30th-siicusp

A short lecture of mine on my scientific initiation project for 30th SIICUSP

git clone: git://git.pablopie.xyz/30th-siicusp
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Commit: 3d891a3b86f33b09e92b5bc7d579335d1538bd8e
Parent: afb84cc570cc022b872a155473b6b35d5e881788
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 17 Oct 2022 21:52:20 +0000

Restructured the last slide

Diffstat

2 files changed, 36 insertions, 19 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	main.tex	54	35	19
Modified	preamble.tex	1	1	0

diff --git a/main.tex b/main.tex
@@ -186,7 +186,7 @@
 \begin{frame}[fragile]{Representações de $\operatorname{SU}_2$}
   \begin{itemize}
     \item Os autovalores de \(h\) em \(V\) formam uma cadeia ininterrupta de
-      inteiros simétrica ao redor de \(0\)
+      inteiros simétrica ao redor de 0
       \begin{center}
         \begin{tabular}{ccccc}
               &    & 0           \\ \noalign{\smallskip\smallskip}
@@ -215,36 +215,52 @@
 \end{frame}
 
 \begin{frame}[fragile]{Representações de Álgebras de Lie Semisimples}
+  \begin{center}
+    \begin{tabular}{rcl}
+      \(\mathfrak{sl}_2 \mathbb{C}\) &
+      \rightsquigarrow &
+      \(\mathfrak{sl}_n \mathbb{C}\) \\
+      \(h\) &
+      \rightsquigarrow &
+      \(\mathfrak{h}
+      = \{X \in \mathfrak{sl}_n \mathbb{C} : X\ \text{diagonal}\}\) \\
+      \(\lambda \in \mathbb{C}\) &
+      \rightsquigarrow &
+      \(\lambda \in \mathfrak{h}^*\)
+    \end{tabular}
+  \end{center}
   \begin{itemize}
-    \item \(\mathfrak{h} \subset \mathfrak{g}\) \emph{subálgebra de Cartan}
-      \begin{gather*}
-        V = \bigoplus_{\lambda \in \mathfrak{h}^*} V_\lambda \\
+    \item \emph{Pesos}: ``autovalores'' da ação de \(\mathfrak{h}\)
+      \[
         V_\lambda =
         \{
           v \in V :
           H \cdot v = \lambda(H) \cdot v
           \quad \forall H \in \mathfrak{h}
         \}
-      \end{gather*}
-
-    \item \emph{Pesos}: ``autovalores'' da ação de \(\mathfrak{h}\)
+      \]
 
     \item Os pesos de \(V\) são congruentes mod um reticulado \(Q \subset
       \mathfrak{h}^*\)
+      \begin{center}
+        \begin{tikzpicture}[scale=1.2]
+          \begin{rootSystem}{A}
+            \wt[mDarkTeal]{0}{0}
+            \wt[mDarkTeal]{-1}{2}
+            \wt[mDarkTeal]{-2}{1}
+            \wt[mDarkTeal]{1}{1}
+            \wt[mDarkTeal]{-1}{-1}
+            \wt[mDarkTeal]{2}{-1}
+            \wt[mDarkTeal]{1}{-2}
+            \filldraw[mDarkTeal] \weight{0}{1}   circle (.75pt);
+            \filldraw[mDarkTeal] \weight{-1}{0}  circle (.75pt);
+            \filldraw[mDarkTeal] \weight{1}{-1}  circle (.75pt);
+            \draw[-latex, mDarkTeal] \weight{-1.5}{-.5} -- \weight{1.5}{.5};
+          \end{rootSystem}
+        \end{tikzpicture}
+      \end{center}
 
     \item \(V\) é determinado por seu maior peso!
-
-    \item Módulos de Verma \(M(\lambda) = \mathcal{U}(\mathfrak{g})
-      \otimes_{\mathcal{U}(\mathfrak{b})} k v_\lambda^+\)
-  \begin{center}
-    \begin{tikzcd}
-      \cdots \arrow[bend left=60]{r}{-10}
-      & M(\lambda)_{-4} \arrow[bend left=60]{r}{0}  \arrow[bend left=60]{l}{1}
-      & M(\lambda)_{-2} \arrow[bend left=60]{r}{2}  \arrow[bend left=60]{l}{1}
-      & M(\lambda)_0    \arrow[bend left=60]{r}{2}  \arrow[bend left=60]{l}{1}
-      & M(\lambda)_2    \arrow[bend left=60]{l}{1}
-    \end{tikzcd}
-  \end{center}
   \end{itemize}
 \end{frame}
 \end{document}

diff --git a/preamble.tex b/preamble.tex
@@ -7,6 +7,7 @@
 \usepackage{pgfplots, tikz, tikz-cd}
 \usepackage{graphicx, wrapfig}
 \usepackage[normalem]{ulem}
+\usepackage{rank-2-roots}
 \usepackage{fontspec}
 
 % Configure graphics