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__ Chamber Systems And Examples -- Buildings And The Origin Of Chamber Systems -- Chamber Systems -- Two Examples Of Buildings -- Exercises -- Coxeter Complexes -- Coxeter Groups And Complexes -- Words And Galleries -- Reduced Words And Homotopy -- Finite Coxeter Complexes -- Self-homotopy -- Exercises -- Buildings -- A Definition Of Buildings -- Generalised M-gons - The Rank 2 Case -- Residues And Apartments -- Exercises -- Local Properties And Coverings -- Chamber Systems Of Type M -- Coverings And The Fundamental Group -- The Universal Cover -- Examples -- Exercises -- Bn - Pairs -- Tits Systems And Buildings -- Parabolic Subgroups -- Exercises -- Buildings Of Spherical Type And Root Groups -- Some Basic Lemmas -- Root Groups And The Moufang Property -- Commutator Relations -- Moufang Buildings - The General Case -- Exercises -- A Construction Of Buildings -- Blueprints -- Natural Labellings Of Moufang Buildings -- Foundations -- Exercises --^ The Classification Of Spherical Buildings -- A3 Blueprints And Foundations -- Diagrams With Single Bonds -- C3 Foundations -- Cn Buildings For N > 4 -- Tits Diagrams And F4 Buildings -- Finite Buildings -- Exercises -- Affine Buildings I -- Affine Coxeter Complexes And Sectors -- The Affine Building An-1 (k,v) -- The Spherical Building At Infinity -- The Proof Of (9.5) -- Exercises -- Affine Buildings Ii -- Apartment Systems, Trees And Projective Valuations -- Trees Associated To Walls And Panels At Infinity -- Root Groups With A Valuation -- Construction Of An Affine Bn-pair -- The Classification -- An Application -- Exercises -- Twin Buildings -- Twin Buildings And Kac-moody Groups -- Twin Trees -- Twin Apartments -- An Example: Affine Twin Buildings -- Residues, Rigidity, And Proj -- 2-spherical Twin Buildings -- The Moufang Property And Root Group Data -- Twin Trees Again -- Appendix 1: Moufang Polygons -- The M-function -- The Natural Labelling For A Moufang Plane --^ 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. Mark Ronan. Includes Bibliographical References (p. [216]-221) And Index. Chamber systems and examples Chamber systems Two examples of buildings Exercises Coxeter complexes Coxeter groups and complexes Words and galleries Reduced words and homotopy Finite coxeter complexes Self-homotopy Exercises Buildings A definition of buildings Generalised m-gons - the rank 2 case Residues and apartments Exercises Local properties and coverings Chamber systems of type m Coverings and the fundamental group The universal cover Examples Exercises Bn - pairs Tits systems and buildings Parabolic subgroups Exercises Buildings of spherical type and root groups Some basic lemmas Root groups and the moufang property Commutator relations Moufang buildings - the general case Exercises A construction of buildings Blueprints Natural labellings of moufang buildings Foundations Exercises The classification of spherical buildings 1.a3 blueprints and foundations Diagrams with single bonds C3 foundations Cn buildings for n > 4 Tits diagrams and f4 buildings Finite buildings Exercises Affine buildings I Affine coxeter complexes and sectors The affine building an-1 (k,v) The spherical building at infinity The proof of (9.5) Exercises Affine buildings II Apartment systems, trees and projective valuations Trees associated to walls and panels at infinity Root groups with a valuation Construction of an affine bn-pair The classification An application Exercises Twin buildings Twin buildings and kac-moody groups Twin trees Twin apartments An example: affine twin buildings Residues, rigidity, and proj 2-spherical twin buildings The moufang property and root group data Twin trees again Appendix 1: moufang polygons The m-function The natural labelling for a moufang plane 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. 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 Contents Introduction to the 2009 Edition Introduction Leitfaden Chapter 1 Chamber syetems and examples Chapter 2 Coxeter complexes Chapter 3 Buildings Chapter 4 Local properties and coverings Chapter 5 BN-pairs Chapter 6 Buildings of spherical types and root groups Chapter 7 A construction of buildings Chapter 8 The classification of spherical buildings Chapter 9 Affine buildings I Chapter 10 Affine buildings II Chapter 11 Twin buildings Appendix 1 Moufang polygons Appendix 2 Diagrams for Moufang polygons Appendix 3 Non-discrete buildings Appendix 4 Topology and Steinberg representation Appendix 5 Finite Coxeter groups (i.e. of spherical type) Bibliography Index of notation Index In mathematics, 'buildings' are geometric structures that represent groups of Lie type over an arbitrary field. This book presents an introduction to mathematical buildings. It is suitable for those doing research or teaching courses on Lie-type groups, on finite groups, or on discrete groups.