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词条 Classification of low-dimensional real Lie algebras
释义
      One-dimensional    Two-dimensional    Three-dimensional    Four-dimensional  

  1. Notes

{{refimprove|date=December 2018}}

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 , .{{clarify|Here and the in the below, I’m not completely sure how the notations work|date=December 2018}} 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

  • , 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 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

  • , 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

1. ^{{harvnb|Mubarakzyanov|1963}}
2. ^{{harvnb|Popovych|2003}}

2 : Lie algebras|Mathematics-related lists
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