低次元実リー代数の分類

• ${\displaystyle {\mathfrak {g}}_{1}}$、アーベル群。

• ${\displaystyle 2{\mathfrak {g}}_{1}}$アーベル群$\displaystyle \mathbb {R} ^{2}}$;
• ${\displaystyle {\mathfrak {g}}_{2.1}}$解決可能${\displaystyle {\mathfrak {aff}}(1)=\left\{{\begin{pmatrix}a&b\\0&0\end{pmatrix}}\,:\,a,b\in \mathbb {R} \right\}}$、

${\displaystyle [e_{1},e_{2}]=e_{1}.}$

• ${\displaystyle 3{\mathfrak {g}}_{1}}$、アーベル群、ビアンキ I ;
• ${\displaystyle {\mathfrak {g}}_{2.1}\oplus {\mathfrak {g}}_{1}}$、分解可能、可解、ビアンキ III;
• ${\displaystyle {\mathfrak {g}}_{3.1}}$、ハイゼンベルク・ワイル代数、冪零、ビアンキII、

${\displaystyle [e_{2},e_{3}]=e_{1};}$

• ${\displaystyle {\mathfrak {g}}_{3.2}}$、解決可能、ビアンキ IV、

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{1}+e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{3.3}}$、解決可能、ビアンキV、

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{3.4}}$、可解、ビアンキ VI、ポアンカレ代数${\displaystyle {\mathfrak {p}}(1,1)}$いつ${\displaystyle \alpha =-1}$、

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=\alpha e_{2},\quad -1\leq \alpha <1,\quad \alpha \neq 0;}$

• ${\displaystyle {\mathfrak {g}}_{3.5}}$、解決可能、ビアンキVII、

${\displaystyle [e_{1},e_{3}]=\beta e_{1}-e_{2},\quad [e_{2},e_{3}]=e_{1}+\beta e_{2},\quad \beta \geq 0;}$

• ${\displaystyle {\mathfrak {g}}_{3.6}}$、シンプル、ビアンキVIII、${\displaystyle {\mathfrak {sl}}(2,\mathbb {R} ),}$

${\displaystyle [e_{1},e_{2}]=e_{1},\quad [e_{2},e_{3}]=e_{3},\quad [e_{1},e_{3}]=2e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{3.7}}$シンプル、ビアンキIX、${\displaystyle {\mathfrak {so}}(3),}$

${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2},\quad [e_{1},e_{2}]=e_{3}.}$

代数${\displaystyle {\mathfrak {g}}_{3.3}}$極端なケースとして考えることができる${\displaystyle {\mathfrak {g}}_{3.5}}$、 いつ${\displaystyle \beta \rightarrow \infty }$リー代数の縮約を形成する。

フィールド全体にわたって${\displaystyle {\mathbb {C} }}$代数${\displaystyle {\mathfrak {g}}_{3.5}}$、${\displaystyle {\mathfrak {g}}_{3.7}}$は同型である${\displaystyle {\mathfrak {g}}_{3.4}}$そして${\displaystyle {\mathfrak {g}}_{3.6}}$、 それぞれ。

• ${\displaystyle 4{\mathfrak {g}}_{1}}$、アーベル群。
• $\displaystyle {\mathfrak {g}}_{2.1}\oplus 2{\mathfrak {g}}_{1}}$、分解可能、解ける、

${\displaystyle [e_{1},e_{2}]=e_{1};}$

• ${\displaystyle 2{\mathfrak {g}}_{2.1}}$、分解可能、解ける、

${\displaystyle [e_{1},e_{2}]=e_{1}\quad [e_{3},e_{4}]=e_{3};}$

• ${\displaystyle {\mathfrak {g}}_{3.1}\oplus {\mathfrak {g}}_{1}}$、分解可能な冪零、

${\displaystyle [e_{2},e_{3}]=e_{1};}$

• ${\displaystyle {\mathfrak {g}}_{3.2}\oplus {\mathfrak {g}}_{1}}$、分解可能、解ける、

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{1}+e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{3.3}\oplus {\mathfrak {g}}_{1}}$、分解可能、解ける、

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{3.4}\oplus {\mathfrak {g}}_{1}}$、分解可能、解ける、

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=\alpha e_{2},\quad -1\leq \alpha <1,\quad \alpha \neq 0;}$

