Classification of low-dimensional real Lie algebras

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

One-dimensional

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

Two-dimensional

$2{\mathfrak {g}}_{1}$ , abelian $\mathbb {R} ^{2}$ ;
${\mathfrak {g}}_{2.1}$ , solvable ${\mathfrak {af}}(1)=\{{\begin{pmatrix}a&b\\0&0\end{pmatrix}}\,:\,a,b\in \mathbb {R} \}$ ,

[e_{1},e_{2}]=e_{1}.

Three-dimensional

$3{\mathfrak {g}}_{1}$ , abelian, Bianchi I;
${\mathfrak {g}}_{2.1}\oplus {\mathfrak {g}}_{1}$ , decomposable solvable, Bianchi III;
${\mathfrak {g}}_{3.1}$ , Heisenberg–Weyl algebra, nilpotent, Bianchi II,

[e_{2},e_{3}]=e_{1};

${\mathfrak {g}}_{3.2}$ , solvable, Bianchi IV,

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

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

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

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

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

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

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

${\mathfrak {g}}_{3.6}$ , simple, Bianchi VIII, ${\mathfrak {sl}}(2,\mathbb {R} ),$

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

${\mathfrak {g}}_{3.7}$ , simple, Bianchi VIII, ${\mathfrak {so}}(3),$

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

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

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

Four-dimensional

$4{\mathfrak {g}}_{1}$ , abelian;
${\mathfrak {g}}_{2.1}\oplus 2{\mathfrak {g}}_{1}$ , decomposable solvable,

[e_{1},e_{2}]=e_{1};

$2{\mathfrak {g}}_{2.1}$ , decomposable solvable,

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

${\mathfrak {g}}_{3.1}\oplus {\mathfrak {g}}_{1}$ , decomposable nilpotent,

[e_{2},e_{3}]=e_{1};

${\mathfrak {g}}_{3.2}\oplus {\mathfrak {g}}_{1}$ , decomposable solvable,

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

${\mathfrak {g}}_{3.3}\oplus {\mathfrak {g}}_{1}$ , decomposable solvable,

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

${\mathfrak {g}}_{3.4}\oplus {\mathfrak {g}}_{1}$ , decomposable solvable,

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

${\mathfrak {g}}_{3.5}\oplus {\mathfrak {g}}_{1}$ , decomposable solvable,

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

${\mathfrak {g}}_{3.6}\oplus {\mathfrak {g}}_{1}$ , unsolvable,

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

${\mathfrak {g}}_{3.7}\oplus {\mathfrak {g}}_{1}$ , unsolvable,

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

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

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

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

[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;

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

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

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

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

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

[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;

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

[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;

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

[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};

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

[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;

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

[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;

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

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

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

Notes

^ Mubarakzyanov 1963
^ Popovych 2003

References

Mubarakzyanov, G.M. (1963). "On solvable Lie algebras". Izv. Vys. Ucheb. Zaved. Matematika (in Russian). 1 (32): 114–123. MR 0153714. Zbl 0166.04104. {{cite journal}}: Invalid |ref=harv (help)
Popovych, R.O.; Boyko, V.M.; Nesterenko, M.O.; Lutfullin, M.W.; et al. (2003). "Realizations of real low-dimensional Lie algebras". J. Phys. A: Math. Gen. 36 (26): 7337–7360. arXiv:math-ph/0301029. Bibcode:2003JPhA...36.7337P. doi:10.1088/0305-4470/36/26/309. {{cite journal}}: Invalid |ref=harv (help)

[1] Mubarakzyanov 1963

[2] Popovych 2003

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