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INTEGRAL REPRESENTATIONS OF THE INFINITE DIHEDRAL GROUP
 

Summary: INTEGRAL REPRESENTATIONS
OF THE INFINITE DIHEDRAL GROUP
JAUME AGUAD´E, CARLES BROTO, AND LAIA SAUMELL
1. Introduction.
We want to study the representations of the infinite dihedral group D in GL2(R),
where R is either the valuation ring Z(p) of rational numbers with denominator prime
to p or the ring of p-adic integers Zp for some prime p.
The motivation for this research comes from the homotopy theory of classifying
spaces of Kac-Moody groups. Associated to each generalized Cartan matrix (see, for
instance, the introduction of [2]), one can define a (not necessarily finite dimensional)
Lie algebra which can be integrated in some way which we will not discuss here (see
[5]) to produce a topological group K called a Kac-Moody group. These topological
groups, and their classifying spaces, have been studied from a homotopical point of
view in several recent papers ([6],[3], [2], [1]). Like in the Lie group case, K has
a maximal torus T and a Weyl group W which acts on the Lie algebra of T as a
cristalographic group. However, in contrast to what happens in the Lie group case,
this Weyl group can be infinite. If we start with a non-singular 2×2 Cartan matrix,
we have a (non-afine) Kac-Moody group of rank two and then the Weyl group is
infinite dihedral and we obtain a representation of D in GL2(Z) associated to K.
In [1] we have investigated the cohomology of the classifying spaces of these rank two

  

Source: Aguadé, Jaume - Departament de Matemàtiques, Universitat Autònoma de Barcelona

 

Collections: Mathematics