Lie algebras were introduced to study the concept of infinitesimal transformations by Sophus Lie in the 1870s, and independently discovered by Wilhelm Killing in … See more

Abelian Lie algebras
Any vector space $${\displaystyle V}$$ endowed with the identically zero Lie bracket becomes a Lie algebra. Such a Lie algebra is called abelian. Every one-dimensional Lie algebra is abelian, by the alternating property … See more

Matrix Lie algebras
A matrix group is a Lie group consisting of invertible matrices, $${\displaystyle G\subset \mathrm {GL} (n,\mathbb {R} )}$$, where the group operation of G is matrix multiplication. The corresponding Lie algebra See more

Lie algebras can be classified to some extent. This is a powerful approach to the classification of Lie groups.
Abelian, nilpotent, and solvable
Analogously to abelian, nilpotent, and solvable groups, one can define abelian, nilpotent, and … See more

A Lie algebra is a vector space $${\displaystyle \,{\mathfrak {g}}}$$ over a field $${\displaystyle F}$$ together with a binary operation See more

Subalgebras, ideals and homomorphisms
The Lie bracket is not required to be associative, meaning that $${\displaystyle [[x,y],z]}$$ need not be equal to $${\displaystyle [x,[y,z]]}$$. Nonetheless, much of the terminology for associative rings See more

Definitions
Given a vector space V, let $${\displaystyle {\mathfrak {gl}}(V)}$$ denote the Lie … See more