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. 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.

Mubarakzyanov's Classification

Let ${\mathfrak {g}}_{n}$ be $n$-dimensional Lie algebra over the field of real numbers with generators $e_{1},\dots ,e_{n}$, $n\leq 4$. 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 ${\mathfrak {aff}}(1)=\left\{{\begin{pmatrix}a&b\\0&0\end{pmatrix}}\,:\,a,b\in \mathbb {R} \right\}$,

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

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 IX, ${\mathfrak {so}}(3),$

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

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.

See also

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

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.