Special Unitary Group
名词解释
\(SU_n(q)\)群，是指“The special unitary group”, 翻译成中文就是“特殊幺正群”。 同理\(SO_n(q)\)群，是指"The special orthogonal group", 翻译成中文就是“特殊正交群”。酉矩阵（unitary matrix）也叫幺正矩阵, 对它取复共轭再转置则等于逆矩阵,也即是幺正矩阵的转置共轭等于它的逆。当矩阵元为实数时也叫正交矩阵（orthogonal matrix），它的共轭就是它自己，所以转置即得到逆矩阵，这也就表现为矩阵的行向量间为正交关系，列向量间也为正交关系。
英文介绍
The special unitary group \(SU_n(q)\) is the set of \(n×n\) unitary matrices with determinant \(+1\) (having \(n^2-1\) independent parameters). SU(2) is homeomorphic with the orthogonal group \(O_3^+(2)\). It is also called the unitary unimodular group and is a Lie group.
Special unitary groups can be represented by matrices
\[\begin{equation}\label{eq:sug0} \begin{bmatrix} a & b \\ -\overline{b} & \overline{a} \end{bmatrix} \end{equation}\]
where \(\overline{a}a+\overline{b}b=1\) and \(a,b\) are the Cayley-Klein parameters. The special unitary group may also be represented by matrices
\[\begin{equation}\label{eq:sug1} U(\xi,\eta,\zeta)= \begin{bmatrix} e^{i\xi}\cos\eta & e^{i\zeta}\sin\eta \\ -e^{-i\zeta}\sin\eta & e^{0i\xi}\cos\eta \end{bmatrix} \end{equation}\]
or the matrices
\[\begin{equation}\label{eq:sug2} U_x(\frac{1}{2}\phi)= \begin{bmatrix} \cos(\frac{1}{2}\phi) & i\sin(\frac{1}{2}\phi) \\ i\sin(\frac{1}{2}\phi) & \cos(\frac{1}{2}\phi) \end{bmatrix} \end{equation}\]
\[\begin{equation}\label{eq:sug3} U_y(\frac{1}{2}\phi)= \begin{bmatrix} \cos(\frac{1}{2}\beta) & \sin(\frac{1}{2}\beta) \\ -\sin(\frac{1}{2}\beta) & \cos(\frac{1}{2}\beta) \end{bmatrix} \end{equation}\]
\[\begin{equation}\label{eq:sug4} U_z(\xi)= \begin{bmatrix} e^{i\xi} & 0 \\ 0 & e^{-i\xi} \end{bmatrix} \end{equation}\]