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Classification of low-dimensional real Lie algebras

## Summary

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This mathematics-related list provides Mubarakzyanov's classification of low-dimensional real Lie algebras, published in Russian in 1963.[1] 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.[2] in 2003.

## Mubarakzyanov's Classification

Let ${\displaystyle {\mathfrak {g}}_{n}}$  be ${\displaystyle n}$ -dimensional Lie algebra over the field of real numbers with generators ${\displaystyle e_{1},\dots ,e_{n}}$ , ${\displaystyle n\leq 4}$ .[clarification needed] For each algebra ${\displaystyle {\mathfrak {g}}}$  we adduce only non-zero commutators between basis elements.

### One-dimensional

• ${\displaystyle {\mathfrak {g}}_{1}}$ , abelian.

### Two-dimensional

• ${\displaystyle 2{\mathfrak {g}}_{1}}$ , abelian ${\displaystyle \mathbb {R} ^{2}}$ ;
• ${\displaystyle {\mathfrak {g}}_{2.1}}$ , solvable ${\displaystyle {\mathfrak {aff}}(1)=\left\{{\begin{pmatrix}a&b\\0&0\end{pmatrix}}\,:\,a,b\in \mathbb {R} \right\}}$ ,
${\displaystyle [e_{1},e_{2}]=e_{1}.}$

### Three-dimensional

• ${\displaystyle 3{\mathfrak {g}}_{1}}$ , abelian, Bianchi I;
• ${\displaystyle {\mathfrak {g}}_{2.1}\oplus {\mathfrak {g}}_{1}}$ , decomposable solvable, Bianchi III;
• ${\displaystyle {\mathfrak {g}}_{3.1}}$ , Heisenberg–Weyl algebra, nilpotent, Bianchi II,
${\displaystyle [e_{2},e_{3}]=e_{1};}$
• ${\displaystyle {\mathfrak {g}}_{3.2}}$ , solvable, Bianchi IV,
${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{1}+e_{2};}$
• ${\displaystyle {\mathfrak {g}}_{3.3}}$ , solvable, Bianchi V,
${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2};}$
• ${\displaystyle {\mathfrak {g}}_{3.4}}$ , solvable, Bianchi VI, Poincaré algebra ${\displaystyle {\mathfrak {p}}(1,1)}$  when ${\displaystyle \alpha =-1}$ ,
${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=\alpha e_{2},\quad -1\leq \alpha <1,\quad \alpha \neq 0;}$
• ${\displaystyle {\mathfrak {g}}_{3.5}}$ , solvable, Bianchi VII,
${\displaystyle [e_{1},e_{3}]=\beta e_{1}-e_{2},\quad [e_{2},e_{3}]=e_{1}+\beta e_{2},\quad \beta \geq 0;}$
• ${\displaystyle {\mathfrak {g}}_{3.6}}$ , simple, Bianchi VIII, ${\displaystyle {\mathfrak {sl}}(2,\mathbb {R} ),}$
${\displaystyle [e_{1},e_{2}]=e_{1},\quad [e_{2},e_{3}]=e_{3},\quad [e_{1},e_{3}]=2e_{2};}$
• ${\displaystyle {\mathfrak {g}}_{3.7}}$ , simple, Bianchi IX, ${\displaystyle {\mathfrak {so}}(3),}$
${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2},\quad [e_{1},e_{2}]=e_{3}.}$

Algebra ${\displaystyle {\mathfrak {g}}_{3.3}}$  can be considered as an extreme case of ${\displaystyle {\mathfrak {g}}_{3.5}}$ , when ${\displaystyle \beta \rightarrow \infty }$ , forming contraction of Lie algebra.

Over the field ${\displaystyle {\mathbb {C} }}$  algebras ${\displaystyle {\mathfrak {g}}_{3.5}}$ , ${\displaystyle {\mathfrak {g}}_{3.7}}$  are isomorphic to ${\displaystyle {\mathfrak {g}}_{3.4}}$  and ${\displaystyle {\mathfrak {g}}_{3.6}}$ , respectively.

