A short lecture of mine on my scientific initiation project for 30th SIICUSP
Added a slide on the general classification of the finite-dimensional irreducible submodules of a complex semisimple Lie algebra
5 files changed, 3875 insertions, 2 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Added | images/cnpq.eps | 182 | 182 | 0 |
| Added | images/cnpq.svg | 62 | 62 | 0 |
| Added | images/ime.eps | 2599 | 2599 | 0 |
| Added | images/ime.svg | 1000 | 1000 | 0 |
| Modified | main.tex | 34 | 32 | 2 |
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diff --git a/main.tex b/main.tex @@ -145,7 +145,7 @@ \end{itemize} \end{frame} -\begin{frame}[fragile]{Representações de \(\operatorname{SU}_2\)} +\begin{frame}[fragile]{Representações de $\operatorname{SU}_2$} \begin{itemize} \item \(\mathbb{C} \otimes \operatorname{Lie}(\operatorname{GL}_n(\mathbb{C})) = \mathfrak{gl}_n @@ -178,7 +178,7 @@ \end{center} \end{frame} -\begin{frame}[fragile]{Representações de \(\operatorname{SU}_2\)} +\begin{frame}[fragile]{Representações de $\operatorname{SU}_2$} \begin{center} \begin{tikzcd} \cdots \arrow[bend left=60]{r} @@ -208,4 +208,34 @@ \end{align*} \end{itemize} \end{frame} + +\begin{frame}[fragile]{Representações de Álgebras de Lie Semisimples} + \begin{itemize} + \item \(\mathfrak{h} \subset \mathfrak{g}\) \emph{subálgebra de Cartan} + \begin{gather*} + V = \bigoplus_{\lambda \in \mathfrak{h}^*} V_\lambda \\ + V_\lambda + = \{v \in V : H v = \lambda(H) \cdot v \; \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 \(Q \subset + \mathfrak{h}^*\) + + \item \(V\) é determinado por seu maior peso! + + \item Módulos de Verma \(M(\lambda) = \mathcal{U}(\mathfrak{g}) + \otimes_{\mathcal{U}(\mathfrak{b})} \mathbb{C}_\lambda\) + \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}
