Loops in SL(2,C) and Factorization

TL;DR

This paper extends the analysis of loops in SU(2,C) to SL(2,C), exploring factorizations and determinants, with new challenges arising from the non-compactness and rational root subgroup coordinates.

Contribution

It generalizes the factorization and determinant results from SU(2,C) to SL(2,C), addressing the complexities introduced by non-compactness and rational functions.

Findings

Root subgroup coordinates are rational functions with an exceptional set.

Loops in SL(2,C) are not necessarily bounded, complicating analysis.

Factorization results extend with modifications to accommodate SL(2,C).

Abstract

In previous work we proved that for a SU(2,C) valued loop having the critical degree of smoothness (one half of a derivative in the L^2 Sobolev sense), the following are equivalent: (1) the Toeplitz and shifted Toeplitz operators associated to the loop are invertible, (2) the loop has a triangular factorization, and (3) the loop has a root subgroup factorization. For a loop g satisfying these conditions, the Toeplitz determinant det(A(g)A(g^{-1})) and shifted Toeplitz determinant det(A_1(g)A_1(g^{-1})) factor as products in root subgroup coordinates. In this paper we observe that, at least in broad outline, there is a relatively simple generalization to loops having values in SL(2,C). The main novel features are that (1) root subgroup coordinates are now rational functions, i.e. there is an exceptional set, and (2) the non-compactness of SL(2,C) entails that loops are no longer automatically bounded, and this (together with the exceptional set) complicates the analysis at the critical exponent.