This mathematics -related list provides Mubarakzyanov's classification of low-dimensional real Lie algebras , published in Russian in 1963.[1] Lie algebra in the area of abstract algebra .
An English version and review of this classification was published by Popovych et al.[2]
Mubarakzyanov's Classification
edit
Let
g
n
{\displaystyle {\mathfrak {g}}_{n}}
n
{\displaystyle n}
Lie algebra over the field of real numbers
with generators
e
1
,
…
,
e
n
{\displaystyle e_{1},\dots ,e_{n}}
n
≤
4
{\displaystyle n\leq 4}
[clarification needed For each algebra
g
{\displaystyle {\mathfrak {g}}}
One-dimensional
edit
g
1
{\displaystyle {\mathfrak {g}}_{1}}
abelian .Two-dimensional
edit
2
g
1
{\displaystyle 2{\mathfrak {g}}_{1}}
R
2
{\displaystyle \mathbb {R} ^{2}}
g
2.1
{\displaystyle {\mathfrak {g}}_{2.1}}
solvable
a
f
f
(
1
)
=
{
(
a
b
0
0
)
:
a
,
b
∈
R
}
{\displaystyle {\mathfrak {aff}}(1)=\left\{{\begin{pmatrix}a&b\\0&0\end{pmatrix}}\,:\,a,b\in \mathbb {R} \right\}}
[
e
1
,
e
2
]
=
e
1
.
{\displaystyle [e_{1},e_{2}]=e_{1}.}
Three-dimensional
edit
3
g
1
{\displaystyle 3{\mathfrak {g}}_{1}}
Bianchi I ;
g
2.1
⊕
g
1
{\displaystyle {\mathfrak {g}}_{2.1}\oplus {\mathfrak {g}}_{1}}
g
3.1
{\displaystyle {\mathfrak {g}}_{3.1}}
[
e
2
,
e
3
]
=
e
1
;
{\displaystyle [e_{2},e_{3}]=e_{1};}
g
3.2
{\displaystyle {\mathfrak {g}}_{3.2}}
[
e
1
,
e
3
]
=
e
1
,
[
e
2
,
e
3
]
=
e
1
+
e
2
;
{\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{1}+e_{2};}
g
3.3
{\displaystyle {\mathfrak {g}}_{3.3}}
[
e
1
,
e
3
]
=
e
1
,
[
e
2
,
e
3
]
=
e
2
;
{\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2};}
g
3.4
{\displaystyle {\mathfrak {g}}_{3.4}}
Poincaré algebra
p
(
1
,
1
)
{\displaystyle {\mathfrak {p}}(1,1)}
α
=
−
1
{\displaystyle \alpha =-1}
[
e
1
,
e
3
]
=
e
1
,
[
e
2
,
e
3
]
=
α
e
2
,
−
1
≤
α
<
1
,
α
≠
0
;
{\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=\alpha e_{2},\quad -1\leq \alpha <1,\quad \alpha \neq 0;}
g
3.5
{\displaystyle {\mathfrak {g}}_{3.5}}
[
e
1
,
e
3
]
=
β
e
1
−
e
2
,
[
e
2
,
e
3
]
=
e
1
+
β
e
2
,
β
≥
0
;
{\displaystyle [e_{1},e_{3}]=\beta e_{1}-e_{2},\quad [e_{2},e_{3}]=e_{1}+\beta e_{2},\quad \beta \geq 0;}
g
3.6
{\displaystyle {\mathfrak {g}}_{3.6}}
s
l
(
2
,
R
)
,
{\displaystyle {\mathfrak {sl}}(2,\mathbb {R} ),}
[
e
1
,
e
2
]
=
e
1
,
[
e
2
,
e
3
]
=
e
3
,
[
e
1
,
e
3
]
=
2
e
2
;
{\displaystyle [e_{1},e_{2}]=e_{1},\quad [e_{2},e_{3}]=e_{3},\quad [e_{1},e_{3}]=2e_{2};}
g
3.7
{\displaystyle {\mathfrak {g}}_{3.7}}
s
o
(
3
)
,
{\displaystyle {\mathfrak {so}}(3),}
[
e
2
,
e
3
]
=
e
1
,
[
e
3
,
e
1
]
=
e
2
,
[
e
1
,
e
2
]
=
e
3
.
