Lie Group: Difference between revisions
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== Definition == | == Definition == | ||
A group G, is a Lie group if G is a [[Manifold | 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 == | == 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 == | == Examples == | ||
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Let $V$ be a vector space. Then $(V, +)$ is a Lie group. | Let $V$ be a vector space. Then $(V, +)$ is a Lie group. | ||
$(\Z, +)$ is a $0$-dimensional Lie group. | $(\Z, +)$ is a $0$-dimensional Lie group. Similarly, any countable group with the discrete topology is a $0$-dimensional Lie group. | ||
$\R / \Z$ 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 === | |||
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 $AA^T=A^TA=1$. | |||
Latest revision as of 08:58, 22 January 2026
Definition
[edit]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
[edit]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
[edit]Let $V$ be a vector space. Then $(V, +)$ is a Lie group.
$(\Z, +)$ is a $0$-dimensional Lie group. Similarly, any countable group with the discrete topology 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
[edit]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 $AA^T=A^TA=1$.