Lie group

A group that is also a smooth manifold, with smooth multiplication and inversion.

Lie group

A Lie group is a group $G$ equipped with the structure of a smooth manifold such that the group operations are smooth maps :

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

Equivalently, $G$ is a smooth manifold for which multiplication and inversion are smooth.

For each $g\in G$ , the left translation $L_g(h)=gh$ and the right translation $R_g(h)=hg$ are diffeomorphisms of $G$ , with inverses $L_{g^{-1}}$ and $R_{g^{-1}}$ . The tangent space at the identity $T_eG$ carries a canonical Lie algebra structure, called the Lie algebra of $G$ , and the exponential map relates this infinitesimal structure to local group behavior near $e$ .

Smooth group homomorphisms between Lie groups are Lie group homomorphisms .

Examples

The additive group $(\mathbb{R}^n,+)$ is a Lie group with its standard smooth structure; multiplication is $x+y$ and inversion is $x\mapsto -x$ .
The circle group $S^1=\{z\in\mathbb{C}:|z|=1\}$ is a $1$ -dimensional Lie group under complex multiplication.
The general linear group $GL(n,\mathbb{R})$ of invertible $n\times n$ real matrices is an open submanifold of $\mathbb{R}^{n^2}$ and is a Lie group under matrix multiplication.