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\begin{center}
Introduction to Lie Theory\\
Homework \#4\\
\end{center}


\begin{enumerate}
\item
For a field $\k$ of characteristic zero, the Lie
algebra
$\mathfrak{so}_3(\k)$ may be defined as the Lie subalgebra of
$\mathfrak{gl}_3(\k)$ consisting of all skew-symmetric matrices,
while $\mathfrak{sl}_2(\k)$ is of course the $2 \times 2$ matrices of
trace zero in $\mathfrak{gl}_2(\k)$.
\begin{enumerate}
\item
Show that $\mathfrak{so}_3(\R)$
is isomorphic to $\R^3$ viewed as a Lie algebra with Lie
bracket being the usual cross product of vectors.
\item
Show that $\mathfrak{so}_3(\R) 
\not\cong \mathfrak{sl}_2(\R)$.
\item Show that
$\mathfrak{so}_3(\C) \cong \mathfrak{sl}_2(\C)$. 
\end{enumerate}
{\em Hint.} For (c) see Q3 below.
\item
In L4-2, I explained how the universal enveloping algebra
$U(\mathfrak{g})$ is a Hopf algebra with comultiplication
$\Delta:U(\mathfrak{g})\rightarrow U(\mathfrak{g})\otimes
U(\mathfrak{g})$, counit
$\eps:U(\mathfrak{g}) \rightarrow \k$ and
antipode $S:U(\mathfrak{g}) \rightarrow U(\mathfrak{g})$
defined so that $\Delta(x) = x \otimes 1 + 1 \otimes x,
\eps(x) = 0$ and $S(x) =-x$ for all $x \in \mathfrak{g}$.
Fill in the details!
\item The comultiplication $\Delta$ on $U(\mathfrak{g})$ is important
  because it means you can define the {\em tensor product} of two
  $\mathfrak{g}$-modules: if $V$ and $W$ are $\mathfrak{g}$-modules
  then the tensor product $V \otimes W$ (over the ground field $\k$)
  is naturally a $U(\mathfrak{g}) \otimes U(\mathfrak{g})$-module with
  $(x_1\otimes x_2) (v \otimes w) = x_1 v \otimes x_2 w$; hence, using the algebra homorphism $\Delta$, it becomes a
$\mathfrak{g}$-module.
\begin{enumerate}
\item
Show that the exterior and symmetric powers $\bigwedge^n V$ and $S^n
V$ of a $\mathfrak{g}$-module $V$ are $\mathfrak{g}$-module 
quotients of $\bigotimes^n V$.
\item
Now suppose that $\mathfrak{g} = \mathfrak{sl}_n(\C)$ for $n \geq 2$ and
let $V$ be the natural $n$-dimensional representation of column vectors.
Show that $\bigwedge^i V\:(1 \leq i \leq n)$ and $S^j V \:(j \geq 0)$
are both irreducible 
$\mathfrak{g}$-modules.
Is $S^2 V$ irreducible over the subalgebra $\mathfrak{so}_n(\C)$?
\item Finally suppose that $n=2$. Show that $V$ and $S^2 V$ possess 
non-degenerate
bilinear forms which are invariant under the action of $\mathfrak{g}$,
i.e., $(xv,w)+(v,xw) = 0$ for all $x
\in \mathfrak{g}$ and all vectors $v,w$.
Deduce that $\mathfrak{sl}_2(\C) \cong \mathfrak{sp}_2(\C)
\cong \mathfrak{so}_3(\C)$.
\end{enumerate}
\item
Let $V$ and $W$ be $\mathfrak{g}$-modules.
\begin{enumerate}
\item
Verify that $\Hom_{\k}(V,W)$ can be made into a $\mathfrak{g}$-module
by setting $(xf)(v) := x f(v) - f(xv)$ for $x \in \mathfrak{g},
f \in \Hom_{\k}(V,W)$ and $v \in V$.
\item
If $V$ is finite-dimensional then $\Hom_{\k}(V,W) \cong V^* \otimes
W$. What does the $\mathfrak{g}$-module structure from (a) correspond
to under this natural isomorphism?
\item
For any $\mathfrak{g}$-module $V$, 
let
$V^{\mathfrak{g}}:=\{v \in V\:|\:x v = 0\text{ for all
}x \in \mathfrak{g}\}$ 
denote the submodule of {\em $\mathfrak{g}$-invariants}.
For $V, W$ as in (a), check that $$
\Hom_{\C}(V,W)^{\mathfrak{g}} = \Hom_{\mathfrak{g}}(V,W).
$$
\item Suppose that $G$ is a connected algebraic group for $\k$ of
  characteristic zero.
If $V$ is a representation of $G$, the
  submodule of {\em $G$-invariants} is
$V^G := \{v \in V\:|\:gv = v\text{ for all }g \in G\}$.
Viewing $V$ as a $\mathfrak{g}$-module via the differential, show that $V^G = V^{\mathfrak{g}}$.
\item
Finally let $V$ and $W$ are representations of $G$.
Explain how to make $\Hom_{\C}(V,W)$ into a
representation of $G$ so that the $\mathfrak{g}$-module structure
from (a) is the naturally induced one.
Deduce that
$\Hom_{\mathfrak{g}}(V,W) = \Hom_G(V,W)$.
\end{enumerate}
(Part (e) proves that the category of representations of $G$ is a full
subcategory of the category of finite-dimensional representations of
$\mathfrak{g}$.)
\item
Let $V$ be a vector space and $T(V)$ be its tensor algebra.
Viewing the associative algebra $T(V)$ as a Lie algebra via the
commutator, let $F(V)$ be the Lie subalgebra of $T(V)$ generated by
$V$.
\begin{enumerate}
\item
Prove that $F(V)$ together with the evident linear map $V \hookrightarrow
F(V)$ satisfies the appropriate universal property making it 
the {\em free Lie algebra on the vector space $V$}.
\item
When $V$ is one-dimensional, $F(V) = V$. What can you say about $F(V)$
in the
case that $V$ is two-dimensional with basis $x,y$?
\item
Now that the free Lie algebra is defined, you can make sense of Lie
algebras defined by generators and relations.
Let $\mathfrak{g}$ be the Lie algebra with two generators $x, y$
subject only to the relations
$$
[x,[x,y]] = [y,[y,x]] = 0.
$$
Prove that $\mathfrak{g}$ is three-dimensional by identifying it with
the Lie algebra of all strictly upper triangular matrices in $\mathfrak{gl}_3(\k)$.
\end{enumerate}
\end{enumerate}
\end{document}