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\title{Smooth Manifolds Fall 2022 - Homework 7}

\begin{document}

\maketitle

\begin{problem}
Let $G$ be a connected Lie group, and $\mf h$ be an ideal of $\Lie(G)$. 

\begin{enumerate}[(a)]
\item Show that there is a well defined action $\overline{\Ad} : G \to GL(\Lie(G) / \mf h)$ defined by

\[ \overline{\Ad}(g)(X+\mf h) = \Ad(g)X + \mf h. \]

\item Show that the kernel of $\overline{\Ad}$ is the set $\set{g \in G : \Ad(g)X \in \mf h \mbox{ for all }X \in \Lie(G)}$. 
\item Give an example of a Lie group and proper ideal for which $\Ad$ is faithful but $\overline{\Ad}$ is trivial.
\end{enumerate}
\end{problem}

\begin{problem}
If $V$ is a vector space of dimension $n$ and $\omega \in \Lambda^k(V)$ is an alternating $k$-multilinear function and $v \in V$, let $\iota_v\omega \in \Lambda^{k-1}(V)$ be the functional $(w_1,\dots,w_{k-1})\mapsto \omega(v,w_1,\dots,w_{k-1})$. Let $\ker \omega = \set{ v \in V : \iota_v\omega = 0}$. 

\begin{enumerate}[(a)]
\item Show that $\ker \omega$ is a subspace of $V$.
\item Prove or find a counterexample: for a fixed $v$, $\iota_v : \Lambda^2(V) \to \Lambda^1(V)$ is never onto.
\item Prove or find a counterexample: if $W \subset \ker \omega$, then $\omega$ induces a well defined alternating $k$-multilinear function on $V / W$.
\end{enumerate}
\end{problem}


\begin{problem}
Let $V$ be an $n$-dimensional vector space and $\omega \in \Lambda^k(V)$. If $W \subset V$ is a vector subspace, let $\pi_W : \Lambda^k(V) \to \Lambda^k(W)$ denote the restriction map, $\pi_W(\alpha) = \alpha|_W$.

\begin{enumerate}[(a)]
\item Show that $\pi_W$ is linear and onto. Use this to compute $\dim(\ker \pi_W)$.
\item Find a basis for $\ker \pi_W$ when $V = \R^5$, $W = \R^3 \times \set{0}$ and $k = 2$.
\end{enumerate}
\end{problem}


\end{document}




