_4^{*},{ϵ}_{k+5}^{*}]=\hfill \\ & -{ϵ}_{k+6}^{*},[{ϵ}_5^{*},{ϵ}_6^{*}]=-{ϵ}_7^{*},[{ϵ}_5^{*},{ϵ}_7^{*}]=-2{ϵ}_8^{*},[{ϵ}_5^{*},{ϵ}_8^{*}]=-3{ϵ}_9^{*},\cdots ,[{ϵ}_5^{*},{ϵ}_{k-1}^{*}]=-(k-6){ϵ}_k^{*},\hfill \\ \hfill & [{ϵ}_5^{*},{ϵ}_k^{*}]=-(k-5){ϵ}_{k+3}^{*},[{ϵ}_6^{*},{ϵ}_7^{*}]=-{ϵ}_9^{*},[{ϵ}_6^{*},{ϵ}_8^{*}]=-2{ϵ}_{10}^{*},[{ϵ}_6^{*},{ϵ}_9^{*}]=-3{ϵ}_{11}^{*},\cdots ,\hfill \\ \hfill & [{ϵ}_6^{*},{ϵ}_{k-2}^{*}]=-(k-8){ϵ}_k^{*},[{ϵ}_6^{*},{ϵ}_{k-1}^{*}]=-(k-7){ϵ}_{k+3}^{*},[{ϵ}_7^{*},{ϵ}_8^{*}]=-{ϵ}_{11}^{*},[{ϵ}_7^{*},{ϵ}_9^{*}]=-2{ϵ}_{12}^{*},\hfill \\ & [{ϵ}_7^{*},{ϵ}_{10}^{*}]=-3{ϵ}_{13}^{*},\cdots ,[{ϵ}_7^{*},{ϵ}_{k-3}^{*}]=-(k-10){ϵ}_k^{*},[{ϵ}_7^{*},{ϵ}_{k-2}^{*}]=-(k-9){ϵ}_{k+3}^{*},\hfill \\ \hfill & ⋮\hfill \\ \hfill & [{ϵ}_{\frac{k+2}{2}}^{*},{ϵ}_{\frac{k+2}{2}+1}^{*}]=-{ϵ}_{k-1}^{*},[{ϵ}_{\frac{k+2}{2}}^{*},{ϵ}_{\frac{k+2}{2}+2}^{*}]=-2{ϵ}_k^{*},[{ϵ}_{\frac{k+2}{2}}^{*},{ϵ}_{\frac{k+2}{2}+3}^{*}]=-3{ϵ}_{k+3}^{*},\hfill \\ & [{ϵ}_{\frac{k+4}{2}}^{*},{ϵ}_{\frac{k+4}{2}+1}^{*}]=-{ϵ}_{k+3}^{*}.\hfill \end{array}$$

Case (b). For $k\ge 9$ and k is odd and, the nilradical $\xi \left(V\right)=\langle {ϵ}_2^{*},{ϵ}_4^{*},{ϵ}_5^{*},{ϵ}_6^{*},\cdots ,{ϵ}_{k+6}^{*}\rangle$ have following multiplication table:

$$\begin{array}{cc}\hfill & [{ϵ}_2^{*},{ϵ}_4^{*}]={ϵ}_k^{*},[{ϵ}_2^{*},{ϵ}_5^{*}]=-2{ϵ}_{k+2}^{*},[{ϵ}_2^{*},{ϵ}_{k+2}^{*}]=-{ϵ}_{k+1}^{*},[{ϵ}_2^{*},{ϵ}_{k+4}^{*}]={ϵ}_{k+1}^{*},[{ϵ}_2^{*},{ϵ}_{k+5}^{*}]=\hfill \\ & {ϵ}_{k+2}^{*}-{ϵ}_{k+4}^{*},[{ϵ}_2^{*},{ϵ}_{k+6}^{*}]={ϵ}_{k+3}^{*},[{ϵ}_4^{*},{ϵ}_5^{*}]=(k-3){ϵ}_{k+6}^{*},[{ϵ}_4^{*},{ϵ}_{k+2}^{*}]=-{ϵ}_{k+3}^{*},[{ϵ}_4^{*},{ϵ}_{k+5}^{*}]=\hfill \\ \hfill & -{ϵ}_{k+6}^{*},[{ϵ}_5^{*},{ϵ}_6^{*}]=-{ϵ}_7^{*},[{ϵ}_5^{*},{ϵ}_7^{*}]=-2{ϵ}_8^{*},[{ϵ}_5^{*},{ϵ}_8^{*}]=-3{ϵ}_9^{*},\cdots ,[{ϵ}_5^{*},{ϵ}_{k-1}^{*}]=-(k-6){ϵ}_k^{*},\hfill \\ \hfill & [{ϵ}_5^{*},{ϵ}_k^{*}]=-(k-5){ϵ}_{k+3}^{*},[{ϵ}_6^{*},{ϵ}_7^{*}]=-{ϵ}_9^{*},[{ϵ}_6^{*},{ϵ}_8^{*}]=-2{ϵ}_{10}^{*},[{ϵ}_6^{*},{ϵ}_9^{*}]=-3{ϵ}_{11}^{*},\cdots ,[{ϵ}_6^{*},{ϵ}_{k-2}^{*}]\hfill \\ \hfill & =-(k-8){ϵ}_k^{*},[{ϵ}_6^{*},{ϵ}_{k-1}^{*}]=-(k-7){ϵ}_{k+3}^{*},[{ϵ}_7^{*},{ϵ}_8^{*}]=-{ϵ}_{11}^{*},[{ϵ}_7^{*},{ϵ}_9^{*}]=-2{ϵ}_{12}^{*},[{ϵ}_7^{*},{ϵ}_{10}^{*}]\hfill \\ \hfill & =-3{ϵ}_{13}^{*},\cdots ,[{ϵ}_7^{*},{ϵ}_{k-3}^{*}]=-(k-10){ϵ}_k^{*},[{ϵ}_7^{*},{ϵ}_{k-2}^{*}]=-(k-9){ϵ}_{k+3}^{*},\hfill \\ \hfill & ⋮\hfill \\ \hfill & [{ϵ}_{\frac{k+1}{2}}^{*},{ϵ}_{\frac{k+1}{2}+1}^{*}]=-{ϵ}_{k-2}^{*},[{ϵ}_{\frac{k+1}{2}}^{*},{ϵ}_{\frac{k+1}{2}+2}^{*}]=-2{ϵ}_{k-1}^{*},[{ϵ}_{\frac{k+1}{2}}^{*},{ϵ}_{\frac{k+1}{2}+3}^{*}]=-3{ϵ}_k^{*},\hfill \\ \hfill & [{ϵ}_{\frac{k+1}{2}}^{*},{ϵ}_{\frac{k+1}{2}+4}^{*}]=-4{ϵ}_{k+3}^{*},[{ϵ}_{\frac{k+3}{2}}^{*},{ϵ}_{\frac{k+3}{2}+1}^{*}]=-{ϵ}_k^{*},[{ϵ}_{\frac{k+3}{2}}^{*},{ϵ}_{\frac{k+3}{2}+2}^{*}]=-2{ϵ}_{k+3}^{*}.\hfill \end{array}$$

