\documentclass{template-jurnal} %Bagian ini diedit oleh editor Jurnal Matematika UNAND \hyphenation{di-tulis-kan di-terang-kan} %Perhatikan aturan penulisan dan ukuran huruf yang digunakan \begin{document} \markboth{E. Kurniadi \& K. Parmikanti} %Jika lebih dari dua penulis, tuliskan sebagai Nama Penulis Pertama dkk. {The Lie Algebra $\mathfrak{su}(3)$ Representation with Respect to Its Basis} %%%%%%%%%%%%%%%%%%%%% Publisher's Area please ignore %%%%%%%%%%%%%%% % \catchline{}{}{}{}{} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{THE LIE ALGEBRA $\mathfrak{su}(3)$ REPRESENTATION WITH RESPECT TO ITS BASIS} \author{EDI KURNIADI$^{a}$\footnote{penulis korespondensi}, KANKAN PARMIKANTI$^{b}$, %NAMA PENULIS KETIGA$^{c}$\\ (ditulis dengan huruf besar, \textbf{TANPA GELAR}) } \address{$^{a}$ Department of Mathematics of Universitas Padjadjaran, Sumedang, Jawa Barat, Indonesia,\\ $^{b}$ Department of Mathematics of Universitas Padjadjaran, Sumedang, Jawa Barat, Indonesia.\\ %$^{c}$ Instansi Penulis Ketiga.\\ email : \email{edi.kurniadi@unpad.ac.id}} \maketitle \setcounter{page}{1} %bagian ini diedit oleh editor Jurnal Matematika UNAND \begin{abstract} \begin{center} Accepted ..... \quad Revised ..... \quad Published ..... %tanggal-tanggal tersebut \textbf{dikosongkan} saja \end{center} \bigskip \textbf{Abstrak}. %Dalam bahasa Indonesia Aljabar Lie berdimensi delapan dari semua matriks anti-Hermit $3\times 3$ dengan \textit{trace}-nya sama dengan nol dinotasikan oleh $\mathfrak{su}(3)$ dan grup Lie-nya dinotasikan oleh $\mathrm{SU}(3)$. Tujuan penelitian ini adalah menentukan semua representasi dari $\mathfrak{su}(3)$ melalui basisnya yang direalisasikan dalam ruang polinom homogen kompleks tiga variabel $\mathbb{P}_1$ berderajat tiga. Langkah pertama adalah mengkonstruksi representasi $\mathrm{SU}(3)$ pada $\mathbb{P}_1$ dan langkah kedua menentukan turunan dari representasi $\mathrm{SU}(3)$. Hasil yang diperoleh adalah delapan rumus eksplisit dari representasi $\mathfrak{su}(3)\curvearrowright \mathbb{P}_1$. \bigskip \textbf{Abstract}. % Dalam bahasa Inggris \textit{The eight-dimensional Lie algebra of $3\times 3$ anti-Hermitian matrices with \changed{its} traces equal to zero is denoted by $\mathfrak{su}(3)$ whose Lie group is denoted by $\mathrm{SU}(3)$. The research aims to provide all representations of $\mathfrak{su}(3)$ with respect to its basis which is realized on the three complex variables homogeneous polynomials $\mathbb{P}_1$ of degree three. The first step is to construct representations of $\mathrm{SU}(3)$ on the space $\mathcal{H}$ and the second step is to find all derived representations of $\mathrm{SU}(3)$. The obtained results are eight explicit formulas of representations $\mathfrak{su}(3)\curvearrowright\mathbb{P}_1$. } \end{abstract} \keywords{Anti-Hermitian Matrices, Derived Representation, $\mathrm{SU}(3)$, $\mathfrak{su}(3)$} \section{Introduction} Representation theory of Lie groups and Lie algebras are widely applied in both mathematics and physics (\cite{Mat}, \cite{Dragt}, \cite{Dai}). In mathematics, Lie groups and Lie algebras contribute in many branches of mathematics such as in biological mathematics and finance mathematics (\cite{Taka}, \cite{Her}). Moreover, representations of Lie algebras are vast studied by many researchers (\cite{Hu}-\cite{Rabi}). In