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 $_n$ be $n$-dimensional Lie algebra over the field of real numbers
with generators $e_1, \dots, e_n$, $n \leq 4$. For each algebra $$ we adduce only non-zero commutators between basis elements.

One-dimensional

* $_1$, abelian.

Two-dimensional

* $2_1$, abelian $\mathbb^2$;
* $_$, solvable $\mathfrak(1)=\left\$,
:: _1, e_2= e_1.

Three-dimensional

* $3_1$, abelian, Bianchi I;
* $_\oplus _1$, decomposable solvable, Bianchi III;
* $_$, Heisenberg–Weyl algebra, nilpotent, Bianchi II,
:: _2, e_3= e_1;
* $_$, solvable, Bianchi IV,
:: _1, e_3= e_1, \quad _2, e_3= e_1 + e_2;
* $_$, solvable, Bianchi V,
:: _1, e_3= e_1, \quad _2, e_3= e_2;
* $_$, solvable, Bianchi VI, Poincaré algebra $\mathfrak(1,1)$ when $\alpha = -1$,
:: _1, e_3= e_1, \quad _2, e_3= \alpha e_2, \quad -1 \leq \alpha < 1, \quad \alpha \neq 0;
* $_$, solvable, Bianchi VII,
:: _1, e_3= \beta e_1 - e_2, \quad _2, e_3= e_1 + \beta e_2, \quad \beta \geq 0;
* $_$, simple, Bianchi VIII, $\mathfrak(2, \mathbb R ),$
:: _1, e_2= e_1, \quad _2, e_3= e_3, \quad _1, e_3= 2 e_2;
* $_$, simple, Bianchi IX, $\mathfrak(3),$
:: _2, e_3= e_1, \quad _3, e_1= e_2, \quad _1, e_2= e_3.
Algebra $_$ can be considered as an extreme case of $_$, when $\beta \rightarrow \infty$, forming contraction of Lie algebra.

Over the field $$ algebras $_$, $_$ are isomorphic to $_$ and $_$, respectively.

Four-dimensional

* $4_1$, abelian;
* $_ \oplus 2_1$, decomposable solvable,
:: _1, e_2= e_1;
* $2_$, decomposable solvable,
:: _1, e_2= e_1 \quad _3, e_4= e_3;
* $_ \oplus _1$, decomposable nilpotent,
:: _2, e_3= e_1;
* $_ \oplus _1$, decomposable solvable,
:: _1, e_3= e_1, \quad _2, e_3= e_1 + e_2;
* $_ \oplus _1$, decomposable solvable,
:: _1, e_3= e_1, \quad _2, e_3= e_2;
* $_ \oplus _1$, decomposable solvable,
:: _1, e_3= e_1, \quad _2, e_3= \alpha e_2, \quad -1 \leq \alpha < 1, \quad \alpha \neq 0;
* $_ \oplus _1$, decomposable solvable,
:: _1, e_3= \beta e_1 - e_2 \quad _2, e_3= e_1 + \beta e_2, \quad \beta \geq 0;
* $_ \oplus _1$, unsolvable,
:: _1, e_2= e_1, \quad _2, e_3= e_3, \quad _1, e_3= 2 e_2;
* $_ \oplus _1$, unsolvable,
:: _1, e_2= e_3, \quad _2, e_3= e_1, \quad _3, e_1= e_2;
* $_$, indecomposable nilpotent,
:: _2, e_4= e_1, \quad _3, e_4= e_2;
* $_$, indecomposable solvable,
:: _1, e_4= \beta e_1, \quad _2, e_4= e_2, \quad _3, e_4= e_2 + e_3, \quad \beta \neq 0;
* $_$, indecomposable solvable,
:: _1, e_4= e_1, \quad _3, e_4= e_2;
* $_$, indecomposable solvable,
:: _1, e_4= e_1, \quad _2, e_4= e_1 + e_2, \quad _3, e_4= e_2+e_3;
* $_$, indecomposable solvable,
:: _1, e_4= \alpha e_1, \quad _2, e_4= \beta e_2, \quad _3, e_4= \gamma e_3, \quad \alpha \beta \gamma \neq 0;
* $_$, indecomposable solvable,
:: _1, e_4= \alpha e_1, \quad _2, e_4= \beta e_2 - e_3, \quad _3, e_4= e_2 + \beta e_3, \quad \alpha > 0;
* $_$, indecomposable solvable,
:: _2, e_3= e_1, \quad _1, e_4= 2 e_1, \quad _2, e_4= e_2, \quad _3, e_4= e_2 + e_3;
* $_$, indecomposable solvable,
:: _2, e_3= e_1, \quad _1, e_4= (1 + \beta)e_1, \quad _2, e_4= e_2, \quad _3, e_4= \beta e_3, \quad -1 \leq \beta \leq 1;
* $_$, indecomposable solvable,
:: _2, e_3= e_1, \quad _1, e_4= 2 \alpha e_1, \quad _2, e_4= \alpha e_2 - e_3, \quad _3, e_4= e_2 + \alpha e_3, \quad \alpha \geq 0;
* $_$, indecomposable solvable,
:: _1, e_3= e_1, \quad _2, e_3= e_2, \quad _1, e_4= -e_2, \quad _2, e_4= e_1.

Algebra $_$ can be considered as an extreme case of $_$, when $\beta \rightarrow 0$, forming contraction of Lie algebra.

Over the field $$ algebras $_ \oplus _1$, $_ \oplus _1$, $_$, $_$, $_$ are isomorphic to $_ \oplus _1$, $_ \oplus _1$, $_$, $_$, $_$, respectively.

See also

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

Notes

References

*
* {{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

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