From multiplication table the type of $D_k$ ($k\ge 8$) singularity is 5 and the nilpotency is $k-5$. The torus T is spanned by

$$\begin{array}{cccc}\hfill & t_1:\xi \left(V\right)⟶\xi \left(V\right)\hfill & \hfill & t_2:\xi \left(V\right)⟶\xi \left(V\right)\hfill \\ \hfill & {ϵ}_2^{*}⟶{ϵ}_2^{*}\hfill & \hfill & {ϵ}_2^{*}⟶0\hfill \\ \hfill & {ϵ}_4^{*}⟶0\hfill & \hfill & {ϵ}_4^{*}⟶\frac{{ϵ}_4^{*}}{k-4}\hfill \\ \hfill & {ϵ}_5^{*}⟶\frac{{ϵ}_5^{*}}{k-4}\hfill & \hfill & {ϵ}_5^{*}⟶\frac{2{ϵ}_5^{*}}{k-4}\hfill \\ \hfill & {ϵ}_6^{*}⟶\frac{2{ϵ}_6^{*}}{k-4}\hfill & \hfill & {ϵ}_6^{*}⟶\frac{3{ϵ}_6^{*}}{k-4}\hfill \\ \hfill & {ϵ}_7^{*}⟶\frac{3{ϵ}_7^{*}}{k-4}\hfill & \hfill & {ϵ}_7^{*}⟶\frac{4{ϵ}_7^{*}}{k-4}\hfill \\ \hfill & ⋮\hfill & \hfill & ⋮\hfill \\ \hfill & {ϵ}_k^{*}⟶{ϵ}_k^{*}\hfill & \hfill & {ϵ}_k^{*}⟶{ϵ}_k^{*}\hfill \\ \hfill & {ϵ}_{k+1}^{*}⟶\frac{(2k-7){ϵ}_{k+1}^{*}}{k-4}\hfill & \hfill & {ϵ}_{k+1}^{*}⟶\frac{{ϵ}_{k+1}^{*}}{k-4}\hfill \\ \hfill & {ϵ}_{k+2}^{*}⟶\frac{(k-3){ϵ}_{k+2}^{*}}{k-4}\hfill & \hfill & {ϵ}_{k+2}^{*}⟶\frac{{ϵ}_{k+2}^{*}}{k-4}\hfill \end{array}$$

$$\begin{array}{cccc}\hfill & {ϵ}_{k+3}^{*}⟶\frac{(k-3){ϵ}_{k+3}^{*}}{k-4}\hfill & \hfill & {ϵ}_{k+3}^{*}⟶\frac{(k-3){ϵ}_{k+3}^{*}}{k-4}\hfill \\ \hfill & {ϵ}_{k+4}^{*}⟶\frac{(k-3){ϵ}_{k+4}^{*}}{k-4}\hfill & \hfill & {ϵ}_{k+4}^{*}⟶\frac{{ϵ}_{k+4}^{*}}{k-4}\hfill \\ \hfill & {ϵ}_{k+5}^{*}⟶\frac{{ϵ}_{k+5}^{*}}{k-4}\hfill & \hfill & {ϵ}_{k+5}^{*}⟶\frac{{ϵ}_{k+5}^{*}}{k-4}\hfill \\ \hfill & {ϵ}_{k+6}^{*}⟶\frac{{ϵ}_{k+6}^{*}}{k-4}\hfill & \hfill & {ϵ}_{k+6}^{*}⟶\frac{(k-3){ϵ}_{k+6}^{*}}{k-4}.\hfill \end{array}$$

After a simple calculation, we obtain

$$\begin{array}{cc}\hfill & \xi \left(V\right)={\xi }^{{\alpha }_1}\oplus {\xi }^{{\alpha }_2}\oplus {\xi }^{\frac{{\alpha }_1+{\alpha }_2}{k-4}}\oplus {\xi }^{\frac{2({\alpha }_1+{\alpha }_2)}{k-4}}\oplus {\xi }^{\frac{3({\alpha }_1+{\alpha }_2)}{k-4}}\oplus \cdots \oplus {\xi }^{{\alpha }_1+{\alpha }_2}\oplus {\xi }^{\frac{(2k-7){\alpha }_1}{k-4}+\frac{{\alpha }_2}{k-4}}\oplus {\xi }^{\frac{(k-3){\alpha }_1}{k-4}+\frac{{\alpha }_2}{k-4}}\hfill \\ \hfill & \oplus {\xi }^{\frac{(k-3){\alpha }_1}{k-4}+\frac{(k-3){\alpha }_2}{k-4}}\oplus {\xi }^{\frac{(k-3){\alpha }_1}{k-4}+\frac{{\alpha }_2}{k-4}}\oplus {\xi }^{\frac{{\alpha }_1}{k-4}+\frac{(k-3){\alpha }_2}{k-4}}\hfill \\ \hfill & =\mathbb{C}{ϵ}_2^{*}\oplus \mathbb{C}{ϵ}_4^{*}\oplus \mathbb{C}({ϵ}_5^{*}\oplus {ϵ}_{k+5}^{*})\oplus \mathbb{C}{ϵ}_6^{*}\oplus \mathbb{C}{ϵ}_7^{*}\oplus \cdots \oplus \mathbb{C}{ϵ}_k^{*}\oplus \mathbb{C}{ϵ}_{k+1}^{*}\oplus \mathbb{C}{ϵ}_{k+2}^{*}\oplus \mathbb{C}{ϵ}_{k+3}^{*}\hfill \\ \hfill & \oplus \mathbb{C}{ϵ}_{k+4}^{*}\oplus \mathbb{C}{ϵ}_{k+5}^{*}\oplus \mathbb{C}{ϵ}_{k+6}^{*}\hfill \end{array}$$

