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

A Lie algebra is a vector space over a field together with a binary operation called the Lie bracket, satisfying the following axioms:
• Bilinearity,
for all scalars in and all elements in .
• The … See more

Any vector space 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 of the Lie bracket. … See more

The Lie bracket is not required to be associative, meaning that need not be equal to . Nonetheless, much of the terminology for associative rings and algebras (and also for groups) has analogs for Lie algebras. A Lie subalge… See more

A matrix group is a Lie group consisting of invertible matrices, , where the group operation of G is matrix multiplication. The corresponding Lie algebra is the space of matrices which are tangent vectors to G inside the li… See more

Given a vector space V, let denote the Lie algebra consisting of all linear maps from V to itself, with bracket given by . A representation of a Lie algebra on V is a Lie algebra homomorphism
That is, sends … See more

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