{\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2},\quad [e_{1},e_{2}]=e_{3}.}
Algebra
g
3.3
{\displaystyle {\mathfrak {g}}_{3.3}}
g
3.5
{\displaystyle {\mathfrak {g}}_{3.5}}
β
→
∞
{\displaystyle \beta \rightarrow \infty }
Over the field
C
{\displaystyle {\mathbb {C} }}
g
3.5
{\displaystyle {\mathfrak {g}}_{3.5}}
g
3.7
{\displaystyle {\mathfrak {g}}_{3.7}}
g
3.4
{\displaystyle {\mathfrak {g}}_{3.4}}
g
3.6
{\displaystyle {\mathfrak {g}}_{3.6}}
Four-dimensional
edit
4
g
1
{\displaystyle 4{\mathfrak {g}}_{1}}
g
2.1
⊕
2
g
1
{\displaystyle {\mathfrak {g}}_{2.1}\oplus 2{\mathfrak {g}}_{1}}
[
e
1
,
e
2
]
=
e
1
;
{\displaystyle [e_{1},e_{2}]=e_{1};}
2
g
2.1
{\displaystyle 2{\mathfrak {g}}_{2.1}}
[
e
1
,
e
2
]
=
e
1
[
e
3
,
e
4
]
=
e
3
;
{\displaystyle [e_{1},e_{2}]=e_{1}\quad [e_{3},e_{4}]=e_{3};}
g
3.1
⊕
g
1
{\displaystyle {\mathfrak {g}}_{3.1}\oplus {\mathfrak {g}}_{1}}
[
e
2
,
e
3
]
=
e
1
;
{\displaystyle [e_{2},e_{3}]=e_{1};}
g
3.2
⊕
g
1
{\displaystyle {\mathfrak {g}}_{3.2}\oplus {\mathfrak {g}}_{1}}
[
e
1
,
e
3
]
=
e
1
,
[
e
2
,
e
3
]
=
e
1
+
e
2
;
{\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{1}+e_{2};}
g
3.3
⊕
g
1
{\displaystyle {\mathfrak {g}}_{3.3}\oplus {\mathfrak {g}}_{1}}
[
e
1
,
e
3
]
=
e
1
,
[
e
2
,
e
3
]
=
e
2
;
{\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2};}
g
3.4
⊕
g
1
{\displaystyle {\mathfrak {g}}_{3.4}\oplus {\mathfrak {g}}_{1}}
[
e
1
,
e
3
]
=
e
1
,
[
e
2
,
e
3
]
=
α
e
2
,
−
1
≤
α
<
1
,
α
≠
0
;
{\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=\alpha e_{2},\quad -1\leq \alpha <1,\quad \alpha \neq 0;}
g
3.5
⊕
g
1
{\displaystyle {\mathfrak {g}}_{3.5}\oplus {\mathfrak {g}}_{1}}
[
e
1
,
e
3
]
=
β
e
1
−
e
2
[
e
2
,
e
3
]
=
e
1
+
β
e
2
,
β
≥
0
;
{\displaystyle [e_{1},e_{3}]=\beta e_{1}-e_{2}\quad [e_{2},e_{3}]=e_{1}+\beta e_{2},\quad \beta \geq 0;}
g
3.6
⊕
g
1
{\displaystyle {\mathfrak {g}}_{3.6}\oplus {\mathfrak {g}}_{1}}
[
e
1
,
e
2
]
=
e
1
,
[
e
2
,
e
3
]
=
e
3
,
[
e
1
,
e
3
]
=
2
e
2
;
{\displaystyle [e_{1},e_{2}]=e_{1},\quad [e_{2},e_{3}]=e_{3},\quad [e_{1},e_{3}]=2e_{2};}
g
3.7
⊕
g
1
{\displaystyle {\mathfrak {g}}_{3.7}\oplus {\mathfrak {g}}_{1}}
[
e
1
,
e
2
]
=
e
3
,
[
e
2
,
e
3
]
=
e
1
,
[
e
3
,
e
1
]
=
e
2
;
{\displaystyle [e_{1},e_{2}]=e_{3},\quad [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2};}
g
4.1
{\displaystyle {\mathfrak {g}}_{4.1}}
[
e
2
,
e
4
]
=
e
1
,
[
e
3
,
e
4
]
=
e
2
;
{\displaystyle [e_{2},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};}
g
4.2
{\displaystyle {\mathfrak {g}}_{4.2}}
[
e
1
,
e
4
]
=
β
e
1
,
[
e
2
,
e
4
]
=
e
2
,
[
e
3
,
e
4
]
=
e
2
+
e
3
,
β
≠
0
;
{\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;}
g
4.3
{\displaystyle {\mathfrak {g}}_{4.3}}
[
e
1
,
e
4
]
=
e
1
,
[
e
3
,
e
4
]
=
e
2
;
{\displaystyle [e_{1},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};}
g
4.4
{\displaystyle {\mathfrak {g}}_{4.4}}
[
e
1
,
e
4
]
=
e
1
,
[
e
2
,
e
4
]
=
e
1
+
e
2
,
[
e
3
,
e
4
]
=
e
2
+
e
3
;
{\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};}
g
4.5
{\displaystyle {\mathfrak {g}}_{4.5}}
[
e
1
,
e
4
]
=
α
e
1
,
[
e
2
,
e
4
]
=
β
e
2
,
[
e
3
,
e
4
]
=
γ
e
3
,
α
β
γ
≠
0
;
{\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;}
g
4.6
{\displaystyle {\mathfrak {g}}_{4.6}}
[
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
;
{\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 >0;}
g
4.7
{\displaystyle {\mathfrak {g}}_{4.7}}
[
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
;
{\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{1},e_{4}]=2e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3};}
g
4.8
{\displaystyle {\mathfrak {g}}_{4.8}}
[
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
≤
β
≤
1
;
{\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;}
g
4.9
{\displaystyle {\mathfrak {g}}_{4.9}}
[
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
≥
0
;
{\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;}
g
4.10
{\displaystyle {\mathfrak {g}}_{4.10}}
[
e
1
,
e
3
]
=
e
1
,
[
e
2
,
e
3
]
=
e
2
,
[
e
1
,
e
4
]
=
−
e
2
,
[
e
2
,
e
4
]
=
e
1
.
