Some algebraic geometry
Zariski and standard topology.
Let F be a eld (almost always R or C). Then we say that a subset X F n
is Zariski closed if there is a set of polynomials S
F [x1 ; :; xn ] such that
X = F (S ) = fx 2
Hilbert-Mumford type theorems.
Basics on group actions.
Algebraic group actions.
Let X be an algebraic variety and let G be an algebraic group both over C.
Then an (algebraic group) action of G on X is a morphism : G X ! X
1. (1; x
Lie groups and algebraic groups.
In this subsection we will introduce the class of groups to be studied. We
rst recall that a Lie group is a group that is also a dierentiable manifold
and multiplication (x; y 7 ! xy ) and inverse (x
,rrr! Qet rr*u
t1r-e- \ x t^.
( ,^) -|,"*re6'G L Ln, ,) a ,b'7
\$e- A- *]^ "f
tat V " M ,CCl l.r.c
o.,^- rt "tcfw_*f
= li x3-l'
&/ \^.>( 4.r"^, J?^* 4rr
Lie Groups and Algebraic
Hermann Weyl, in his famous book (Weyl ), gave the name classical groups
to certain families of matrix groups. In this chapter we introduce these groups and
develop the basic ideas of Lie groups, Lie algebra
Manifolds and Lie Groups
The purpose of this appendix is to collect the essential parts of manifold and Lie
group theory in a convenient form for the body of the book. The philosophy of this
appendix is to give the main denitions and to prove m
We develop the aspects of algebraic geometry needed for the study of algebraic
groups over C in this book. Although we give self-contained proofs of almost all
of the results stated, we do not attempt to give an introduction