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
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.
One-dimensional
- ,
abelian.
Two-dimensional
- , abelian ;
- ,
solvable ,
Three-dimensional
- , 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.
Four-dimensional
- , 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.
See also
Notes
References