diff options
Diffstat (limited to 'vorlesungen/slides/7/liealgebra.tex')
-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 59c9121..574467b 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 |