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\noindent
\parbox{2.25in}{\bf Math 4010-101}
\hfill {\Large \bf Homework \#7} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Nov. $15^\mathrm{th}$, 2017}

\vspace{0.2in}

Let $\mathfrak{g}$ be a Lie algebra over a field $\mathbb{F}$.

\begin{description}
\item[\#1 Dualing Problem:] Consider $\beta = \{ (1,0,1), (1,2,0), (0,1,-1) \}$ (this is a basis for $\mathbb{R}^3$). 

\begin{enumerate}[(a)]
   \item Find the dual basis $\beta^*$. [Give formulas for each dual vector. For example: $f(x,y,z)=2x+5y-z$. {\it Unnecessary note:} $f$ is {\bf not} one of the elements of $\beta^*$.]
   
\vspace{0.1in}   
   
   \item Explain why $f(x,y,z)=x^2$ {\it is not} in $(\mathbb{R}^3)^*$.
   
\vspace{0.1in}   

   \item Explain why $f(x,y,z)=x$ {\it is} in $(\mathbb{R}^3)^*$ and compute $[f]_{\beta^*}$ (this is $f$'s $\beta^*$-coordinate vector).
   
\vspace{0.1in}   

   \item Find the change of basis matrices from $\mathrm{std}^* = \{ \B i^*, \B j^*, \B k^* \}$ (the standard dual basis) to $\beta^*$ (i.e. $[I]_{\mathrm{std}^*}^{\beta^*}$). Also, find  the change of basis matrices from to $\beta^*$ to $\mathrm{std}^*$ (i.e. $[I]_{\beta^*}^{\mathrm{std}^*}$).
   
\vspace{0.1in}   

   \item What is the relationship between $[I]_{\mathrm{std}^*}^{\beta^*}$ and $[I]_{\mathrm{std}}^{\beta}$?
   
    Given two arbitrary bases $\alpha$ and $\gamma$ for an arbitrary finite dimensional vector space (over some field $\mathbb{F}$), conjecture a relationship between $[I]_\alpha^\gamma$ and $[I]_{\alpha^*}^{\gamma^*}$.
    
\vspace{0.1in}   

    \item {}[{\bf Grad Students}] Prove your conjecture from part (e).
   
\end{enumerate}

\vspace{0.2in}   


\item[\#2 Look at me, Mom. I'm a module too!] Let $V$ and $W$ be $\mathfrak{g}$-modules. 

\begin{enumerate}[(a)]

\item The space of linear maps from $V$ to $W$ can be turned into a $\mathfrak{g}$-module as follows:

          Let $T \in \mathrm{Hom}(V,W) = \{ S:V \to W \;|\; S \mbox{ is linear} \}$ and $g \in \mathfrak{g}$. Then for all $\B v \in V$, define $$(x \bdot T)(\B v) = x \bdot T(\B v)-T(x \bdot \B v).$$ 
          
          First, show $x \bdot T \in \mathrm{Hom}(V,W)$. Then show that this turns $\mathrm{Hom}(V,W)$ into a $\mathfrak{g}$-module (don't forget to check bilinearity).
          
\vspace{0.1in}   

\item We can turn $\mathbb{F}$ into a $\mathfrak{g}$-module via the trivial action: $x \bdot s = 0$ for all $s \in \mathbb{F}$. Explain how this then allows us to define a dual module $V^*$ for any $\mathfrak{g}$-module $V$ [{\it Hint:} Use part (a).] What is the action of $\mathfrak{g}$ on $V^*$?

\end{enumerate}   

\vspace{0.2in}   

\item[\#3 Like Mt. Fuji Reflected in a Lake:] Recall that $V(m)$ is the $\mathfrak{sl}_2(\mathbb{C})$ module with highest weight $m$ (where $m$ is a non-negative integer).

\begin{enumerate}[(a)]

\item Prove that $V(1)^* \cong V(1)$.

\vspace{0.1in}   

\item {}[{\bf Grad Students}] Prove $V(m)^* \cong V(m)$.

\vspace{0.1in}   

\item {}[{\bf Everyone}] Assuming the grad student problem (part (c)) and the fact that as $\mathfrak{g}$-modules: $\displaystyle \left(\bigoplus_{i=1}^\ell W_i \right)^{\hspace*{-0.05in} *} \cong \bigoplus_{i=1}^\ell W_i^*$ (for any finite dimensional $\mathfrak{g}$-modules $W_i$), prove that $V^* \cong V$ for any finite dimensional $\mathfrak{sl}_2(\mathbb{C})$-module $V$.  {\bf Slogan:} $\mathfrak{sl}_2$-modules are self dual!

\end{enumerate}

\end{description}

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