What is "lie algebra"

In mathematics, a Lie algebra (, not ) is a vector space together with a non-associative multiplication called "Lie bracket" [x, y]. When an algebraic product is defined on the space, the Lie bracket is the commutator [x, y] = xy − yx.

Lie algebras were introduced to study the concept of infinitesimal transformations. Hermann Weyl introduced the term "Lie algebra" (after Sophus Lie) in the 1930s. In older texts, the name "infinitesimal group" is used.

Lie algebras are closely related to Lie groups which are groups that are also smooth manifolds, with the property that the group operations of multiplication and inversion are smooth maps. Any Lie group gives rise to a Lie algebra. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to covering ( Lie's third theorem). This correspondence between Lie groups and Lie algebras allows one to study Lie groups in terms of Lie algebras.