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href="https://nforum.ncatlab.org/search/?PostBackAction=Search&Type=Topics&Tag=universal" class="TagLink">universal</a></span> <span style="font-size:53%"><a href="https://nforum.ncatlab.org/search/?PostBackAction=Search&Type=Topics&Tag=variational-calculus" class="TagLink">variational-calculus</a></span></div><p id="AboutVanilla"><a href="http://getvanilla.com">Vanilla 1.1.10</a> is a product of <a href="http://lussumo.com">Lussumo</a>. More Information: <a href="http://lussumo.com/docs">Documentation</a>, <a href="http://lussumo.com/community">Community Support</a>.</p></div> <div id="Content"><div id="NoticeCollector" class="Notices"><div class="Notice"><strong>Welcome to nForum</strong> <br />If you want to take part in these discussions either <a href="/people.php?ReturnUrl=http%3A%2F%2Fnforum.ncatlab.org%2Fdiscussion%2F10043%2F">sign in now</a> (if you have an account), <a href="https://nforum.ncatlab.org/people/?PostBackAction=ApplyForm">apply for one now</a> (if you don't).</div></div><div class="ContentInfo Top"> <h1><a href="https://nforum.ncatlab.org/5/"><a href="https://nforum.ncatlab.org/18/">nLab</a> > </a> <a href="https://nforum.ncatlab.org/5/">Latest Changes</a>: orbit</h1> <a href="#pgbottom">Bottom of Page</a> <div class="PageInfo"> <p>1 to 4 of 4</p> <ol class="PageList PageListEmpty"> <li> </li> </ol> </div> </div> <div id="ContentBody"> <script type="text/javascript"> //<![CDATA[ function toggle_source(id) { var mysrc = document.getElementById("CommentBody_" + id).firstChild; if (mysrc.className == "source") { if (mysrc.style.display == "none") { mysrc.style.display = "block"; } else { mysrc.style.display = "none"; } } } var commentIds = new Array(0); function hide_sources() { for (i = 0; i < commentIds.length; i++) { var myself = document.getElementById("Source" + commentIds[i]); var mycmt = document.getElementById("CommentBody_" + commentIds[i]); if (mycmt.firstChild.className != "source") { myself.style.display = "none"; } } } window.onload = hide_sources; //]]> </script> <ol id="Comments"><li id="Comment_78572"> <a id="Item_1"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>1.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/356/">Dmitri Pavlov</a></li> <li><span>CommentTime</span>Jun 16th 2019</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/10043/orbit/?Focus=78572#Comment_78572">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_78572"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/356/">Dmitri Pavlov</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Added an alternative description of the category of orbits via normalizers. <a href="https://ncatlab.org/nlab/revision/diff/orbit/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/orbit/7">v7</a>, <a href="https://ncatlab.org/nlab/show/orbit">current</a></code></div><div> <p>Added an alternative description of the category of orbits via normalizers.</p> <p><a href="https://ncatlab.org/nlab/revision/diff/orbit/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/orbit/7">v7</a>, <a href="https://ncatlab.org/nlab/show/orbit">current</a></p> </div> </div> </li><li id="Comment_91265" class="Alternate"> <a id="Item_2"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>2.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Apr 8th 2021</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/10043/orbit/?Focus=91265#Comment_91265">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_91265"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>added pointer to: * [[Glen Bredon]], Section I.4 of: _[[Introduction to compact transformation groups]]_, Academic Press 1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf)) <a href="https://ncatlab.org/nlab/revision/diff/orbit/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/orbit/10">v10</a>, <a href="https://ncatlab.org/nlab/show/orbit">current</a></code></div><div> <p>added pointer to:</p> <ul> <li><a href="https://ncatlab.org/nlab/show/Glen Bredon">Glen Bredon</a>, Section I.4 of: <em><a href="https://ncatlab.org/nlab/show/Introduction to compact transformation groups">Introduction to compact transformation groups</a></em>, Academic Press 1972 (<a href="https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1">ISBN 9780080873596</a>, <a href="http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf">pdf</a>)</li> </ul> <p><a href="https://ncatlab.org/nlab/revision/diff/orbit/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/orbit/10">v10</a>, <a href="https://ncatlab.org/nlab/show/orbit">current</a></p> </div> </div> </li><li id="Comment_91335"> <a id="Item_3"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>3.