1
1.1
1.1.1
Some algebraic geometry
Basic denitions.
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
n
n
X = F (S ) = fx 2
3
3.1
3.1.1
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
satisfying:
1. (1; x
2
2.1
Lie groups and algebraic groups.
Basic Denitions.
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
t
-Tb
tr*
bg ,Jt
Lt><,t.^->t*;!l
(ri
L
I
gfa^"^*'
,rrr! Qet rr*u
t1r-e- \ x t^.
*&^CU
( ,^) -|,"*re6'G L Ln, ,) a ,b'7
4
vvu,6,
t/t^
\$e- A- *]^ "f
"-e
f*e W
tat V " M ,CCl l.r.c
'vt
tu
iI*
o.,^- rt "tcfw_*f
u Lul
= li x3-l'
fl'><
&/ \^.>( 4.r"^, J?^* 4rr
Chapter 1
Lie Groups and Algebraic
Groups
Hermann Weyl, in his famous book (Weyl [1946]), 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
Appendix D
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
Appendix A
Algebraic Geometry
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