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\parbox{2.25in}{\bf Math 4010/5530}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Sept. $13^\mathrm{th}$, 2017}

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Unless otherwise specified, $V$ is a vector space over a field $\mathbb{F}$.

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\begin{description}
\item[\#1 Coordinate Matrices] Recall that when $L$ is a Lie algebra and $x,y \in L$, we define $\mathrm{ad}_x(y)=[x,y]$ (i.e. the adjoint map $\mathrm{ad}_x$ is left multiplication by $x$ in $L$).

\begin{enumerate}[(a)]
\item Let $x \in L$. Show that $\mathrm{ad}_x \in \mathfrak{gl}(L)$ (i.e. show that $\mathrm{ad}_x$ is a linear operator on $L$). 
\item Let $\alpha = \{ E_{11}, E_{12}, E_{21}, E_{22} \}$ (in that order) be the standard basis for $\mathfrak{gl}_2(\mathbb{F})$. Find $[\mathrm{ad}_{E_{ij}}]_\alpha$ for all $i,j$ (i.e. compute the coordinate matrices for the adjoint operators relative to the standard basis).
\item Let $\beta = \{ E_{12}, E_{13}, E_{23} \}$ (in that order) be the standard basis for $\mathfrak{st}_3(\mathbb{F})$ (strictly upper triangular $3 \times 3$ matrices). Find $[\mathrm{ad}_{E_{ij}}]_\beta$ for $(i,j)=(1,2), (1,3), (2,3)$.
\end{enumerate}

\item[\#2 Matrix Fun] (Misra 1.27 part iv) Let $L = \mathrm{span} \{ E_{12}, E_{31}, E_{32} \} \subseteq \mathfrak{gl}_3(\mathbb{F})$.

\begin{enumerate}[(a)]
\item Show that $L$ is a subalgebra of $\mathfrak{gl}_3(\mathbb{F})$.
\item Let $A \in L$. Show that $A$ is nilpotent: $A^m=0$ for $m \gg 0$ ($m \gg 0$ means ``for $m$ sufficiently large'').
\end{enumerate}

\item[\#3 Differential Operators] Let $V=\mathbb{F}[x]$ (i.e. polynomials with coefficients in $\mathbb{F}$). Note that $V$ is a vector space (over $\mathbb{F}$). Define the following
linear operators on $V$ (i.e. endomorphisms on $V$):
$$I[f(x)] = f(x) \qquad L_x[f(x)] = xf(x) \qquad D_x[f(x)] = \dfrac{d}{dx}\bigg[f(x)\bigg]$$

\begin{enumerate}[(a)]
\item Let $L = \mathrm{span} \{ I,L_x,D_x \}$. Show that $L$ is a subalgebra of $\mathfrak{gl}(V)$.
\item If we replace $L_x$ with $\displaystyle K_x[f(x)] = \int_0^x f(t)\,dt$ (i.e. integrate and set constant equal to zero), would this still be a subalgebra? Explain.
\end{enumerate}

\item[\#4 Transpose] ({\it for graduate students}) Let $T:\mathfrak{gl}_n(\mathbb{C}) \to \mathfrak{gl}_n(\mathbb{C})$ be defined by $T(A)=A^T$ for all $A \in \mathfrak{gl}_n(\mathbb{C})$.

\begin{enumerate}[(a)]
\item Show that the only eigenvalues of $T$ are $\lambda = \pm 1$. 
\item Prove that the $(-1)$-eigenspace of $T$ forms a subalgebra of $\mathfrak{gl}_n(\mathbb{C})$. What is its dimension? 
\item Is the 1-eigenspace a subalgebra? Explain.
\end{enumerate}


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