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常用的矩阵李群

所有矩阵都定义在\(\mathbb{C}\)上。其中,

\[g=\begin{bmatrix}

I 0\\

0 -I

\ end { b matrix } \ qquad \ Omega=\ begin { b matrix }

0 I\\

-我0

\end{bmatrix}\]

名字

定义

紧凑

连通性

一般线性群\(\text{GL}(n)\)

\(n\乘以n \)可逆矩阵

连接

特殊组\(\text{SL}(n)\)

秩为1的可逆矩阵

简单连接

酉群\(\text{U}(n)\)

酉矩阵:\(n\乘以n \)可逆矩阵\(A\)满足\ (a * a=i \)

连接

特殊酉群\(\text{SU}(n)\)

秩为1的\(n\乘以n \)酉矩阵

简单连接

正交群\(\text{O}(n)\)

正交矩阵:\(n\乘以n \)可逆矩阵\(A\)满足\ (a \ top a=I \)

断开

广义正交群\(\text{O}(n,k)\)

\(n\乘以n \)正交矩阵满足\ (a \ top g a=g \)

断开

正交群\(\text{SO}(n)\)

秩为1的\(n\乘以n \)正交矩阵

连接

Xin \(\text{Sp}(n)\)

辛矩阵:\(n\乘以n \)可逆矩阵\(A\ top \ omega a=\ omega \)。

连接

压缩组\(\text{USp}(n)\)

(n \n \n)可逆矩阵(A \u\u\u\u\u\u\u\u\u\u\u\u\u\u\u\u\u\u\u\u\\)既酉又是辛矩阵

简单连接

欧几里得群\(\text{E}(n)\)

一种形式为\(\ begin { b matrix } r x \ \ 01 \ end { b matrix } \)的矩阵,其中\ (x \ in \ mathbb {r} n,r \ in o (n) \)

断开

庞加莱群\(\text{P}(n,k)\)

一种形式为\(\ begin { b matrix } a x \ \ 01 \ end { b matrix } \)的矩阵,其中\ (x \ in \ mathbb {r} n,a \ in o (n,k) \)

断开

海森堡集团

一种形式为\(\ begin { b matrix } 1ab \ \ 01c \ \ 001 \ end { b matrix } \)的矩阵

连接

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