Taking place at Université Pierre et Marie Curie, 8th of June, 2017.

Ralf Köhl (Universität Giessen)

## Kac-Moody symmetric spaces

In this talk I discuss a class of topological reflection spaces that has been introduced in collaboration with Freyn, Hartnick and Horn and which we call Kac-Moody symmetric spaces. These generalize Riemannian symmetric spaces of non-compact type. A non-spherical Kac-Moody symmetric space is non-geodesic but nonetheless geodesically connected. Maximal flats in Kac-Moody symmetric spaces correspond to tori of the underlying split real Kac-Moody group.
Unlike Riemannian symmetric spaces, non-spherical non-affine irreducible Kac-Moody symmetric spaces admit an invariant causal structure. The future and past boundaries of Kac-Moody symmetric spaces carry a natural simplicial structure and are simplicially isomorphic to the halves of the geometric realization of the twin buildings of the underlying split real Kac-Moody group.
Every automorphism of the symmetric space is uniquely determined by the induced simplicial 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. For many Kac-Moody symmetric spaces including the E_n-series, n at least 10, Kostant convexity allows one to establish that this pre-order is actually a partial order, which might be of interest for supergravity.

Slides shown in this talk

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