Lie Group: Difference between revisions
No edit summary |
|||
| Line 10: | Line 10: | ||
$\R / \Z$ is a $1$-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 === | === Matrix Lie Groups === | ||
$GL_n(\R)$ | $GL_n(\R)$ | ||
Revision as of 09:41, 19 January 2026
Definition
Morphisms
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
$GL_n(\R)$