Loops in SL (2, C) and root subgroup factorization

Estelle Basor, Douglas M Pickrell

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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 L2 Sobolev sense), the following are equivalent: (1) the Toeplitz and shifted Toeplitz operators associated to the loop are invertible, (2) the loop has a unique triangular factorization, and (3) the loop has a unique root subgroup factorization. For a loop g satisfying these conditions, the Toeplitz determinant det(A(g)A(g-1)) and shifted Toeplitz determinant det(A1(g)A1(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 associated uniqueness issues, and (2) the noncompactness 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.

Original languageEnglish (US)
Article number1850008
JournalRandom Matrices: Theory and Application
Volume7
Issue number3
DOIs
StatePublished - Jul 1 2018

Keywords

  • Loop groups
  • Toeplitz operators
  • root subgroup and triangular factorization

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Discrete Mathematics and Combinatorics

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