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This mathematics -related list provides Mubarakzyanov's classification of low-dimensional real Lie algebras , published in Russian in 1963. It complements the article on Lie algebra in the area of abstract algebra .
An English version and review of this classification was published by Popovych et al. in 2003.
[ edit ] Let g n {\displaystyle {\mathfrak {g}}_{n}} n {\displaystyle n} Lie algebra over the field of real numbers with generators e 1 , … , e n {\displaystyle e_{1},\dots ,e_{n}} n ≤ 4 {\displaystyle n\leq 4} g {\displaystyle {\mathfrak {g}}}
g 1 {\displaystyle {\mathfrak {g}}_{1}} abelian .
2 g 1 {\displaystyle 2{\mathfrak {g}}_{1}} R 2 {\displaystyle \mathbb {R} ^{2}} g 2.1 {\displaystyle {\mathfrak {g}}_{2.1}} solvable a f f ( 1 ) = { ( a b 0 0 ) : a , b ∈ R } {\displaystyle {\mathfrak {aff}}(1)=\left\{{\begin{pmatrix}a&b\\0&0\end{pmatrix}}\,:\,a,b\in \mathbb {R} \right\}} [ e 1 , e 2 ] = e 1 . {\displaystyle [e_{1},e_{2}]=e_{1}.}
3 g 1 {\displaystyle 3{\mathfrak {g}}_{1}} Bianchi I ; g 2.1 ⊕ g 1 {\displaystyle {\mathfrak {g}}_{2.1}\oplus {\mathfrak {g}}_{1}} g 3.1 {\displaystyle {\mathfrak {g}}_{3.1}} [ e 2 , e 3 ] = e 1 ; {\displaystyle [e_{2},e_{3}]=e_{1};}
g 3.2 {\displaystyle {\mathfrak {g}}_{3.2}} [ e 1 , e 3 ] = e 1 , [ e 2 , e 3 ] = e 1 + e 2 ; {\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{1}+e_{2};}
g 3.3 {\displaystyle {\mathfrak {g}}_{3.3}} [ e 1 , e 3 ] = e 1 , [ e 2 , e 3 ] = e 2 ; {\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2};}
g 3.4 {\displaystyle {\mathfrak {g}}_{3.4}} Poincaré algebra p ( 1 , 1 ) {\displaystyle {\mathfrak {p}}(1,1)} α = − 1 {\displaystyle \alpha =-1} [ e 1 , e 3 ] = e 1 , [ e 2 , e 3 ] = α e 2 , − 1 ≤ α < 1 , α ≠ 0 ; {\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=\alpha e_{2},\quad -1\leq \alpha <1,\quad \alpha \neq 0;}
g 3.5 {\displaystyle {\mathfrak {g}}_{3.5}} [ e 1 , e 3 ] = β e 1 − e 2 , [ e 2 , e 3 ] = e 1 + β e 2 , β ≥ 0 ; {\displaystyle [e_{1},e_{3}]=\beta e_{1}-e_{2},\quad [e_{2},e_{3}]=e_{1}+\beta e_{2},\quad \beta \geq 0;}
g 3.6 {\displaystyle {\mathfrak {g}}_{3.6}} s l ( 2 , R ) , {\displaystyle {\mathfrak {sl}}(2,\mathbb {R} ),} [ e 1 , e 2 ] = e 1 , [ e 2 , e 3 ] = e 3 , [ e 1 , e 3 ] = 2 e 2 ; {\displaystyle [e_{1},e_{2}]=e_{1},\quad [e_{2},e_{3}]=e_{3},\quad [e_{1},e_{3}]=2e_{2};}
