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\begin{center}
Introduction to Lie Theory\\
Homework \#2\\
\end{center}

\bigskip
\begin{enumerate}
\item
Calculate:
\begin{enumerate}
\item The differential 
$dm_{(e,e)}:\mathfrak{g}\oplus \mathfrak{g} \rightarrow \mathfrak{g}$
of the multiplication map $m:G \times G \rightarrow G$ for any
algebraic group $G$.
\item The differential
$d i_e: \mathfrak{g} \rightarrow \mathfrak{g}$
of the group inversion map.
\item The differential
$d(\det)_e:\mathfrak{gl}_n(\k) \rightarrow \k$
of the determinant morphism $\det:GL_n(\k) \rightarrow \mathbb{G}_m$.
\end{enumerate}
\item
Assume in this question that $p \neq 2$.
\begin{enumerate}
\item
In class, we identified the Lie algebra
of the group $Sp_{2n}(\k)$ with a subalgebra
$\mathfrak{sp}_{2n}(\k)$ of $\mathfrak{gl}_{2n}(\k)$, and used this to
deduce that $\dim Sp_{2n}(\k) = 2n^2 + n$.
Do the same thing with the group $SO_n(\k)$.
\item
Now let $S$ be the affine 
variety of $n \times n$ symmetric matrices.
Note that
$G := SL_n(\k)$ acts on $S$ via the map
$G \times S \rightarrow S, (g,x)
\mapsto g^T x g$ (a morphism of varieties).
Use this plus the following general fact to 
calculate
the dimension of $SO_n(\k)$ in a different way: 
the dimension of an orbit of a
connected algebraic group is equal to the dimension of the group
minus the dimension of the identity component of the stabilizer
of a point in the orbit.
\end{enumerate}
\item 
Recall the notion of a representation of an
algebraic group $G$ from HW1, Q5.
A {\em right $\k[G]$-comodule}
is a vector space $V$ together with a {\em structure map}
$\eta :V \rightarrow V \otimes \k[G]$, that is, a linear map
such that $(\id \otimes \Delta) \circ \eta  = (\eta \otimes \id) \circ
\eta$ and $(\id \bar \otimes \eps) \circ \eta = \id$,
where $\Delta$ and $\eps$ denote the comultiplication and counit on
the coordinate algebra $\k[G]$, respectively.
\begin{enumerate}
\item
For a finite-dimensional 
right $\k[G]$-comodule $V$, show that there is a well-defined
representation $\rho:G \rightarrow GL(V)$
such that 
$$
\rho(g)(v) = \sum_{i=1}^n f_i(g) v_i
$$ 
for $g \in G$ and 
$v \in V$ with $\eta(v) = \sum_{i=1}^n v_i \otimes f_i$.
\item
Let $\operatorname{Rep}(G)$ and
$\operatorname{comod}_{\operatorname{fd}}\operatorname{-}\k[G]$ denote
the categories of 
representations of $G$ and of finite-dimensional right
$\k[G]$-comodules, respectively, with the obvious notion of morphisms
in each case.
Show that these categories are {\em isomorphic}.
\end{enumerate}
\item
Let $G := \mathbb{G}_m$ with coordinate algebra $\k[G] =
\k[T,T^{-1}]$.
Let $\rho:G \rightarrow V$ be a representation.
For $n \in \Z$ let 
$$
V_n:= \{v \in V\:|\:\rho(t)(v) = t^n v\:\text{for all }t \in G\}.
$$
Prove that $V = \bigoplus_{n \in \Z} V_n$.
Hence, the Abelian 
category $\operatorname{Rep}(G)$ is semisimple, i.e., all 
representations decompose into a direct sum of irreducible
representations.

({\em Hint.} One way to proceed is to let
$\eta:V \rightarrow \k[T,T^{-1}]$ be its comodule structure map. Assuming this 
sends $v \in V$ to $\sum_{n \in \Z} v_n \otimes T^n$,
show that $v = \sum_{n \in \Z} v_n$ and $v_n \in V_n$.)
\item
An algebraic group $G$ is said to be {\em linearly reductive} if
the category $\operatorname{Rep}(G)$ is semisimple.
For example, by Maschke's theorem, a finite group $G$
(viewed as an algebraic
group over our usual field $\k$ of characteristic $p$)
is linearly reductive if and only if either $p=0$ or
$p \nmid |G|$.
\begin{enumerate}
\item
Show that $G \times H$ is linearly reductive if and only if both $G$
and $H$ are linearly reductive. Deduce from this and the previous
question that any {\em torus} $T \cong \mathbb{G}_m \times \cdots \times
\mathbb{G}_m$ ($n$ times) is linearly reductive.
\item
Show that $G$ is linearly reductive if and only if both $G^\circ$ and
$G / G^\circ$ are linearly reductive.
\end{enumerate}
(It is known that a connected algebraic group is linearly reductive if
and only if $p > 0$ and $G$ is a torus, or $p = 0$
and $G$ is {reductive}, i.e., $G = T G'$ for a torus 
$T \leq Z(G)$ and a semisimple group $G'\leq G$.)
\end{enumerate}
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