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SUMMARY:Geometric construction of buildings for hyperbolic Kac-Moody group
s
DTSTART;VALUE=DATE-TIME:20150526T121500Z
DTEND;VALUE=DATE-TIME:20150526T131500Z
DTSTAMP;VALUE=DATE-TIME:20210414T012648Z
UID:indico-event-810@indico.math.cnrs.fr
DESCRIPTION:A twin building is a simplicial complex associated to a group
G with a twin BN-pair. A complex hyperbolic Kac-Moody (KM) group G is asso
ciated with a hyperbolic KM Lie algebra g = g(A)\, where A is a hyperbolic
type Cartan matrix. The invariant symmetric bilinear form (. \, .) on the
standard Cartan subalgebra h in g has signature (n-1\,1) on the split rea
l form of h\, providing a Lorentzian geometry. The Cartan-Chevalley involu
tion on g gives a ``compact" real form k of g\, a real Lie algebra whose c
omplexification is g\, whose Cartan subalgebra t also has Lorentzian form
(. \, .)\, and there is also a corresponding compact real group K. We are
able to embed the twin building for G inside the union of all ``lightcones
" {x in k | (x\,x) <= 0} which is in the union of all K conjugates of t.
This provides a geometrical realization of the twin building of G closel
y related to the structure of all Cartan subalgebras in k\, and sheds ligh
t on the geometry of the infinite dimensional groups G and K. This is espe
cially interesting in the case of rank 3 hyperbolic algebras whose Weyl gr
oups are hyperbolic triangle groups\, so that the building is a union of c
opies of the tesselated Poincaré disk with certain boundary lines identif
ied. This is joint work with Lisa Carbone (Rutgers University) and Walter
Freyn (TU-Darmstadt).\n\nhttps://indico.math.cnrs.fr/event/810/
LOCATION:IHES Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/810/
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