%% boi-6.tex (short version)
%% Chapter 6 of the Book of Involutions

\chapter{Algebraic Groups} 
\label{6.chap.alg.grp} 

It turns out that most of the groups which have occurred thus far in
the book are groups of points of certain algebraic group schemes.
Moreover, many constructions described previously are related to
algebraic groups.  For instance, the Clifford algebra and the
discriminant algebra are nothing but Tits algebras for certain
semisimple algebraic groups; the equivalences of categories considered
in Chapter~\ref{4.chap.alg.four}, for example of central simple
algebras of degree~$6$ with a quadratic pair and central simple
algebras of degree~$4$ with a unitary involution over an \'etale
quadratic extension (see \S \ref{ADthree.subsec}), reflect the fact
that certain semisimple groups have the same Dynkin diagram
($D_3\simeq A_3$ in this example).

The aim of this chapter is to give the classification of semisimple
algebraic groups of classical type without any field characteristic
assumption, and also to study the Tits algebras of semisimple groups.

In the study of linear algebraic groups (more generally, affine group
schemes) we use a functorial approach equivalent to the study of Hopf
algebras. The advantage of such an approach is that nilpotents in
algebras of functions are allowed (and they really do appear when
considering centers of simply connected groups over fields of positive
characteristic); moreover many constructions like kernels,
intersections of subgroups, are very natural. A basic reference for
this approach is Waterhouse~\cite{Waterhouse}. The classical view of
an algebraic group as a variety with a regular group structure is
equivalent to what we call a smooth algebraic group scheme.

The classical theory (mostly over an algebraically closed field) can
be found in Borel~\cite{Borel}, Humphreys~\cite{Humphreys}, or
Springer~\cite{Springergps}. We also refer to Springer's survey
article~\cite{Springersurvey}.  (The new (1998) edition
of~\cite{Springergps} will contain the theory of algebraic groups over
non algebraically closed fields.) We use some results in commutative
algebra which can be found in Bourbaki \cite{Boucom1-4},
\cite{Boucom.5-7},~\cite{Boucom8-9}, and in the book of
Matsumura~\cite{Matsumura}.

The first three sections of the chapter are devoted to the general
theory of group schemes. In \S\ref{automorphisms} we define the
families of algebraic groups related to an algebra with involution, a
quadratic form, and an algebra with a quadratic pair. After a short
interlude (root systems, in \S\ref{roots}) we come to the
classification of split semisimple groups over an arbitrary field. In
fact, this classification does not depend on the ground field $F$, and
is essentially equivalent to the classification over the algebraic
closure $F_{\alg}$\index{B}{Falg@{$F_{\alg}$ (algebraic closure)}}
(see Tits~\cite{Tits3}, Borel-Tits~\cite{BorelTits}).

The central section of this chapter, \S\ref{classificationss},
gives the classification of adjoint semisimple groups over arbitrary
fields.  It is based on the observation of Weil~\cite{We} that (in
characteristic different from~$2$) a classical adjoint semisimple
group is the connected component of the automorphism group of some
algebra with involution. In arbitrary characteristic the notion of
orthogonal involution has to be replaced by the notion of a quadratic
pair which has its origin in the fundamental paper~\cite{Tits} of
Tits.  Groups of type $G_2$ and~$F_4$ which are related to Cayley
algebras (Chapter~\ref{8.chap.comp.tria}) and exceptional Jordan
algebras (Chapter~\ref{9.chap.cub.jord}), are also briefly discussed.

In the last section we define and study Tits algebras of semisimple
groups. It turns out that for the classical groups the nontrivial Tits
algebras are the $\lambda$-powers of a central simple algebra, the
discriminant algebra of a simple algebra with a unitary involution,
and the Clifford algebra of a central simple algebra with an
orthogonal pair---exactly those algebras which have been studied in
the book (and nothing more!).

\section{Hopf Algebras and Group Schemes}
\label{hopfgroup}
This section is mainly expository. We refer to
Waterhouse~\cite{Waterhouse} for proofs and more details.

\subsubsection{Hopf algebras}
Let~$F$ be a field and let~$A$ be a commutative (unital, associative)
$F$-algebra with multiplication $m\col A \otimes_F A \to A$.  Assume
we have $F$-algebra homomorphisms
\begin{alignat*}{3}
  c%
  \col & A \to A \otimes_F A &\quad&
  \text{(\emph{comultiplication})} \\
  i%
  \col & A \to A && \text{(\emph{co-inverse})} \\
  u%
  \col & A \to F && \text{(\emph{co-unit})}
\end{alignat*}
which satisfy the following:
\begin{enumerateT}
\item
  The diagram
%%\begin{gather*}
%%  A\xrightarrow c A\tens_FA \xrightarrow{c\tens\Id}A\tens_FA\tens_FA\\ 
%%  A\xrightarrow c A\tens_FA \xrightarrow{\Id\tens c}A\tens_FA\tens_FA
%%\end{gather*}
  \[
  \begin{CD}
    A @>c>> A \tens_F A \\
    @V c VV @ VV c\tens\Id V\\
    A \tens_FA @>\Id\tens c>> A \tens_F A \tens_F A
  \end{CD}
  \]
  commutes.
\item
  The map
  \[
  A \xrightarrow{c} A \tens_F A \xrightarrow{u\tens\Id } F \tens_F A=A
  \]
%%\begin{gather*}
%%  A \xrightarrow{c} A \tens_F A \xrightarrow{u\tens\Id } F \tens_F A=
%%  A\\ A \xrightarrow\Id A
%%\end{gather*}
  equals the identity map $\Id\col A\to A$.
\item
  The two maps
  \begin{gather*}
    A \xrightarrow{c} A \tens_F A \xrightarrow{i\tens\Id } A \tens_F A
    \xrightarrow{m}A
    \\
    A \xrightarrow{u} F \xrightarrow{\cdot 1 } A
  \end{gather*}
  coincide.
\end{enumerateT}
An $F$-algebra~$A$ together with maps $c$,~$i$, and~$u$ as above is
called a \emph{\upp(commutative\upp) Hopf \index{A}{Hopf algebra}
  algebra over~$F$}.  A \index{A}{Hopf algebra homomorphism}\emph{Hopf
  algebra homomorphism} $f \col A \to B$ is an $F$-algebra
homomorphism preserving $c$,~$i$, and~$u$, i.e., $(f\tens f)\circ c_A
= c_B \circ f$, $f\circ i_A = i_B \circ f$, and $u_A = u_B \circ f$.
Hopf algebras and homomorphisms of Hopf algebras form a category.

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\noindent
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is the \index{A}{cone of dominant weights}\emph{cone of dominant
  weights} in $\Lambda$ (relative to~$\Pi$). We introduce a partial
ordering on~$\Lambda$: $\chi > \chi'$ if $\chi-\chi'$ is sum of simple
roots.  For any $\lambda \in \Lambda/\Lambda_r$ there exists a unique
\index{A}{minimal dominant weight}\emph{minimal dominant weight} $\chi
(\lambda)\index{B}{xhi(lam)@{$\chi (\lambda)$}} \in \Lambda_+$ in the
coset $\lambda$. Clearly, $\chi (0) = 0$.  

\input{boi-6r}


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