# Lie algebras of matrix groups

The Lie algebra associated with a matrix group is denoted by the same abbreviation as the Lie group, but with lowercase letters; e.g. the Lie algebra of $${GL(n,\mathbb{R})}$$ is denoted $${gl(n,\mathbb{R})}$$. $${gl(n,\mathbb{R})}$$ is easily seen to be the set of all real $${n\times n}$$ matrices under the Lie commutator, and in general the Lie algebra associated with a matrix group can be expressed as matrices with entries in the same division algebra as the matrix group.

 Δ It is important to remember that the multiplication operation on the matrices of a Lie algebra is that of the Lie commutator using matrix multiplication.

If an element of $${GL(n,\mathbb{R})}$$ is considered to be a linear transformation on $${\mathbb{R}^{n}}$$, an element of $${gl(n,\mathbb{R})}$$ is an infinitesimal generator of a linear transformation. Thus an element $${A}$$ of $${gl(n,\mathbb{R})}$$ can be viewed as a vector field on $${\mathbb{R}^{n}}$$ that “points in the direction of a linear transformation,” i.e. as a matrix it linearly transforms a vector $${v}$$ into the tangent to the path in $${\mathbb{R}^{n}}$$ traced by the one-parameter subgroup $${\phi_{A}\left(t\right)}$$ applied to $${v}$$.

 Δ This view of the element $${A\in gl(n,\mathbb{R})}$$ as a vector field on $${\mathbb{R}^{n}}$$ should not be confused with the view of $${A}$$ as a vector field on $${GL(n,\mathbb{R})}$$ as a submanifold of $${\mathbb{R}^{n^{2}}}$$.

An element of $${GL(n,\mathbb{R})}$$ is a linear transformation on $${\mathbb{R}^{n}}$$, while an element $${A}$$ of the associated Lie algebra $${gl(n,\mathbb{R})}$$ is a vector field on $${\mathbb{R}^{n}}$$ that “points in the direction” of the element $${e^{A}\in GL\left(n,\mathbb{R}\right)}$$.