• ${\displaystyle {\mathfrak {g}}_{3.5}\oplus {\mathfrak {g}}_{1}}$、分解可能、解ける、

${\displaystyle [e_{1},e_{3}]=\beta e_{1}-e_{2}\quad [e_{2},e_{3}]=e_{1}+\beta e_{2},\quad \beta \geq 0;}$

• ${\displaystyle {\mathfrak {g}}_{3.6}\oplus {\mathfrak {g}}_{1}}$解決不可能、

${\displaystyle [e_{1},e_{2}]=e_{1},\quad [e_{2},e_{3}]=e_{3},\quad [e_{1},e_{3}]=2e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{3.7}\oplus {\mathfrak {g}}_{1}}$解決不可能、

${\displaystyle [e_{1},e_{2}]=e_{3},\quad [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{4.1}}$、分解不可能な冪零、

${\displaystyle [e_{2},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{4.2}}$、分解不可能、可解、

${\displaystyle [e_{1},e_{4}]=\beta e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3},\quad \beta \neq 0;}$

• ${\displaystyle {\mathfrak {g}}_{4.3}}$、分解不可能、可解、

${\displaystyle [e_{1},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};}$

• ${\displaystyle {\mathfrak {g}}_{4.4}}$、分解不可能、可解、

${\displaystyle [e_{1},e_{4}]=e_{1},\quad [e_{2},e_{4}]=e_{1}+e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3};}$

• ${\displaystyle {\mathfrak {g}}_{4.5}}$、分解不可能、可解、

${\displaystyle [e_{1},e_{4}]=\alpha e_{1},\quad [e_{2},e_{4}]=\beta e_{2},\quad [e_{3},e_{4}]=\gamma e_{3},\quad \alpha \beta \gamma \neq 0;}$

• ${\displaystyle {\mathfrak {g}}_{4.6}}$、分解不可能、可解、

${\displaystyle [e_{1},e_{4}]=\alpha e_{1},\quad [e_{2},e_{4}]=\beta e_{2}-e_{3},\quad [e_{3},e_{4}]=e_{2}+\beta e_{3},\quad \alpha >0;}$

• ${\displaystyle {\mathfrak {g}}_{4.7}}$、分解不可能、可解、

${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{1},e_{4}]=2e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3};}$

• ${\displaystyle {\mathfrak {g}}_{4.8}}$、分解不可能、可解、

${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{1},e_{4}]=(1+\beta )e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=\beta e_{3},\quad -1\leq \beta \leq 1;}$

• ${\displaystyle {\mathfrak {g}}_{4.9}}$、分解不可能、可解、

${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{1},e_{4}]=2\alpha e_{1},\quad [e_{2},e_{4}]=\alpha e_{2}-e_{3},\quad [e_{3},e_{4}]=e_{2}+\alpha e_{3},\quad \alpha \geq 0;}$

• ${\displaystyle {\mathfrak {g}}_{4.10}}$、分解不可能、可解、

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2},\quad [e_{1},e_{4}]=-e_{2},\quad [e_{2},e_{4}]=e_{1}.}$

代数${\displaystyle {\mathfrak {g}}_{4.3}}$極端なケースとして考えることができる${\displaystyle {\mathfrak {g}}_{4.2}}$、 いつ${\displaystyle \beta \rightarrow 0}$リー代数の縮約を形成する。

フィールド全体にわたって${\displaystyle {\mathbb {C} }}$代数${\displaystyle {\mathfrak {g}}_{3.5}\oplus {\mathfrak {g}}_{1}}$、${\displaystyle {\mathfrak {g}}_{3.7}\oplus {\mathfrak {g}}_{1}}$、${\displaystyle {\mathfrak {g}}_{4.6}}$、${\displaystyle {\mathfrak {g}}_{4.9}}$、${\displaystyle {\mathfrak {g}}_{4.10}}$は同型である${\displaystyle {\mathfrak {g}}_{3.4}\oplus {\mathfrak {g}}_{1}}$、${\displaystyle {\mathfrak {g}}_{3.6}\oplus {\mathfrak {g}}_{1}}$、${\displaystyle {\mathfrak {g}}_{4.5}}$、${\displaystyle {\mathfrak {g}}_{4.8}}$、${\displaystyle {2{\mathfrak {g}}}_{2.1}}$、 それぞれ。