# The Lie groups of a Lie algebra

Recall that on a differentiable manifold, it is not possible to use a tangent vector $${v}$$ to “transport a point in the direction $${v}$$” in a coordinate-independent way, since there is no special curve on $${M}$$ among the many that have $${v}$$ as a tangent. On a Lie group this is possible, since the left-invariant vector fields provide a unique flow in the direction of $${v}$$.

A one-parameter subgroup of $${G}$$ is a homomorphism $${\phi\colon\mathbb{R}\to G}$$. Given a left-invariant vector field $${A}$$, there is a unique one-parameter subgroup $${\phi_{A}}$$ such that $${\phi_{A}\left(0\right)=e}$$ and $${\dot{\phi}_{A}\left(t\right)=A}$$ for all $${t}$$ (i.e. $${\phi_{A}\left(t\right)}$$ is the local flow from the Lie derivative, but being defined for all $${t}$$ it is called simply the flow of $${A}$$). We can then define the exponential map $${\textrm{exp}\colon\mathfrak{g}\to G}$$ by

$$\displaystyle \mathrm{exp}(A)\equiv e^{A}\equiv \phi_{A}\left(1\right).$$

Since scaling the parametrization scales the tangent vectors, we have $${\phi_{A}\left(t\right)=\phi_{tA}\left(1\right)=\mathrm{exp}(tA)}$$.

In particular, the elements of $${G}$$ infinitesimally close to the identity can be written $${e+\varepsilon A}$$. The exponential map is a generalization of familiar exponential functions: if $${G=\mathbb{R}^{+}}$$, the positive reals under multiplication, $${\mathfrak{g}=\mathbb{R}}$$ and $${\textrm{exp}}$$ is the normal exponential function for real numbers; if $${G}$$ is the non-zero complex numbers under multiplication, $${\mathfrak{g}=\mathbb{C}}$$ and $${\textrm{exp}}$$ is the normal complex exponential function; and if $${G=GL(n,\mathbb{R})}$$, the real invertible $${n\times n}$$ matrices under matrix multiplication, $${\mathfrak{g}=gl(n,\mathbb{R})}$$, the real $${n\times n}$$ matrices, and $${\textrm{exp}}$$ is matrix exponentiation, defined by

$$\displaystyle e^{A}\equiv\overset{\infty}{\underset{k=0}{\sum}}\frac{1}{k!}A^{k}.$$

The multiplication of matrix exponentials does not follow the scalar rule, instead being given by the Baker-Campbell-Hausdorff formula:

$$\displaystyle e^{A}e^{B}\equiv e^{A+B+\frac{1}{2}\left[A,B\right]+\dotsb}$$

The terms that continue the series are all expressed in terms of Lie commutators, and as Lie brackets hold for the exponential maps of any Lie algebra; however, the series may not converge, limiting validity to a neighborhood of the identity. The terms shown above comprise the entire series if both matrices commute with the commutator, i.e. if $${\left[A,\left[A,B\right]\right]=\left[B,\left[A,B\right]\right]=0}$$.