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\begin{document}
George McNinch
Math 651 -- Topics in Algebra --
Fall 1999
Linear Algebraic Groups
\noindent \textbf{Text:} \emph{Linear Algebraic Groups}, 2nd edition, Tonny A.
Springer, Progress In Mathematics Vol 9, Birkh\"auser, 1998.
\bigskip\noindent \textbf{Overview:} In this course, the objects of study are
groups which have some geometric structure. A group which is also a
smooth (real or complex) manifold (with certain natural stipulations
that the group operations respect the geometry) is called a \emph{Lie
group}. In this course, we study objects analogous to Lie groups;
the groups to be considered are \emph{algebraic varieties} over an
algebraically closed field, and are known as algebraic groups.
Algebraic groups play an important role in a number of different areas
of mathematics. The much-studied ``classical groups'' (e.g.
orthogonal, symplectic, unitary, general linear,... groups) are all
examples of algebraic groups. From the point of view of finite group theory,
algebraic groups are significant in that many of the now-classified
finite simple groups arise from certain linear algebraic groups
defined over fields of positive characteristic. Furthermore, there
are numerous problems in geometry, number theory and algebra for which
the study of algebraic groups has proved crucial.
The course will investigate the structure of algebraic groups. A
connected linear algebraic group turns out to have a closed normal
subgroup for which the quotient group is semisimple. (Actually, the
normal subgroup can be taken connected and solvable as well.) Thus, to
understand linear algebraic groups, one needs to study connected
solvable groups and semisimple groups; the latter case turns out to be
the most interesting. As in the study of compact Lie groups, or of
semisimple complex Lie groups, there is a classification theorem for
semisimple linear algebraic groups. (An algebraic group is linear
when it is a closed subgroup of the group of all invertible $n \times
n$ matrices for some $n >0$; the notion of semisimplicity is a bit
more complex to describe so I'll make you wait for the course!)
In fact, the classification of the three classes of groups just
mentioned is basically the same: one associates to the group a
combinatorial gadget called a \emph{root system}, one proves that the
root system more-or-less determines the group, and one classifies the
possible root systems (indecomposable roots systems fall into 4
infinite families, with 5 exceptional diagrams. In particular, the
classification of compact Lie groups, semisimple complex Lie groups,
and semisimple algebraic groups is essentially the same).
The main goal of the course will be to work towards the classification
theorem. There is quite a bit of preliminary work to do; fundamental
results about algebraic groups must first be established. Once the
preliminary results are obtained, we will begin work on the
classification theorem. However, at this point in the course time
constraints will make us prefer an emphasis on examples and
explanation rather more than a complete proof which lacks examples.
If time permits, we will discuss (as does Springer's text) some aspects
of the theory of linear algebraic groups defined over fields which are
not algebraically closed.
\bigskip \noindent\textbf{Expectations:}
A student who has completed the first year algebra course should be
adequately prepared for the course. Some knowledge of basic algebraic
geometry would be helpful, but is not required.
There will be no exams in the course. You are expected to attend class
and to hand in solutions to the homework problems assigned
periodically during the term.
\newpage
\bigskip \noindent\textbf{Course Outline:}
A nice feature of Springer's text is that it gives a self-contained
account of the results from algebraic geometry and commutative algebra
needed for the theory.
The following topics will be considered:
\begin{itemize}
\item Introduction to algebraic geometry.
\item First properties of linear algebraic groups.
\item Commutative algebraic groups.
\item The Lie algebra of an algebraic group.
\item Morphisms and quotients.
\item Solvable groups, Parabolic subgroups.
\item Weyl group, rank one groups.
\item Reductive groups.
\item classification issues
\item (?) $F$-groups, $F$ an arbitrary field.
\end{itemize}
\bigskip \noindent\textbf{Additional References:}
\bibliographystyle{amsalpha}
(Caveat: No claim is made that this list is at all complete!!)
\begin{itemize}
\item General references for algebraic groups:
\cite{Bor1}
\cite{Hu3}
\cite{SGA3}
\cite{springer98:_linear_algeb_group}
\cite{borel-tits}
\cite{Steinberg}
\item Related Finite Groups:
\cite{Steinberg}
\cite{Carter1}
\cite{DigneMichel}
\item Relations with Number Theory:
\cite{SerreGC}
\cite{Platonov}
\item Representation Theory
\cite{JRAG}
\end{itemize}
%\bibliography{MathBib}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{Hum75}
\bibitem[Bor91]{Bor1}
Armand Borel, \emph{Linear algebraic groups}, 2nd ed., Grad. Texts in Math.,
no. 129, Springer Verlag, 1991.
\bibitem[BT65]{borel-tits}
Armand Borel and Jacques Tits, \emph{Groupes r{\'e}ductifs}, Publ. Math.
I.H.E.S. (1965), no.~27, 55--150.
\bibitem[Car85]{Carter1}
Roger~W. Carter, \emph{Finite groups of {L}ie type: Conjugacy classes and
complex characters}, Pure and Applied Mathematics, Wiley, Chichester, New
York, Brisbane, 1985.
\bibitem[DG70]{SGA3}
M.~Demazure and A.~Grothendieck, \emph{Sch{\'e}mas en groupes, s{\'e}minaire de
g{\'e}om{\'e}trie du {B}ois {M}arie 1962/64 ({SGA}3)}, Lecture Notes in
Math., no. 151-153, Springer-Verlag, 1970.
\bibitem[DM91]{DigneMichel}
F.~Digne and J.~Michel, \emph{Representations of finite groups of {L}ie type},
London Math. Soc. Student Texts, vol.~21, Cambridge University Press,
Cambridge, 1991.
\bibitem[Hum75]{Hu3}
James Humphreys, \emph{Linear algebraic groups}, Grad. Texts in Math., no.~21,
Springer Verlag, 1975.
\bibitem[Jan87]{JRAG}
Jens~C. Jantzen, \emph{Representations of algebraic groups}, Pure and Applied
Mathematics, vol. 131, Academic Press, Orlando, FL, 1987.
\bibitem[PR94]{Platonov}
Vladimir Platonov and Andrei Rapinchuk, \emph{Algebraic groups and number
theory}, Pure and Applied Mathematics, vol. 139, Academic Press, 1994,
English translation.
\bibitem[Ser97]{SerreGC}
Jean-Pierre Serre, \emph{Galois cohomology}, Springer Verlag, 1997, New edition
of ``Cohomologie Galoisienne''.
\bibitem[Spr98]{springer98:_linear_algeb_group}
T.A. Springer, \emph{Linear algebraic groups}, 2nd ed., Progr. in Math.,
vol.~9, Birh{\"a}user, Boston, 1998.
\bibitem[Ste68]{Steinberg}
Robert Steinberg, \emph{Lectures on {C}hevalley groups}, Yale University, 1968.
\end{thebibliography}
\end{document}
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