- Commit
- d0557d93c7d04e7b36068a7c748b275c30990347
- Parent
- 813881862d8b87f7304ac18a9deea8b8e5207df7
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Made some adjustments on the sections on representations of SU(2)
30th-siicusp
A short lecture of mine on my scientific initiation project for 30th SIICUSP
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 \(h\)