Therefore, the T-minimal system of generators are $({ϵ}_2^{*},{ϵ}_4^{*},{ϵ}_5^{*},{ϵ}_{k+5}^{*},{ϵ}_6^{*})$. So the generalized Cartan matrix is

• Case (a). For $k\ge 8$ and k is even we have

  $$C_2^2\left(D_k\right)=\left(\begin{array}{ccccc}2& -1& -2& -2& 0\\ -1& 2& -1& -1& 0\\ -1& -1& 2& 0& -k+5\\ -1& -1& 0& 2& -k+5\\ 0& 0& \frac{-k+6}{2}& \frac{-k+6}{2}& 2\end{array}\right).$$
• Case (b). For $k\ge 9$ and k is odd we have

  $$C_2^2\left(D_k\right)=\left(\begin{array}{ccccc}2& -1& -2& -2& 0\\ -1& 2& -1& -1& 0\\ -1& -1& 2& 0& -k+5\\ -1& -1& 0& 2& -k+5\\ 0& 0& \frac{-k+5}{2}& \frac{-k+5}{2}& 2\end{array}\right).$$

□

Proposition 3. 

Let $V=\{(x_1,x_2)\in {\mathbb{C}}^2:x_1^3+x_2^4=0\}$ be the ${ϵ}_6^{*}$ singularity. Then

$$C_2^2\left(E_6\right)=\left(\begin{array}{ccc}2& -1& -2\\ -1& 2& -2\\ -2& -3& 2\end{array}\right)$$

Proof. 

The algebra ${\mathcal{M}}_2^2\left(V\right)$ has a basis of:

$$\begin{array}{cc}\hfill & \{1,x_1,x_2,x_2^2,x_2^3,x_1{x}_2,x_1{x}_2^2,x_1^2,x_1^2{x}_2\}.\hfill \end{array}$$

Basis of Lie algebra ${\mathcal{L}}_2^2\left(V\right)$ define as:

$$\begin{array}{cc}\hfill {ϵ}_1^{*}& =x_2{\partial }_2;{ϵ}_2^{*}=x_1{x}_2{\partial }_1-x_2^2{\partial }_2,{ϵ}_3^{*}=x_1{\partial }_1-2x_2{\partial }_2,{ϵ}_4^{*}=x_2^2{\partial }_1,{ϵ}_5^{*}=x_1{\partial }_2,{ϵ}_6^{*}=x_1^2{\partial }_1,\hfill \\ & {ϵ}_7^{*}=x_2^2{\partial }_2,{ϵ}_8^{*}=x_1^2{\partial }_2,{ϵ}_9^{*}=-x_1^2{\partial }_1+x_1{x}_2{\partial }_2,{ϵ}_{10}^{*}=x_1^2{x}_2{\partial }_2,{ϵ}_{11}^{*}=x_1{x}_2^2{\partial }_2,{ϵ}_{12}^{*}=x_2^3{\partial }_2,\hfill \\ & {ϵ}_{13}^{*}=x_1^2{x}_2{\partial }_1,{ϵ}_{14}^{*}=x_1{x}_2^2{\partial }_1,{ϵ}_{15}^{*}=x_2^3{\partial }_1.\hfill \end{array}$$

The nilradical $\xi \left(V\right)=\langle {ϵ}_2^{*},{ϵ}_4^{*},{ϵ}_5^{*},{ϵ}_6^{*},\cdots ,{ϵ}_{15}^{*}\rangle$ have the following multiplication table:

$$\begin{array}{cc}\hfill & [{ϵ}_2^{*},{ϵ}_4^{*}]=3{ϵ}_{15}^{*},[{ϵ}_2^{*},{ϵ}_5^{*}]=-2{ϵ}_6^{*}-3{ϵ}_9^{*},[{ϵ}_2^{*},{ϵ}_6^{*}]=-{ϵ}_{13}^{*},[{ϵ}_2^{*},{ϵ}_7^{*}]={ϵ}_{14}^{*},[{ϵ}_2^{*},{ϵ}_8^{*}]=-4{ϵ}_{10}^{*},\hfill \\ & [{ϵ}_2^{*},{ϵ}_9^{*}]=-2{ϵ}_{11}^{*}+2{ϵ}_{13}^{*},[{ϵ}_4^{*},{ϵ}_5^{*}]=2{ϵ}_2^{*}+{ϵ}_7^{*},[{ϵ}_4^{*},{ϵ}_6^{*}]=-2{ϵ}_{14}^{*},[{ϵ}_4^{*},{ϵ}_7^{*}]=2{ϵ}_{15}^{*},\hfill \\ \hfill & [{ϵ}_4^{*},{ϵ}_8^{*}]=-2{ϵ}_{11}^{*}+2{ϵ}_{13}^{*},[{ϵ}_4^{*},{ϵ}_9^{*}]=-{ϵ}_{12}^{*}+4{ϵ}_{14}^{*},[{ϵ}_5^{*},{ϵ}_6^{*}]={ϵ}_8^{*},[{ϵ}_5^{*},{ϵ}_7^{*}]=-2{ϵ}_6^{*}-2{ϵ}_9^{*},\hfill \\ & [{ϵ}_5^{*},{ϵ}_9^{*}]=-2{ϵ}_8^{*},[{ϵ}_5^{*},{ϵ}_{11}^{*}]=-2{ϵ}_{10}^{*},[{ϵ}_5^{*},{ϵ}_{12}^{*}]=-3{ϵ}_{11}^{*},[{ϵ}_5^{*},{ϵ}_{13}^{*}]={ϵ}_{10}^{*},[{ϵ}_5^{*},{ϵ}_{14}^{*}]={ϵ}_{11}^{*}-2{ϵ}_{13}^{*},\hfill \\ \hfill & [{ϵ}_5^{*},{ϵ}_{15}^{*}]={ϵ}_{12}^{*}-3{ϵ}_{14}^{*},[{ϵ}_6^{*},{ϵ}_9^{*}]=-{ϵ}_{10}^{*},[{ϵ}_7^{*},{ϵ}_8^{*}]=2{ϵ}_{10}^{*},[{ϵ}_7^{*},{ϵ}_9^{*}]={ϵ}_{11}^{*}.\hfill \end{array}$$

