Classification of low-dimensional real Lie algebras

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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.

Notes

References

  • Mubarakzyanov, G.M. (1963). "On solvable Lie algebras". Izv. Vys. Ucheb. Zaved. Matematika (in Russian). 1 (32): 114–123. MR 0153714. Zbl 0166.04104.
  • Popovych, R.O.; Boyko, V.M.; Nesterenko, M.O.; Lutfullin, M.W.; et al. (2003). "Realizations of real low-dimensional Lie algebras". J. Phys. A: Math. Gen. 36 (26): 7337–7360. arXiv:math-ph/0301029. Bibcode:2003JPhA...36.7337P. doi:10.1088/0305-4470/36/26/309. S2CID 9800361.