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: d0557d93c7d04e7b36068a7c748b275c30990347
Parent: 813881862d8b87f7304ac18a9deea8b8e5207df7
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 17 Oct 2022 20:23:10 +0000

Made some adjustments on the sections on representations of SU(2)

Diffstat

1 file changed, 13 insertions, 14 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	main.tex	27	13	14

diff --git a/main.tex b/main.tex
@@ -126,8 +126,7 @@
     \item Funtorialidade
       \begin{center}
         \begin{tabular}{ccccc}
-          \(G \to H\) &
-          \rightsquigarrow &
+          \(G \to H\) & \rightsquigarrow &
           \(T_1 G \to T_1 H\) &
           \rightsquigarrow &
           \(\operatorname{Lie}(G) \to \operatorname{Lie}(H)\)
@@ -154,11 +153,11 @@
 \begin{frame}[fragile]{Representações de $\operatorname{SU}_2$}
   \begin{itemize}
     \item \(\mathbb{C} \otimes
-      \operatorname{Lie}(\operatorname{GL}_n(\mathbb{C})) = \mathfrak{gl}_n
+      \operatorname{Lie}(\operatorname{GL}_n(\mathbb{C})) \cong \mathfrak{gl}_n
       \mathbb{C}\)
 
-    \item \(\mathbb{C} \otimes \operatorname{Lie}(\operatorname{SU}_n) =
-      \mathfrak{sl}_n \mathbb{C}\) é a subálgebra dos \(X \in \mathfrak{gl}_n
+    \item \(\mathbb{C} \otimes \operatorname{Lie}(\operatorname{SU}_2) \cong
+      \mathfrak{sl}_2 \mathbb{C}\) é a subálgebra dos \(X \in \mathfrak{gl}_2
       \mathbb{C}\) com \(\operatorname{Tr}(X) = 0\)
       \begin{align*}
         e & = \begin{pmatrix} 0 & 1 \\ 0 &  0 \end{pmatrix} &
@@ -185,15 +184,6 @@
 \end{frame}
 
 \begin{frame}[fragile]{Representações de $\operatorname{SU}_2$}
-  \begin{center}
-    \begin{tikzcd}
-      \cdots \arrow[bend left=60]{r}
-      & V_{\lambda - 2} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l}
-      & V_{\lambda} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l}{f}
-      & V_{\lambda + 2} \arrow[bend left=60]{r} \arrow[bend left=60]{l}{f}
-      & \cdots \arrow[bend left=60]{l}
-    \end{tikzcd}
-  \end{center}
   \begin{itemize}
     \item Os autovalores de \(h\) em \(V\) formam uma cadeia ininterrupta de
       inteiros simétrica ao redor de \(0\)
@@ -204,6 +194,15 @@
            -2 &    & 0 &    & +2 \\ \noalign{\smallskip\smallskip}
         \end{tabular}
       \end{center}
+      \begin{center}
+        \begin{tikzcd}
+            V_{- 4} \arrow [bend left=60]{r}{e}
+          & V_{- 2} \arrow [bend left=60]{r}{e} \arrow [bend left=60]{l}{f}
+          & V_0     \arrow [bend left=60]{r}{e} \arrow [bend left=60]{l}{f}
+          & V_2     \arrow [bend left=60]{r}{e} \arrow [bend left=60]{l}{f}
+          & V_4     \arrow [bend left=60]{l}{f}
+        \end{tikzcd}
+      \end{center}
 
     \item \(V\) é completamente caracterizada pelo maior autovalor \(\lambda\)
       de \(h\)