EQUIVARIANT TRIANGULATIONS OF TORI OF COMPACT LIE GROUPS AND HYPERBOLIC EXTENSION TO NON-CRYSTALLOGRAPHIC COXETER GROUPS - Université de Picardie Jules Verne Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2021

EQUIVARIANT TRIANGULATIONS OF TORI OF COMPACT LIE GROUPS AND HYPERBOLIC EXTENSION TO NON-CRYSTALLOGRAPHIC COXETER GROUPS

Arthur Garnier

Résumé

Given a simple connected compact Lie group $K$ and a maximal torus $T$ of $K$, the Weyl group $W=N_K(T)/T$ naturally acts on $T$. First, we use the combinatorics of the (extended) affine Weyl group to provide an explicit $W$-equivariant triangulation of $T$. We describe the associated cellular homology chain complex and give a formula for the cup product on its dual cochain complex, making it a $\mathbb{Z}[W]$-dg-algebra. Next, remarking that the combinatorics of this dg-algebra is still valid for Coxeter groups, we associate a closed compact manifold $\mathbf{T}(W)$ to any finite irreducible Coxeter group $W$, which coincides with a torus if $W$ is a Weyl group and is hyperbolic in other cases. Of course, we focus our study on non-crystallographic groups, which are $I_2(m)$ with $m=5$ or $m\ge 7$, $H_3$ and $H_4$. The manifold $\mathbf{T}(W)$ comes with a $W$-action and an equivariant triangulation, whose related $\mathbb{Z}[W]$-dg-algebra is the one mentioned above. We finish by computing the homology of $\mathbf{T}(W)$, as a representation of $W$.
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Dates et versions

hal-03214218 , version 1 (01-05-2021)
hal-03214218 , version 2 (09-10-2021)

Identifiants

  • HAL Id : hal-03214218 , version 2

Citer

Arthur Garnier. EQUIVARIANT TRIANGULATIONS OF TORI OF COMPACT LIE GROUPS AND HYPERBOLIC EXTENSION TO NON-CRYSTALLOGRAPHIC COXETER GROUPS. 2021. ⟨hal-03214218v2⟩

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