Classification of low-dimensional real Lie algebras

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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.

## Contents

* 1 Mubarakzyanov's Classification
* 1.1 One-dimensional
* 1.2 Two-dimensional
* 1.3 Three-dimensional
* 1.4 Four-dimensional
* 2 See also
* 3 Notes
* 4 References

## Mubarakzyanov's Classification

Let 𝔤n be n-dimensional Lie algebra over the field of real numbers with generators e1,…,en, n≤4.[clarification needed] For each algebra 𝔤 we adduce only non-zero commutators between basis elements.

### One-dimensional

* 𝔤1, abelian.

### Two-dimensional

* 2𝔤1, abelian ℝ2;
* 𝔤2.1, solvable 𝔞𝔣𝔣(1)={(ab00):a,b∈ℝ},

[e1,e2]=e1.

### Three-dimensional

* 3𝔤1, abelian, Bianchi I;
* 𝔤2.1⊕𝔤1, decomposable solvable, Bianchi III;
* 𝔤3.1, Heisenberg–Weyl algebra, nilpotent, Bianchi II,

[e2,e3]=e1;

* 𝔤3.2, solvable, Bianchi IV,

[e1,e3]=e1,[e2,e3]=e1+e2;

* 𝔤3.3, solvable, Bianchi V,

[e1,e3]=e1,[e2,e3]=e2;

* 𝔤3.4, solvable, Bianchi VI, Poincaré algebra 𝔭(1,1) when α=−1,

[e1,e3]=e1,[e2,e3]=αe2,−1≤α<1,α≠0;

* 𝔤3.5, solvable, Bianchi VII,

[e1,e3]=βe1−e2,[e2,e3]=e1+βe2,β≥0;

* 𝔤3.6, simple, Bianchi VIII, 𝔰𝔩(2,ℝ),

[e1,e2]=e1,[e2,e3]=e3,[e1,e3]=2e2;

* 𝔤3.7, simple, Bianchi IX, 𝔰𝔬(3),

[e2,e3]=e1,[e3,e1]=e2,[e1,e2]=e3.

Algebra 𝔤3.3 can be considered as an extreme case of 𝔤3.5, when β->∞, forming contraction of Lie algebra.

Over the field ℂ algebras 𝔤3.5, 𝔤3.7 are isomorphic to 𝔤3.4 and 𝔤3.6, respectively.

### Four-dimensional

* 4𝔤1, abelian;
* 𝔤2.1⊕2𝔤1, decomposable solvable,

[e1,e2]=e1;

* 2𝔤2.1, decomposable solvable,

[e1,e2]=e1[e3,e4]=e3;

* 𝔤3.1⊕𝔤1, decomposable nilpotent,

[e2,e3]=e1;

* 𝔤3.2⊕𝔤1, decomposable solvable,

[e1,e3]=e1,[e2,e3]=e1+e2;

* 𝔤3.3⊕𝔤1, decomposable solvable,

[e1,e3]=e1,[e2,e3]=e2;

* 𝔤3.4⊕𝔤1, decomposable solvable,

[e1,e3]=e1,[e2,e3]=αe2,−1≤α<1,α≠0;

* 𝔤3.5⊕𝔤1, decomposable solvable,

[e1,e3]=βe1−e2[e2,e3]=e1+βe2,β≥0;

* 𝔤3.6⊕𝔤1, unsolvable,

[e1,e2]=e1,[e2,e3]=e3,[e1,e3]=2e2;

* 𝔤3.7⊕𝔤1, unsolvable,

[e1,e2]=e3,[e2,e3]=e1,[e3,e1]=e2;

* 𝔤4.1, indecomposable nilpotent,

[e2,e4]=e1,[e3,e4]=e2;

* 𝔤4.2, indecomposable solvable,

[e1,e4]=βe1,[e2,e4]=e2,[e3,e4]=e2+e3,β≠0;

* 𝔤4.3, indecomposable solvable,

[e1,e4]=e1,[e3,e4]=e2;

* 𝔤4.4, indecomposable solvable,

[e1,e4]=e1,[e2,e4]=e1+e2,[e3,e4]=e2+e3;

* 𝔤4.5, indecomposable solvable,

[e1,e4]=αe1,[e2,e4]=βe2,[e3,e4]=γe3,αβγ≠0;

* 𝔤4.6, indecomposable solvable,

[e1,e4]=αe1,[e2,e4]=βe2−e3,[e3,e4]=e2+βe3,α>0;

* 𝔤4.7, indecomposable solvable,

[e2,e3]=e1,[e1,e4]=2e1,[e2,e4]=e2,[e3,e4]=e2+e3;

* 𝔤4.8, indecomposable solvable,

[e2,e3]=e1,[e1,e4]=(1+β)e1,[e2,e4]=e2,[e3,e4]=βe3,−1≤β≤1;

* 𝔤4.9, indecomposable solvable,

[e2,e3]=e1,[e1,e4]=2αe1,[e2,e4]=αe2−e3,[e3,e4]=e2+αe3,α≥0;

* 𝔤4.10, indecomposable solvable,

[e1,e3]=e1,[e2,e3]=e2,[e1,e4]=−e2,[e2,e4]=e1.

Algebra 𝔤4.3 can be considered as an extreme case of 𝔤4.2, when β->0, forming contraction of Lie algebra.

Over the field ℂ algebras 𝔤3.5⊕𝔤1, 𝔤3.7⊕𝔤1, 𝔤4.6, 𝔤4.9, 𝔤4.10 are isomorphic to 𝔤3.4⊕𝔤1, 𝔤3.6⊕𝔤1, 𝔤4.5, 𝔤4.8, 2𝔤2.1, respectively.

## See also

* Table of Lie groups
* Simple Lie group

## Notes

1. ↑ Mubarakzyanov 1963
2. ↑ Popovych 2003

## 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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