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\parbox{2.25in}{\bf Math 4010/5530}
\hfill {\Large \bf Homework \#3} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Sept. $25^\mathrm{th}$, 2017}

\vspace{0.2in}

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

\vspace{0.1in}

\begin{description}
\item[\#1 More Eigenfun] Let $A, B, C, D, P \in \mathfrak{gl}_n(\mathbb{F})$. Suppose that $D$ is a diagonal matrix with with diagonal entries $\lambda_1,\dots,\lambda_n$ and that $A$ is diagonalizable with eigenvalues $\lambda_1,\dots,\lambda_n$. In addition, suppose that $P$ is invertible.

\begin{enumerate}[(a)]
\item Show that $E_{ij}$ is an eigenvector for $\mathrm{ad}_D$ for each $i,j=1,\dots,n$. What are $\mathrm{ad}_D$'s eigenvalues? Why is $\mathrm{ad}_D$ diagonalizable?
\item Let $A=PDP^{-1}$ and $B=PCP^{-1}$. Show that $[A,B] = P[D,C]P^{-1}$.
\item {\bf (Grad Students)} Show that $\mathrm{ad}_A$ is diagonalizable.
\end{enumerate}

\item[\#2 Direct Sums] Recall the definition of a vector space direct sum: $L=A \oplus B$ for some subspaces $A$ and $B$ if $L=A+B$ (i.e. for every $\B  v \in L$ there exists some $\B a \in A$ and $\B b \in B$ such that $\B v = \B a + \B b$) and $A \cap B = \{ \B 0 \}$. 

If in addition $L$ is a Lie algebra, $A$ and $B$ are subalgebras, and $[A,B]=\{ \B 0\}$, then we say $L=A \oplus B$ as Lie algebras (i.e. $L$ is a direct sum of its subalgebras $A$ and $B$).

Let $\mathbb{F}$ be field of characteristic $0$. Show that $\mathrm{gl}_n(\mathbb{F}) = \mathfrak{sl}_n(\mathbb{F}) \oplus \mathbb{F}I_n$ as Lie algebras (note: $\mathbb{F}I_n$ is the set of scalar multiples of the identity matrix).

\item[\#3 An Ideal Problem] Let $X \subseteq L$ ($X$ is just a subset). Recall that $C_L(X) = \{ \B v \in L \;|\; [\B v, \B x]=\B 0 \mathrm{\ for\  all\  } \B x \in X \}$ is the centralizer of $X$ in $L$.

\begin{enumerate}[(a)]
\item Show that $C_L(X)$ is a subalgebra of $L$ (don't forget to show it is a subspace as well).
\item Assume $X \triangleleft L$. Show that $C_L(X) \triangleleft L$.
\end{enumerate}

\item[\#4 Automorphism via Conjugation] For some fixed $J \in \mathfrak{gl}_n(\mathbb{F})$, recall that $\mathfrak{gl}^J_n(\mathbb{F}) = \{ X \in \mathfrak{gl}_n(\mathbb{F}) \;|\; JX+X^TJ=0 \}$. Suppose that $A$ is an invertible $n \times n$ matrix such that $A^TJA=J$. Show that $\varphi:\mathfrak{gl}^J(\mathbb{F}) \to \mathfrak{gl}^J(\mathbb{F})$ defined by $\varphi(X)=A^{-1}XA$ is an automorphism of $\mathfrak{gl}^J(\mathbb{F})$. 

Note: The first thing you should verify is that $\varphi$ actually maps from $\mathfrak{gl}^J(\mathbb{F})$ to itself.

\item[\#5 Derivations] Recall $\mathrm{Der}(\mathbb{F}[x]) = \{ \partial:\mathbb{F}[x] \to \mathbb{F}[x] \;|\; \partial \mbox{ is linear and } \partial(fg) = \partial(f)\,g + f\,\partial(g) \mbox{ for all } f,g \in \mathbb{F}[x] \}$ where $\mathbb{F}[x]$ is the algebra of polynomials with coefficients in $\mathbb{F}[x]$.

\begin{enumerate}[(a)]
\item Let $\partial \in \mathrm{Der}(\mathbb{F}[x])$. Show that $\partial(1)=0$ and thus any derivation applied to a constant yields 0.
\item Let $p(x) \in \mathbb{F}[x]$. Define $\partial(f(x)) = p(x)f'(x)$ (i.e. $\partial = p(x)\frac{d}{dx}$). Show that $\partial$ is a derivation on $\mathbb{F}[x]$.
\item Compute $[p(x)\frac{d}{dx}, q(x)\frac{d}{dx}]$ where $p(x),q(x) \in \mathbb{F}[x]$.
\item ({\bf Grad Students}) Show that $\mathrm{Der}(\mathbb{F}[x]) = \{ p(x)\frac{d}{dx} \;|\; p(x) \in \mathbb{F}[x] \}$ (i.e. all derivations on $\mathbb{F}[x]$ are like those defined in part (b)).
\end{enumerate}

\end{description}

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