Lie Algebra

Related Concepts

Lie groups Rigid motions Geometric transformations Intrinsic methods

Definitions and Properties

Lie algebras can be interpreted in a number of ways: the set of infinitesimal transformations, tangent space about the identity element of a group, and a linear space with special properties. Formally, a Lie algebra (denoted $\mathfrak {g}$) is a vector space equipped with a bilinear, non-associative map $\left [.,. \right ] : \mathfrak {g} \times \mathfrak {g} \rightarrow \mathfrak {g}$ known as the Lie bracket. The Lie bracket satisfies the following relationships:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} [\mathbf{x},\mathbf{y}] &\displaystyle =&\displaystyle -[\mathbf{y},\mathbf{x}] \,(\text{\textit{anti-symmetry}}), \\ {} [\mathbf{x}+\mathbf{y},\mathbf{z}] &\displaystyle =&\displaystyle [\mathbf{x},\mathbf{z}] + [\mathbf{y},\mathbf{z}] \, (\textit{bilinearity}),\\ {} [\mathbf{x},{[\mathbf{y},\mathbf{z}]}] &\displaystyle +&\displaystyle...