On topological split Kac-Moody groups and their twin buildings

Hartnick, Tobias; Köhl, Ralf; Mars, Andreas

doi:10.2140/iig.2013.13.1

Cited by 7 publications

(15 citation statements)

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“…In case G admits a finite Weyl group it is abstractly isomorphic to an algebraically simply connected semisimple split real Lie group and the thus defined topology coincides with its Lie group topology by an open-mapping theorem, cf. Proposition 2.2 of [6] and Corollary 7.16 of [7].…”

Section: Semisimple Lie Groups and Kac-moody Groupsmentioning

confidence: 94%

“…Now let δ denote the diagram given by the G i and G {i,j} and these inclusion maps. The Kac-Moody group G is now defined as the colimit of δ in the category of topological groups and, by [7], turns out to be a Hausdorff topological group.…”

Section: Semisimple Lie Groups and Kac-moody Groupsmentioning

confidence: 99%

“…As noted in the introduction, an algebraically simply connected 2-spherical split real Kac-Moody group may be defined as a certain colimit. This approach is due to Abramenko-Mühlherr [1] as a generalization of Tits original observation [11] concerning semisimple split algebraic groups/Lie groups, and has been topologized as Theorem 7.22 of [7].…”

Section: Semisimple Lie Groups and Kac-moody Groupsmentioning

confidence: 99%

“…Recall that any algebraically simply connected semisimple split real Lie group or any algebraically simply connected two-spherical split real Kac-Moody group G may be defined as the colimit (in the category of Hausdorff topological groups) of a diagram δ given by copies of SL 2 (R) and algebraically simply connected rank-2 split real Lie groups, cf. Theorem 7.22 of [7].…”

Section: Introductionmentioning

confidence: 99%

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An extension theorem for non-compact split embedded Riemannian symmetric spaces and an application to their universal property

Ocak

2019

Advances in Geometry

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By [5] it is known that a geodesic γ in an abstract reflection space X (in the sense of Loos, without any assumption of differential structure) canonically admits an action of a 1-parameter subgroup of the group of transvections of X. In this article, we modify these arguments in order to prove an analog of this result stating that, if X contains an embedded hyperbolic plane H ⊂ X, then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL2(R). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact type Y lying in X and can be used to prove that a Riemannian symmetric space and, more generally, the Kac-Moody symmetric space G/K for an algebraically simply connected two-spherical Kac-Moody group G, as defined in [5], satisfies a universal property similar to the universal property that the group G satisfies itself.

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Section: Semisimple Lie Groups and Kac-moody Groupsmentioning

confidence: 94%

Section: Semisimple Lie Groups and Kac-moody Groupsmentioning

confidence: 99%

Section: Semisimple Lie Groups and Kac-moody Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An extension theorem for non-compact split embedded Riemannian symmetric spaces and an application to their universal property

Ocak

2019

Advances in Geometry

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“…In the case of a hyperbolic Kac-Moody group, Theorem 1.15 can be seen as a global version of the lightcone embedding of the twin building as described in [CFF16]. The analogous construction of a twin building at infinity for masures can be found in [Rou11, Section 3]; by [CMR17, Theorem 1] this twin building at infinity of a masure actually carries a natural topology that turns it into a weak topological twin building in the sense of [HKM13].…”

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confidence: 99%

Kac-Moody symmetric spaces

Freyn,

Hartnick,

Horn

et al. 2017

Preprint

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In the present article we introduce and study a class of topological reflection spaces that we call Kac-Moody symmetric spaces. These are associated with split real Kac-Moody groups and generalize Riemannian symmetric spaces of non-compact split type.Based on work by the third-named author we observe that in a non-spherical Kac-Moody symmetric space there exist pairs of points that do not lie on a common geodesic; however, any two points can be connected by a chain of geodesic segments. We moreover classify maximal flats in Kac-Moody symmetric spaces and study their intersection patterns, leading to a classification of global and local automorphisms. Some of our methods apply to general topological reflection spaces beyond the Kac-Moody setting.Unlike Riemannian symmetric spaces, non-spherical non-affine irreducible Kac-Moody symmetric spaces also admit an invariant causal structure. For causal and anti-causal geodesic rays with respect to this structure we find a notion of asymptoticity, which allows us to define a future and past boundary of such Kac-Moody symmetric space. We show that these boundaries carry a natural polyhedral cell structure and are cellularly isomorphic to geometric realizations of the two halves of the twin buildings of the underlying split real Kac-Moody group. We also show that every automorphism of the symmetric space is uniquely determined by the induced cellular automorphism of the future and past boundary.The invariant causal structure on a non-spherical non-affine irreducible Kac-Moody symmetric space gives rise to an invariant pre-order on the underlying space, and thus to a subsemigroup of the Kac-Moody group.We conclude that while in some aspects Kac-Moody symmetric spaces closely resemble Riemannian symmetric spaces, in other aspects they behave similarly to masures, their non-Archimedean cousins. Acknowledgements:The authors thank the Mathematisches Forschungsinstitut Oberwolfach for the hospitality during the mini-workshops Symmetric varieties and involutions of algebraic groups in 2008 and Generalizations of symmetric varieties in 2012 and, furthermore, the Lorentz Center Leiden and the Center for Mathematical Sciences at the Technion, Haifa for hosting the subsequent international workshops on that topic in 2013 and 2015. The authors express their gratitude to the participants of these workshops for numerous discussions on the topic of this article; they particularly thank Bernhard M ühlherr for a wealth of very helpful conversations and Pierre-Emmanuel Caprace and Guy Rousseau for several deep discussions concerning the question whether Kac-Moody symmetric spaces admit a meaningful canonical pre-order. The authors also thank an anonymous referee and Guy Rousseau for extremely helpful very detailed criticism on preliminary versions of this article; these comments tremendously helped improve the quality of the structure and of the results of this article. In addition the authors thank Julius Gr üning and Timothée Marquis for further very useful comments on one of these prelimi...

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Fundamental Groups of Split Real Kac-Moody Groups and Generalized Real Flag Manifolds

Harring¹,

Köhl²

2022

Transformation Groups

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We determine the fundamental groups of symmetrizable algebraically simply connected split real Kac-Moody groups endowed with the Kac-Peterson topology. In analogy to the finite-dimensional situation, because of the Iwasawa decomposition G = KAU+, the embedding K ,↪ G is a weak homotopy equivalence, in particular π1(G) = π1(K). It thus suffices to determine π1(K), which we achieve by investigating the fundamental groups of generalized ag varieties. Our results apply in all cases in which the Bruhat decomposition of the generalized ag variety is a CW decomposition- in particular, we cover the complete symmetrizable situation; furthermore, the results concerning only the structure of π1(K) actually also hold in the nonsymmetrizable two-spherical case.

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On topological split Kac-Moody groups and their twin buildings

Cited by 7 publications

References 28 publications

An extension theorem for non-compact split embedded Riemannian symmetric spaces and an application to their universal property

An extension theorem for non-compact split embedded Riemannian symmetric spaces and an application to their universal property

Kac-Moody symmetric spaces

Fundamental Groups of Split Real Kac-Moody Groups and Generalized Real Flag Manifolds

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