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
[edit]
Let
be
-dimensional Lie algebra over the field of real numbers
with generators
,
.[clarification needed] For each algebra
we adduce only non-zero commutators between basis elements.
, abelian.
, abelian
;
, solvable
,
![{\displaystyle [e_{1},e_{2}]=e_{1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/963764b5b09e26abd719db414ac0b5072adef3dc)
, abelian, Bianchi I;
, decomposable solvable, Bianchi III;
, Heisenberg–Weyl algebra, nilpotent, Bianchi II,
![{\displaystyle [e_{2},e_{3}]=e_{1};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c182b4694187494050b54e40dad06c1c430ab54)
, solvable, Bianchi IV,
![{\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)
, solvable, Bianchi V,
![{\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)
, solvable, Bianchi VI, Poincaré algebra
when
,
![{\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)
, solvable, Bianchi VII,
![{\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)
, simple, Bianchi VIII, ![{\displaystyle {\mathfrak {sl}}(2,\mathbb {R} ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b8889cee03f099bc80ec80733e58d1d4b33cf30)
![{\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)
, simple, Bianchi IX, ![{\displaystyle {\mathfrak {so}}(3),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0b216599a18df448fe258d3e160a969c64c7dc2)
![{\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)
Algebra
can be considered as an extreme case of
, when
, forming contraction of Lie algebra.
Over the field
algebras
,
are isomorphic to
and
, respectively.
, abelian;
, decomposable solvable,
![{\displaystyle [e_{1},e_{2}]=e_{1};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c536c4d95b5fa8a2d43c9bdd7c5d9becd5f68c79)
, decomposable solvable,
![{\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)
, decomposable nilpotent,
![{\displaystyle [e_{2},e_{3}]=e_{1};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c182b4694187494050b54e40dad06c1c430ab54)
, decomposable solvable,
![{\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)
, decomposable solvable,
![{\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)
, decomposable solvable,
![{\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)
, decomposable solvable,
![{\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)
, unsolvable,
![{\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)
, unsolvable,
![{\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)
, indecomposable nilpotent,
![{\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)
, indecomposable solvable,
![{\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)
, indecomposable solvable,
![{\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)
, indecomposable solvable,
![{\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)
, indecomposable solvable,
![{\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)
, indecomposable solvable,
![{\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)
, indecomposable solvable,
![{\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)
, indecomposable solvable,
![{\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)
, indecomposable solvable,
![{\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)
, indecomposable solvable,
![{\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)
Algebra
can be considered as an extreme case of
, when
, forming contraction of Lie algebra.
Over the field
algebras
,
,
,
,
are isomorphic to
,
,
,
,
, respectively.