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.{{harvnb|Mubarakzyanov|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.{{harvnb|Popovych|2003}} in 2003.

Mubarakzyanov's Classification

Let ${\mathfrak g}_n$ be $n$ -dimensional Lie algebra over the field of real numbers

with generators $e_1, \dots, e_n$ , $n \leq 4$ .{{clarify|Here and the in the below, I’m not completely sure how the notations work|date=December 2018}} 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, $\mathfrak{sl}(2, \mathbb R ),$

:: $[e_1, e_2] = e_1, \quad [e_2, e_3] = e_3, \quad [e_1, e_3] = 2 e_2;$

${\mathfrak g}_{3.7}$ , 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,

:: $[e_1, e_2] = e_1, \quad [e_2, e_3] = e_3, \quad [e_1, e_3] = 2 e_2;$

${\mathfrak g}_{3.7} \oplus {\mathfrak g}_1$ , 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,

:: $[e_2, e_3] = e_1, \quad [e_1, e_4] = 2 e_1, \quad [e_2, e_4] = e_2, \quad [e_3, e_4] = e_2 + e_3;$

${\mathfrak g}_{4.8}$ , 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.

Notes

References

{{cite journal

| last1=Mubarakzyanov

| first1=G.M.

| title=On solvable Lie algebras

| journal=Izv. Vys. Ucheb. Zaved. Matematika

| volume=1

| issue=32

| year=1963

| pages=114–123

| mr=153714

| zbl=0166.04104

| url=http://mi.mathnet.ru/eng/ivm2141

| language=Russian

}}

{{cite journal

| last1=Popovych

| first1=R.O.

| last2=Boyko

| first2=V.M.

| last3=Nesterenko

| first3=M.O.

| last4=Lutfullin

| first4=M.W.

| display-authors=etal

| title=Realizations of real low-dimensional Lie algebras

| journal=J. Phys. A: Math. Gen.

| volume=36

| issue=26

| year=2003

| pages=7337–7360

| doi=10.1088/0305-4470/36/26/309

| arxiv=math-ph/0301029

| bibcode=2003JPhA...36.7337P

| s2cid=9800361 |ref={{harvid|Popovych|2003}}

}}

Category:Lie algebras

Category:Mathematics-related lists

Category:Mathematical classification systems