addition, matrix Lie groups can be used as a model for understanding representation theory. The set of all $n\times n$ matrices whose their adjoints are equal to their inverse and their determinant are equals one is denoted by $\mathrm{SU}(n)$. In other words, $\mathrm{SU}(n)$ is the collections of $n\times n$ unitary matrices with determinant \changed{equal to} one. We write it as $\mathrm{SU}(n)=\{S\in \mathrm{U}(n) ; \mathrm{det}(S)=1\}$. The dimension of $\mathrm{SU}(n)$ is $n^2-1$. In case of $n=2$, the unitary-irreducible representation of $\mathrm{SU}(2)\curvearrowright \mathbb{P}_n$ where $\mathbb{P}_n$ is homogeneous polynomials of two-complex variables can be found in (\cite{Bern}, \cite{Brian}). In this research, we shall take for case $n=3$ in which the representation of the matrix Lie group $\mathrm{SU}(3)$ can be applied to the field of particle theory \cite{Brian}. Different from previous results, we give \changed{explicit} formulas for representations of $\mathrm{SU}(3)$. The derived representations of $\mathrm{SU}(3)$ arise to the notion of representation of the Lie algebra of $\mathrm{SU}(3)$ which is denoted by $\mathfrak{su}(3)$. It is well known that the Lie algebra $\mathfrak{su}(3)$ consists of $3\times 3$ anti-Hermitian matrices whose traces of them equals zero, namely, $\mathfrak{su}(3)=\{P\in \mathrm{M}(3,\mathbb{C}) ; P^*+P=O, \mathrm{tr}(P)=0\}$. Roughly speaking, it is also well known that $\mathfrak{su}(3)$ contains all smooth left-invariant vector fields of $\mathrm{SU}(3)$. Furthermore, the representations of the Lie algebra $\mathfrak{su}(3)$ is important since it gives the best model in studying representation of matrix Lie groups. The obtained result of representation of the Lie algebra $\mathfrak{su}(3)$ in this research can be developed to the general case $\mathfrak{su}(n) \curvearrowright \mathbb{P}_n$ where $\mathbb{P}_n$ is homogeneous polynomials of $n$-complex variables. We organize this paper as follows: In part 1, we introduce the motivation and state of art of our research, in part 2, we deliver some of the relevant \changed{backgrounds}: The Lie group $\mathrm{SU}(3)$, the Lie algebra $\mathfrak{su}(3)$ and its basis, representation of matrix Lie groups, and derived representations, part 3, we state the main result, particularly, in Propotition \ref{prop2} equipped with complete proofs. \section{Preliminaries} In this section, we shall briefly review some of the relevant materials which shall be used in main results: $\mathfrak{su}(3)$ notion and its basis, representation theory, derived representations. As mentioned before the Lie algebra $\mathfrak{su}(3)$ can be witten in the following form: \begin{eqnarray} \mathfrak{su}(3)=\{P\in \mathrm{M}(3,\mathbb{C}) ; P^*+P=O, \mathrm{tr}(P)=0\}. \end{eqnarray} \begin{definition}\label{defi1}\cite{sun} %\cite{B} mengacu ke referensi no. 2 di Daftar Pustaka. \ref{th1} mengacu kepada nomor teorema ini. The Lie algebra $\mathfrak{su}(3)$ has the basis $\mathcal{A}=\{\tau_k=-\frac{i}{2}\zeta_k, \quad i^2=-1$ \mbox{and} $k=1,2,\ldots,8\}$ where $\zeta_k$ are given in the following $3\times 3$ Hermitian matrices: \begin{eqnarray*} \zeta_1=\left(\begin{array}{ccc}0&1&0\\1&0&0\\0&0&0\end{array}\right),\,\, \zeta_2=\left(\begin{array}{crc}0&-i&0\\i&0&0\\0&0&0\end{array}\right), \,\, \zeta_3=\left(\begin{array}{crc}1&0&0\\0&-1&0\\0&0&0\end{array}\right), \,\, \zeta_4=\left(\begin{array}{ccc}0&0&1\\0&0&0\\1&0&0\end{array}\right), \\ \zeta_5=\left(\begin{array}{ccr}0&0&-i\\0&0&0\\i&0&0\end{array}\right), \,\, \zeta_6=\left(\begin{array}{ccc}0&0&0\\0&0&1\\0&1&0\end{array}\right), \,\, \zeta_7=\left(\begin{array}{ccr}0&0&0\\0&0&-i\\0&i&0\end{array}\right), \,\, \zeta_8=\frac{1}{\sqrt{3}}\left(\begin{array}{ccr}1&0&0\\0&1&0\\0&0&-2\end{array}\right). \end{eqnarray*} \end{definition} We recall that \changed{the} matrix Lie group $H$ is a subgroup of $\mathrm{GL}_n(\mathbb{C})$ and closed subgroup as well. A representation of $H$ is given by the following definition \begin{definition}\label{defi2}\cite{Brian} A finite complex representation of a matrix Lie group $H$ on a finite complex vector space $\mathcal{V}$ or $H \curvearrowright \mathcal{V}$ is a Lie group homomorphism given by \begin{equation} \Phi : H\rightarrow \mathrm{GL}(\mathcal{V}). \end{equation} \end{definition} \changed{In the context of representation theory}, Definition \ref{defi2} is equivalent to the following definition: \begin{definition}\cite{Brian} A finite complex representation of a matrix Lie group $H$ on a finite complex vector space $\mathcal{V}$ or $H \curvearrowright \mathcal{V}$ is a linear action of $H$ on $\mathcal{V}$. Namely, for all $x\in H$ we have \begin{equation} \Phi(g) : \changed{\mathcal{V}\ni v}\mapsto \Phi(g)v:=g.v\in \mathcal{V}. \end{equation} \end{definition} \begin{proposition} Let $\Phi:H \curvearrowright \mathcal{V}$ be a representation of a matrix Lie group on the complex vector space $\mathcal{V}$. Then there exists a unique $\pi:\mathfrak{h}\curvearrowright\mathcal{V}$ \begin{equation} \pi(A)=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta A})|_{\delta=0} \end{equation} where $\Phi(e^{\delta A})=e^{\pi(A)}$. \end{proposition} The representation $\pi:\mathfrak{h}\curvearrowright\mathcal{V}$ is named by \emph{derived representation} associated to $\Phi:H \curvearrowright \mathcal{V}$. Indeed, if $H$ is simply connected then the representation $\pi:\mathfrak{h}\curvearrowright\mathcal{V}$ can be obtained from the representation $\Phi:H \curvearrowright \mathcal{V}$. In our case, $\mathrm{SU}(3)$ is topologically simply connected. Therefore, we can obtain the $\pi:\mathfrak{su}(3)\curvearrowright\mathbb{P}_1$ from $\Phi:\mathrm{SU}(3) \curvearrowright \mathbb{P}_1$ where $\mathbb{P}_1$ is three complex variable homogeneous polynoms. \section{Results and Discussion} Firstly, we compute all exponential matrices in the basis $\mathcal{A}=\{\tau_k=-\frac{i}{2}\zeta_k, \quad i^2=-1$ \mbox{and} $k=1,2,\ldots,8\}$ of $\mathfrak{su}(3)$ as written in Definition \ref{defi1}. All computations we state in the following proposition. \begin{proposition} \label{prop1} Let $\mathcal{A}=\{\tau_k=-\frac{i}{2}\zeta_k, \quad i^2=-1$ \mbox{and} $k=1,2,\ldots,8\}$ be a basis of $\mathfrak{su}(3)$ as written in Definition \ref{defi1}. Then the exponential matrices of elements of $\mathcal{A}$ can be listed in the following forms: \begin{eqnarray*} e^{\delta\tau_1}=\left(\begin{array}{ccc}\cos \frac{\delta}{2}&-i\sin \frac{\delta}{2}&0\\-i\sin \frac{\delta}{2} &\cos \frac{\delta}{2}&0\\0&0&1\end{array}\right),\,\, e^{\delta\tau_2}=\left(\begin{array}{crc}0\cos \frac{\delta}{2}&-\sin \frac{\delta}{2}&0\\\sin \frac{\delta}{2} &\cos \frac{\delta}{2}&0\\0&0&1\end{array}\right), \\ e^{\delta\tau_3}=\left(\begin{array}{ccc}e^{-i/2}&0&0\\0 &e^{i/2}&0\\0&0&1\end{array}\right),\,\, e^{\delta\tau_4}=\left(\begin{array}{rcr}\cos \frac{\delta}{2}&0&-i\sin \frac{\delta}{2}\\0 &1 &0\\-i\sin \frac{\delta}{2}&0&\cos \frac{\delta}{2}\end{array}\right),\\ e^{\delta\tau_5}=\left(\begin{array}{ccr}\cos \frac{\delta}{2}&0&-i\sin \frac{\delta}{2}\\0 &1&0\\\sin \frac{\delta}{2}&0&\cos \frac{\delta}{2}\end{array}\right),\,\, e^{\delta\tau_6}=\left(\begin{array}{crr}1&0&0\\0&\cos \frac{\delta}{2}&-i\sin \frac{\delta}{2}\\0&-i\sin \frac{\delta}{2}&\cos \frac{\delta}{2}\end{array}\right),\\ e^{\delta\tau_7}=\left(\begin{array}{ccr}1&0&0\\0 &\cos \frac{\delta}{2}&-\sin \frac{\delta}{2}\\0&\sin \frac{\delta}{2}&\cos \frac{\delta}{2}\end{array}\right),\,\, e^{\delta\tau_8}=\left(\begin{array}{ccc}e^{\frac{-i}{2\sqrt{3}}}&0&0\\0&e^{\frac{-i}{2\sqrt{3}}}&0\\0&0&e^{\frac{i}{\sqrt{3}}}\end{array}\right). \end{eqnarray*} \end{proposition} \begin{proof} We just compute exponential matrices for $\delta\tau_1$. The exponential matrices of other elements can be computed in similar ways. We can observe that the matrix $\tau_1$ has three linear independent eigenvectors \begin{equation} s_1=\left(\begin{array}{c}0\\0\\1\end{array}\right),\quad s_2=\left(\begin{array}{r}-1\\1\\0\end{array}\right), s_3=\left(\begin{array}{c}1\\1\\0\end{array}\right). \end{equation} corresponding to the eigenvalues $\lambda_1=0, \lambda_2=i/2$, and $\lambda_3=-i/2$. By using \changed{Proposition} 2.3 in \cite{Brian}, $e^{\delta\tau_1}$ can be computed as $e^{\delta\tau_1}=Pe^DP^{-1}$ where P is an invertible matrix contained all linearly independent eigenvectors of $\tau_1$ and $D$ is the diagonal matrix contained all $\delta$-multiples of eigenvalues of $\tau_1$. In other words, we get: \begin{equation*} e^{\delta\tau_1}=Pe^DP^{-1}=\left(\begin{array}{crc}0&-1&1\\0 &1&1\\1&0&0\end{array}\right)\left(\begin{array}{ccc}1&0&0\\0 &e^{i/2}&0\\0&0&e^{-i/2}\end{array}\right)\left(\begin{array}{crc}0&-1&1\\0 &1&1\\1&0&0\end{array}\right)^{-1}. \end{equation*} Then we have the exponential matrix of $\delta\tau_1$ or \begin{equation*} e^{\delta\tau_1}=\left(\begin{array}{ccc}\cos \frac{\delta}{2}&-i\sin \frac{\delta}{2}&0\\-i\sin \frac{\delta}{2} &\cos \frac{\delta}{2}&0\\0&0&1\end{array}\right). \end{equation*} as required. \end{proof} Secondly, we let $\mathrm{SU}(3)$ acts on the space $\mathbb{P}_1$ by $\Phi$. Thirdly, derived representation of $\Phi$ is the representation $\pi$ of $\mathfrak{su}(3)$ realized on the space $\mathbb{P}_1$. Now we are ready to state \changed{our} main result in the following proposition: \begin{proposition} \label{prop2} Let $\mathcal{A}=\{\tau_k=-\frac{i}{2}\zeta_k, \quad i^2=-1$ \mbox{and} $k=1,2,\ldots,8\}$ be a basis for $\mathfrak{su}(3)$ as written in Definition \ref{defi1} and let \begin{equation} \begin{split} \mathcal{B}=\{v_1=x^3, v_2=y^3, v_3=z^3, v_4=x^2y, v_5=x^2z, v_6=xy^2, v_7=xz^2,\\ \changed{v_8=y^2z, v_9=yz^2, v_{10}=xyz} \} \end{split} \end{equation} be a basis for three-complex homogeneous polynomial space $\mathbb{P}_1$ of degree 3. Then the representations of $\pi:\mathfrak{su}(3)\curvearrowright\mathbb{P}_1$ with respect to the basis of $\mathcal{A}$ are of the forms: \begin{eqnarray} \hat{\tau}_1v_1&=\pi(\tau_1)v_1&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_1})v_1|_{\delta=0}=\frac{3}{2}iv_4\\ \hat{\tau}_2v_2&=\pi(\tau_2)v_2&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_2})v_2|_{\delta=0}=-\frac{3}{2}v_6\\ \hat{\tau}_3v_3&=\pi(\tau_3)v_3&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_3})v_3|_{\delta=0}=0\\ \hat{\tau}_4v_4&=\pi(\tau_4)v_4&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_4})v_4|_{\delta=0}=iv_{10}\\ \hat{\tau}_5v_5&=\pi(\tau_5)v_5&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_5})v_5|_{\delta=0}=v_{10}-\frac{1}{2}v_1\\ \hat{\tau}_6v_6&=\pi(\tau_6)v_6&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_6})v_6|_{\delta=0}=iv_{10}\\ \hat{\tau}_7v_7&=\pi(\tau_7)v_7&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_7})v_7|_{\delta=0}=-v_{10}\\ \hat{\tau}_8v_{10}&=\pi(\tau_8)v_{10}&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_8})v_{10}|_{\delta=0}=0. \end{eqnarray} \end{proposition} \begin{proof} Let $x$ be any element of $\mathrm{SU}(3)$ and $v$ be element of $\mathbb{P}_1$. The map $[\Phi(x)v](z)=v(x^{-1}z), z\in \mathbb{C}^3$ is a actually a representation. To see that, let us compute: \begin{eqnarray*} \Phi(x)[\Phi(y)v](z)&=[\Phi(y)v](x^{-1}z)\\ &=v(y^{-1}x^{-1}z)\\ &=\Phi(xy)v(z). \end{eqnarray*} We just compute for the cases $\hat{\tau}_1v_1$, $\hat{\tau}_2v_2$, and the last $\hat{\tau}_2v_3$. Another case can be computed in the similar ways. Firstly, from Proposition \ref{prop1} we have \begin{equation*} e^{\delta\tau_1}=\left(\begin{array}{ccc}\cos \frac{\delta}{2}&-i\sin \frac{\delta}{2}&0\\-i\sin \frac{\delta}{2} &\cos \frac{\delta}{2}&0\\0&0&1\end{array}\right). \end{equation*} By direct computation, we have the inverse of $e^{\delta\tau_1}$ is of the form \begin{equation*} e^{\delta\tau_1}=\left(\begin{array}{ccc}\cos \frac{\delta}{2}&i\sin \frac{\delta}{2}&0\\i\sin \frac{\delta}{2} &\cos \frac{\delta}{2}&0\\0&0&1\end{array}\right). \end{equation*} In addition, we can compute the representations $\Phi(e^{\delta \tau_1})v_1$ as follows: \begin{eqnarray*} \Phi(e^{\delta \tau_1})v_1(z)&&=v_1(e^{-\delta \tau_1}z)\\ &&=v_1\left[\left(\begin{array}{ccc}\cos \frac{\delta}{2}&i\sin \frac{\delta}{2}&0\\i\sin \frac{\delta}{2} &\cos \frac{\delta}{2}&0\\0&0&1\end{array}\right)\left(\begin{array}{c}x\\y\\z\end{array}\right)\right]\\ &&=v_1\left[\left(\begin{array}{c}x\cos \frac{\delta}{2}+iy\sin \frac{\delta}{2}\\xi\sin \frac{\delta}{2}+y\cos \frac{\delta}{2}\\z\end{array}\right)\right]\\ &&=(x\cos \frac{\delta}{2}+iy\sin \frac{\delta}{2})^3. \end{eqnarray*} The derived representation of $\Phi(e^{\delta \tau_1})v_1$ is given by $\hat{\tau}_1v_1$ which is nothing but the representation $\pi$ of $\mathfrak{su}(3)$ represented on $\mathbb{P}_1$. By direct computations, then we get an explicit formula in the following form: \begin{eqnarray*} \hat{\tau}_1v_1&=\pi(\tau_1)v_1&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_1})v_1|_{\delta=0}\\ &&=\frac{\mathrm{d}}{\mathrm{d}\delta}(x\cos \frac{\delta}{2}+iy\sin \frac{\delta}{2})^3|_{\delta=0}\\ &&=3(x\cos \frac{\delta}{2}+iy\sin \frac{\delta}{2})^2(-x/2 \sin \frac{\delta}{2} + iy/2 \cos \frac{\delta}{2})|_{\delta=0}\\ &&=3(x)^2(iy/2)=\frac{3}{2}i x^2y\\ &&=\frac{3}{2}i v_4. \end{eqnarray*} Secondly, we can compute inverse of $e^{\delta \tau_2}$, namely we have: \begin{equation*} e^{-\delta \tau_2}=\left(\begin{array}{crc}0\cos \frac{\delta}{2}&\sin \frac{\delta}{2}&0\\-\sin \frac{\delta}{2} &\cos \frac{\delta}{2}&0\\0&0&1\end{array}\right). \end{equation*} We can observe that \begin{eqnarray*} \Phi(e^{\delta \tau_2})v_2(z)&&=v_2(e^{-\delta \tau_2}z)\\ &&=(-x\sin \frac{\delta}{2}+y\cos \frac{\delta}{2})^3. \end{eqnarray*} Similarly we have \begin{eqnarray*} \hat{\tau}_2v_2&=\pi(\tau_2)v_2&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_1})v_1|_{\delta=0}\\ &&=\frac{\mathrm{d}}{\mathrm{d}\delta}(-x\sin \frac{\delta}{2}+y\cos \frac{\delta}{2})^3|_{\delta=0}\\ &&=3(x\cos \frac{\delta}{2}+iy\sin \frac{\delta}{2})^2(-x/2 \sin \frac{\delta}{2} + iy/2 \cos \frac{\delta}{2})|_{\delta=0}\\ &&=3(y)^2(-x/2)=-\frac{3}{2}i xy^2\\ &&=-\frac{3}{2} v_6. \end{eqnarray*} Thirdly, since $e^{\delta \tau_2}$ is the diagonal matrix then the inverse of $e^{\delta \tau_2}$ is easy to consider. \changed{Then} we have \changed{the} representation $\hat{\tau}_3v_3=0$ as desired. \end{proof} \section{Conclusion} We concluded that the representations of $\pi:\mathfrak{su}(3)\curvearrowright\mathbb{P}_1$ with respect to the basis of $\mathcal{A}$ are of the forms: \begin{eqnarray} \hat{\tau}_1v_1&=\pi(\tau_1)v_1&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_1})v_1|_{\delta=0}=\frac{3}{2}iv_4\\ \hat{\tau}_2v_2&=\pi(\tau_2)v_2&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_2})v_2|_{\delta=0}=-\frac{3}{2}v_6\\ \hat{\tau}_3v_3&=\pi(\tau_3)v_3&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_3})v_3|_{\delta=0}=0\\ \hat{\tau}_4v_4&=\pi(\tau_4)v_4&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_4})v_4|_{\delta=0}=iv_{10}\\ \hat{\tau}_5v_5&=\pi(\tau_5)v_5&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_5})v_5|_{\delta=0}=v_{10}-\frac{1}{2}v_1\\ \hat{\tau}_6v_6&=\pi(\tau_6)v_6&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_6})v_6|_{\delta=0}=iv_{10}\\ \hat{\tau}_7v_7&=\pi(\tau_7)v_7&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_7})v_7|_{\delta=0}=-v_{10}\\ \hat{\tau}_8v_{10}&=\pi(\tau_8)v_{10}&=\frac{\mathrm{d}}{\mathrm{d}\delta}\Phi(e^{\delta \tau_8})v_{10}|_{\delta=0}=0. \end{eqnarray} We can generalize to the case representations of $\pi:\mathfrak{su}(N)\curvearrowright\mathbb{P}_k$ where $\mathbb{P}_k$ is $N$ complex variables of degree $N$. \section{Acknowledgement} Authors would like to thank reviewers for valuable comments and suggestions for this article. \newpage \begin{thebibliography}{0} \bibitem{Mat} Matsuo, Y., Noshita, G., and Zhu, R.-D., 2024, Quantum toroidal algebras and solvable structures in gauge/string theory, \emph{Physics Report}, \textbf{1055} : 1 -- 14 \bibitem{Dragt} Dragt, A.J., and Forest,E., 1986, Lie Algebraic Theory of Charged-Particle Optics and Electron Microscopes, \emph{Advances in Electronics and Electron Phsyics}, \textbf{67} : 65 -- 120 \bibitem{Dai} Dai, W.-S., and Xie, M., 2004, A representation of angular momentum (SU(2)) algebra, \emph{Physica A: Statistical Mechanics and its Applications}, \textbf{331} : 497 -- 504 \bibitem{Taka} TAKAHASHI, L., MAIDANA, N., FERREIRAJR, W., PULINO, P., and YANG, H., 2005, Mathematical models for the dispersal dynamics: travelling waves by wing and wind, \emph{Bull Math Biol}, \textbf{67} : 509–528 \bibitem{Her} Hernández, I., Mateos, C., Núñez, J., and Tenorio, Á. 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