{\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}.}
Algebra
g
4.3
{\displaystyle {\mathfrak {g}}_{4.3}}
g
4.2
{\displaystyle {\mathfrak {g}}_{4.2}}
β
→
0
{\displaystyle \beta \rightarrow 0}
Over the field
C
{\displaystyle {\mathbb {C} }}
g
3.5
⊕
g
1
{\displaystyle {\mathfrak {g}}_{3.5}\oplus {\mathfrak {g}}_{1}}
g
3.7
⊕
g
1
{\displaystyle {\mathfrak {g}}_{3.7}\oplus {\mathfrak {g}}_{1}}
g
4.6
{\displaystyle {\mathfrak {g}}_{4.6}}
g
4.9
{\displaystyle {\mathfrak {g}}_{4.9}}
g
4.10
{\displaystyle {\mathfrak {g}}_{4.10}}
g
3.4
⊕
g
1
{\displaystyle {\mathfrak {g}}_{3.4}\oplus {\mathfrak {g}}_{1}}
g
3.6
⊕
g
1
{\displaystyle {\mathfrak {g}}_{3.6}\oplus {\mathfrak {g}}_{1}}
g
4.5
{\displaystyle {\mathfrak {g}}_{4.5}}
g
4.8
{\displaystyle {\mathfrak {g}}_{4.8}}
2
g
2.1
{\displaystyle {2{\mathfrak {g}}}_{2.1}}
See also
edit
References
edit
![{\displaystyle [e_{1},e_{2}]=e_{1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/963764b5b09e26abd719db414ac0b5072adef3dc)
![{\displaystyle [e_{2},e_{3}]=e_{1};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c182b4694187494050b54e40dad06c1c430ab54)
![{\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{1}+e_{2};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5935ebf5ca03ea4b1ca634f2e3af78acbbe33639)
![{\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/974c2e0bcfb71d97cb9953e7bca3b2712ba90942)
![{\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=\alpha e_{2},\quad -1\leq \alpha <1,\quad \alpha \neq 0;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98b285ff018491e6c493addc20965f5a6c1ed216)
![{\displaystyle [e_{1},e_{3}]=\beta e_{1}-e_{2},\quad [e_{2},e_{3}]=e_{1}+\beta e_{2},\quad \beta \geq 0;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b961cad12339fbcd9f00d26349b56de57370715c)
![{\displaystyle [e_{1},e_{2}]=e_{1},\quad [e_{2},e_{3}]=e_{3},\quad [e_{1},e_{3}]=2e_{2};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab1ff7eac20f65d2ea43d92ee5e9530a5d74f779)
![{\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2},\quad [e_{1},e_{2}]=e_{3}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f1e2825a50b65a724c05fee39324b42de770ac)
![{\displaystyle [e_{1},e_{2}]=e_{1};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c536c4d95b5fa8a2d43c9bdd7c5d9becd5f68c79)
![{\displaystyle [e_{1},e_{2}]=e_{1}\quad [e_{3},e_{4}]=e_{3};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e3cc0ba7dbd31672835b738f0d23d481d0dc6b8)
![{\displaystyle [e_{1},e_{3}]=\beta e_{1}-e_{2}\quad [e_{2},e_{3}]=e_{1}+\beta e_{2},\quad \beta \geq 0;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60c92c7ac9f1cae0ead75eb62f316d5185bcd639)
![{\displaystyle [e_{1},e_{2}]=e_{3},\quad [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/177d2292b9f04fba52105937252652503ca8aac7)
![{\displaystyle [e_{2},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3234d4a8d3180d65f53d821cdc21ca6f45fc827)
![{\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;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83aba1442f700da3e4278c930585757a277b660d)
![{\displaystyle [e_{1},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b2b0ab15fcc0d68f7b4dfb2942e9560ffc762b4)
![{\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};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ebcc9ae5c931d8859ead56d35b3c434801408a1)
![{\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;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b57f7c115385ddd05c393cf69c12a9e028b540cb)
![{\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 >0;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2205ea932e8b731d0aceb604cd3fb9af8233ab)
![{\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{1},e_{4}]=2e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb05c0b3c9090a357a556fcf9cd79b2723651b15)
![{\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;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/941282388a6088ed8e4a431d6632e3074b01cf59)
![{\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;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc56449e914e919a9bd8df0afed266450b8e86e2)
![{\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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6beef23a27c53047793aab1a5f974b8987bd253b)
