Abelian Lie algebra

Abelian Lie algebras are a term from the mathematical theory of Lie groups and Lie algebras .

A Lie algebra is Abelian if the Lie bracket is identical to zero.

Every vector space forms an Abelian Lie algebra if one defines every Lie bracket as zero.

If the Lie algebra of the Lie group is an Abelian Lie algebra, then as a semi-direct product ${\displaystyle G}$ ${\displaystyle G}$

${\displaystyle G=A\rtimes \Gamma }$

decompose from an Abelian Lie group and a discrete group . ${\displaystyle A}$ ${\displaystyle \Gamma }$