# 리군과 리대수

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## (고전적) 리 군

• 일반선형군 General linear $\operatorname{GL}_{n}\mathbb{R}$
• 특수선형군 Special linear $\operatorname{SL}_{n}\mathbb{R}=\{A\in \operatorname{GL}_{n}\mathbb{R}|\det A=1\}$
• 직교군 Orthogonal $\mathit{O}(n)=\{A\in \operatorname{GL}_{n}\mathbb{R}|AA^{t}=I\}$
• 특수직교군 Special orthogonal $\mathit{SO}(n)=\{A\in \mathit{O}(n)|\det A=1\}$
• 유니타리군 Unitary $\mathit{U}(n)=\{A\in \operatorname{GL}_{n}\mathbb{C}|A\bar{A}^t=I\}$
• 특수유니타리군 Special unitary $\mathit{SU}(n)=\{A\in \mathit{U}(n)|\det A=1\}$
• 사교군 Symplectic $\mathit{Sp}(n)=\{A\in \mathit{U}(2n)|A^tJ=JA^{-1}\}$ 여기서 $J =\begin{pmatrix}0 & I_n \\-I_n & 0 \\\end{pmatrix}$

## 테이블

\begin{array}{l|l|l|I|I} & \text{rank} & \text{dim} & \text{Coxeter} & \text{dual Coxeter} \\ \hline A_n & n & n^2+2n & n+1 & n+1\\ B_n & n & 2n^2+n & 2n & 2n-1 \\ C_n & n & 2n^2+n & 2n & n+1\\ D_n & n & 2n^2-n & 2n-2 & 2n-2\\ E_6 & 6 & 78 & 12 & 12\\ E_7 & 7 & 133 & 18 & 18\\ E_8 & 8 & 248 & 30 & 30\\ F_4 & 4 & 52 & 12 & 9\\ G_2 & 2 & 14 & 6 & 4 \end{array}

## 메모

• A_n SL_{n+1}(C)
• B_n O_{2n+1}(C)
• C_n Sp_{2n}(C)
• D_n O_{2n}(C)
• Manivel, Laurent. “On the Variety of Four Dimensional Lie Algebras.” arXiv:1506.02871 [math], June 9, 2015. http://arxiv.org/abs/1506.02871.

## 관련도서

• Faraut, Jacques. 2008. Analysis on Lie Groups: An Introduction. Cambridge University Press.