Loops in noncompact groups and 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.

The first version of this paper was split into two parts. The first part, concerning finite dimensional results, was expanded and became arXiv:1503.08461. The remainder constitutes this revision. arXiv admin note: text overlap with arXiv:0905.2911

References in corpus (3)

• Homogeneous Poisson Structures on Loop Spaces of Symmetric Spaces
• Loops in SU(2), Riemann Surfaces, and Factorization, I
• Non-compact groups of inner type and factorization