"Abstract" homomorphisms of split Kac-Moody groups / Caprace P.E. - Providence: AMS, 2009. - xv, 84 p.: ill. - (Memoirs of the American Mathematical Society; Vol.198, N 924) - ISBN 978-0-8218-4258-4; ISSN 0065-9266  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Contents

Introduction ................................................... ix Acknowledgements ............................................... xv Chapter 1. The objects: Kac-Moody groups, root data and Tits buildings ....................................... 1 1.1. Kac-Moody groups and Tits functors ......................... 1 1.2. Root data .................................................. 3 1.3. Tits buildings ............................................. 6 1.4. Twin root data and twin buildings: a short dictionary ...... 8 Chapter 2. Basic tools from geometric group theory ............. 11 2.1. CAT(0) geometry ........................................... 11 2.2. Rigidity of algebraic-group-actionsontrees ................ 14 Chapter 3. Kac-Moody groups and algebraic groups ............... 17 3.1. Bounded subgroups ......................................... 17 3.2. Adjoint representation of Tits functors ................... 19 3.3. A few facts from the theory of algebraic groups ........... 23 Chapter 4. Isomorphisms of Kac-Moody groups: an overview ....... 27 4.1. The isomorphism theorem ................................... 27 4.2. Diagonalizable subgroups and their centralizers ........... 30 4.3. Completely reducible subgroups and their centralizers ..... 35 4.4. Basic recognition of the ground field ..................... 39 4.5. Detecting rank one subgroups of Kac-Moody groups .......... 40 4.6. Images of diagonalizable subgroups under Kac-Moody group isomorphisms ........................................ 43 4.7. A technical auxiliary to the isomorphism theorem .......... 43 Chapter 5. Isomorphisms of Kac-Moody groups in characteristic zero ................................. 45 5.1. Rigidity of SL2 (Q)-actions on CAT(0) polyhedral complexes ................................................. 45 5.2. Homomorphisms of Chevalley groups over Q to Kac-Moody groups .................................................... 48 5.3. Regularity of diagonalizable subgroups .................... 51 5.4. Proof of the isomorphism theorem .......................... 52 Chapter 6. Isomorphisms of Kac-Moody groups in positive characteristic ...................................... 53 6.1. On bounded subgroups of Kac-Moody groups .................. 53 6.2. Homomorphisms of certain algebraic groups to Kac-Moody groups .................................................... 54 6.3. Images of certain small subgroups under Kac-Moody group isomorphisms .............................................. 59 6.4. Proof of the isomorphism theorem .......................... 62 Chapter 7. Homomorphisms of Kac-Moody groups to algebraic groups .............................................. 63 7.1. The non-linearity theorem ................................. 63 7.2. A combinatorial characterization of affine Coxeter groups .................................................... 63 7.3. On infinite root Systems .................................. 66 7.4. Proof of the non-linearity theorem ........................ 70 Chapter 8. Unitary forms of Kac-Moody groups ................... 73 8.1. Introduction .............................................. 73 8.2. Definitions ............................................... 73 8.3. Isomorphisms of unitary forms ............................. 75 8.4. Non-linearity ............................................. 78 Bibliography ................................................... 79 Index .......................................................... 83