Loop groups and twin buildings

Details Loop groups and twin buildings Loop groups and twin buildings

2001-09-19

arxiv.org/abs/math/0109128v1

Mathematics

Geometric Topology

Dedicated to John Stallings on the occasion of his 65th birthday. To appear in Geometriae Dedicata

Scientific paper

We describe some buildings related to complex Kac-Moody groups. First we describe the spherical building of SLn(C) (i.e. the projective geometry PG(Cn)) and its Veronese representation. Next we recall the construction of the affine building associated to a discrete valuation on the rational function field $C(z)$. Then we describe the same building in terms of complex Laurent polynomials, and introduce the Veronese representation, which is an equivariant embedding of the building into an affine Kac-Moody algebra. Next, we introduce topological twin buildings. These buildings can be used for a proof - which is a variant of the proof by Quillen and Mitchell - of Bott periodicity which uses only topological geometry. At the end we indicate very briefly that the whole process works also for affine real almost split Kac-Moody groups.

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