From multiplication table the type of ${ϵ}_6^{*}$ singularity is 3 and the nilpotency is 5. The torus T is spanned by

$$\begin{array}{cccc}\hfill & t_1:\xi \left(V\right)⟶\xi \left(V\right)\hfill & \hfill & t_2:\xi \left(V\right)⟶\xi \left(V\right)\hfill \\ \hfill & {ϵ}_2^{*}⟶{ϵ}_2^{*}\hfill & \hfill & {ϵ}_2^{*}⟶0\hfill \\ \hfill & {ϵ}_4^{*}⟶0\hfill & \hfill & {ϵ}_4^{*}⟶{ϵ}_4^{*}\hfill \\ \hfill & {ϵ}_5^{*}⟶{ϵ}_5^{*}\hfill & \hfill & {ϵ}_5^{*}⟶-{ϵ}_5^{*}\hfill \\ \hfill & {ϵ}_6^{*}⟶2{ϵ}_6^{*}\hfill & \hfill & {ϵ}_6^{*}⟶-{ϵ}_6^{*}\hfill \\ \hfill & {ϵ}_7^{*}⟶{ϵ}_7^{*}\hfill & \hfill & {ϵ}_7^{*}⟶0\hfill \\ \hfill & {ϵ}_8^{*}⟶3{ϵ}_8^{*}\hfill & \hfill & {ϵ}_8^{*}⟶-2{ϵ}_8^{*}\hfill \\ \hfill & {ϵ}_9^{*}⟶2{ϵ}_9^{*}\hfill & \hfill & {ϵ}_9^{*}⟶-{ϵ}_9^{*}\hfill \\ \hfill & {ϵ}_{10}^{*}⟶4{ϵ}_{10}^{*}\hfill & \hfill & {ϵ}_{10}^{*}⟶-2{ϵ}_{10}^{*}\hfill \\ \hfill & {ϵ}_{11}^{*}⟶3{ϵ}_{11}^{*}\hfill & \hfill & {ϵ}_{11}^{*}⟶-{ϵ}_{11}^{*}\hfill \\ \hfill & {ϵ}_{12}^{*}⟶2{ϵ}_{12}^{*}\hfill & \hfill & {ϵ}_{12}^{*}⟶0\hfill \\ \hfill & {ϵ}_{13}^{*}⟶3{ϵ}_{13}^{*}\hfill & \hfill & {ϵ}_{13}^{*}⟶-{ϵ}_{13}^{*}\hfill \\ \hfill & {ϵ}_{14}^{*}⟶2{ϵ}_{14}^{*}\hfill & \hfill & {ϵ}_{14}^{*}⟶0\hfill \\ \hfill & {ϵ}_{15}^{*}⟶{ϵ}_{15}^{*}\hfill & \hfill & {ϵ}_{15}^{*}⟶{ϵ}_{15}^{*}.\hfill \end{array}$$

After a simple calculation, we obtain

$$\begin{array}{cc}\hfill & \xi \left(V\right)={\xi }^{{\alpha }_1}\oplus {\xi }^{{\alpha }_2}\oplus {\xi }^{{\alpha }_1-{\alpha }_2}\oplus {\xi }^{2{\alpha }_1-{\alpha }_2}\oplus {\xi }^{3{\alpha }_1-2{\alpha }_2}\oplus {\xi }^{4{\alpha }_1-2{\alpha }_2}\oplus {\xi }^{3{\alpha }_1-{\alpha }_2}\oplus {\xi }^{2{\alpha }_1}\oplus {\xi }^{{\alpha }_1+{\beta }_2}\hfill \\ \hfill & =\mathbb{C}({ϵ}_2^{*}\oplus {ϵ}_7^{*})\oplus \mathbb{C}{ϵ}_4^{*}\oplus \mathbb{C}{ϵ}_5^{*}\oplus \mathbb{C}({ϵ}_6^{*}\oplus {ϵ}_9^{*})\oplus \mathbb{C}{ϵ}_7^{*}\oplus \mathbb{C}{ϵ}_{10}^{*}\oplus \mathbb{C}({ϵ}_{11}^{*}\oplus {ϵ}_{13}^{*})\oplus \mathbb{C}({ϵ}_{12}^{*}\oplus {ϵ}_{14}^{*})\hfill \\ \hfill & \oplus \mathbb{C}{ϵ}_{15}^{*}.\hfill \end{array}$$

Therefore, T-minimal system of generators are $({ϵ}_2^{*},{ϵ}_4^{*},{ϵ}_5^{*})$. So the generalized Cartan matrix is

$$C_2^2\left(E_6\right)=\left(\begin{array}{ccc}2& -1& -2\\ -1& 2& -2\\ -2& -3& 2\end{array}\right).$$

□

Proposition 4. 

Let $V=\{(x_1,x_2)\in {\mathbb{C}}^2:x_1^3+x_2^3{x}_1=0\}$ be the ${ϵ}_7^{*}$ singularity. Then

$$C_2^2\left(E_7\right)=\left(\begin{array}{ccc}2& -2& -2\\ -1& 2& -3\\ -2& -5& 2\end{array}\right).$$

Proof. 

The basis of algebra ${\mathcal{M}}_2^2\left(V\right)$ is defined as:

$$\begin{array}{cc}\hfill & \{1,x_1,x_2,x_2^2,x_2^3,x_2^4,x_1{x}_2,x_1{x}_2^2,x_1^2,x_1^2{x}_1\}.\hfill \end{array}$$

After these simple calculations, the basis of the Lie algebra ${\mathcal{L}}_2^2\left(V\right)$ is defined as:

$$\begin{array}{cc}\hfill {ϵ}_1^{*}& =x_2{\partial }_2,{ϵ}_2^{*}=x_1{x}_2{\partial }_1-x_2^2{\partial }_2,{ϵ}_3^{*}=x_1{\partial }_1-2x_2{\partial }_2,{ϵ}_4^{*}=x_2^2{\partial }_1,{ϵ}_5^{*}=x_1{\partial }_2,{ϵ}_6^{*}=x_1^2{\partial }_1,\hfill \\ & {ϵ}_7^{*}=x_2^2{\partial }_2,{ϵ}_8^{*}=x_2^3{\partial }_1,{ϵ}_9^{*}=x_1^2{\partial }_2,{ϵ}_{10}^{*}=-x_1^2{\partial }_1+x_1{x}_2{\partial }_2,{ϵ}_{11}^{*}=x_2^3{\partial }_2,{ϵ}_{12}^{*}=x_1^2{x}_2{\partial }_2,\hfill \\ & {ϵ}_{13}^{*}=x_1{x}_2^2{\partial }_2,{ϵ}_{14}^{*}=x_2^4{\partial }_2,{ϵ}_{15}^{*}=x_1^2{x}_2{\partial }_1,{ϵ}_{16}^{*}=x_1{x}_2^2{\partial }_1,{ϵ}_{17}^{*}=x_2^4{\partial }_1.\hfill \end{array}$$

