Book contents
- Frontmatter
- Contents
- Introduction
- Terminology, conventions, and notation
- Part I Constructions, examples, and structure theory
- Part II Standard presentations and their applications
- Part III General classification and applications
- Part IV Appendices
- A Background in linear algebraic groups
- B Tits' work on unipotent groups in nonzero characteristic
- C Rational conjugacy in connected groups
- References
- Index
B - Tits' work on unipotent groups in nonzero characteristic
from Part IV - Appendices
Published online by Cambridge University Press: 05 July 2011
- Frontmatter
- Contents
- Introduction
- Terminology, conventions, and notation
- Part I Constructions, examples, and structure theory
- Part II Standard presentations and their applications
- Part III General classification and applications
- Part IV Appendices
- A Background in linear algebraic groups
- B Tits' work on unipotent groups in nonzero characteristic
- C Rational conjugacy in connected groups
- References
- Index
Summary
In our study of pseudo-reductive groups, we need to use some fundamental results of Tits concerning the structure of smooth connected unipotent groups and torus actions on such groups over an arbitrary (especially imperfect) ground field of nonzero characteristic. These results were presented by Tits in a course at Yale University in 1967, and lecture notes [Ti1] for that course were circulated but never published. Much of the course was concerned with general results on linear algebraic groups that are available now in many standard references (such as [Bo2], [Hum2], and [Spr]).
The only previously available written account (with proofs) of Tits' structure theory of unipotent groups is his unpublished Yale lecture notes, though a summary of the statements of his results is given in [Oes, Ch. V]. In this appendix we give a self-contained development of the theory, with complete proofs, and in many parts we have simply reproduced arguments from Tits' lecture notes. We go beyond what we need in order that this appendix may serve as a useful general reference on Tits' work on this topic.
Throughout this appendix, k is an arbitrary field with characteristic p > 0.
Subgroups of vector groups
A smooth connected solvable k-group G is k-split if it admits a composition series whose successive quotients are k-isomorphic to Ga or GL1.
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- Chapter
- Information
- Pseudo-reductive Groups , pp. 473 - 493Publisher: Cambridge University PressPrint publication year: 2010