Lectures on Buildings
Updated and Revised
Lectures on Buildings
Updated and Revised
In mathematics, “buildings” are geometric structures that represent groups of Lie type over an arbitrary field. This concept is critical to physicists and mathematicians working in discrete mathematics, simple groups, and algebraic group theory, to name just a few areas.
Almost twenty years after its original publication, Mark Ronan’s Lectures on Buildings remains one of the best introductory texts on the subject. A thorough, concise introduction to mathematical buildings, it contains problem sets and an excellent bibliography that will prove invaluable to students new to the field. Lectures on Buildings will find a grateful audience among those doing research or teaching courses on Lie-type groups, on finite groups, or on discrete groups.
“Ronan’s account of the classification of affine buildings [is] both interesting and stimulating, and his book is highly recommended to those who already have some knowledge and enthusiasm for the theory of buildings.”—Bulletin of the London Mathematical Society
Table of Contents
introduction to the 2009 edition
introduction
leitfaden
chapter 1 - chamber systems and examples
1. chamber systems
2. two examples of buildings
exercises
chapter 2 - coxeter complexes
1. coxeter groups and complexes
2. words and galleries
3. reduced words and homotopy
4. finite coxeter complexes
5. self-homotopy
exercises
chapter 3 - buildings
1. a definition of buildings
2. generalised m-gons - the rank 2 case
3. residues and apartments
exercises
chapter 4 - local properties and coverings
1. chamber systems of type m
2. coverings and the fundamental group
3. the universal cover
4. examples
exercises
chapter 5 - bn - pairs
1. tits systems and buildings
2. parabolic subgroups
exercises
chapter 6 - buildings of spherical type and root groups
1. some basic lemmas
2. root groups and the moufang property
3. commutator relations
4. moufang buildings - the general case
exercises 80
chapter 7 - a construction of buildings
1. blueprints
2. natural labellings of moufang buildings
3. foundations
exercises
chapter 8 - the classification of spherical buildings
1.a3 blueprints and foundations
2. diagrams with single bonds
3. c3 foundations
4. cn buildings for n > 4
5. tits diagrams and f4 buildings
6. finite buildings
exercises
chapter 9 - affine buildings I
1. affine coxeter complexes and sectors
2. the affine building an-1 (k,v)
3. the spherical building at infinity
4. the proof of (9.5)
exercises
chapter 10 - affine buildings II
1. apartment systems, trees and projective valuations
2. trees associated to walls and panels at infinity
3. root groups with a valuation
4. construction of an affine bn-pair
5. the classification
6. an application
exercises
chapter 11 - twin buildings
1. twin buildings and kac-moody groups
2. twin trees
3. twin apartments
4. an example: affine twin buildings
5. residues, rigidity, and proj.
6. 2-spherical twin buildings
7. the moufang property and root group data
8. twin trees again
appendix 1 - moufang polygons
1. the m-function
2. the natural labelling for a moufang plane
3. the non-existence theorem
appendix 2 - diagrams for moufang polygons
appendix 3 - non-discrete buildings
appendix 4 - topology and the steinberg representation
appendix 5 - finite coxeter groups
appendix 6 finite buildings and groups of lie type
bibliography
index of notation
index