g 3.7 {\displaystyle {\mathfrak {g}}_{3.7}} s o ( 3 ) , {\displaystyle {\mathfrak {so}}(3),} [ e 2 , e 3 ] = e 1 , [ e 3 , e 1 ] = e 2 , [ e 1 , e 2 ] = e 3 . {\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2},\quad [e_{1},e_{2}]=e_{3}.} Algebra g 3.3 {\displaystyle {\mathfrak {g}}_{3.3}} g 3.5 {\displaystyle {\mathfrak {g}}_{3.5}} β → ∞ {\displaystyle \beta \rightarrow \infty }
Over the field C {\displaystyle {\mathbb {C} }} g 3.5 {\displaystyle {\mathfrak {g}}_{3.5}} g 3.7 {\displaystyle {\mathfrak {g}}_{3.7}} g 3.4 {\displaystyle {\mathfrak {g}}_{3.4}} g 3.6 {\displaystyle {\mathfrak {g}}_{3.6}}
4 g 1 {\displaystyle 4{\mathfrak {g}}_{1}} g 2.1 ⊕ 2 g 1 {\displaystyle {\mathfrak {g}}_{2.1}\oplus 2{\mathfrak {g}}_{1}} [ e 1 , e 2 ] = e 1 ; {\displaystyle [e_{1},e_{2}]=e_{1};}
2 g 2.1 {\displaystyle 2{\mathfrak {g}}_{2.1}} [ e 1 , e 2 ] = e 1 [ e 3 , e 4 ] = e 3 ; {\displaystyle [e_{1},e_{2}]=e_{1}\quad [e_{3},e_{4}]=e_{3};}
g 3.1 ⊕ g 1 {\displaystyle {\mathfrak {g}}_{3.1}\oplus {\mathfrak {g}}_{1}} [ e 2 , e 3 ] = e 1 ; {\displaystyle [e_{2},e_{3}]=e_{1};}
g 3.2 ⊕ g 1 {\displaystyle {\mathfrak {g}}_{3.2}\oplus {\mathfrak {g}}_{1}} [ e 1 , e 3 ] = e 1 , [ e 2 , e 3 ] = e 1 + e 2 ; {\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{1}+e_{2};}
g 3.3 ⊕ g 1 {\displaystyle {\mathfrak {g}}_{3.3}\oplus {\mathfrak {g}}_{1}} [ e 1 , e 3 ] = e 1 , [ e 2 , e 3 ] = e 2 ; {\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2};}
g 3.4 ⊕ g 1 {\displaystyle {\mathfrak {g}}_{3.4}\oplus {\mathfrak {g}}_{1}} [ e 1 , e 3 ] = e 1 , [ e 2 , e 3 ] = α e 2 , − 1 ≤ α < 1 , α ≠ 0 ; {\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=\alpha e_{2},\quad -1\leq \alpha <1,\quad \alpha \neq 0;}
g 3.5 ⊕ g 1 {\displaystyle {\mathfrak {g}}_{3.5}\oplus {\mathfrak {g}}_{1}} [ e 1 , e 3 ] = β e 1 − e 2 [ e 2 , e 3 ] = e 1 + β e 2 , β ≥ 0 ; {\displaystyle [e_{1},e_{3}]=\beta e_{1}-e_{2}\quad [e_{2},e_{3}]=e_{1}+\beta e_{2},\quad \beta \geq 0;}
g 3.6 ⊕ g 1 {\displaystyle {\mathfrak {g}}_{3.6}\oplus {\mathfrak {g}}_{1}} [ e 1 , e 2 ] = e 1 , [ e 2 , e 3 ] = e 3 , [ e 1 , e 3 ] = 2 e 2 ; {\displaystyle [e_{1},e_{2}]=e_{1},\quad [e_{2},e_{3}]=e_{3},\quad [e_{1},e_{3}]=2e_{2};}
g 3.7 ⊕ g 1 {\displaystyle {\mathfrak {g}}_{3.7}\oplus {\mathfrak {g}}_{1}} [ e 1 , e 2 ] = e 3 , [ e 2 , e 3 ] = e 1 , [ e 3 , e 1 ] = e 2 ; {\displaystyle [e_{1},e_{2}]=e_{3},\quad [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2};}
g 4.1 {\displaystyle {\mathfrak {g}}_{4.1}} [ e 2 , e 4 ] = e 1 , [ e 3 , e 4 ] = e 2 ; {\displaystyle [e_{2},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};}
