# Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field: 170 (Cambridge Studies in Advanced Mathematics) (Cambridge Studies in Advanced Mathematics, Series Number 170)

Cambridge University Press, 9/21/2017

EAN 9781107167483, ISBN10: 1107167485

Hardcover, 660 pages, 23.4 x 16 x 4.1 cm

Language: English

Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the BarsottiÃ¢â‚¬â€œChevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the BorelÃ¢â‚¬â€œChevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.

Introduction

1. Definitions and basic properties

2. Examples and basic constructions

3. Affine algebraic groups and Hopf algebras

4. Linear representations of algebraic groups

5. Group theory

the isomorphism theorems

6. Subnormal series

solvable and nilpotent algebraic groups

7. Algebraic groups acting on schemes

8. The structure of general algebraic groups

9. Tannaka duality

Jordan decompositions

10. The Lie algebra of an algebraic group

11. Finite group schemes

12. Groups of multiplicative type

linearly reductive groups

13. Tori acting on schemes

14. Unipotent algebraic groups

15. Cohomology and extensions

16. The structure of solvable algebraic groups

17. Borel subgroups and applications

18. The geometry of algebraic groups

19. Semisimple and reductive groups

20. Algebraic groups of semisimple rank one

21. Split reductive groups

22. Representations of reductive groups

23. The isogeny and existence theorems

24. Construction of the semisimple groups

25. Additional topics

Appendix A. Review of algebraic geometry

Appendix B. Existence of quotients of algebraic groups

Appendix C. Root data

Bibliography

Index.