“Abstract” homomorphisms of split Kac-Moody groups
About this Title
Publication: Memoirs of the American Mathematical Society
Publication Year 2009: Volume 198, Number 924
ISBNs: 978-0-8218-4258-4 (print); 978-1-4704-0530-4 (online)
MathSciNet review: 2499773
MSC (2000): Primary 20G44; Secondary 20E42, 20F55, 20F65, 51E24
This work is devoted to the isomorphism problem for split Kac-Moody groups over arbitrary fields. This problem turns out to be a special case of a more general problem, which consists in determining homomorphisms of isotropic semisimple algebraic groups to Kac-Moody groups, whose image is bounded. Since Kac-Moody groups possess natural actions on twin buildings, and since their bounded subgroups can be characterized by fixed point properties for these actions, the latter is actually a rigidity problem for algebraic group actions on twin buildings. The author establishes some partial rigidity results, which we use to prove an isomorphism theorem for Kac-Moody groups over arbitrary fields of cardinality at least $4$. In particular, he obtains a detailed description of automorphisms of Kac-Moody groups. This provides a complete understanding of the structure of the automorphism group of Kac-Moody groups over ground fields of characteristic $0$.
The same arguments allow to treat unitary forms of complex Kac-Moody groups. In particular, the author shows that the Hausdorff topology that these groups carry is an invariant of the abstract group structure.
Finally, the author proves the non-existence of cocentral homomorphisms of Kac-Moody groups of indefinite type over infinite fields with finite-dimensional target. This provides a partial solution to the linearity problem for Kac-Moody groups.
Table of Contents
- Chapter 1. The objects: Kac-Moody groups, root data and Tits buildings
- Chapter 2. Basic tools from geometric group theory
- Chapter 3. Kac-Moody groups and algebraic groups
- Chapter 4. Isomorphisms of Kac-Moody groups: an overview
- Chapter 5. Isomorphisms of Kac-Moody groups in characteristic zero
- Chapter 6. Isomorphisms of Kac-Moody groups in positive characteristic
- Chapter 7. Homomorphisms of Kac-Moody groups to algebraic groups
- Chapter 8. Unitary forms of Kac-Moody groups