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 <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Representations of  U_q(mathfraksl_2(K)) · TensorCategories.jl</title><meta name="title" content="Representations of  U_q(mathfraksl_2(K)) · TensorCategories.jl"/><meta property="og:title" content="Representations of  U_q(mathfraksl_2(K)) · TensorCategories.jl"/><meta property="twitter:title" content="Representations of  U_q(mathfraksl_2(K)) · TensorCategories.jl"/><meta name="description" content="Documentation for TensorCategories.jl."/><meta property="og:description" content="Documentation for TensorCategories.jl."/><meta property="twitter:description" content="Documentation for TensorCategories.jl."/><meta property="og:url" content="https://juliadocs.github.io/Documenter.jl/stable/ConcreteExamples/UqSl2/"/><meta property="twitter:url" content="https://juliadocs.github.io/Documenter.jl/stable/ConcreteExamples/UqSl2/"/><link rel="canonical" 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href="../../Constructions/Centralizer/">The Drinfeld Centralizer</a></li><li><a class="tocitem" href="../../Constructions/ModuleCategories/">Internal Module Categories</a></li></ul></li><li><a class="tocitem" href="../../References/">References</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>Representations of  <span>$U_q(\mathfrak{sl}_2(K))$</span></a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Representations of  <span>$U_q(\mathfrak{sl}_2(K))$</span></a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/FabianMaeurer/TensorCategories.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/FabianMaeurer/TensorCategories.jl/blob/master/docs/src/ConcreteExamples/UqSl2.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Representations-of-\\mathfrak{sl}_2(\\mathbb-k)"><a class="docs-heading-anchor" href="#Representations-of-\\mathfrak{sl}_2(\\mathbb-k)">Representations of <span>$\mathfrak{sl}_2(\mathbb k)$</span></a><a id="Representations-of-\\mathfrak{sl}_2(\\mathbb-k)-1"></a><a class="docs-heading-anchor-permalink" href="#Representations-of-\\mathfrak{sl}_2(\\mathbb-k)" title="Permalink"></a></h1><p>The representation category of <span>$\mathfrak{sl}_2(\mathbb k)$</span> has countably infinte simple objects <span>$\{V_i \mid i \in \mathbb N_0\}$</span> where the tensor product is given by the  Clebsch-Gordon rule</p><p class="math-container">\[V_i \otimes V_j = ⨁\limits_{l = 0}^{\min{(i,j)}} V_{i+j - 2l}.\]</p><p>We can construct the category and specify at any value for <span>$q$</span> that is either 1 or not  a rot of unity.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="TensorCategories.sl2_representations-ConcreteExamples-UqSl2" href="#TensorCategories.sl2_representations-ConcreteExamples-UqSl2"><code>TensorCategories.sl2_representations</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">sl2_representations(F::Ring)
-sl2_representations(F::Ring, q::RingElem)</code></pre><p>Construct a skeletal category equivalent to the category of representations of <span>$𝔰𝔩₂(F)$</span> specialized at <span>$q$</span>. <span>$q$</span> defaults to <span>$1$</span>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/FabianMaeurer/TensorCategories.jl/blob/1bef28dbd6a76d9096581e0661f3e632cccc39b5/src/Examples/UqSL2Representations/RepresentationsSL2.jl#L33-L40">source</a></section></article><h2 id="Verlinde-type-categories"><a class="docs-heading-anchor" href="#Verlinde-type-categories">Verlinde type categories</a><a id="Verlinde-type-categories-1"></a><a class="docs-heading-anchor-permalink" href="#Verlinde-type-categories" title="Permalink"></a></h2><p>When specifying at a root of unity we arrive at the Verlinde category.  These categoryies have <span>$n$</span> simple objects <span>$V_0,...,V_{n-1}$</span>and fusion rule </p><p class="math-container">\[V_i \otimes V_j = ⨁\limits_{l = 0}^{\min{(i,j)}} V_{i+j - 2l}.\]</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../SixJCategories/6JExamples/">« Fusion Categories with 6j-Symbols</a><a class="docs-footer-nextpage" href="../../Constructions/Center/">The Drinfeld Center »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Wednesday 3 July 2024 20:04">Wednesday 3 July 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+sl2_representations(F::Ring, q::RingElem)</code></pre><p>Construct a skeletal category equivalent to the category of representations of <span>$𝔰𝔩₂(F)$</span> specialized at <span>$q$</span>. <span>$q$</span> defaults to <span>$1$</span>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/FabianMaeurer/TensorCategories.jl/blob/5c30ebc83cf4ee77fc227c15196ea648ac0f2315/src/Examples/UqSL2Representations/RepresentationsSL2.jl#L33-L40">source</a></section></article><h2 id="Verlinde-type-categories"><a class="docs-heading-anchor" href="#Verlinde-type-categories">Verlinde type categories</a><a id="Verlinde-type-categories-1"></a><a class="docs-heading-anchor-permalink" href="#Verlinde-type-categories" title="Permalink"></a></h2><p>When specifying at a root of unity we arrive at the Verlinde category.  These categoryies have <span>$n$</span> simple objects <span>$V_0,...,V_{n-1}$</span>and fusion rule </p><p class="math-container">\[V_i \otimes V_j = ⨁\limits_{l = 0}^{\min{(i,j)}} V_{i+j - 2l}.\]</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../SixJCategories/6JExamples/">« Fusion Categories with 6j-Symbols</a><a class="docs-footer-nextpage" href="../../Constructions/Center/">The Drinfeld Center »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Wednesday 16 October 2024 16:49">Wednesday 16 October 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>