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Apr 12th 2021</li><li><em>(edited Apr 12th 2021)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/10043/orbit/?Focus=91335#Comment_91335">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_91335"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I am hereby removing the following text from the section "Related concepts", and instead replaced it by but a pointer to the *[[orbit method]]*. Whatever parts of this text that is missing at *[[orbit category]]* should be moved to there: *** Begin extracted text *** ### Orbit method The __[[orbit method]]__ is a method in [[representation theory]] introduced by Kirillov, Kostant and Souriau; it is a special case of [[geometric quantization]]. The orbit method is based on the study of the [[representations]] constructed from the [[coadjoint orbits]] with Kirillov symplectic structure. The terminology 'geometric quantization' allows for more general underlying spaces. Given a [[compact space|compact]] [[Lie group]] $K$ with [[complexification]] $G$ and a [[unitary representation]] $\rho$ of $K$ on a finite-dimensional complex space $V$, the real orbits of the highest weight vector agrees with the complex orbits, i.e. the orbits of the extension of this representation to the representation of the complexification. These are the [[coherent state]] orbits; there is also an infinite-dimensional version for reductive groups and representations which allows them (so-called coherent state representations). *** End extracted text *** <a href="https://ncatlab.org/nlab/revision/diff/orbit/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/orbit/12">v12</a>, <a href="https://ncatlab.org/nlab/show/orbit">current</a></code></div><div> <p>I am hereby removing the following text from the section “Related concepts”, and instead replaced it by but a pointer to the <em><a href="https://ncatlab.org/nlab/show/orbit method">orbit method</a></em>.</p> <p>Whatever parts of this text that is missing at <em><a href="https://ncatlab.org/nlab/show/orbit category">orbit category</a></em> should be moved to there:</p> <hr/> <p>Begin extracted text</p> <hr/> <h3>Orbit method</h3> <p>The <strong><a href="https://ncatlab.org/nlab/show/orbit method">orbit method</a></strong> is a method in <a href="https://ncatlab.org/nlab/show/representation theory">representation theory</a> introduced by Kirillov, Kostant and Souriau; it is a special case of <a href="https://ncatlab.org/nlab/show/geometric quantization">geometric quantization</a>. The orbit method is based on the study of the <a href="https://ncatlab.org/nlab/show/representations">representations</a> constructed from the <a href="https://ncatlab.org/nlab/show/coadjoint orbits">coadjoint orbits</a> with Kirillov symplectic structure. The terminology ’geometric quantization’ allows for more general underlying spaces.</p> <p>Given a <a href="https://ncatlab.org/nlab/show/compact space">compact</a> <a href="https://ncatlab.org/nlab/show/Lie group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> with <a href="https://ncatlab.org/nlab/show/complexification">complexification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and a <a href="https://ncatlab.org/nlab/show/unitary representation">unitary representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> on a finite-dimensional complex space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, the real orbits of the highest weight vector agrees with the complex orbits, i.e. the orbits of the extension of this representation to the representation of the complexification. These are the <a href="https://ncatlab.org/nlab/show/coherent state">coherent state</a> orbits; there is also an infinite-dimensional version for reductive groups and representations which allows them (so-called coherent state representations).</p> <hr/> <p>End extracted text</p> <hr/> <p><a href="https://ncatlab.org/nlab/revision/diff/orbit/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/orbit/12">v12</a>, <a href="https://ncatlab.org/nlab/show/orbit">current</a></p> </div> </div> </li><li id="Comment_91568" class="Alternate"> <a id="Item_4"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>4.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Apr 18th 2021</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/10043/orbit/?Focus=91568#Comment_91568">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_91568"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Added the example of a permutation cycle. <a href="https://ncatlab.org/nlab/revision/diff/orbit/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/orbit/13">v13</a>, <a href="https://ncatlab.org/nlab/show/orbit">current</a></code></div><div> <p>Added the example of a permutation cycle.</p> <p><a href="https://ncatlab.org/nlab/revision/diff/orbit/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/orbit/13">v13</a>, <a href="https://ncatlab.org/nlab/show/orbit">current</a></p> </div> </div> </li></ol> </div><div class="ContentInfo Middle"> <div class="PageInfo"> <p>1 to 4 of 4</p> <ol class="PageList PageListEmpty"> <li> </li> </ol> </div> </div></div> <a id="pgbottom" > </a> </div> </div></body> </html>