g 4.2 {\displaystyle {\mathfrak {g}}_{4.2}} [ e 1 , e 4 ] = β e 1 , [ e 2 , e 4 ] = e 2 , [ e 3 , e 4 ] = e 2 + e 3 , β ≠ 0 ; {\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;}
g 4.3 {\displaystyle {\mathfrak {g}}_{4.3}} [ e 1 , e 4 ] = e 1 , [ e 3 , e 4 ] = e 2 ; {\displaystyle [e_{1},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};}
g 4.4 {\displaystyle {\mathfrak {g}}_{4.4}} [ e 1 , e 4 ] = e 1 , [ e 2 , e 4 ] = e 1 + e 2 , [ e 3 , e 4 ] = e 2 + e 3 ; {\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};}
g 4.5 {\displaystyle {\mathfrak {g}}_{4.5}} [ e 1 , e 4 ] = α e 1 , [ e 2 , e 4 ] = β e 2 , [ e 3 , e 4 ] = γ e 3 , α β γ ≠ 0 ; {\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;}
g 4.6 {\displaystyle {\mathfrak {g}}_{4.6}} [ e 1 , e 4 ] = α e 1 , [ e 2 , e 4 ] = β e 2 − e 3 , [ e 3 , e 4 ] = e 2 + β e 3 , α > 0 ; {\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;}
g 4.7 {\displaystyle {\mathfrak {g}}_{4.7}} [ e 2 , e 3 ] = e 1 , [ e 1 , e 4 ] = 2 e 1 , [ e 2 , e 4 ] = e 2 , [ e 3 , e 4 ] = e 2 + e 3 ; {\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};}
g 4.8 {\displaystyle {\mathfrak {g}}_{4.8}} [ e 2 , e 3 ] = e 1 , [ e 1 , e 4 ] = ( 1 + β ) e 1 , [ e 2 , e 4 ] = e 2 , [ e 3 , e 4 ] = β e 3 , − 1 ≤ β ≤ 1 ; {\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;}
g 4.9 {\displaystyle {\mathfrak {g}}_{4.9}} [ e 2 , e 3 ] = e 1 , [ e 1 , e 4 ] = 2 α e 1 , [ e 2 , e 4 ] = α e 2 − e 3 , [ e 3 , e 4 ] = e 2 + α e 3 , α ≥ 0 ; {\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;}
g 4.10 {\displaystyle {\mathfrak {g}}_{4.10}} [ e 1 , e 3 ] = e 1 , [ e 2 , e 3 ] = e 2 , [ e 1 , e 4 ] = − e 2 , [ e 2 , e 4 ] = e 1 . {\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}.} Algebra g 4.3 {\displaystyle {\mathfrak {g}}_{4.3}} g 4.2 {\displaystyle {\mathfrak {g}}_{4.2}} β → 0 {\displaystyle \beta \rightarrow 0}
Over the field C {\displaystyle {\mathbb {C} }} g 3.5 ⊕ g 1 {\displaystyle {\mathfrak {g}}_{3.5}\oplus {\mathfrak {g}}_{1}} g 3.7 ⊕ g 1 {\displaystyle {\mathfrak {g}}_{3.7}\oplus {\mathfrak {g}}_{1}} g 4.6 {\displaystyle {\mathfrak {g}}_{4.6}} g 4.9 {\displaystyle {\mathfrak {g}}_{4.9}} g 4.10 {\displaystyle {\mathfrak {g}}_{4.10}} g 3.4 ⊕ g 1 {\displaystyle {\mathfrak {g}}_{3.4}\oplus {\mathfrak {g}}_{1}} g 3.6 ⊕ g 1 {\displaystyle {\mathfrak {g}}_{3.6}\oplus {\mathfrak {g}}_{1}} g 4.5 {\displaystyle {\mathfrak {g}}_{4.5}} g 4.8 {\displaystyle {\mathfrak {g}}_{4.8}} 2 g 2.1 {\displaystyle {2{\mathfrak {g}}}_{2.1}}
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