The nilradical $\xi \left(V\right)=\langle {ϵ}_2^{*},{ϵ}_4^{*},{ϵ}_5^{*},{ϵ}_6^{*},\cdots ,{ϵ}_{17}^{*}\rangle$ have following multiplication table:

$$\begin{array}{c}[{ϵ}_2^{*},{ϵ}_5^{*}]=-3{ϵ}_{10}^{*}-2{ϵ}_6^{*},[{ϵ}_2^{*},{ϵ}_6^{*}]=-{ϵ}_{15}^{*},[{ϵ}_2^{*},{ϵ}_7^{*}]={ϵ}_{16}^{*},[{ϵ}_2^{*},{ϵ}_8^{*}]=4{ϵ}_{17}^{*},[{ϵ}_2^{*},{ϵ}_9^{*}]=\hfill \\ -4{ϵ}_{12}^{*},[{ϵ}_2^{*},{ϵ}_{10}^{*}]=-2{ϵ}_{13}^{*}+2{ϵ}_{15}^{*},[{ϵ}_2^{*},{ϵ}_{11}^{*}]={ϵ}_{14}^{*},[{ϵ}_4^{*},{ϵ}_5^{*}]=2{ϵ}_2^{*}+2{ϵ}_7^{*},[{ϵ}_4^{*},{ϵ}_6^{*}]=\hfill \\ -2{ϵ}_{16}^{*},[{ϵ}_4^{*},{ϵ}_7^{*}]=2{ϵ}_8^{*},[{ϵ}_4^{*},{ϵ}_9^{*}]=-2{ϵ}_{13}^{*}+2{ϵ}_{15}^{*},[{ϵ}_4^{*},{ϵ}_{10}^{*}]=-{ϵ}_{11}^{*}+4{ϵ}_{16}^{*},[{ϵ}_4^{*},{ϵ}_{11}^{*}]=\hfill \\ 2{ϵ}_{17}^{*},[{ϵ}_4^{*},{ϵ}_{13}^{*}]=-{ϵ}_{14}^{*},[{ϵ}_4^{*},{ϵ}_{16}^{*}]=-{ϵ}_{17}^{*},[{ϵ}_5^{*},{ϵ}_6^{*}]={ϵ}_9^{*},[{ϵ}_5^{*},{ϵ}_7^{*}]=-2{ϵ}_{10}^{*}-2{ϵ}_6^{*},[{ϵ}_5^{*},{ϵ}_8^{*}]=\hfill \\ {ϵ}_{11}^{*}-3{ϵ}_{16}^{*},[{ϵ}_5^{*},{ϵ}_{10}^{*}]=-2{ϵ}_9^{*},[{ϵ}_5^{*},{ϵ}_{11}^{*}]=-3{ϵ}_{13}^{*},[{ϵ}_5^{*},{ϵ}_{13}^{*}]=-2{ϵ}_{12}^{*},[{ϵ}_5^{*},{ϵ}_{15}^{*}]=\hfill \\ {ϵ}_{12}^{*},[{ϵ}_5^{*},{ϵ}_{16}^{*}]={ϵ}_{13}^{*}-2{ϵ}_{15}^{*},[{ϵ}_5^{*},{ϵ}_{17}^{*}]={ϵ}_{14}^{*}[{ϵ}_6^{*},{ϵ}_{10}^{*}]=-{ϵ}_{12}^{*},[{ϵ}_7^{*},{ϵ}_8^{*}]=-3{ϵ}_{17}^{*},[{ϵ}_7^{*},{ϵ}_9^{*}]=\hfill \\ 2{ϵ}_{12}^{*},[{ϵ}_7^{*},{ϵ}_{10}^{*}]={ϵ}_{13}^{*},[{ϵ}_7^{*},{ϵ}_{11}^{*}]=-{ϵ}_{14}^{*},[{ϵ}_8^{*},{ϵ}_{10}^{*}]=-{ϵ}_{14}^{*}.\hfill \end{array}$$

From multiplication table the type of ${ϵ}_7^{*}$ singularity is 3 and the nilpotency is 5. The torus T is spanned by

$$\begin{array}{cccc}\hfill & t_1:\xi \left(V\right)⟶\xi \left(V\right)\hfill & \hfill & t_2:\xi \left(V\right)⟶\xi \left(V\right)\hfill \\ \hfill & {ϵ}_2^{*}⟶{ϵ}_2^{*}\hfill & \hfill & {ϵ}_2^{*}⟶0\hfill \\ \hfill & {ϵ}_4^{*}⟶0\hfill & \hfill & {ϵ}_4^{*}⟶{ϵ}_4^{*}\hfill \\ \hfill & {ϵ}_5^{*}⟶{ϵ}_5^{*}\hfill & \hfill & {ϵ}_5^{*}⟶-{ϵ}_5^{*}\hfill \\ \hfill & {ϵ}_6^{*}⟶2{ϵ}_6^{*}\hfill & \hfill & {ϵ}_6^{*}⟶-{ϵ}_6^{*}\hfill \\ \hfill & {ϵ}_7^{*}⟶{ϵ}_7^{*}\hfill & \hfill & {ϵ}_7^{*}⟶0\hfill \\ \hfill & {ϵ}_8^{*}⟶{ϵ}_8^{*}\hfill & \hfill & {ϵ}_8^{*}⟶{ϵ}_8^{*}\hfill \\ \hfill & {ϵ}_9^{*}⟶3{ϵ}_9^{*}\hfill & \hfill & {ϵ}_9^{*}⟶-2{ϵ}_9^{*}\hfill \\ \hfill & {ϵ}_{10}^{*}⟶2{ϵ}_{10}^{*}\hfill & \hfill & {ϵ}_{10}^{*}⟶-{ϵ}_{10}^{*}\hfill \\ \hfill & {ϵ}_{11}^{*}⟶2{ϵ}_{11}^{*}\hfill & \hfill & {ϵ}_{11}^{*}⟶0\hfill \\ \hfill & {ϵ}_{12}^{*}⟶4{ϵ}_{12}^{*}\hfill & \hfill & {ϵ}_{12}^{*}⟶-2{ϵ}_{12}^{*}\hfill \\ \hfill & {ϵ}_{13}^{*}⟶3{ϵ}_{13}^{*}\hfill & \hfill & {ϵ}_{13}^{*}⟶-{ϵ}_{13}^{*}\hfill \\ \hfill & {ϵ}_{14}^{*}⟶3{ϵ}_{14}^{*}\hfill & \hfill & {ϵ}_{14}^{*}⟶0\hfill \\ \hfill & {ϵ}_{15}^{*}⟶3{ϵ}_{15}^{*}\hfill & \hfill & {ϵ}_{15}^{*}⟶-{ϵ}_{15}^{*}\hfill \\ \hfill & {ϵ}_{16}^{*}⟶2{ϵ}_{16}^{*}\hfill & \hfill & {ϵ}_{16}^{*}⟶0\hfill \\ \hfill & {ϵ}_{17}^{*}⟶2{ϵ}_{17}^{*}\hfill & \hfill & {ϵ}_{17}^{*}⟶{ϵ}_{17}^{*}.\hfill \end{array}$$

