Solving for Root Subgroup Coordinates: The SU(2) Case

TL;DR

This paper explores the relationship between root subgroup coordinates and triangular factorization in the SU(2) case, showing they are rational functions of each other, and conjecturing a broader algebraic relation in general.

Contribution

It demonstrates that in the SU(2) case, root subgroup coordinates are rational functions of triangular coordinates, advancing understanding of Lie group factorizations.

Findings

Root subgroup coordinates are rational functions of triangular coordinates in SU(2).

Unique triangular and root subgroup factorizations are equivalent for loops in compact Lie groups.

Conjecture that root subgroup coordinates are algebraic functions in the general case.

Abstract

Previously we showed that a loop in a simply connected compact Lie group K has a unique 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 we show that in the K=SU(2) case, root subgroup coordinates are rational functions (with positive denominators) of the triangular factorization coordinates. We conjecture that in general they are algebraic functions.