For this printing, I have corrected some errors and made numerous minor changes in the interest of clarity. The most significant corrections occur in Sections 4.2, 4.3, 5.5, 30.3, 32.1, and 32.3. I have also updated the bibliography to some extent. Thanks are due to a number of readers who took the trouble to point out errors, or obscurities; especially helpful were the detailed comments of Jose Antonio Vargas.

Ⅰ.Algebraic Geometry

 0.Some Commutative Algebra

 1.Affine and Projective Varieties

1.1 Ideals and Afline Varieties

1.2 Zariski Topology on Affine Space

1.3 Irreducible Components

1.4 Products of Affine Varieties

1.5 Affine Algebras and Morphisms

1.6 Projective Varieties

1.7 Products of Projective Varieties

1.8 Flag Varieties

 2.Varieties

2.1 Local Rings

2.2 Prevarieties

2.3 Morphisms

2.4 Products

2.5 Hausdorff Axiom

 3.Dimension

3.1 Dimension of a Variety

3.2 Dimension of a Subvariety

3.3 Dimension Theorem

3.4 Consequences

 4.Morphisms

4.1 Fibres of a Morphism

4.2 Finite Morphisms

4.3 Image ofa Morphism

4.4 Constructible Sets

4.5 Open Morphisms

4.6 Bijective Morphisms

4.7 Birational Morphisms

 5.Tangent Spaces

5.1 Zariski Tangent Space

5.2 Existence of Simple Points

5.3 Local Ring of a Simple Point

5.4 Differential of a Morphism.

5.5 Differential Criterion for Separability

 6.Complete Varieties

6.1 Basic Properties

6.2 Completeness of Projective Varieties

6.3 Varieties Isomorphic to P

6.4 Automorphisms of P

Ⅱ.Afline Algebraic Groups

 7.Basic Concepts and Examples

7.1 The Notion of Algebraic Group

7.2 Some Classical Groups

7.3 Identity Component

7.4 Subgroups and Homomorphisms.

7.5 Generation by Irreducible Subsets

7.6 HopfAIgebras

 8.Actions of Algebraic Groups on Varieties

8.1 Group Actions

8.2 Actions of Algebraic Groups

8.3 Closed Orbits

8.4 Semidirect Products

8.5 Translation of Functions

8.6 Linearization of Affine Groups

Ⅲ.Lie Algebras

 9.Lie Algebra of an Algebraic Group

9.1 Lie Algebras and Tangent Spaces

9.2 Convolution

9.3 Examples.

9.4 Subgroups and Lie Subalgebras

9.5 Dual Numbers

 10.Differentiation.

10.1 Some Elementary Formulas

10.2 Differential of Right Translation

10.3 The Adjoint Representation

10.4 Differential of Ad

10.5 Commutators

10.6 Centralizers

10.7 Automorphisms and Derivations

Ⅳ.Homogeneous Spaces

 11.Construction of Certain Representations

11.1 Action on Exterior Powers

11.2 A Theorem of Chevalley

11.3 Passage to Projective Space

11.4 Characters and Semi-lnvariants.

11.5 Normal Subgroups

 12.Quotients

Ⅴ.Characteristic O Theory

Ⅵ.Semisimple and Unipotent Elements

Ⅶ.Solvable Groups

Ⅷ.Borel Subgroups

Ⅸ.Centralizers of Tori

Ⅹ.Structure of Reductive Groups

Ⅺ.Representatins and Classfication of Semisimple Groups

Ⅻ.Survey of Rationality Properties

Appendix Root Systems

Bibliography

Index of Terminology

Index of Symbols