Book contents
- Frontmatter
- Contents
- Introduction
- Terminology, conventions, and notation
- Part I Constructions, examples, and structure theory
- 1 Overview of pseudo-reductivity
- 2 Root groups and root systems
- 3 Basic structure theory
- Part II Standard presentations and their applications
- Part III General classification and applications
- Part IV Appendices
- References
- Index
2 - Root groups and root systems
from Part I - Constructions, examples, and structure theory
Published online by Cambridge University Press: 05 July 2011
- Frontmatter
- Contents
- Introduction
- Terminology, conventions, and notation
- Part I Constructions, examples, and structure theory
- 1 Overview of pseudo-reductivity
- 2 Root groups and root systems
- 3 Basic structure theory
- Part II Standard presentations and their applications
- Part III General classification and applications
- Part IV Appendices
- References
- Index
Summary
The category of pseudo-reductive groups over a field k has weak stability properties under basic operations on group schemes, such as extension of the base field and passage to quotients (even by finite central multiplicative or smooth normal subgroups). Fortunately, pseudo-reductive groups do admit many properties generalizing those of connected reductive groups, and this chapter establishes such properties for convenient use later on.
Following Tits, we use an interesting construction of unipotent groups to construct a theory of root systems and root groups for pseudo-reductive groups over any separably closed field, or more generally for pseudo-split pseudoreductive groups, i.e., the ones which contain a split maximal torus (over an arbitrary ground field). Tits developed a version of this theory without requiring a split maximal torus, but it will suffice for our needs to have this theory for pseudo-split groups. In [Spr, Ch. 13-15] an exposition of Tits' theory is given with an arbitrary base field. We provide an exposition in order to make our work more self-contained and to incorporate some scheme-theoretic improvements into the theory.
In contrast with the connected reductive case, root groups in pseudo-split pseudo-reductive groups over a field k may have dimension larger than 1, though they turn out to always be vector groups over k (i.e., k-isomorphic to a power of Ga).
- Type
- Chapter
- Information
- Pseudo-reductive Groups , pp. 43 - 85Publisher: Cambridge University PressPrint publication year: 2010