\documentclass[12pt]{article}

% There are loads of latex packages. Here are a few...

\usepackage{amsmath,amsthm,amssymb,amsfonts,amscd,gensymb,graphicx,mathtools,enumitem,mathdots,euscript,tikz,pgfplots}

% Put your user define shorthands here...

\newcommand{\C}{\mathbb{R}}  % Short for the real numbers
\renewcommand{\k}{\Bbbk}  % Short for the real numbers
\newcommand{\eps}{\varepsilon}

\begin{document}

\bigskip

\begin{center}
Introduction to Lie Theory\\
Homework \#1\\
\end{center}

\bigskip
\begin{enumerate}
\item
Let $\mathbb{G}_a$ and $\mathbb{G}_m$ be the {\em additive group}
and the {\em multiplicative group}, respectively, that is, the groups
$(\k,+)$ and 
$(\k^\times, \cdot)$ with the obvious affine variety structures.
\begin{itemize}
\item[(a)]
Show that the coodinate algebra $\k[\mathbb{G}_a]$ is isomorphic to
the algebra $\k[T]$ viewed
as a Hopf algebra with
comultiplication 
$T \mapsto T \otimes 1 + 1 \otimes T$, 
counit $T \mapsto 0$
and antipode $T \mapsto -T$.
\item[(b)]
Give a similarly explicit description of
  $\k[\mathbb{G}_m]$.
\item[(c)] Show that $\mathbb{G}_a$ is isomorphic to the closed subgroup
  $U$ of $GL_2(\k)$ consisting of 
the upper unitriangular matrices.
\item[(d)] Show that the closed subgroup $T$ of
$GL_n(\k)$ consisting of diagonal invertible matrices is isomorphic to
$\mathbb{G}_m \times \cdots \times \mathbb{G}_m$ ($n$ times).
\end{itemize}
\item Let $X_i\:(i \in I)$ be a family of irreducible affine varieties
and $\phi_i:X_i \rightarrow G$ be morphisms to an algebraic group $G$.
Assume that $e \in \phi_i(X_i)$ for each $i$\footnote{This hypothesis
  is necessary: see Exercise 2.2.9(2) of ``Linear Algebraic Groups''
  by T. A. Springer.}.
Let $H$
be the subgroup
of $G$ generated by the images of all $\phi_i$.
Show that $H$ is a closed, connected subgroup of $G$.

\noindent
({\em Hints.} 
Let $Y_i = \phi_i(X_i)$.
Choose $i_1, \dots,i_n \in I$ and $\eps_1,\dots,\eps_n \in \{\pm 1\}$ so that
the irreducible closed subset $\overline{Y_{i_1}^{\eps_1}\cdots Y_{i_n}^{\eps_n}}$
of $G$ is of maximal dimension amongst all such subsets. Then show that
$H = \overline{Y_{i_1}^{\eps_1} \cdots Y_{i_n}^{\eps_n}}$.
You will also need to use the fact 
that an algebraic group is equal to the set-wise product $UV$
of any dense open subsets $U$ and $V$,
and that the image of a morphism contains a
non-empty open subset of its closure.)

\item
Let $V$ be a symplectic vector space, that is, it is
a finite-dimensional vector space equipped with a non-degenerate
skew-symmetric bilinear form $(\cdot,\cdot)$.
The {\em symplectic group} is 
the isometry group: 
$$
Sp(V) := \{g \in GL(V)\:|\:(gv,gw) = (v,w)\text{ for all
}v,w \in V\}.
$$
\begin{itemize}
\item[(a)] Explain why $Sp(V)$ is a closed subgroup of $GL(V)$. Hence,
  it is an algebraic group.
\item[(b)] Let $\det:GL(V) \rightarrow \mathbb{G}_m$ be the
morphism of algebraic groups defined by  determinant.
Show that $Sp(V) \subseteq SL(V) := \ker \det$.
({\em Hint:} Pfaffians.)
\item[(c)]
For $0 \neq a \in V$ and $t \in \k$,
let $u_a(t):V \rightarrow V$ be the {\em transvection}
$v \mapsto v +t(v,a)a$. Check that $u_a(t) \in Sp(V)$
and that $u_a(t) u_a(t') = u_a(t+t')$.
\item[(d)]
Let $\phi_a:\mathbb{G}_a \rightarrow Sp(V), t \mapsto u_a(t)$.
Show that this is a morphism of algebraic groups.
Use question 2 and a fact from group theory, namely, that $Sp(V)$ is generated by all
transvections, to prove that $Sp(V)$ is connected.
\end{itemize}
(For an orthogonal vector space $V$ and $\operatorname{char} \k\neq 2$, the isometry group is the {\em
  orthogonal group} $O(V)$; in characteristic 2 the definition of
$O(V)$ is slightly more complicated.
Unlike for $Sp(V)$, elements of $O(V)$ have can determinant either
$+1$ or $-1$.
You get the {\em special orthogonal group} by taking just the ones of
determinant $+1$. In fact, $SO(V)$ is the connected component of the
identity in $O(V)$. This can be proved similarly to the
above\footnote{See also Exercise 2.2.2(2) of Springer for another
  approach in characteristic $\neq 2$.}.)
\item
Let $G$ be a connected algebraic group. Use question 2 to prove that the derived
subgroup $G'$, that is, the subgroup generated by all commutators
$ghg^{-1} h^{-1}$ is a closed, connected subgroup of $G$.
Hence, all of the subgroups in the derived series of $G$ are closed
and connected.

Give an explicit description of the derived series of the closed subgroup $B$ of $GL_n(\k)$
consisting of all upper triangular invertible matrices.
\item 
A {\em representation} of an algebraic group $G$ is a
finite-dimensional vector space $V$ plus a 
morphism of algebraic groups
$\rho:G \rightarrow GL(V)$.
\begin{itemize}
\item[(a)]
Suppose that $V$ is a representation of $G$ with basis 
$v_1,\dots,v_n$.
By considering the comorphism $\rho^*$, or otherwise, show that the
functions $f_{i,j}:G \rightarrow \k$ defined from
$g v_j = \sum_{i=1}^n f_{i,j}(g) v_i$ belong to $\k[G]$.
Moreover, the comultiplication of $\k[G]$ sends
$f_{i,j} \mapsto 
\sum_{k=1}^n f_{i,k} \otimes f_{k,j}$, and its
counit sends $f_{i,j} \mapsto \delta_{i,j}$.
\item[(b)]
Suppose you are given
$f_{i,j} \in \k[G]$ for $1 \leq i,j \leq n$ such that the
comultiplication and counit of $\k[G]$ have the properties 
formulated in (a). Show that the function $\rho:G \rightarrow GL_n(\k),
g \mapsto \left(f_{i,j}(g)\right)_{1 \leq i,j \leq n}$
is a morphism of algebraic groups.
\end{itemize}
(This question hints at the notion of a {\em $\k[G]$-comodule} which is
the appropriate algebraic gadget that corresponds to representations
of $G$.)
\end{enumerate}

\end{document}