After a simple calculation, we obtain

$$\begin{array}{c}\hfill \xi \left(V\right)={\xi }^{{\alpha }_1}\oplus {\xi }^{{\alpha }_2}\oplus {\xi }^{{\alpha }_1-{\alpha }_2}\oplus {\xi }^{2{\alpha }_1-{\alpha }_2}\oplus {\xi }^{{\alpha }_1+{\alpha }_2}\oplus {\xi }^{3{\alpha }_1-2{\alpha }_2}\oplus {\xi }^{4{\alpha }_1-2{\alpha }_2}\oplus {\xi }^{2{\alpha }_1}\oplus \\ \hfill {\xi }^{3{\alpha }_1-{\alpha }_2}\oplus {\xi }^{2{\alpha }_1+{\alpha }_2}\oplus {\xi }^{3{\alpha }_1}=\mathbb{C}({ϵ}_2^{*}\oplus {ϵ}_7^{*})\oplus \mathbb{C}{ϵ}_4^{*}\oplus \mathbb{C}{ϵ}_5^{*}\oplus \mathbb{C}({ϵ}_6^{*}\oplus {ϵ}_{10}^{*})\oplus \mathbb{C}{ϵ}_8^{*}\oplus \\ \hfill \mathbb{C}{ϵ}_9^{*}\oplus \mathbb{C}({ϵ}_{11}^{*}\oplus {ϵ}_{16}^{*})\oplus \mathbb{C}({ϵ}_{13}^{*}\oplus {ϵ}_{15}^{*})\oplus \mathbb{C}{ϵ}_{12}^{*}\oplus \mathbb{C}{ϵ}_{14}^{*}\oplus \mathbb{C}{ϵ}_{17}^{*}.\end{array}$$

Therefore, T-minimal system of generators are $({ϵ}_2^{*},{ϵ}_4^{*},{ϵ}_5^{*})$. So the generalized Cartan matrix is

$$C_2^2\left(E_7\right)=\left(\begin{array}{ccc}2& -2& -2\\ -1& 2& -3\\ -2& -5& 2\end{array}\right).$$

□

Proposition 5. 

Let $V=\{(x_1,x_2)\in {\mathbb{C}}^2:x_1^3+x_2^5=0\}$ be the ${ϵ}_8^{*}$ singularity. Then

$$C_2^2\left(E_8\right)=\left(\begin{array}{cccc}2& -2& -1& -4\\ -2& 2& -1& -4\\ -1& -1& 2& -1\\ -2& -2& -3& 2\end{array}\right).$$

Proof. 

The basis of algebra ${\mathcal{M}}_2^2\left(V\right)$ define as:

$$\begin{array}{cc}\hfill & \{1,x_1,x_2,x_2^2,x_2^3,x_2^4,x_1{x}_2,x_1{x}_2^2,x_1{x}_2^3,x_1^2,x_1^2{x}_2,x_1^2{x}_2^2\}.\hfill \end{array}$$

After some calculations we obtain the following basis of Lie algebra ${\mathcal{L}}_2^2\left(V\right)$:

$$\begin{array}{c}{ϵ}_1^{*}=x_1{\partial }_1-2x_2{\partial }_2,{ϵ}_2^{*}=-x_2^2{\partial }_2,{ϵ}_3^{*}=-x_1{\partial }_1+x_2{\partial }_2,{ϵ}_4^{*}=x_1{x}_2^2{\partial }_1+x_2^3{\partial }_2,{ϵ}_5^{*}=\hfill \\ x_1{x}_2{\partial }_1+2x_2^2{\partial }_2,{ϵ}_6^{*}=-x_2^3{\partial }_1,{ϵ}_7^{*}=-x_1{\partial }_2,{ϵ}_8^{*}=x_1^2{\partial }_1-x_1{x}_2{\partial }_2,{ϵ}_9^{*}=-x_1^2{x}_2{\partial }_1,{ϵ}_{10}^{*}=\hfill \\ -2x_1^2{\partial }_1+x_1{x}_2{\partial }_2,{ϵ}_{11}^{*}=-x_2^3{\partial }_2,{ϵ}_{12}^{*}=-x_1^2{\partial }_2,{ϵ}_{13}^{*}=-x_1^2{x}_2{\partial }_2,{ϵ}_{14}^{*}=x_1^2{x}_2{\partial }_1-.\hfill \\ x_1{x}_2^2{\partial }_2,{ϵ}_{15}^{*}=-x_1^2{x}_2^2{\partial }_2,{ϵ}_{16}^{*}=-x_1{x}_2^3{\partial }_2,{ϵ}_{17}^{*}=-x_2^4{\partial }_2,{ϵ}_{18}^{*}=-x_1^2{x}_2^2{\partial }_1,{ϵ}_{19}^{*}=\hfill \\ -x_1{x}_2^3{\partial }_1,{ϵ}_{20}^{*}=-x_2^4{\partial }_1\hfill \end{array}$$

