Lie Group

A smooth manifold equipped with a group structure for which multiplication and inversion are smooth.

Lie Group

A Lie group is a smooth manifold $G$ together with a group structure such that the maps

\mu: G \times G \to G,\quad (g,h)\mapsto gh, \qquad \iota: G \to G,\quad g\mapsto g^{-1}

Equivalently, a Lie group is a topological group whose underlying space is a smooth manifold and whose structure maps are smooth.

Basic examples

$(\mathbb{R}^n,+)$ and tori $T^n=\mathbb{R}^n/\mathbb{Z}^n$ (see abelian Lie group ).
Matrix groups such as $\operatorname{GL}(n,\mathbb{R})$ , $\operatorname{SO}(n)$ , $\operatorname{SU}(n)$ .
connected Lie groups and compact Lie groups are important special classes.

Every Lie group $G$ has an associated Lie algebra $\mathfrak{g}=T_eG$ , the tangent space at the identity, with a canonical Lie bracket .

The exponential map $\exp:\mathfrak{g}\to G$ relates $\mathfrak{g}$ to one-parameter subgroups , and the Lie correspondence explains how much of $G$ is determined by $\mathfrak{g}$ .