### Four-dimensional

• ${\displaystyle 4{\mathfrak {g}}_{1}}$ , abelian;
• ${\displaystyle {\mathfrak {g}}_{2.1}\oplus 2{\mathfrak {g}}_{1}}$ , decomposable solvable,
${\displaystyle [e_{1},e_{2}]=e_{1};}$
• ${\displaystyle 2{\mathfrak {g}}_{2.1}}$ , decomposable solvable,
${\displaystyle [e_{1},e_{2}]=e_{1}\quad [e_{3},e_{4}]=e_{3};}$
• ${\displaystyle {\mathfrak {g}}_{3.1}\oplus {\mathfrak {g}}_{1}}$ , decomposable nilpotent,
${\displaystyle [e_{2},e_{3}]=e_{1};}$
• ${\displaystyle {\mathfrak {g}}_{3.2}\oplus {\mathfrak {g}}_{1}}$ , decomposable solvable,
${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{1}+e_{2};}$
• ${\displaystyle {\mathfrak {g}}_{3.3}\oplus {\mathfrak {g}}_{1}}$ , decomposable solvable,
${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2};}$
• ${\displaystyle {\mathfrak {g}}_{3.4}\oplus {\mathfrak {g}}_{1}}$ , decomposable solvable,
${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=\alpha e_{2},\quad -1\leq \alpha <1,\quad \alpha \neq 0;}$
• ${\displaystyle {\mathfrak {g}}_{3.5}\oplus {\mathfrak {g}}_{1}}$ , decomposable solvable,
${\displaystyle [e_{1},e_{3}]=\beta e_{1}-e_{2}\quad [e_{2},e_{3}]=e_{1}+\beta e_{2},\quad \beta \geq 0;}$
• ${\displaystyle {\mathfrak {g}}_{3.6}\oplus {\mathfrak {g}}_{1}}$ , unsolvable,
${\displaystyle [e_{1},e_{2}]=e_{1},\quad [e_{2},e_{3}]=e_{3},\quad [e_{1},e_{3}]=2e_{2};}$
• ${\displaystyle {\mathfrak {g}}_{3.7}\oplus {\mathfrak {g}}_{1}}$ , unsolvable,
${\displaystyle [e_{1},e_{2}]=e_{3},\quad [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2};}$
• ${\displaystyle {\mathfrak {g}}_{4.1}}$ , indecomposable nilpotent,
${\displaystyle [e_{2},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};}$
• ${\displaystyle {\mathfrak {g}}_{4.2}}$ , indecomposable solvable,
${\displaystyle [e_{1},e_{4}]=\beta e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3},\quad \beta \neq 0;}$
• ${\displaystyle {\mathfrak {g}}_{4.3}}$ , indecomposable solvable,
${\displaystyle [e_{1},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};}$
• ${\displaystyle {\mathfrak {g}}_{4.4}}$ , indecomposable solvable,
${\displaystyle [e_{1},e_{4}]=e_{1},\quad [e_{2},e_{4}]=e_{1}+e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3};}$
• ${\displaystyle {\mathfrak {g}}_{4.5}}$ , indecomposable solvable,
${\displaystyle [e_{1},e_{4}]=\alpha e_{1},\quad [e_{2},e_{4}]=\beta e_{2},\quad [e_{3},e_{4}]=\gamma e_{3},\quad \alpha \beta \gamma \neq 0;}$
• ${\displaystyle {\mathfrak {g}}_{4.6}}$ , indecomposable solvable,
${\displaystyle [e_{1},e_{4}]=\alpha e_{1},\quad [e_{2},e_{4}]=\beta e_{2}-e_{3},\quad [e_{3},e_{4}]=e_{2}+\beta e_{3},\quad \alpha >0;}$
• ${\displaystyle {\mathfrak {g}}_{4.7}}$ , indecomposable solvable,
${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{1},e_{4}]=2e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3};}$
• ${\displaystyle {\mathfrak {g}}_{4.8}}$ , indecomposable solvable,
${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{1},e_{4}]=(1+\beta )e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=\beta e_{3},\quad -1\leq \beta \leq 1;}$
• ${\displaystyle {\mathfrak {g}}_{4.9}}$ , indecomposable solvable,
${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{1},e_{4}]=2\alpha e_{1},\quad [e_{2},e_{4}]=\alpha e_{2}-e_{3},\quad [e_{3},e_{4}]=e_{2}+\alpha e_{3},\quad \alpha \geq 0;}$
• ${\displaystyle {\mathfrak {g}}_{4.10}}$ , indecomposable solvable,
${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2},\quad [e_{1},e_{4}]=-e_{2},\quad [e_{2},e_{4}]=e_{1}.}$

Algebra ${\displaystyle {\mathfrak {g}}_{4.3}}$  can be considered as an extreme case of ${\displaystyle {\mathfrak {g}}_{4.2}}$ , when ${\displaystyle \beta \rightarrow 0}$ , forming contraction of Lie algebra.

Over the field ${\displaystyle {\mathbb {C} }}$  algebras ${\displaystyle {\mathfrak {g}}_{3.5}\oplus {\mathfrak {g}}_{1}}$ , ${\displaystyle {\mathfrak {g}}_{3.7}\oplus {\mathfrak {g}}_{1}}$ , ${\displaystyle {\mathfrak {g}}_{4.6}}$ , ${\displaystyle {\mathfrak {g}}_{4.9}}$ , ${\displaystyle {\mathfrak {g}}_{4.10}}$  are isomorphic to ${\displaystyle {\mathfrak {g}}_{3.4}\oplus {\mathfrak {g}}_{1}}$ , ${\displaystyle {\mathfrak {g}}_{3.6}\oplus {\mathfrak {g}}_{1}}$ , ${\displaystyle {\mathfrak {g}}_{4.5}}$ , ${\displaystyle {\mathfrak {g}}_{4.8}}$ , ${\displaystyle {2{\mathfrak {g}}}_{2.1}}$ , respectively.