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\noindent
\parbox{2.25in}{\bf Math 4010/5530}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Mar. $27^\mathrm{th}$, 2023}

\vspace{0.2in}

Let $\mathfrak{g}$ be a Lie algebra over a field $\mathbb{F}$ and let $V$ be a $\mathfrak{g}$-module.

\begin{description}
\item[\#1 Submodule Basics] A warm up.
   \begin{enumerate}[(a)]
   \item Let $W_1$ and $W_2$ be submodules of $V$. Show that $W_1 \cap W_2$ and $W_1+W_2$ are submodules.\\ \force \hfill Is $W_1 \cup W_2$ a submodule? Discuss.
   \item Let $\varphi:V \to W$ be a $\mathfrak{g}$-module homomorphism. Also, let $U$ be a submodule of $W$. Show that the inverse image of $U$, $\varphi^{-1}(U) = \{ \B v \in V \;|\; \varphi(\B v) \in U \}$, is a submodule of $V$ which contains the kernel.
   \end{enumerate}

\item[\#2 \sout{Meth} Math Lab] Let $\mathfrak{h} = \mathrm{span} \{ \B x, \B y, \B z \}$ be the Heisenberg Lie algebra (over $\mathbb{C}$) whose bracket structure is defined by: $[\B x, \B y]= \B z$ and $\B z$ is central (i.e., $[\B x, \B z]=[\B y, \B z]=0$). Let $\mathbb{C}[t]$ be the algebra of polynomials with complex coefficients (and indeterminate $t$). For $f(t) \in \mathbb{C}[t]$ define $\B x \bdot f(t)=f'(t)$ (differentiation), $\B y \bdot f(t) = tf(t)$ (multiplication by $t$), $\B z \bdot f(t) = f(t)$ (multiplication by $1$), and extend linearly. Assuming bilinearity of this action, show $\mathbb{C}[t]$ is indeed a $\mathfrak{h}$-module. Is this an irreducible module?

\item[\#3 Irreducible] Let $V \not= \{0\}$. Show $V$ is irreducible if and only if $V$ is generated by any of its non-zero elements (i.e., for any $0 \not= \B v \in V$ we have that the submodule generated by $\B v$ is all of $V$). 

{\it Note:} The submodule generated by $\B v$ is the span of all elements of the form:
\[ \B x_1 \bdot (\B x_2 \bdot ( \cdots (\B x_\ell \bdot \B v) \cdots ))  \qquad \qquad \mbox{where} \quad \B x_1, \dots, \B x_\ell \in \mathfrak{g} \quad \mbox{and} \quad \ell \geq 0. \]
By the way, $\ell=0$ gives us the empty product (no elements acting on $\B v$) which is just $\B v$ itself.

\end{description}

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