The nilradical $\xi \left(V\right)=\langle {ϵ}_2^{*},{ϵ}_4^{*},{ϵ}_5^{*},{ϵ}_6^{*},\cdots ,{ϵ}_{20}^{*}\rangle$ have the following multiplication table:

$$\begin{array}{c}[{ϵ}_2^{*},{ϵ}_4^{*}]=-{ϵ}_{17}^{*}+2{ϵ}_{19}^{*},[{ϵ}_2^{*},{ϵ}_5^{*}]={ϵ}_{11}^{*}+{ϵ}_4^{*},[{ϵ}_2^{*},{ϵ}_6^{*}]=3{ϵ}_{20}^{*},[{ϵ}_2^{*},{ϵ}_7^{*}]=-2{ϵ}_{10}^{*}-\hfill \\ 4{ϵ}_8^{*},[{ϵ}_2^{*},{ϵ}_8^{*}]=-{ϵ}_{14}^{*}-{ϵ}_9^{*},[{ϵ}_2^{*},{ϵ}_9^{*}]={ϵ}_{18}^{*},[{ϵ}_2^{*},{ϵ}_{10}^{*}]={ϵ}_{14}^{*}+{ϵ}_9^{*},[{ϵ}_2^{*},{ϵ}_{11}^{*}]=\hfill \\ {ϵ}_{17}^{*},[{ϵ}_2^{*},{ϵ}_{12}^{*}]=-2{ϵ}_{13}^{*},[{ϵ}_2^{*},{ϵ}_{13}^{*}]=-{ϵ}_{15}^{*},[{ϵ}_2^{*},{ϵ}_{14}^{*}]=-{ϵ}_{18}^{*},[{ϵ}_4^{*},{ϵ}_5^{*}]=-2{ϵ}_{17}^{*}+\hfill \\ 3{ϵ}_{19}^{*},[{ϵ}_4^{*},{ϵ}_7^{*}]=4{ϵ}_{14}^{*}+2{ϵ}_9^{*},[{ϵ}_4^{*},{ϵ}_8^{*}]=3{ϵ}_{16}^{*}-3{ϵ}_{18}^{*},[{ϵ}_4^{*},{ϵ}_{10}^{*}]=-3{ϵ}_{16}^{*}+4{ϵ}_{18}^{*},[{ϵ}_4^{*},{ϵ}_{12}^{*}]=\hfill \\ 5{ϵ}_{15}^{*},[{ϵ}_5^{*},{ϵ}_6^{*}]=-7{ϵ}_{20}^{*},[{ϵ}_5^{*},{ϵ}_7^{*}]=4{ϵ}_{10}^{*}+9{ϵ}_8^{*},[{ϵ}_5^{*},{ϵ}_8^{*}]=3{ϵ}_{14}^{*}+{ϵ}_9^{*},[{ϵ}_5^{*},{ϵ}_9^{*}]=\hfill \\ -{ϵ}_{18}^{*},[{ϵ}_5^{*},{ϵ}_{10}^{*}]=-3{ϵ}_{14}^{*},[{ϵ}_5^{*},{ϵ}_{11}^{*}]=-2{ϵ}_{17}^{*}-{ϵ}_{19}^{*},[{ϵ}_5^{*},{ϵ}_{12}^{*}]=6{ϵ}_{13}^{*},[{ϵ}_5^{*},{ϵ}_{13}^{*}]=\hfill \\ 4{ϵ}_{15}^{*},[{ϵ}_5^{*},{ϵ}_{14}^{*}]={ϵ}_{16}^{*},[{ϵ}_6^{*},{ϵ}_7^{*}]=-2{ϵ}_{11}^{*}-3{ϵ}_4^{*},[{ϵ}_6^{*},{ϵ}_8^{*}]={ϵ}_{17}^{*}-5{ϵ}_{19}^{*},[{ϵ}_6^{*},{ϵ}_{10}^{*}]=\hfill \\ -{ϵ}_{17}^{*}+7{ϵ}_{19}^{*},[{ϵ}_6^{*},{ϵ}_{12}^{*}]=2{ϵ}_{16}^{*}-3{ϵ}_{18}^{*},[{ϵ}_7^{*},{ϵ}_8^{*}]=2{ϵ}_{12}^{*},[{ϵ}_7^{*},{ϵ}_9^{*}]=-{ϵ}_{13}^{*},[{ϵ}_7^{*},{ϵ}_{10}^{*}]=\hfill \\ -3{ϵ}_{12}^{*},[{ϵ}_7^{*},{ϵ}_{11}^{*}]=3{ϵ}_{14}^{*}+3{ϵ}_9^{*},[{ϵ}_7^{*},{ϵ}_{14}^{*}]=3{ϵ}_{13}^{*},[{ϵ}_7^{*},{ϵ}_{16}^{*}]=3{ϵ}_{15}^{*},[{ϵ}_7^{*},{ϵ}_{17}^{*}]=\hfill \\ 4{ϵ}_{16}^{*},[{ϵ}_7^{*},{ϵ}_{18}^{*}]=-{ϵ}_{15}^{*},[{ϵ}_7^{*},{ϵ}_{19}^{*}]=-{ϵ}_{16}^{*}+3{ϵ}_{18}^{*},[{ϵ}_7^{*},{ϵ}_{20}^{*}]=-{ϵ}_{17}^{*}+4{ϵ}_{19}^{*},[{ϵ}_8^{*},{ϵ}_9^{*}]=\hfill \\ -{ϵ}_{15}^{*},[{ϵ}_8^{*},{ϵ}_{10}^{*}]=-{ϵ}_{13}^{*},[{ϵ}_8^{*},{ϵ}_{11}^{*}]=2{ϵ}_{16}^{*},[{ϵ}_8^{*},{ϵ}_{14}^{*}]={ϵ}_{15}^{*},[{ϵ}_9^{*},{ϵ}_{10}^{*}]=-{ϵ}_{15}^{*},[{ϵ}_{10}^{*},{ϵ}_{11}^{*}]=\hfill \\ -2{ϵ}_{16}^{*},[{ϵ}_{11}^{*},{ϵ}_{12}^{*}]=-3{ϵ}_{15}^{*}.\hfill \end{array}$$

