Classification of low-dimensional real Lie algebras

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In mathematics, there is a classification of low-dimensional real Lie algebras.

Let be -dimensional Lie algebra over the field of real numbers with generators , .[clarification needed] Below we give Mubarakzyanov's classification[1] and numeration of these algebras. For review see also Popovych et al.[2] For each algebra we adduce only non-zero commutators between basis elements.

One-dimensional[edit]

  • , abelian.

Two-dimensional[edit]

  • , abelian;
  • , solvable,

Three-dimensional[edit]

  • , 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 VIII,

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[edit]

  • , 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[edit]