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 $𝔤_{n}$ 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 $𝔤$ we adduce only non-zero commutators between basis elements.

One-dimensional

$𝔤_{1}$ , abelian.

Two-dimensional

$2 𝔤_{1}$ , abelian $ℝ^{2}$ ;
$𝔤_{2.1}$ , solvable $𝔞 𝔣 𝔣 (1) = {(\begin{matrix} a & b \\ 0 & 0 \end{matrix}) : a, b \in ℝ}$ ,

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

Three-dimensional

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

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

$𝔤_{3.2}$ , solvable, Bianchi IV,

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

$𝔤_{3.3}$ , solvable, Bianchi V,

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

$𝔤_{3.4}$ , solvable, Bianchi VI, Poincaré algebra $𝔭 (1, 1)$ when $α = - 1$ ,

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

$𝔤_{3.5}$ , solvable, Bianchi VII,

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

$𝔤_{3.6}$ , simple, Bianchi VIII, $𝔰 𝔩 (2, ℝ),$

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

$𝔤_{3.7}$ , simple, Bianchi IX, $𝔰 𝔬 (3),$

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

Algebra $𝔤_{3.3}$ can be considered as an extreme case of $𝔤_{3.5}$ , when $β \to \infty$ , forming contraction of Lie algebra.

Over the field $ℂ$ algebras $𝔤_{3.5}$ , $𝔤_{3.7}$ are isomorphic to $𝔤_{3.4}$ and $𝔤_{3.6}$ , respectively.

Four-dimensional

$4 𝔤_{1}$ , abelian;
$𝔤_{2.1} \oplus 2 𝔤_{1}$ , decomposable solvable,

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

$2 𝔤_{2.1}$ , decomposable solvable,

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

$𝔤_{3.1} \oplus 𝔤_{1}$ , decomposable nilpotent,

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

$𝔤_{3.2} \oplus 𝔤_{1}$ , decomposable solvable,

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

$𝔤_{3.3} \oplus 𝔤_{1}$ , decomposable solvable,

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

$𝔤_{3.4} \oplus 𝔤_{1}$ , decomposable solvable,

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

$𝔤_{3.5} \oplus 𝔤_{1}$ , decomposable solvable,

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

$𝔤_{3.6} \oplus 𝔤_{1}$ , unsolvable,

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

$𝔤_{3.7} \oplus 𝔤_{1}$ , unsolvable,

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

$𝔤_{4.1}$ , indecomposable nilpotent,

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

$𝔤_{4.2}$ , indecomposable solvable,

[e_{1}, e_{4}] = β e_{1}, [e_{2}, e_{4}] = e_{2}, [e_{3}, e_{4}] = e_{2} + e_{3}, β \neq 0;

$𝔤_{4.3}$ , indecomposable solvable,

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

$𝔤_{4.4}$ , indecomposable solvable,

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

$𝔤_{4.5}$ , indecomposable solvable,

[e_{1}, e_{4}] = α e_{1}, [e_{2}, e_{4}] = β e_{2}, [e_{3}, e_{4}] = γ e_{3}, α β γ \neq 0;

$𝔤_{4.6}$ , indecomposable solvable,

[e_{1}, e_{4}] = α e_{1}, [e_{2}, e_{4}] = β e_{2} - e_{3}, [e_{3}, e_{4}] = e_{2} + β e_{3}, α > 0;

$𝔤_{4.7}$ , indecomposable solvable,

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

$𝔤_{4.8}$ , indecomposable solvable,

[e_{2}, e_{3}] = e_{1}, [e_{1}, e_{4}] = (1 + β) e_{1}, [e_{2}, e_{4}] = e_{2}, [e_{3}, e_{4}] = β e_{3}, - 1 \leq β \leq 1;

$𝔤_{4.9}$ , indecomposable solvable,

[e_{2}, e_{3}] = e_{1}, [e_{1}, e_{4}] = 2 α e_{1}, [e_{2}, e_{4}] = α e_{2} - e_{3}, [e_{3}, e_{4}] = e_{2} + α e_{3}, α \geq 0;

$𝔤_{4.10}$ , indecomposable solvable,

[e_{1}, e_{3}] = e_{1}, [e_{2}, e_{3}] = e_{2}, [e_{1}, e_{4}] = - e_{2}, [e_{2}, e_{4}] = e_{1} .

Algebra $𝔤_{4.3}$ can be considered as an extreme case of $𝔤_{4.2}$ , when $β \to 0$ , forming contraction of Lie algebra.

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

References

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[1] Mubarakzyanov 1963

[2] Popovych 2003

[1]

[2]

Classification of low-dimensional real Lie algebras

Contents

Mubarakzyanov's Classification

One-dimensional

Two-dimensional

Three-dimensional

Four-dimensional

See also

Notes

References

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Classification of low-dimensional real Lie algebras

Mubarakzyanov's Classification

One-dimensional

Two-dimensional

Three-dimensional

Four-dimensional

See also

Notes

References

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