Isomorphisms of Kac-Moody groups

Caprace, Pierre–Emmanuel; Mühlherr, Bernhard

doi:10.1007/s00222-004-0430-z

Cited by 11 publications

(19 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the proofs in both cases follow a similar global strategy, and the purpose of the present chapter is to collect all the technical preparations that these two proofs have in common. An important idea, which was actually at the basis of [CM05a], is that the structure of a Kac-Moody group is roughly 'controlled' by its rank one Levi subgroups. One is thus interested in obtaining an abstract characterization of these Levi subgroups; such a characterization is provided by Proposition 4.17 below and rests heavily on Tits' rigidity theorem for actions on trees.…”

Section: Chapter 4 Isomorphisms Of Kac-moody Groups: An Overviewmentioning

confidence: 99%

“…In the latter, one considers homomorphisms of a Fisotropic reductive F group to a split Kac-Moody group whose restriction to the center of the reductive group is injective. The main idea, which was at the heart of [CM05a], is to study the action of the semisimple part on the fixed point set of the abelian part in the twin building. Combining the aforementioned result of M. Bridson, a fixed point theorem for automorphism groups of twin buildings by B. Mühlherr [Müh94] and Borel-Tits' description of centralizers of tori in reductive groups [BT65], one shows essentially that the image of the center centralizes a subgroup of G(K) which is of Kac-Moody type but not necessarily split.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

“Abstract” homomorphisms of split Kac-Moody groups

Caprace¹

2009

Memoirs of the AMS

Self Cite

View full text Add to dashboard Cite

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. We establish some partial rigidity results, which we use to prove an isomorphism theorem for Kac-Moody groups over arbitrarily fields of cardinality at least 4. In particular, we obtain 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, we show that the Hausdorff topology that these groups carry is an invariant of the abstract group structure. Finally, we prove 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.

show abstract

Section: Chapter 4 Isomorphisms Of Kac-moody Groups: An Overviewmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“Abstract” homomorphisms of split Kac-Moody groups

Caprace¹

2009

Memoirs of the AMS

Self Cite

View full text Add to dashboard Cite

show abstract

“…Groups appearing in (v) were discussed in [We2]. For groups in (vi), a classification of automorphisms is available [Ca], see also [CMu1], [CMu2]; the case of finite ground fields should be similar to [FS]. Perhaps one can also treat unitary forms of Kac-Moody groups over C. As to B-rigidity, one can also use the approach of [BT] (apparently, to treat (iv)-(vi), it is to be generalized in an appropriate way).…”

Section: 2mentioning

confidence: 99%

Local–global invariants of finite and infinite groups: Around Burnside from another side

Kunyavskiı̆

2013

Expositiones Mathematicae

View full text Add to dashboard Cite

show abstract

“…This notion may be regarded as the group theoretical counterpart of the Kac-Moody algebra. We refer the reader to the lecture notes of P.-E. Caprace and B. Rémy [6] as an excellent introductory source, see also [5,27].…”

Section: Affine Lie Algebrasmentioning

confidence: 99%

On generalizing generalized polygons

Woldar

2009

Innov. Incidence Geom.

View full text Add to dashboard Cite

The purpose of this paper is to reveal in geometric terms a decade-old construction of certain families of graphs with nice extremal properties. Construction of the graphs in question is motivated by the way in which regular generalized polygons may be embedded in their Lie algebras, so that point-line incidence corresponds to the vanishing Lie product. The only caveat is that the generalized polygons are greatly limited in number. By performing successive truncations on an infinite root system of type e A 1 , we are able to obtain an infinite series of incidence structures which approximate the behavior of generalized polygons. Indeed, the first two members of the series are exactly the affine parts of the generalized polygons of type A 2 and B 2 .

show abstract

Isomorphisms of Kac-Moody groups

Cited by 11 publications

References 14 publications

“Abstract” homomorphisms of split Kac-Moody groups

“Abstract” homomorphisms of split Kac-Moody groups

Local–global invariants of finite and infinite groups: Around Burnside from another side

On generalizing generalized polygons

Contact Info

Product

Resources

About