Classification of low-dimensional real Lie algebras

An English version and review of this classification was published by Popovych et al. in 2003.

Mubarakzyanov's Classification

Let ${\displaystyle {\mathfrak{g}}_n}$ be ${\displaystyle n}$-dimensional Lie algebra over the field of real numbers with generators ${\displaystyle e_1,\dots ,e_n}$, ${\displaystyle n\le 4}$. 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 ${\displaystyle \mathfrak{a}\mathfrak{f}\mathfrak{f}(1)=\left\{\left(\begin{array}{cc}a& b\\ 0& 0\end{array}\right):a,b\in \mathbb{R}\right\}}$,

${\displaystyle [e_1,e_2]=e_1.}$

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,

${\displaystyle [e_1,e_3]=e_1,[e_2,e_3]=e_1+e_2;}$

• ${\displaystyle {\mathfrak{g}}_{3.3}}$, solvable, Bianchi V,

${\displaystyle [e_1,e_3]=e_1,[e_2,e_3]=e_2;}$

• ${\displaystyle {\mathfrak{g}}_{3.4}}$, solvable, Bianchi VI, Poincaré algebra ${\displaystyle \mathfrak{p}(1,1)}$ when ${\displaystyle \alpha =-1}$,

${\displaystyle [e_1,e_3]=e_1,[e_2,e_3]=\alpha e_2,-1\le \alpha <1,\alpha \ne 0;}$

• ${\displaystyle {\mathfrak{g}}_{3.5}}$, solvable, Bianchi VII,

${\displaystyle [e_1,e_3]=\beta e_1-e_2,[e_2,e_3]=e_1+\beta e_2,\beta \ge 0;}$

• ${\displaystyle {\mathfrak{g}}_{3.6}}$, simple, Bianchi VIII, ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{R}),}$

${\displaystyle [e_1,e_2]=e_1,[e_2,e_3]=e_3,[e_1,e_3]=2e_2;}$

• ${\displaystyle {\mathfrak{g}}_{3.7}}$, simple, Bianchi IX, ${\displaystyle \mathfrak{s}\mathfrak{o}(3),}$

${\displaystyle [e_2,e_3]=e_1,[e_3,e_1]=e_2,[e_1,e_2]=e_3.}$

Algebra ${\displaystyle {\mathfrak{g}}_{3.3}}$ can be considered as an extreme case of ${\displaystyle {\mathfrak{g}}_{3.5}}$, when ${\displaystyle \beta \to \mathrm{\infty }}$, 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,

${\displaystyle [e_1,e_2]=e_1[e_3,e_4]=e_3;}$

• ${\displaystyle {\mathfrak{g}}_{3.1}\oplus {\mathfrak{g}}_1}$, decomposable nilpotent,

${\displaystyle [e_2,e_3]=e_1;}$

• ${\displaystyle {\mathfrak{g}}_{3.2}\oplus {\mathfrak{g}}_1}$, decomposable solvable,

${\displaystyle [e_1,e_3]=e_1,[e_2,e_3]=e_1+e_2;}$

• ${\displaystyle {\mathfrak{g}}_{3.3}\oplus {\mathfrak{g}}_1}$, decomposable solvable,

${\displaystyle [e_1,e_3]=e_1,[e_2,e_3]=e_2;}$

• ${\displaystyle {\mathfrak{g}}_{3.4}\oplus {\mathfrak{g}}_1}$, decomposable solvable,

${\displaystyle [e_1,e_3]=e_1,[e_2,e_3]=\alpha e_2,-1\le \alpha <1,\alpha \ne 0;}$

• ${\displaystyle {\mathfrak{g}}_{3.5}\oplus {\mathfrak{g}}_1}$, decomposable solvable,

${\displaystyle [e_1,e_3]=\beta e_1-e_2[e_2,e_3]=e_1+\beta e_2,\beta \ge 0;}$

• ${\displaystyle {\mathfrak{g}}_{3.6}\oplus {\mathfrak{g}}_1}$, unsolvable,

${\displaystyle [e_1,e_2]=e_1,[e_2,e_3]=e_3,[e_1,e_3]=2e_2;}$

• ${\displaystyle {\mathfrak{g}}_{3.7}\oplus {\mathfrak{g}}_1}$, unsolvable,

${\displaystyle [e_1,e_2]=e_3,[e_2,e_3]=e_1,[e_3,e_1]=e_2;}$

• ${\displaystyle {\mathfrak{g}}_{4.1}}$, indecomposable nilpotent,

${\displaystyle [e_2,e_4]=e_1,[e_3,e_4]=e_2;}$

• ${\displaystyle {\mathfrak{g}}_{4.2}}$, indecomposable solvable,

${\displaystyle [e_1,e_4]=\beta e_1,[e_2,e_4]=e_2,[e_3,e_4]=e_2+e_3,\beta \ne 0;}$

• ${\displaystyle {\mathfrak{g}}_{4.3}}$, indecomposable solvable,

${\displaystyle [e_1,e_4]=e_1,[e_3,e_4]=e_2;}$

• ${\displaystyle {\mathfrak{g}}_{4.4}}$, indecomposable solvable,

${\displaystyle [e_1,e_4]=e_1,[e_2,e_4]=e_1+e_2,[e_3,e_4]=e_2+e_3;}$

• ${\displaystyle {\mathfrak{g}}_{4.5}}$, indecomposable solvable,

${\displaystyle [e_1,e_4]=\alpha e_1,[e_2,e_4]=\beta e_2,[e_3,e_4]=\gamma e_3,\alpha \beta \gamma \ne 0;}$

• ${\displaystyle {\mathfrak{g}}_{4.6}}$, indecomposable solvable,

${\displaystyle [e_1,e_4]=\alpha e_1,[e_2,e_4]=\beta e_2-e_3,[e_3,e_4]=e_2+\beta e_3,\alpha >0;}$

• ${\displaystyle {\mathfrak{g}}_{4.7}}$, indecomposable solvable,

${\displaystyle [e_2,e_3]=e_1,[e_1,e_4]=2e_1,[e_2,e_4]=e_2,[e_3,e_4]=e_2+e_3;}$

• ${\displaystyle {\mathfrak{g}}_{4.8}}$, indecomposable solvable,

${\displaystyle [e_2,e_3]=e_1,[e_1,e_4]=(1+\beta )e_1,[e_2,e_4]=e_2,[e_3,e_4]=\beta e_3,-1\le \beta \le 1;}$

• ${\displaystyle {\mathfrak{g}}_{4.9}}$, indecomposable solvable,

${\displaystyle [e_2,e_3]=e_1,[e_1,e_4]=2\alpha e_1,[e_2,e_4]=\alpha e_2-e_3,[e_3,e_4]=e_2+\alpha e_3,\alpha \ge 0;}$

• ${\displaystyle {\mathfrak{g}}_{4.10}}$, indecomposable solvable,

${\displaystyle [e_1,e_3]=e_1,[e_2,e_3]=e_2,[e_1,e_4]=-e_2,[e_2,e_4]=e_1.}$

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