Loops in noncompact groups and factorization

TL;DR

This paper explores Birkhoff and root subgroup factorizations for loops in noncompact semisimple Lie groups, extending prior results from compact groups and examining the relationship with Toeplitz determinant factorizations.

Contribution

It investigates the existence and uniqueness of Birkhoff and root subgroup factorizations in noncompact Lie groups, highlighting differences from the compact case and analyzing related Toeplitz determinant factorizations.

Findings

Root subgroup factorization implies Birkhoff factorization in noncompact groups.

Obstacles prevent the converse from always holding.

Root subgroup factorization relates to Toeplitz determinant factorization.

Abstract

In [11] we showed that a loop in a simply connected compact Lie group $\dot{U}$ has a unique Birkhoff (or triangular) factorization if and only if the loop has a unique root subgroup factorization (relative to a choice of a reduced sequence of simple reflections in the affine Weyl group). In this paper our main purpose is to investigate Birkhoff and root subgroup factorization for loops in a noncompact type semisimple Lie group $\dot G_0$ of inner type. In [4] we showed that for an element of $\dot G_0$, i.e. a constant loop, there is a unique Birkhoff factorization if and only if there is a root subgroup factorization. However for loops in $\dot G_0$, while a root subgroup factorization implies a unique Birkhoff factorization, there are several obstacles to the converse. As in the compact case, root subgroup factorization is intimately related to factorization of Toeplitz determinants.