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\title{Smooth Manifolds Fall 2022 - Homework 6}

\begin{document}

\maketitle

\begin{problem}
Let $G$ be a connected Lie group. Show that $Z(G) = \ker(\Ad)$, where $Z(G) = \set{g \in G : gh = hg \mbox{ for all }h \in G}$.
\end{problem}

\begin{problem}
Classify the 2-dimensional connected Lie subgroups of $\Heis = \set{\begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{pmatrix} : x,y,z \in \R}$.
\end{problem}

\begin{problem}
Let $G$ be the group of transformations of $\R^2$ obtained by compositions of tranlsations and homotheties $x \mapsto \lambda x$ for $\lambda \in \R_+$. 

\begin{enumerate}
\item Show that $G$ is center-free (ie, that $Z(G) = \set{e}$).
\item Find  vector fields on $\R^2$ generating the actions by homotheties and translations.
\item Show that the vector fields of the previous part form the basis of a Lie algebra (ie, that they span a space closed under Lie brackets).
\item Compute the adjoint representation of its Lie algebra in the basis of the previous part.
\item Build a matrix group $H$ isomorphic to $G$.
\end{enumerate}
\end{problem}


{\noindent \color{red} \it Non-Graded.}

\begin{problem}
If $G$ is a Lie group, let $\bar{P}$ denote the set of continuous paths $\gamma : [0,1] \to G$ such that $\gamma(0) = e$. Define a multiplication

\[ (\gamma_1 * \gamma_2)(t) = \left\lbrace \begin{array}{ll} \gamma_2(2t), & t \in[0,1/2] \\ \gamma_1(2t-1)\gamma_2(1), & t \in [1/2,1] \end{array}\right. \]

Define a relation that $\gamma_1 \sim \gamma_2$ if and only if $(\gamma_1 * \gamma_2)(1) = e$, and $\gamma_1 * \gamma_2$ is trivial in $\pi_1(G,e)$. Show that $\sim$ is an equivalence relation, $*$ descends to $P = \bar{P} / \sim$ as a well-defined group operation, and that the map $\pi : \bar{P} \to G$ defined by $\pi(\gamma) = \gamma(1)$ is a group homomorphism. Furthermore, build a topology on $P$ so that $\pi$ is a covering map. (This shows that the universal cover of a Lie group is a Lie group!)
\end{problem}

\begin{problem}
Find all 2-dimensional connected Lie subgroups of $SL(2,\R)$ up to conjugacy.
\end{problem}

\begin{problem} *
Find all 2-dimensional connected Lie subgroups of $SL(3,\R)$ up to conjugacy.
\end{problem}

\end{document}




