math: lie groups

Notes/closed_subgroup_theorem.md

@@ -0,0 +1,8 @@
+---
+title: Closed Subgroup Theorem
+date: 2024-09-14T14:19:07-04:00
+references: []
+tags: [In_Progress]
+---
+
+# Motivation & Statement

Notes/lie_group.md

@@ -15,7 +15,7 @@ A *Lie group* is a smooth manifold $G$ equipped with a group structure $(G,\mu,\
 
 ## Lie Algebras
 
-## The Exponential Map
+## The {{< link file="the_exponential_map.md" display="Exponential Map" type="constructions" >}}
 
 # General Techniques
 

Notes/submanifold.md

@@ -0,0 +1,8 @@
+---
+title: Submanifold
+date: 2024-09-14T14:55:11-04:00
+references: []
+tags: [In_Progress]
+---
+
+# Motivation & Definition

Notes/the_exponential_map.md

@@ -0,0 +1,12 @@
+---
+title: The Exponential Map
+date: 2024-09-14T14:20:03-04:00
+references: []
+tags: [In_Progress]
+---
+
+# Motivation & Definition
+
+# Applications
+
+## {{< link file="closed_subgroup_theorem.md" display="Closed Subgroup Theorem" type="applications" >}}

README.md

@@ -40,4 +40,4 @@ I wrote some bash scripts to (try) maintain the consistency of my notes; [main.s
 * Basic stats of the site (and updating them below) is handled by [stats.sh](https://github.com/zhaoshenzhai/mathwiki/blob/master/.scripts/stats.sh).
 * Displaying all links that have yet to be created and all notes that have those links, are handled by [ghost.sh](https://github.com/zhaoshenzhai/mathwiki/blob/master/.scripts/ghost.sh).
 
-Some stats: 66 notes, 276 links (ratio: 4.181), 8 images ([updated](https://github.com/zhaoshenzhai/mathwiki/blob/master/.scripts/stats.sh) every commit).
+Some stats: 69 notes, 278 links (ratio: 4.028), 8 images ([updated](https://github.com/zhaoshenzhai/mathwiki/blob/master/.scripts/stats.sh) every commit).

docs/abelian_group/index.html

@@ -58,7 +58,7 @@ <h1 id="motivation--definition">Motivation &amp; Definition</h1>
 <p>Here, we consider a class of groups which $-$ despite being very general $-$ exhibit very useful properties absent in general groups.</p>
 <div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b>A group $G$ is said to be <em>Abelian</em> if $gh=hg$ for all $g,h\in G$.</div>
 
-<h2 id="abelian-groups-as-hahahugoshortcode4s1hbhb">Abelian Groups as <a href=https://zhaoshenzhai.github.io/mathwiki/module.md class="internalLink references" title="Module" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/module.md&#34;, {&#34;Date&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Module&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/module&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Module&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/module&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/module.md&#34;, {&#34;Date&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Module&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/module&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">$\Z$-modules</a></h2>
+<h2 id="abelian-groups-as-hahahugoshortcode6s1hbhb">Abelian Groups as <a href=https://zhaoshenzhai.github.io/mathwiki/module.md class="internalLink references" title="Module" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/module.md&#34;, {&#34;Date&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Module&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/module&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Module&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/module&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/module.md&#34;, {&#34;Date&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Module&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/module&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">$\Z$-modules</a></h2>
 <p>We denote the <a href=https://zhaoshenzhai.github.io/mathwiki/subcategory.md class="internalLink references ghostLink" title="subcategory" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/subcategory.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/subcategory.md&#34;, &#34;nopPage&#34;);">subcategory</a> of Abelian groups as $\catabgrp$, which is one of the most well-behaved categories. Indeed, <span style="color:red">???</span>.</p>
 <ul>
 <li>We have a forgetful functor $\catmod[\Z]\to\catabgrp$, which is an isomorphism of categories <span style="color:red">???</span>. In this way, the notions of <a href=https://zhaoshenzhai.github.io/mathwiki/torsion.md class="internalLink constructions ghostLink" title="torsion" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/torsion.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/torsion.md&#34;, &#34;nopPage&#34;);">torsion</a>, <a href=https://zhaoshenzhai.github.io/mathwiki/rank.md class="internalLink constructions ghostLink" title="rank" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/rank.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/rank.md&#34;, &#34;nopPage&#34;);">rank</a>, etc, all carry forward.</li>
@@ -75,7 +75,7 @@ <h2 id="structure-theory">Structure Theory</h2>
 
 <p>On the other hand, Abelian groups that are <em>not</em> finitely-generated are much wilder, and many set-theoretical issues crop up.</p>
 <h1 id="examples">Examples</h1>
-<h2 id="hahahugoshortcode4s8hbhb"><a href=https://zhaoshenzhai.github.io/mathwiki/cyclic_group.md class="internalLink examples" title="Cyclic Group" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/cyclic_group.md&#34;, {&#34;Date&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Cyclic Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/cyclic_group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Cyclic Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/cyclic_group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/cyclic_group.md&#34;, {&#34;Date&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Cyclic Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/cyclic_group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">Cyclic Groups</a></h2>
+<h2 id="hahahugoshortcode6s8hbhb"><a href=https://zhaoshenzhai.github.io/mathwiki/cyclic_group.md class="internalLink examples" title="Cyclic Group" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/cyclic_group.md&#34;, {&#34;Date&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Cyclic Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/cyclic_group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Cyclic Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/cyclic_group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/cyclic_group.md&#34;, {&#34;Date&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Cyclic Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/cyclic_group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">Cyclic Groups</a></h2>
 <h1 id="generalizations">Generalizations</h1>
 <h2 id="abelian-categories">Abelian Categories</h2>