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\parbox{2.25in}{\bf Math 4010/5530}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Feb. $3^\mathrm{rd}$, 2023}

\vspace{0.2in}

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

\vspace{0.1in}

\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 $\mathfrak{gl}(V)$.

\vspace{0.05in}

{\color{blue}Oops! Did this in class.} {\bf Proof:} We see that $L$ is a subspace of $\mathfrak{gl}(V)$ since spans are subspaces. Thus we only need to check closure under the (commutator) bracket. As mentioned, because of bilinearity and alternation, we need only check on ordered pairs of basis elements. So it suffices to check:
$[I,L_x] = I \circ L_x - L_x \circ I = L_x-L-x = 0 \in L$, $[I,D_x]=I \circ D_x-D_x \circ I = D_x-D_x = 0 \in L$, and finally $[L_x,D_x]=L_x \circ D_x - D_x \circ L_x = -I \in L$ where the last equality is justified by considering: For all $f(x) \in V=\mathbb{F}[x]$, we have $(L_x \circ D_x)(f(x))-(D_x \circ L_x)(f(x)) = L_x(D_x(f(x)))-D_x(L_x(f(x)))=xf'(x)-D_x(xf(x)) = xf'(x)-f(x)-xf'(x) =-f(x) = -I(f(x))$.

\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),\\ \force \hfill would this still be a subalgebra? Explain.

{\it Hint:} Compute $M=[D_x,K_x]$. Consider what this does to $1$ and $x$ versus what a general element $N=aI+bK_x+cD_x$ ($a,b,c \in \mathbb{F}$) does. Can you make $M$ and $N$ match?

\end{enumerate}


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

{\it Note:} The $\lambda$-eigenspace of a linear operator $T:V \to V$ is $E_\lambda = \mathrm{ker}(T-\lambda I) = \{ \B x \in V \;|\; T(\B x)=\lambda \B x\}$ (i.e., all eigenvectors with eigenvalue $\lambda$ along with the zero vector). For the operator above, an $n \times n$ matrix $A$ is in the $\lambda$-eigenspace if and only if $A^T = T(A) = \lambda A$.

\begin{enumerate}[(a)]
\item ({\it for graduate students}) Show that the only eigenvalues of $T$ are $\lambda = \pm 1$. \quad {\it Hint:} Consider $T^2$.\\ \force \hfill What do we call the matrices in $E_1$? What about $E_{-1}$?
\item Prove that the $(-1)$-eigenspace of $T$ forms a subalgebra of $\mathfrak{gl}_n(\mathbb{C})$.\\ \force \hfill {}[{\it for graduate students}: What is its dimension?]
\item ({\it for graduate students}) Is the 1-eigenspace a subalgebra? Explain.
\end{enumerate}


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