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
\[\begin{equation}\label{eq:sug2}
U_x(\frac{1}{2}\phi)=
\begin{bmatrix}
\cos(\frac{1}{2}\phi) & i\sin(\frac{1}{2}\phi) \\
i\sin(\frac{1}{2}\phi) & \cos(\frac{1}{2}\phi)
\end{bmatrix}
\end{equation}\]
\[\begin{equation}\label{eq:sug3}
U_y(\frac{1}{2}\phi)=
\begin{bmatrix}
\cos(\frac{1}{2}\beta) & \sin(\frac{1}{2}\beta) \\
-\sin(\frac{1}{2}\beta) & \cos(\frac{1}{2}\beta)
\end{bmatrix}
\end{equation}\]
\[\begin{equation}\label{eq:sug4}
U_z(\xi)=
\begin{bmatrix}
e^{i\xi} & 0 \\
0 & e^{-i\xi}
\end{bmatrix}
\end{equation}\]
参考引文