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Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/slides/7/liealgebra.tex | 170 |
1 files changed, 85 insertions, 85 deletions
diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex index 574467b..59c9121 100644 --- a/vorlesungen/slides/7/liealgebra.tex +++ b/vorlesungen/slides/7/liealgebra.tex @@ -1,85 +1,85 @@ -% -% 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} -\ifthenelse{\boolean{presentation}}{\vspace{-15pt}}{\vspace{-8pt}} -\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 -\uncover<2->{ -\operatorname{GL}_n(\mathbb{R}) -& beliebige Matrizen -& M_n(\mathbb{R}) -} -\\ -\uncover<3->{ -\operatorname{O(n)} -& antisymmetrische Matrizen -& \operatorname{o}(n) -} -\\ -\uncover<4->{ -\operatorname{SL}_n(\mathbb{R}) -& spurlose Matrizen -& \operatorname{sl}_2(\mathbb{R}) -} -\\ -\uncover<5->{ -\operatorname{U(n)} -& antihermitesche Matrizen -& \operatorname{u}(n) -} -\\ -\uncover<6->{ -\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} -\uncover<7->{% -\begin{block}{Lie-Klammer} -Kommutator: $[A,B] = AB-BA$ -\end{block}} -\uncover<8->{% -\begin{block}{Nachprüfen} -$[A,B]\in LG$ -für $A,B\in LG$ -\end{block}} -\end{column} -\begin{column}{0.56\textwidth} -\uncover<9->{% -\begin{block}{Algebraische Eigenschaften} -\begin{itemize} -\item<10-> antisymmetrisch: $[A,B]=-[B,A]$ -\item<11-> Jacobi-Identität -\[ -[A,[B,C]]+ -[B,[C,A]]+ -[C,[A,B]] -= 0 -\] -\end{itemize} -\vspace{-13pt} -\uncover<12->{% -{\usebeamercolor[fg]{title} -Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$ -} -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +%
+% 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}
+\ifthenelse{\boolean{presentation}}{\vspace{-15pt}}{\vspace{-8pt}}
+\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
+\uncover<2->{
+\operatorname{GL}_n(\mathbb{R})
+& beliebige Matrizen
+& M_n(\mathbb{R})
+}
+\\
+\uncover<3->{
+\operatorname{O(n)}
+& antisymmetrische Matrizen
+& \operatorname{o}(n)
+}
+\\
+\uncover<4->{
+\operatorname{SL}_n(\mathbb{R})
+& spurlose Matrizen
+& \operatorname{sl}_2(\mathbb{R})
+}
+\\
+\uncover<5->{
+\operatorname{U(n)}
+& antihermitesche Matrizen
+& \operatorname{u}(n)
+}
+\\
+\uncover<6->{
+\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}
+\uncover<7->{%
+\begin{block}{Lie-Klammer}
+Kommutator: $[A,B] = AB-BA$
+\end{block}}
+\uncover<8->{%
+\begin{block}{Nachprüfen}
+$[A,B]\in LG$
+für $A,B\in LG$
+\end{block}}
+\end{column}
+\begin{column}{0.56\textwidth}
+\uncover<9->{%
+\begin{block}{Algebraische Eigenschaften}
+\begin{itemize}
+\item<10-> antisymmetrisch: $[A,B]=-[B,A]$
+\item<11-> Jacobi-Identität
+\[
+[A,[B,C]]+
+[B,[C,A]]+
+[C,[A,B]]
+= 0
+\]
+\end{itemize}
+\vspace{-13pt}
+\uncover<12->{%
+{\usebeamercolor[fg]{title}
+Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
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