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 [math]\displaystyle{ {\mathfrak g}_n }[/math] be [math]\displaystyle{ n }[/math]-dimensional Lie algebra over the field of real numbers with generators [math]\displaystyle{ e_1, \dots, e_n }[/math], [math]\displaystyle{ n \leq 4 }[/math].^{[clarification needed]} For each algebra [math]\displaystyle{ {\mathfrak g} }[/math] we adduce only non-zero commutators between basis elements.

One-dimensional

• [math]\displaystyle{ {\mathfrak g}_1 }[/math], abelian.

Two-dimensional

• [math]\displaystyle{ 2{\mathfrak g}_1 }[/math], abelian [math]\displaystyle{ \mathbb{R}^2 }[/math];
• [math]\displaystyle{ {\mathfrak g}_{2.1} }[/math], solvable [math]\displaystyle{ \mathfrak{aff}(1)=\left\{\begin{pmatrix} a&b \\ 0&0 \end{pmatrix}\,:\,a,b\in\mathbb{R}\right\} }[/math],

[math]\displaystyle{ [e_1, e_2] = e_1. }[/math]

Three-dimensional

• [math]\displaystyle{ 3{\mathfrak g}_1 }[/math], abelian, Bianchi I;
• [math]\displaystyle{ {\mathfrak g}_{2.1}\oplus {\mathfrak g}_1 }[/math], decomposable solvable, Bianchi III;
• [math]\displaystyle{ {\mathfrak g}_{3.1} }[/math], Heisenberg–Weyl algebra, nilpotent, Bianchi II,

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

• [math]\displaystyle{ {\mathfrak g}_{3.2} }[/math], solvable, Bianchi IV,

[math]\displaystyle{ [e_1, e_3] = e_1, \quad [e_2, e_3] = e_1 + e_2; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{3.3} }[/math], solvable, Bianchi V,

[math]\displaystyle{ [e_1, e_3] = e_1, \quad [e_2, e_3] = e_2; }[/math]

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

[math]\displaystyle{ [e_1, e_3] = e_1, \quad [e_2, e_3] = \alpha e_2, \quad -1 \leq \alpha \lt 1, \quad \alpha \neq 0; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{3.5} }[/math], solvable, Bianchi VII,

[math]\displaystyle{ [e_1, e_3] = \beta e_1 - e_2, \quad [e_2, e_3] = e_1 + \beta e_2, \quad \beta \geq 0; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{3.6} }[/math], simple, Bianchi VIII, [math]\displaystyle{ \mathfrak{sl}(2, \mathbb R ), }[/math]

[math]\displaystyle{ [e_1, e_2] = e_1, \quad [e_2, e_3] = e_3, \quad [e_1, e_3] = 2 e_2; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{3.7} }[/math], simple, Bianchi IX, [math]\displaystyle{ \mathfrak{so}(3), }[/math]

[math]\displaystyle{ [e_2, e_3] = e_1, \quad [e_3, e_1] = e_2, \quad [e_1, e_2] = e_3. }[/math]

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

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

Four-dimensional

• [math]\displaystyle{ 4{\mathfrak g}_1 }[/math], abelian;
• [math]\displaystyle{ {\mathfrak g}_{2.1} \oplus 2{\mathfrak g}_1 }[/math], decomposable solvable,

[math]\displaystyle{ [e_1, e_2] = e_1; }[/math]

• [math]\displaystyle{ 2{\mathfrak g}_{2.1} }[/math], decomposable solvable,

[math]\displaystyle{ [e_1, e_2] = e_1 \quad [e_3, e_4] = e_3; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{3.1} \oplus {\mathfrak g}_1 }[/math], decomposable nilpotent,

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

• [math]\displaystyle{ {\mathfrak g}_{3.2} \oplus {\mathfrak g}_1 }[/math], decomposable solvable,

[math]\displaystyle{ [e_1, e_3] = e_1, \quad [e_2, e_3] = e_1 + e_2; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{3.3} \oplus {\mathfrak g}_1 }[/math], decomposable solvable,

[math]\displaystyle{ [e_1, e_3] = e_1, \quad [e_2, e_3] = e_2; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{3.4} \oplus {\mathfrak g}_1 }[/math], decomposable solvable,

[math]\displaystyle{ [e_1, e_3] = e_1, \quad [e_2, e_3] = \alpha e_2, \quad -1 \leq \alpha \lt 1, \quad \alpha \neq 0; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{3.5} \oplus {\mathfrak g}_1 }[/math], decomposable solvable,

[math]\displaystyle{ [e_1, e_3] = \beta e_1 - e_2 \quad [e_2, e_3] = e_1 + \beta e_2, \quad \beta \geq 0; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{3.6} \oplus {\mathfrak g}_1 }[/math], unsolvable,

[math]\displaystyle{ [e_1, e_2] = e_1, \quad [e_2, e_3] = e_3, \quad [e_1, e_3] = 2 e_2; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{3.7} \oplus {\mathfrak g}_1 }[/math], unsolvable,

[math]\displaystyle{ [e_1, e_2] = e_3, \quad [e_2, e_3] = e_1, \quad [e_3, e_1] = e_2; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{4.1} }[/math], indecomposable nilpotent,

[math]\displaystyle{ [e_2, e_4] = e_1, \quad [e_3, e_4] = e_2; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{4.2} }[/math], indecomposable solvable,

[math]\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; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{4.3} }[/math], indecomposable solvable,

[math]\displaystyle{ [e_1, e_4] = e_1, \quad [e_3, e_4] = e_2; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{4.4} }[/math], indecomposable solvable,

[math]\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; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{4.5} }[/math], indecomposable solvable,

[math]\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; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{4.6} }[/math], indecomposable solvable,

[math]\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 \gt 0; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{4.7} }[/math], indecomposable solvable,

[math]\displaystyle{ [e_2, e_3] = e_1, \quad [e_1, e_4] = 2 e_1, \quad [e_2, e_4] = e_2, \quad [e_3, e_4] = e_2 + e_3; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{4.8} }[/math], indecomposable solvable,

[math]\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; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{4.9} }[/math], indecomposable solvable,

[math]\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; }[/math]

• [math]\displaystyle{ {\mathfrak g}_{4.10} }[/math], indecomposable solvable,

[math]\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. }[/math]

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

Over the field [math]\displaystyle{ {\mathbb C} }[/math] algebras [math]\displaystyle{ {\mathfrak g}_{3.5} \oplus {\mathfrak g}_1 }[/math], [math]\displaystyle{ {\mathfrak g}_{3.7} \oplus {\mathfrak g}_1 }[/math], [math]\displaystyle{ {\mathfrak g}_{4.6} }[/math], [math]\displaystyle{ {\mathfrak g}_{4.9} }[/math], [math]\displaystyle{ {\mathfrak g}_{4.10} }[/math] are isomorphic to [math]\displaystyle{ {\mathfrak g}_{3.4} \oplus {\mathfrak g}_1 }[/math], [math]\displaystyle{ {\mathfrak g}_{3.6} \oplus {\mathfrak g}_1 }[/math], [math]\displaystyle{ {\mathfrak g}_{4.5} }[/math], [math]\displaystyle{ {\mathfrak g}_{4.8} }[/math], [math]\displaystyle{ {2\mathfrak g}_{2.1} }[/math], respectively.

Notes

References

• Mubarakzyanov, G.M. (1963). "On solvable Lie algebras" (in Russian). Izv. Vys. Ucheb. Zaved. Matematika 1 (32): 114–123. http://mi.mathnet.ru/eng/ivm2141. 
• Popovych, R.O. et al. (2003). "Realizations of real low-dimensional Lie algebras". J. Phys. A: Math. Gen. 36 (26): 7337–7360. doi:10.1088/0305-4470/36/26/309. Bibcode: 2003JPhA...36.7337P. 

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