From 1fb743df08b0734932d510c6b11405d0a2dbbe47 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 10 Apr 2021 19:57:13 +0200 Subject: new slides --- vorlesungen/slides/7/liealgebra.tex | 69 +++++++++++++++++++++++++++++++++++++ 1 file changed, 69 insertions(+) create mode 100644 vorlesungen/slides/7/liealgebra.tex (limited to 'vorlesungen/slides/7/liealgebra.tex') 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 -- cgit v1.2.1 From 4d96632a4cb5dce985c92bc5daebe272dcbc1410 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 10 Apr 2021 23:01:39 +0200 Subject: typos --- vorlesungen/slides/7/liealgebra.tex | 1 - 1 file changed, 1 deletion(-) (limited to 'vorlesungen/slides/7/liealgebra.tex') diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex index 16a7aa0..892216e 100644 --- a/vorlesungen/slides/7/liealgebra.tex +++ b/vorlesungen/slides/7/liealgebra.tex @@ -8,7 +8,6 @@ \setlength{\abovedisplayskip}{5pt} \setlength{\belowdisplayskip}{5pt} \frametitle{Lie-Algebra} -\vspace{-20pt} \begin{block}{Vektorraum} Tangentialvektoren im Punkt $I$: \begin{center} -- cgit v1.2.1 From 15881729aa3f1293d546a1692a02094ed3f24e2b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 11 Apr 2021 10:30:05 +0200 Subject: phases --- vorlesungen/slides/7/liealgebra.tex | 27 ++++++++++++++++++++++----- 1 file changed, 22 insertions(+), 5 deletions(-) (limited to 'vorlesungen/slides/7/liealgebra.tex') diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex index 892216e..574467b 100644 --- a/vorlesungen/slides/7/liealgebra.tex +++ b/vorlesungen/slides/7/liealgebra.tex @@ -8,50 +8,64 @@ \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} +\end{block}} +\uncover<8->{% \begin{block}{Nachprüfen} $[A,B]\in LG$ für $A,B\in LG$ -\end{block} +\end{block}} \end{column} \begin{column}{0.56\textwidth} +\uncover<9->{% \begin{block}{Algebraische Eigenschaften} \begin{itemize} -\item antisymmetrisch: $[A,B]=-[B,A]$ -\item Jacobi-Identität +\item<10-> antisymmetrisch: $[A,B]=-[B,A]$ +\item<11-> Jacobi-Identität \[ [A,[B,C]]+ [B,[C,A]]+ @@ -59,9 +73,12 @@ für $A,B\in LG$ = 0 \] \end{itemize} +\vspace{-13pt} +\uncover<12->{% {\usebeamercolor[fg]{title} Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$ -\end{block} +} +\end{block}} \end{column} \end{columns} \end{frame} -- cgit v1.2.1