From multiplication table the type of ${ϵ}_8^{*}$ singularity is 4 and the nilpotency is 4. The torus T is spanned by

$$\begin{array}{cccc}\hfill & t_1:\xi \left(V\right)⟶\xi \left(V\right)\hfill & \hfill & t_2:\xi \left(V\right)⟶\xi \left(V\right)\hfill \\ \hfill & {ϵ}_2^{*}⟶{ϵ}_2^{*}\hfill & \hfill & {ϵ}_2^{*}⟶0\hfill \\ \hfill & {ϵ}_4^{*}⟶2{ϵ}_4^{*}\hfill & \hfill & {ϵ}_4^{*}⟶0\hfill \\ \hfill & {ϵ}_5^{*}⟶{ϵ}_5^{*}\hfill & \hfill & {ϵ}_5^{*}⟶0\hfill \\ \hfill & {ϵ}_6^{*}⟶0\hfill & \hfill & {ϵ}_6^{*}⟶{ϵ}_6^{*}\hfill \\ \hfill & {ϵ}_7^{*}⟶2{ϵ}_7^{*}\hfill & \hfill & {ϵ}_7^{*}⟶-{ϵ}_7^{*}\hfill \\ \hfill & {ϵ}_8^{*}⟶3{ϵ}_8^{*}\hfill & \hfill & {ϵ}_8^{*}⟶-{ϵ}_8^{*}\hfill \\ \hfill & {ϵ}_9^{*}⟶4{ϵ}_9^{*}\hfill & \hfill & {ϵ}_9^{*}⟶-{ϵ}_9^{*}\hfill \\ \hfill & {ϵ}_{10}^{*}⟶3{ϵ}_{10}^{*}\hfill & \hfill & {ϵ}_{10}^{*}⟶-{ϵ}_{10}^{*}\hfill \\ \hfill & {ϵ}_{11}^{*}⟶2{ϵ}_{11}^{*}\hfill & \hfill & {ϵ}_{11}^{*}⟶0\hfill \\ \hfill & {ϵ}_{12}^{*}⟶5{ϵ}_{12}^{*}\hfill & \hfill & {ϵ}_{12}^{*}⟶-2{ϵ}_{12}^{*}\hfill \\ \hfill & {ϵ}_{13}^{*}⟶6{ϵ}_{13}^{*}\hfill & \hfill & {ϵ}_{13}^{*}⟶-2{ϵ}_{13}^{*}\hfill \\ \hfill & {ϵ}_{14}^{*}⟶4{ϵ}_{14}^{*}\hfill & \hfill & {ϵ}_{14}^{*}⟶-{ϵ}_{14}^{*}\hfill \\ \hfill & {ϵ}_{15}^{*}⟶7{ϵ}_{15}^{*}\hfill & \hfill & {ϵ}_{15}^{*}⟶-2{ϵ}_{15}^{*}\hfill \\ \hfill & {ϵ}_{16}^{*}⟶5{ϵ}_{16}^{*}\hfill & \hfill & {ϵ}_{16}^{*}⟶-{ϵ}_{16}^{*}\hfill \\ \hfill & {ϵ}_{17}^{*}⟶3{ϵ}_{17}^{*}\hfill & \hfill & {ϵ}_{17}^{*}⟶0\hfill \\ \hfill & {ϵ}_{18}^{*}⟶5{ϵ}_{18}^{*}\hfill & \hfill & {ϵ}_{18}^{*}⟶-{ϵ}_{18}^{*}\hfill \\ \hfill & {ϵ}_{19}^{*}⟶3{ϵ}_{19}^{*}\hfill & \hfill & {ϵ}_{19}^{*}⟶0\hfill \\ \hfill & {ϵ}_{20}^{*}⟶{ϵ}_{20}^{*}\hfill & \hfill & {ϵ}_{20}^{*}⟶{ϵ}_{20}^{*}.\hfill \end{array}$$

After a simple calculation, we obtain

$$\begin{array}{cc}\hfill & \xi \left(V\right)={\xi }^{{\alpha }_1}\oplus {\xi }^{2{\alpha }_1}\oplus {\xi }^{{\alpha }_2}\oplus {\xi }^{2{\alpha }_1-{\alpha }_2}\oplus {\xi }^{3{\alpha }_1-{\alpha }_2}\oplus {\xi }^{4{\alpha }_1-2{\alpha }_2}\oplus {\xi }^{5{\alpha }_1-2{\alpha }_2}\oplus {\xi }^{6{\alpha }_1-2{\alpha }_2}\oplus {\xi }^{7{\alpha }_1-2{\alpha }_2}\hfill \\ & \oplus {\xi }^{5{\alpha }_1-{\alpha }_2}\oplus {\xi }^{3{\alpha }_1}\oplus {\xi }^{{\alpha }_1+{\alpha }_2}\hfill \\ \hfill & =\mathbb{C}({ϵ}_2^{*}\oplus {ϵ}_5^{*})\oplus \mathbb{C}({ϵ}_4^{*}\oplus {ϵ}_{11}^{*})\oplus \mathbb{C}{ϵ}_6^{*}\oplus \mathbb{C}{ϵ}_7^{*}\oplus \mathbb{C}({ϵ}_8^{*}\oplus {ϵ}_{10}^{*})\oplus \mathbb{C}({ϵ}_9^{*}\oplus {ϵ}_{14}^{*})\oplus \mathbb{C}{ϵ}_{12}^{*}\oplus \mathbb{C}{ϵ}_{13}^{*}\oplus \mathbb{C}{ϵ}_{15}^{*}\hfill \\ \hfill & \oplus \mathbb{C}({ϵ}_{16}^{*}\oplus {ϵ}_{18}^{*})\oplus \mathbb{C}({ϵ}_{17}^{*}\oplus {ϵ}_{19}^{*})\oplus \mathbb{C}{ϵ}_{20}^{*}.\hfill \end{array}$$

Therefore, T-minimal system of generators are $({ϵ}_2^{*},{ϵ}_5^{*},{ϵ}_6^{*},{ϵ}_7^{*})$. So the generalized Cartan matrices is

$$C_2^2\left(E_8\right)=\left(\begin{array}{cccc}2& -2& -1& -4\\ -2& 2& -1& -4\\ -1& -1& 2& -1\\ -2& -2& -3& 2\end{array}\right).$$

□

Proof of Theorem 1. 

The Proof Theorem 1 is an immediate corollary of Proposition 1, Proposition 2, Proposition 3, Proposition 4, and Proposition 5. □

4. Conclusions

The ${\mathcal{L}}_k^l\left(V\right)$ is a new analytic invariant of singularities. In this paper, we proposed a conjecture to characterize the ADE singularities using Lie algebra ${\mathcal{L}}_k^l\left(V\right)$ and we verify it for ${\mathcal{L}}_2^2\left(V\right)$. We also compute many other invariants namely, dimension of Lie algebra ${\mathcal{L}}_2^2\left(V\right)$, maximal torus, nilpotency, type of singularity, and generalized Cartan matrix for ADE singularities. Before this work, two previous defined Lie algebra $L\left(V\right)$ and $L^{*}\left(V\right)$ can not completely characterized the ADE singularities. The novelty of this paper is complete characterization of ADE singularities using generalized Cartan matrix that arising from Lie algebra ${\mathcal{L}}_2^2\left(V\right)$.