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 polyno
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
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 ma
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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
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 thi
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 res