{"paper":{"title":"Decompositions of Kac-Moody groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Max Horn","submitted_at":"2017-08-18T11:30:39Z","abstract_excerpt":"Let $G$ be a split (minimal) Kac-Moody group over $\\mathbb{R}$ or $\\mathbb{C}$ with maximal torus $T$, and let $\\theta$ be a Cartan-Chevalley involution of $G$, twisted by complex conjugation, and satisfying that $\\theta(T)=T$. Furthermore, let $K$ be the subgroup fixed by $\\theta$, and $\\tau:G\\to G, g\\mapsto g\\theta(g)^{-1}$. Let $A:=\\tau(T)$.\n  In this note, we show resp. revisit that $G$ admits a (refined) Iwasawa decompositions $G=UAK$. We also show that if $G$ is of non-spherical type, then it never admits a polar decomposition $G=\\tau(G)K$ nor a Cartan decompositions $G=KAK$. This has im"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.05566","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}