2023/09/05 by Ralf Köhl, Köhl, Ralf, Christian Vock +1
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2309.02176
openalex publication_date 2023/09/05 · openalex created_date 2023/09/09 · openalex updated_date 2026/08/01
Kac-Moody symmetric spaces have been introduced by Freyn, Hartnick, Horn and the first-named author for centered Kac-Moody groups, that is, Kac-Moody groups that are generated by their root subgroups. In the case of non-invertible generalized Cartan matrices this leads to complications that -- within the approach proposed originally -- cannot be repaired in the affine case. In the present article we propose an alternative approach to Kac-Moody symmetric spaces which for invertible generalized Cartan matrices provides exactly the same concept, which for the non-affine non-invertible case provides alternative Kac-Moody symmetric spaces, and which finally provides Kac-Moody symmetric spaces for affine Kac-Moody groups. In a nutshell, the original intention by Freyn, Hartnick, Horn and Köhl was to construct symmetric spaces that likely lead to primitive actions of the Kac-Moody groups; this, of course, cannot work in the affine case as affine Kac-Moody groups are far from simple. Additionally, we study the Galois descent to almost split real Kac-Moody symmetric spaces based on the theory of almost split Kac-Moody groups developed by Rémy 2002.