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diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex new file mode 100644 index 0000000..16a7aa0 --- /dev/null +++ b/vorlesungen/slides/7/liealgebra.tex @@ -0,0 +1,69 @@ +% +% liealgebra.tex -- Lie-Algebra +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lie-Algebra} +\vspace{-20pt} +\begin{block}{Vektorraum} +Tangentialvektoren im Punkt $I$: +\begin{center} +\begin{tabular}{>{$}c<{$}|p{6cm}|>{$}c<{$}} +\text{Lie-Gruppe $G$}&Tangentialvektoren&\text{Lie-Algebra $LG$} \\ +\hline +\operatorname{GL}_n(\mathbb{R}) +& beliebige Matrizen +& M_n(\mathbb{R}) +\\ +\operatorname{O(n)} +& antisymmetrische Matrizen +& \operatorname{o}(n) +\\ +\operatorname{SL}_n(\mathbb{R}) +& spurlose Matrizen +& \operatorname{sl}_2(\mathbb{R}) +\\ +\operatorname{U(n)} +& antihermitesche Matrizen +& \operatorname{u}(n) +\\ +\operatorname{SU(n)} +& spurlose, antihermitesche Matrizen +& \operatorname{su}(n) +\end{tabular} +\end{center} +\end{block} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{block}{Lie-Klammer} +Kommutator: $[A,B] = AB-BA$ +\end{block} +\begin{block}{Nachprüfen} +$[A,B]\in LG$ +für $A,B\in LG$ +\end{block} +\end{column} +\begin{column}{0.56\textwidth} +\begin{block}{Algebraische Eigenschaften} +\begin{itemize} +\item antisymmetrisch: $[A,B]=-[B,A]$ +\item Jacobi-Identität +\[ +[A,[B,C]]+ +[B,[C,A]]+ +[C,[A,B]] += 0 +\] +\end{itemize} +{\usebeamercolor[fg]{title} +Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$ +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup |