Abelian Lie algebra

Summary

An abelian Lie algebra is a Lie algebra in which every bracket is zero. Any vector space becomes an abelian Lie algebra with the zero bracket. Two abelian Lie algebras are isomorphic exactly when they have the same dimension. Abelian Lie algebras are the Lie algebras of abelian connected Lie groups, such as tori and $\R^n$. They are the simplest solvable and nilpotent Lie algebras.