Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Decompositions of Kac-Moody groups   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 $A:=\tau(T)$.
  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 implications for the geometrical structure of the Kac-Moody
symmetric space $G/K \cong \tau(G)$.

36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
19
The categories of this paper are: eess
23
The categories of this paper are: solv-int
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph