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 ,
- , abelian, Bianchi I;
- , decomposable solvable, Bianchi III;
- , Heisenberg–Weyl algebra, nilpotent, Bianchi II,
- , solvable, Bianchi IV,
- , solvable, Bianchi V,
- , solvable, Bianchi VI, Poincaré algebra when ,
- , solvable, Bianchi VII,
- , simple, Bianchi VIII,
- , simple, Bianchi IX,
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,
- , decomposable solvable,
- , decomposable nilpotent,
- , decomposable solvable,
- , decomposable solvable,
- , decomposable solvable,
- , decomposable solvable,
- , unsolvable,
- , unsolvable,
- , indecomposable nilpotent,
- , indecomposable solvable,
- , indecomposable solvable,
- , indecomposable solvable,
- , indecomposable solvable,
- , indecomposable solvable,
- , indecomposable solvable,
- , indecomposable solvable,
- , indecomposable solvable,
- , indecomposable solvable,
Algebra can be considered as an extreme case of , when , forming contraction of Lie algebra.
Over the field algebras , , , , are isomorphic to , , , , , respectively.