Lie Group

Definition

A group G, is a Lie group if G is a smooth manifold such that group multiplication $m:G\times G\to G$ and inversion $i:G\to G$ are smooth maps.

On every Lie group there are maps $L_a, R_a, C_a$ defined by left multiplying by $a$, right multiplying by $a$, and conjugation by $a$ ($x\mapsto axa^{-1}$) respectively. The maps $L_a$ and $R_a$ are diffeomorphisms.

Morphisms

A morphism of Lie groups $\phi:G\to H$ is a smooth group homomorphism. A Lie group morphism is an isomorphism if it is a diffeomorphism that is also a group homomorphism.

Lie groups and Lie group morphisms form a category.

Examples

Let $V$ be a vector space. Then $(V, +)$ is a Lie group.

$(\Z, +)$ is a $0$-dimensional Lie group.

$\R / \Z$ is a $1$-dimensional Lie group.

$S^3 = \{(x_1, y_1, x_2, y_2) \in \R^4: \Vert-\Vert = 1\}$ with group structure $(z_1, z_2) \cdot (z_3, z_4) := (z_1 z_3 - z_2 \overline{z_4}, z_1 z_4 - z_2 \overline{z_3})$ where we treat $\R^4$ as $\C^2$.

Matrix Lie Groups

For $k=\R$ or $k=\C$

$GL_n(k)$ invertible matrices with entries in $k$.

$SL_n(k)$ matrices with determinant 1.

$O_n(k)$ matricies such that $AA^T=A^TA=1$.