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 ${\displaystyle {\mathfrak {g}}_{n}}$ 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.

See also

• Table of Lie groups
• Simple Lie group#Full classification

Notes

1. Mubarakzyanov 1963
2. 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.
• 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. S2CID 9800361.